GRADUATE STUDIES I N M AT H E M AT I C S
188
Introduction to Algebraic Geometry Steven Dale Cutkosky
GRADUATE STUDIES I N M AT H E M AT I C S
188
Introduction to Algebraic Geometry
Steven Dale Cutkosky
EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staﬃlani Jeﬀ A. Viaclovsky 2010 Mathematics Subject Classiﬁcation. Primary 1401.
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Library of Congress CataloginginPublication Data Names: Cutkosky, Steven Dale, author. Title: Introduction to algebraic geometry / Steven Dale Cutkosky. Other titles: Algebraic geometry Description: Providence, Rhode Island : American Mathematical Society, [2018]  Series: Graduate studies in mathematics ; volume 188  Includes bibliographical references and index. Identiﬁers: LCCN 2017045552  ISBN 9781470435189 (alk. paper) Subjects: LCSH: Geometry, Algebraic.  AMS: Algebraic geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc Classiﬁcation: LCC QA564 .C8794 2018  DDC 516.3/5–dc23 LC record available at https://lccn.loc.gov/2017045552
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23 22 21 20 19 18
To Hema, Ashok, and Maya
Contents
Preface Chapter 1. A Crash Course in Commutative Algebra
xi 1
§1.1. Basic algebra
1
§1.2. Field extensions
6
§1.3. Modules
8
§1.4. Localization
9
§1.5. Noetherian rings and factorization
10
§1.6. Primary decomposition
13
§1.7. Integral extensions
16
§1.8. Dimension
19
§1.9. Depth
20
§1.10. Normal rings and regular rings
22
Chapter 2. Aﬃne Varieties
27
§2.1. Aﬃne space and algebraic sets
27
§2.2. Regular functions and regular maps of aﬃne algebraic sets
33
§2.3. Finite maps
40
§2.4. Dimension of algebraic sets
42
§2.5. Regular functions and regular maps of quasiaﬃne varieties
48
§2.6. Rational maps of aﬃne varieties
58
Chapter 3. Projective Varieties §3.1. Standard graded algebras
63 63 v
vi
Contents
§3.2. Projective varieties
67
§3.3. Grassmann varieties
73
§3.4. Regular functions and regular maps of quasiprojective varieties
74
Chapter 4. Regular and Rational Maps of Quasiprojective Varieties
87
§4.1. Criteria for regular maps
87
§4.2. Linear isomorphisms of projective space
90
§4.3. The Veronese embedding
91
§4.4. Rational maps of quasiprojective varieties
93
§4.5. Projection from a linear subspace
95
Chapter 5. Products §5.1. Tensor products
99 99
§5.2. Products of varieties
101
§5.3. The Segre embedding
105
§5.4. Graphs of regular and rational maps
106
Chapter 6. The Blowup of an Ideal
111
§6.1. The blowup of an ideal in an aﬃne variety
111
§6.2. The blowup of an ideal in a projective variety
120
Chapter 7. Finite Maps of Quasiprojective Varieties
127
§7.1. Aﬃne and ﬁnite maps
127
§7.2. Finite maps
131
§7.3. Construction of the normalization
135
Chapter 8. Dimension of Quasiprojective Algebraic Sets
139
§8.1. Properties of dimension
139
§8.2. The theorem on dimension of ﬁbers
141
Chapter 9. Zariski’s Main Theorem
147
Chapter 10. Nonsingularity
153
§10.1. Regular parameters
153
§10.2. Local equations
155
§10.3. The tangent space
156
§10.4. Nonsingularity and the singular locus
159
§10.5. Applications to rational maps
165
Contents
§10.6. Factorization of birational regular maps of nonsingular surfaces §10.7. Projective embedding of nonsingular varieties §10.8. Complex manifolds
vii
168 170 175
Chapter 11. Sheaves §11.1. Limits §11.2. Presheaves and sheaves §11.3. Some sheaves associated to modules §11.4. Quasicoherent and coherent sheaves §11.5. Constructions of sheaves from sheaves of modules §11.6. Some theorems about coherent sheaves
181 181 185 196 200 204 209
Chapter 12. Applications to Regular and Rational Maps §12.1. Blowups of ideal sheaves §12.2. Resolution of singularities §12.3. Valuations in algebraic geometry §12.4. Factorization of birational maps §12.5. Monomialization of maps
221 221 225 228 232 236
Chapter 13. Divisors §13.1. Divisors and the class group §13.2. The sheaf associated to a divisor §13.3. Divisors associated to forms §13.4. Calculation of some class groups §13.5. The class group of a curve §13.6. Divisors, rational maps, and linear systems §13.7. Criteria for closed embeddings §13.8. Invertible sheaves §13.9. Transition functions
239 240 242 249 249 254 259 264 269 271
Chapter 14. Diﬀerential Forms and the Canonical Divisor §14.1. Derivations and K¨ahler diﬀerentials §14.2. Diﬀerentials on varieties §14.3. nforms and canonical divisors
279 279 283 286
Chapter 15. Schemes §15.1. Subschemes of varieties, schemes, and Cartier divisors §15.2. Blowups of ideals and associated graded rings of ideals
289 289 293
viii
Contents
§15.3. Abstract algebraic varieties
295
§15.4. Varieties over nonclosed ﬁelds
296
§15.5. General schemes
296
Chapter 16. The Degree of a Projective Variety
299
Chapter 17. Cohomology
307
§17.1. Complexes
307
§17.2. Sheaf cohomology ˇ §17.3. Cech cohomology
308
§17.4. Applications
312
§17.5. Higher direct images of sheaves
320
§17.6. Local cohomology and regularity
325
Chapter 18. Curves
310
333
§18.1. The RiemannRoch inequality
334
§18.2. Serre duality
335
§18.3. The RiemannRoch theorem
340
§18.4. The RiemannRoch problem on varieties
343
§18.5. The Hurwitz theorem
345
§18.6. Inseparable maps of curves
348
§18.7. Elliptic curves
351
§18.8. Complex curves
358
§18.9. Abelian varieties and Jacobians of curves
360
Chapter 19. An Introduction to Intersection Theory
365
§19.1. Deﬁnition, properties, and some examples of intersection numbers
366
§19.2. Applications to degree and multiplicity
375
Chapter 20. Surfaces
379
§20.1. The RiemannRoch theorem and the Hodge index theorem on a surface
379
§20.2. Contractions and linear systems
383
´ Chapter 21. Ramiﬁcation and Etale Maps
391
§21.1. Norms and Traces
392
§21.2. Integral extensions
393
§21.3. Discriminants and ramiﬁcation
398
Contents
ix
§21.4. Ramiﬁcation of regular maps of varieties
406
§21.5. Completion
408
§21.6. Zariski’s main theorem and Zariski’s subspace theorem
413
§21.7. Galois theory of varieties
421
§21.8. Derivations and K¨ahler diﬀerentials redux ´ §21.9. Etale maps and uniformizing parameters
424 426
§21.10. Purity of the branch locus and the AbhyankarJung theorem
433
§21.11. Galois theory of local rings
438
§21.12. A proof of the AbhyankarJung theorem
441
Chapter 22. Bertini’s Theorems and General Fibers of Maps
451
§22.1. Geometric integrality
452
§22.2. Nonsingularity of the general ﬁber
454
§22.3. Bertini’s second theorem
457
§22.4. Bertini’s ﬁrst theorem
458
Bibliography
469
Index
477
Preface
This book is an introductory course in algebraic geometry, proving most of the fundamental classical results of algebraic geometry. Algebraic geometry combines the intuition of geometry with the precision of algebra. Starting with geometric concepts, we introduce machinery as necessary to model important ideas from algebraic geometry and to prove fundamental results. Emphasis is put on developing facility with connecting geometric and algebraic concepts. Examples are constructed or cited illustrating the scope of these results. The theory in this book is developed in increasing sophistication, giving (and reﬁning) deﬁnitions as required to accommodate new geometric ideas. We work as much as possible with quasiprojective varieties over an algebraically closed ﬁeld of arbitrary characteristic. This allows us to interpret varieties through their function ﬁelds. This approach and the use of methods of algebraic number theory in algebraic geometry have been central to algebraic geometry at least since the time of Dedekind and Weber (Theorie der algebraischen Functionen einer Ver¨ anderlichen [46], translated in [47]). By interpreting the geometric concept of varieties through their regular functions, we are able to use the techniques of commutative algebra. Diﬀerences between the theory in characteristic 0 and positive characteristic are emphasized in this book. We extend our view to schemes, allowing rings with nilpotents, to study ﬁbers of regular maps and to develop intersection theory. We discuss the cases of nonclosed ground ﬁelds and nonseparated schemes and some of the extra considerations which appear in these situations. A list of exercises is given at the end of many sections and chapters.
xi
xii
Preface
The classic textbooks Basic Algebraic Geometry [136] by Shafarevich, Introduction to Algebraic Geometry [116] by Mumford, and Algebraic Geometry [73] by Hartshorne, as well as the works of Zariski, Abhyankar, Serre, and Grothendieck, have been major inﬂuences on this book. The necessary commutative algebra is introduced and reviewed, beginning with Chapter 1, “A Crash Course in Commutative Algebra”. We state deﬁnitions and theorems, explain concepts, and give examples from commutative algebra for everything that we will need, proving some results and giving a few examples, but mostly giving references to books on commutative algebra for proofs. As such, this book is intended to be selfcontained, although a reader may be curious about the proofs for some cited results in commutative algebra and will want to either derive them or look up the references. We give references to several books, mostly depending on which book has the exact statement we require. A reader should be familiar with the material through Section 1.6 on primary decomposition before beginning Chapter 2 on aﬃne varieties. Depending on the background of students, the material in Chapter 1 can be skipped, quickly reviewed at the beginning of a course, or be used as an outline of a semesterlong course in commutative algebra before beginning the study of geometry in Chapter 2. Chapters 2–10 give a onesemester introduction to algebraic geometry, through aﬃne and projective varieties. The sections on integral extensions, dimension, depth, and normal and regular local rings in Chapter 1 can be referred to as necessary as these concepts are encountered within a geometric context. Chapters 11–20 provide a secondsemester course, which includes sheaves, schemes, cohomology, divisors, intersection theory, and the application of these concepts to curves and surfaces. Chapter 21 (on ramiﬁcation and ´etale maps) and Chapter 22 (on Bertini’s theorems and general ﬁbers of maps) could be the subject of a topics course for a thirdsemester course. The distinctions between characteristic 0 and positive characteristic are especially explored in these last chapters. These two chapters could be read any time after the completion of Chapter 14. I thank the students of my classes, especially Razieh Ahmadian, Navaneeth Chenicheri Chathath, Suprajo Das, Arpan Dutta, Melissa Emory, Kyle Maddox, Smita Praharaj, Thomas Polstra, Soumya Sanyal, Pham An Vinh, and particularly Roberto N´ un ˜ez, for their feedback and helpful comments on preliminary versions of this book. I thank Maya Cutkosky for help with the ﬁgures.
Chapter 1
A Crash Course in Commutative Algebra
In this chapter we review some basics of commutative algebra which will be assumed in this book. All rings will be commutative (with identity 1). The natural numbers, {0, 1, 2, . . .}, will be denoted by N. The positive integers, {1, 2, . . .}, will be denoted by Z+ . Throughout this book, k will be an algebraically closed ﬁeld (of arbitrary characteristic) unless speciﬁed otherwise.
1.1. Basic algebra The starting point of commutative algebra is the fact that every ring R has a maximal ideal and thus has at least one prime ideal [13, Theorem1.3]. We will say that a ring R is a local ring if R has a unique maximal ideal. We will denote the maximal ideal of the local ring R by mR . If φ : R → S is a ring homomorphism and I is an ideal in S, then φ−1 (I) is an ideal in R. If P is a prime ideal in S, then φ−1 (P ) is a prime ideal in R. Suppose that R, S are local domains with maximal ideals mR , mS , respectively. We will say that S dominates R if R ⊂ S and mS ∩ R = mR . We will write QF(R) for the quotient ﬁeld of a domain R. A fundamental fact is the following theorem. Lemma 1.1. Let π : R → S be a surjective ring homomorphism, with kernel K. 1) Suppose that I is an ideal in S. Then π −1 (I) is an ideal in R containing K. 1
2
1. A Crash Course in Commutative Algebra
2) Suppose that J is an ideal in R such that J contains K. Then π(J) is an ideal in S. 3) The map I → π −1 (I) is a 11 correspondence between the set of ideals in S and the set of ideals in R which contain K. The inverse map is J → π(J). 4) The correspondence is order preserving: for ideals I1 , I2 in S, I1 ⊂ I2 if and only if π −1 (I1 ) ⊂ π −1 (I2 ). 5) For an ideal I in S, I is a prime ideal if and only if π −1 (I) is a prime ideal in R. 6) For an ideal I in S, I is a maximal ideal if and only if π −1 (I) is a maximal ideal in R. In the case when S = R/K and π : R → R/K is the map π(x) = x + K for x ∈ R, we have that π(J) = J/K for J an ideal of R containing K. Proof. [84, Theorem 2.6].
A ring S is an Ralgebra if there is a given ring homomorphism φ : R → S. This gives us a multiplication rs = φ(r)s for r ∈ R and s ∈ S. Suppose that S is an Ralgebra by a homomorphism φ : R → S and T is an Ralgebra by a homomorphism ψ : R → T . Then a ring homomorphism σ : S → T is an Ralgebra homomorphism if σ(φ(r)) = ψ(r) for all r ∈ R. A proof of the following universal property of polynomial rings can be found in [84, Theorem 2.11]. Theorem 1.2. Suppose that R and S are rings and R[x1 , . . . , xn ] is a polynomial ring over R. Suppose that φ : R → S is a ring homomorphism and t1 , . . . , tn ∈ S. Then there exists a unique ring homomorphism Φ : R[x1 , . . . , xn ] → S such that Φ(r) = φ(r) for r ∈ R and Φ(xi ) = ti for 1 ≤ i ≤ n. A polynomial ring over a ring R is naturally an Ralgebra. With the notation of the previous theorem, S is an Ralgebra by the homomorphism φ, making φ an Ralgebra homomorphism. Suppose that R ⊂ S is a subring and Λ ⊂ S is a subset. Then R[Λ] is deﬁned to be the smallest subring of S containing R and Λ. For n ∈ N, letting R[x1 , . . . , xn ] be a polynomial ring, we have that R[Λ] = {f (t1 , . . . , tn )  t1 , . . . , tn ∈ Λ and f ∈ R[x1 , . . . , xn ]}. If Λ = {t1 , . . . , tn }, write R[Λ] = R[t1 , . . . , tn ]. Suppose that we have a surjective ring homomorphism Φ : R[x1 , . . . , xn ] → S from the polynomial ring R[x1 , . . . , xn ]. Letting I be the kernel of Φ,
1.1. Basic algebra
3
we have an induced isomorphism R[x1 , . . . , xn ]/I ∼ = S. Letting R = Φ(R) ∼ = R/I ∩ R, we have that S = R[t1 , . . . , tn ] where ti = Φ(xi ). More abstractly, if I is an ideal in the polynomial ring R[x1 , . . . , xn ], let S = R[x1 , . . . , xn ]/I. Let R = R/(I ∩ R) ⊂ S and xi = xi + I in S. Then S = R[x1 , . . . , xn ]. An element x ∈ R is a zero divisor if x = 0 and there exists 0 = y ∈ R such that xy = 0. An element x ∈ R is nilpotent if x = 0 and there exists n ∈ N such that xn = 0. The radical of an ideal I in R is √ I = {f ∈ R  f n ∈ I for some n ∈ N}. A ring R is reduced if whenever f ∈ R is such that f n = 0 for some positive integer n, we have that f = 0. Suppose that I is an ideal in a ring √ R. The ring R/I is reduced if and only if I = I. An Ralgebra A is ﬁnitely generated if A is generated by a ﬁnite number of elements as an Ralgebra, so that A is a quotient of a polynomial ring over R in ﬁnitely many variables. If A is nonzero and is generated by u1 , . . . , un as a Ralgebra, then R = R1A is a subring of A and A = R[u1 , . . . , un ]. In particular, A is a quotient of a polynomial ring over R. If K is a ﬁeld and A is a nonzero Kalgebra, then we can view K as a subring of A by identifying K with K1A . The following lemma will be useful in some of the problems in Chapter 2. Lemma 1.3. Suppose that K is a ﬁeld, K[x1 , . . . , xn , z] is a polynomial ring over K, and f1 , . . . , fr , g ∈ K[x1 , . . . , xn ]. Then A = K[x1 , . . . , xn , z]/(f1 (x1 , . . . , xn ), . . . , fr (x1 , . . . , xn ), z − g(x1 , . . . , xn )) ∼ = K[x1 , . . . , xn ]/(f1 (x1 , . . . , xn ), . . . , fr (x1 , . . . , xn )). Proof. Let x1 , . . . , xn , z be the classes of x1 , . . . , xn , z in A. We have that A is generated by x1 , . . . , xn and z as a Kalgebra, and z = g(x1 , . . . , xn ), so A = K[x1 , . . . , xn ] is generated by x1 , . . . , xn as a Kalgebra. By the universal property of polynomial rings, we have a Kalgebra homomorphism Φ : K[x1 , . . . , xn ] → A deﬁned by Φ(xi ) = xi for 1 ≤ i ≤ n. Since A = K[x1 , . . . , xn ], Φ is surjective. We now compute the kernel of Φ. The elements fi (x1 , . . . , xn ) are in Kernel(Φ) since Φ(fi ) = fi (x1 , . . . , xn ) = 0. Suppose h(x1 , . . . , xn ) ∈ Kernel(Φ). Then Φ(h(x1 , . . . , xn )) = h(x1 , . . . , xn ) = 0 in A, and so h(x1 , . . . , xn ) is in the ideal (f1 (x1 , . . . , xn ), . . . , fr (x1 , . . . , xn ), z − g(x1 , . . . , xn ))
4
1. A Crash Course in Commutative Algebra
of K[x1 , . . . , xn , z] and so h(x1 , . . . , xn ) = a1 (x1 , . . . , xn , z)f1 (x1 , . . . , xn ) + · · · + ar (x1 , . . . , xn , z)fr (x1 , . . . , xn ) + b(x1 , . . . , xn , z)(z − g(x1 , . . . , xn )) for some ai , b ∈ K[x1 , . . . , xn , z]. Setting z = g(x1 , . . . , xn ), we have h(x1 , . . . , xn ) = a1 (x1 , . . . , xn , g(x1 , . . . , xn ))f1 (x1 , . . . , xn ) + · · · + ar (x1 , . . . , xn , g(x1 , . . . , xn ))fr (x1 , . . . , xn ). Thus h is in the ideal (f1 , . . . , fr ) in K[x1 , . . . , xn ], and so Kernel(Φ) = (f1 , . . . , fr ). Thus A ∼ = K[x1 , . . . , xn ]/(f1 , . . . , fr ). The following theorem justiﬁes the common identiﬁcation of polynomials and polynomial functions over an inﬁnite ﬁeld. Theorem 1.4. Suppose that L is an inﬁnite ﬁeld and f ∈ L[x1 , . . . , xn ] is a nonzero polynomial. Then there exist elements a1 , . . . , an ∈ L such that f (a1 , . . . , an ) = 0. Proof. We prove the theorem by induction on n. A nonzero polynomial f (x) ∈ L[x] has at most ﬁnitely many roots so, since L is inﬁnite, there exists a ∈ L such that f (a) = 0. Assume that n > 1 and the theorem is true for n − 1 indeterminates. Expand f (x1 , . . . , xn ) = B0 + B1 xn + · · · + Bd xdn where Bi ∈ L[x1 , . . . , xn−1 ] for all i and Bd = 0. By induction, there exist ai ∈ L such that Bd (a1 , . . . , an−1 ) = 0. Thus f (a1 , . . . , an−1 , xn ) = B0 (a1 , . . . , an−1 ) + B1 (a1 , . . . , an−1 )xn + · · · + Bd (a1 , . . . , an−1 )xdn is a nonzero polyomial in L[xn ]. Hence we can choose an ∈ L such that f (a1 , . . . , an−1 , an ) = 0. Theorem 1.5 (Chinese remainder theorem). Let A be a ring and I1 , . . . , In be ideals in A such that Ii + Ij = A for i = j (Ii and Ij are coprime). Given elements x1 , . . . , xn ∈ A, there exists x ∈ A such that x ≡ xi mod Ii for all i. Proof. [95, page 94].
1.1. Basic algebra
5
Corollary 1.6. Let A be a ring and I1 , . . . , In be ideals in A. Assume that Ii + Ij = A for i = j. Let f :A→
n
A/Ii
i=1
be homomorphism induced by the canonical maps of A onto each factor A/Ii . Then the kernel of f is I1 ∩ I 2 ∩···∩ In = I1 I2 · · · In and f is surjective, so ∼ we have an isomorphism A/ Ii = A/Ii .
Proof. [95, page 95].
Exercise 1.7. Suppose that R is a domain and 0 = f ∈ R. Show that R[x]/(xf − 1) ∼ = R[ f1 ]. Hint: Start by using the universal property of polynomial rings to get an Ralgebra homomorphism φ : R[x] → K where K is the quotient ﬁeld of R and φ(x) = f1 . Exercise 1.8. Let K be a ﬁeld and R = K[x, y]/(y 2 − x3 ) = K[x, y] where x, y are the classes of x and y in R. Show that R is a domain. Let R1 be the subring R1 = R[ xy ] of the quotient ﬁeld of R. Show that R2 = R[t]/(xt − y) is not a domain, so that the Kalgebras R1 and R2 are not isomorphic. Exercise 1.9. Let K be a ﬁeld and R = K[x, y] be a polynomial ring in the variables x and y. Let R1 be the subring R1 = R[ xy ] of the quotient ﬁeld of R. Let R2 = R[t]/(xt − y). Show that the Kalgebras R1 and R2 are isomorphic and that R1 = K[x, xy ] is a polynomial ring in the variables x and xy . Exercise 1.10. Suppose that R is a domain and f, g ∈ R with g =
0. Show f ∼ that R[ g ] = R[t]/(tg − f ) if and only if (tg − f ) is a prime ideal in R[t]. Exercise 1.11. Let A be a ring and X be the set of all prime ideals in A. For each subset E of A, let V (E) be the set of all prime ideals in A which contain E. Prove that: √ a) If I is the ideal generated by E, then V (E) = V (I) = V ( I). b) V (0) = X and V (1) = ∅. c) If {Es }s∈S is any family of subsets of A, then Es = V (Es ). V s∈S
s∈S
d) V (I ∩ J) = V (IJ) = V (I) ∪ V (J) for any ideals I, J of A This exercise shows that the sets V (E) satisfy the axioms for closed sets in a topological space. We call this topology on X the Zariski topology and write Spec(A) for this topological space.
6
1. A Crash Course in Commutative Algebra
Exercise 1.12. Suppose that R is a local ring and f1 , . . . , fr ∈ R generate an ideal I of R. Suppose that I is a principal ideal. Show that there exists an index i such that I = (fi ). Give an example to show that this is false in a polynomial ring k[x]. Exercise 1.13. Let κ be a ﬁeld, and deﬁne Map(κn , κ) to be the set of maps (of sets) from κn to κ. Since κ is a κalgebra, Map(κn , κ) is a κalgebra, with the operations (φ + ψ)(α) = φ(α) + ψ(α), (φψ)(α) = φ(α)ψ(α), and (cφ)(α) = cφ(α) for φ, ψ ∈ Map(κn , κ), α ∈ κn , and c ∈ κ. a) Let κ[x1 , . . . , xn ] be a polynomial ring over κ and deﬁne Λ : κ[x1 , . . . , xn ] → Map(κn , κ) by Λ(f )(α) = f (α) for f ∈ k[x1 , . . . , xn ] and α ∈ κn . Show that Λ is a κalgebra homomorphism. The image, Λ(κ[x1 , . . . , xn ]), is a subring of Map(κn , κ) which is called the ring of polynomial functions on κn . b) Show that Λ is an isomorphism onto the polynomial functions of κn if and only if κ is an inﬁnite ﬁeld.
1.2. Field extensions Suppose that K is a ﬁeld and A is a Kalgebra. Suppose that Λ is a subset of A. The set Λ is said to be algebraically independent over K if whenever we have a relation f (t1 , . . . , tn ) = 0 for some distinct t1 , . . . , tn ∈ Λ and a polynomial f in the polynomial ring K[x1 , . . . , xn ], we have that f = 0 (all the coeﬃcients of f are zero). Suppose that K is a subﬁeld of a ﬁeld L and Λ is a subset of L. The subﬁeld K(Λ) of L is the smallest subﬁeld of L which contains K and Λ. A subset Λ of L which is algebraically independent over K and is maximal with respect to inclusions is called a transcendence basis of L over K. Transcendence bases always exist. Any set of algebraically independent elements in L over K can be extended to a transcendence basis of L over K. Any two transcendence bases of L over K have the same cardinality ([95, Theorem 1.1, page 356] or [160, Theorem 25, page 99]). This cardinality is called the transcendence degree of the ﬁeld L over K and is written as trdegK L. Suppose that L ⊂ M ⊂ N is a tower of ﬁelds. Then (1.1)
trdegL N = trdegM N + trdegL M
by [160, Theorem 26, page 100].
1.2. Field extensions
7
An algebraic function ﬁeld over a ﬁeld K is a ﬁnitely generated ﬁeld extension L = K(y1 , . . . , ym ) of K. After possibly permuting y1 , . . . , ym , there exists an integer r with 0 ≤ r ≤ m such that y1 , . . . , yr is a transcendence basis of L over K. The ﬁeld L is then said to be an rdimensional algebraic function ﬁeld. We have that L is ﬁnite over K(y1 , . . . , yr ). The ﬁeld K(y1 , . . . , yr ) is isomorphic as a Kalgebra to the quotient ﬁeld of a polynomial ring over K in r variables, so K(y1 , . . . , yr ) is called a rational function ﬁeld over K. The ﬁeld L is said to be separably generated over K if there exists a transcendence basis z1 , . . . , zn of L over K such that L is separably algebraic over K(z1 , . . . , zn ). The set of elements z1 , . . . , zn is then called a separating transcendence basis of L over K. Theorem 1.14. If K is a perfect ﬁeld (K has characteristic 0, or K has characteristic p > 0 and all elements of K have a pth root in K), then all ﬁnitely generated ﬁeld extensions over K are separably generated over K.
Proof. [160, Theorem 31, page 105].
In any algebraic extension of ﬁelds, there is a maximal separable extension. Theorem 1.15. Suppose that L is an algebraic extension of a ﬁeld K. Then there exits a maximal subﬁeld M of L which is separable algebraic over K and such that L is purely inseparable over M .
Proof. [95, Theorem 4.5, page 241].
The M of the conclusions of Theorem 1.15 is called the separable closure of K in L. With the notation of the above theorem, we deﬁne (1.2)
[L : K]s = [M : K]
and
[L : K]i = [L : M ].
The primitive element theorem gives a nice description of ﬁnite separable extensions. Theorem 1.16 (Primitive element theorem). Suppose that L is ﬁnite extension ﬁeld of a ﬁeld K. There exists an element α ∈ L such that L = K(α) if and only if there exist only a ﬁnite number of ﬁelds F such that K ⊂ F ⊂ L. If L is separable over K then there exists such an element α. Proof. [95, Theorem 4.6, page 243].
Suppose that L is a ﬁnite extension ﬁeld of a ﬁeld K. We will write Aut(L/K) for the group of Kautomorphisms of L. In the case that L is Galois over K, we will write G(L/K) for the Galois group Aut(L/K).
8
1. A Crash Course in Commutative Algebra
Exercise 1.17. Suppose that κ is a perfect ﬁeld of characteristic p > 0 and L = κ(s, t) is a rational function ﬁeld over κ. Let K = {f p  f ∈ L}. Show that K = κ(sp , tp ), L is ﬁnite algebraic over K, and L is not a primitive extension of K.
1.3. Modules [13, Chapter 2] and [95, Chapters III and X] are good introductions to the theory of modules over a ring. An Rmodule M is a ﬁnitely generated Rmodule if there exist n ∈ Z+ and f1 , . . . , fn ∈ M such that M = {r1 f1 + · · · + rn fn  r1 , . . . , rn ∈ R}. The following is Nakayama’s lemma. Lemma 1.18. Suppose that R is a ring, I is an ideal of R which is contained in all maximal ideals of R, M is a ﬁnitely generated Rmodule, and N is a submodule. If M = N + IM , then M = N . Proof. [95, Chapter X, Section 4] or [13, Proposition 2.6].
We will use the following lemma to determine the minimal number of generators of an ideal. Lemma 1.19. Suppose that R is a local ring with maximal ideal m and M is a ﬁnitely generated Rmodule. Then the minimal number of elements μ(M ) of M which generate M as an Rmodule is the R/mvector space dimension μ(M ) = dimR/m M/mM. Proof. Observe that M/mM is an R/mvector space by the welldeﬁned map R/m × M/mM → M/mM given by mapping the classes [x] in R/m of x ∈ R and [y] ∈ M/mM of y ∈ M to the class [xy] of xy in M/mM . Suppose that a1 , . . . , ar generate M as an Rmodule. Then the classes [a1 ], . . . , [ar ] ∈ M/mM generate M/mM as an R/mvector space. Thus dimR/m M/mM ≤ μ(M ). Suppose that a1 , . . . , ar ∈ M are such that the classes [a1 ], . . . , [ar ] ∈ M/mM generate M/mM as an R/mvector space. Let N be the submodule of M generated by a1 , . . . , ar . Then N + mM = M so N = M by Lemma 1.18. Thus μ(M ) ≤ dimR/m M/mM. A chain of submodules of a module M is a sequence of submodules (1.3)
0 = Mn ⊂ · · · ⊂ M1 ⊂ M0 = M.
The length of (1.3) is n. If each module Mi /Mi+1 has no submodules other than 0 and Mi /Mi+1 , then (1.3) is called a composition series. If M has a
1.4. Localization
9
composition series, then every composition series of M has length n, and every chain of submodules of M can be extended to a composition series [13, Proposition 6.7]. We deﬁne the length R (M ) of an Rmodule M to be the length of a composition series if a composition series exists, and we deﬁne R (M ) = ∞ if a composition series does not exist. If R is a local ring with maximal ideal mR containing a ﬁeld κ such that R/mR ∼ = κ, then any Rmodule M is naturally a κvector space, and R (M ) = dimκ M.
1.4. Localization A multiplicatively closed (multiplicative) subset S of a ring R is a subset of R such that 1 ∈ S and S is closed under multiplication. Deﬁne an equivalence relation ≡ on R × S by (a, s) ≡ (b, t) if and only if (at − bs)u = 0 for some u ∈ S. The localization of R with respect to S, denoted by S −1 R, is the set of equivalence classes R × S/ ≡. The equivalence class of (a, s) is denoted by as . The localization S −1 R is a ring with addition deﬁned by at + bs a b + = s t st and multiplication deﬁned by
a b s
t
=
ab . st
This deﬁnition extends to localization S −1 M of Rmodules M , in particular for ideals in R [13, page 38]. r 1
There is a natural ring homomorphism φ : R → S −1 R deﬁned by φ(r) = for r ∈ R.
We summarize a few facts from [13, Proposition 3.11]. The ideals in S −1 R are the ideals S −1 I = I(S −1 R) such that I is an ideal of R. We have that S −1 I = S −1 R if and only if S ∩ I = ∅. The prime ideals of S −1 R are in 11 correspondence with the prime ideals of R which are disjoint from S. Suppose that f ∈ R. Then S = {f n  n ∈ N} is a multiplicatively closed set. The localization S −1 R is denoted by Rf . Suppose that p is a prime ideal in R. Then S = R \ p is a multiplicatively closed set. The localization S −1 R is denoted by Rp . If p is a prime ideal in a ring R, then Rp is a local ring with maximal ideal pp = pRp .
10
1. A Crash Course in Commutative Algebra
If R is a domain, the quotient ﬁeld of R is deﬁned as QF(R) = Rp where p is the zero ideal of R. More basic properties of localization and localization of homomorphisms are established in [13, Chapter 3]. Exercise 1.20. Suppose that R is a domain and I is an ideal in R. Let RI be the subset of the quotient ﬁeld of R deﬁned by
f  f ∈ R, g ∈ R \ I . RI = g Show that RI is a ring if and only if I is a prime ideal in R. Exercise 1.21. Suppose that S is a multiplicative set in a ring R. Show that the kernel of the natural homomorphism φ : R → S −1 R is the ideal {g ∈ R  gs = 0 for some s ∈ S}. Give an example of a ring R and 0 = f ∈ R such that the kernel of R → Rf is nonzero. Exercise 1.22. Suppose that S and T are multiplicatively closed subsets of a ring R. Let U be the image of T in S −1 R. Show that (ST )−1 R ∼ = U −1 (S −1 R), where ST = {st  s ∈ S and t ∈ T }. Exercise 1.23. Suppose that R is a ring and P is a prime ideal in R. Show that RP is a local ring with maximal ideal P RP = PP . Let Λ : R → RP be the natural homomorphism deﬁned by Λ(f ) = f1 for f ∈ R. Show that Λ−1 (P RP ) = P .
1.5. Noetherian rings and factorization Noetherian rings enjoy many good properties. They are ubiquitous throughout algebraic geometry. Deﬁnition 1.24. A ring R is Noetherian if every ascending chain of ideals I1 ⊂ I2 ⊂ · · · ⊂ In ⊂ · · · is stationary (there exists n0 such that In = In0 for n ≥ n0 ). Proposition 1.25. A ring R is Noetherian if and only if every ideal I in R is ﬁnitely generated; that is, there exist f1 , . . . , fn ∈ I for some n ∈ Z+ such that I = (f1 , . . . , fn ) = f1 R + · · · + fn R. Proof. [13, Proposition 6.3].
1.5. Noetherian rings and factorization
11
We have the following fundamental theorem. Theorem 1.26 (Hilbert’s basis theorem). If R is a Noetherian ring, then the polynomial ring R[x] is Noetherian. Proof. [13, Theorem 7.5, page 81].
Corollary 1.27. A polynomial ring over a ﬁeld is Noetherian. A quotient of a Noetherian ring is Noetherian. A localization of a Noetherian ring is Noetherian. The following lemma will simplify some calculations. Lemma 1.28. Suppose that R is a Noetherian ring, m is a maximal ideal of R, and N is an Rmodule such that ma N = 0 for some positive integer a. Then Nm ∼ = N. Proof. Suppose that f ∈ R \ m. We will ﬁrst prove that for any r ∈ Z+ , there exists e ∈ R such that f e ≡ 1 mod mr . The ring R/m is a ﬁeld, and the residue of f in R/m is nonzero. Thus for any h ∈ R, there exists g ∈ R such that f g ≡ h mod m. Taking h = 1, we obtain that there exists e0 ∈ R such that f e0 ≡ 1 mod m. Suppose that we have found e ∈ R such that f e ≡ 1 mod mr . Let x1 , . . . , xn be a set of generators of m. There exists hi1 ,...,in ∈ R such that f e − 1 = i1 +···+in =r hi1 ,...,in xi11 · · · xinn . There exist gi1 ,...,in ∈ R such that f gi1 ,...,in ≡ hi1 ,...,in mod m. Thus f gi1 ,...,in xi11 · · · xinn ≡ hi1 ,...,in xi11 · · · xinn mod mr+1 . i1 +···+in =r
Set e = e −
i1 +···+in =r
gi1 ,...,in xi11 · · · xinn to get f e ≡ 1 mod mr+1 .
Consider the natural homomorphism Φ : N → Nm deﬁned by Φ(n) = n1 for n ∈ N . We will show that Φ is an isomorphism. Suppose that Φ(n) = 0. Then there exists f ∈ R \ m such that f n = 0. By the ﬁrst part of the proof, there exists e ∈ R such that f e ≡ 1 mod ma . Thus n = ef n = 0. Suppose n a f ∈ Nm . Then there exists e ∈ R \ m such that ef = 1 + h with h ∈ m , and n Φ(ne) = f . Suppose R is a domain. A nonzero element f ∈ R is called irreducible if f is not a unit and whenever we have a factorization f = gh with g and h in R, then g is a unit or h is a unit. Since every ascending chain of principal ideals is stationary in a Noetherian ring, we have the following proposition.
12
1. A Crash Course in Commutative Algebra
Proposition 1.29. Suppose R is a Noetherian domain. Then every nonzero nonunit f ∈ R has a factorization f = g1 · · · gr for some positive integer r and irreducible elements g1 , . . . , gr ∈ R. Suppose R is a domain. A nonzero element f ∈ R is called a prime if the ideal (f ) ⊂ R is a prime ideal. Proposition 1.30. Suppose R is a domain and f ∈ R. Then: 1) If f is prime, then f is irreducible. 2) If R is a unique factorization domain (UFD), then f is a prime if and only if f is irreducible. Proof. [84, Theorem 2.21].
Proposition 1.31. Suppose that A is a UFD. Let K be the quotient ﬁeld of A. Then the ring of polynomials in n variables A[x1 , . . . , xn ] is a UFD. Its units are precisely the units of A, and its prime elements are either primes of A or polynomials which are irreducible in K[x1 , . . . , xn ] and have content 1 (the greatest common divisor of the coeﬃcients in A of the polynomial is 1). Proof. [95, Corollary 2.4, page 183].
Suppose that R is a ring and R[x1 , . . . , xn ] is a polynomial ring over R. If f ∈ R[x1 , . . . , xn ], then f has a unique expansion f= ai1 ,...,in xi11 · · · xinn with ai1 ,...,in ∈ R. If f is nonzero, the (total) degree deg f of f is deﬁned to be deg f = max{i1 + · · · + in  ai1 ,...,in = 0}. The polynomial f is homogeneous of degree d if ai1 ,...,in = 0 if i1 +· · ·+in = d. Suppose that A = K[x, y, z, w] is a polynomial ring over a ﬁeld K. The units in A are the nonzero elements of K. Let f = xy−zw ∈ A. Suppose that f = gh with g, h ∈ A nonunits. Since f is homogeneous of degree 2, we have that g and h are both homogeneous of degree 1, so g = a0 x+a1 y +a2 z +a3 w and h = b0 x + b1 y + b2 z + b3 w with a0 , . . . , a3 , b0 , . . . , b3 ∈ K. We verify by expanding gh that there do not exist a0 , . . . , a3 , b0 , . . . , b3 ∈ K such that gh = f . Thus xy − zw is irreducible in A. Since A is a UFD, we have that (f ) is a prime ideal in A, and thus R = A/(f ) is a domain. For u ∈ A, let u denote the class of u in R. Then R = K[x, y, z, w] where x, y, z, w are the classes of x, y, z, w. Since f is homogeneous, the function deg g = deg g if 0 = g is welldeﬁned on R (we will see that R is graded in Section 3.1). The units of R are the nonzero elements of K (they have
1.6. Primary decomposition
13
degree 0) and since x, y, z, w are all nonzero and they have degree 1, they must be irreducible in R. We have that the ideal (f, x) = (zw, x) in A, so R/(x) ∼ = A/(zw, x) ∼ = K[y, z, w]/zw by Lemma 1.3, which is not a domain (the classes of z and w are zero divisors). In particular, x is an irreducible element of R which is not a prime. We see from Proposition 1.30 that R is not a UFD. We also have that xy = zw gives two factorizations in R by irreducible elements, none of which are associates, showing directly that R is not a UFD. Exercise 1.32. Suppose that K is a ﬁeld and K[x1 , . . . , xn ] is a polynomial ring over K. Let f ∈ K[x1 , . . . , xn ] be nonzero and homogeneous, and suppose that g, h ∈ K[x1 , . . . , xn ] are such that f = gh. Show that g and h are homogeneous and deg g + deg h = deg f . Exercise 1.33. Suppose that K is an algebraically closed ﬁeld and K[x1 , x2 ] is a polynomial ring over K. Suppose that f ∈ K[x1 , x2 ] is homogeneous of positive degree. Show that f is a product of homogeneous polynomials of degree 1. Show that this is false if K is not algebraically closed. Exercise 1.34. Let K be a ﬁeld and K[x, y, z] be a polynomial ring over K. Let f = y 3 − x3 + xz 2 ∈ K[x, y, z]. Show that f is irreducible and that R = K[x, y, z]/(f ) is a domain. Show that R is not a UFD. Exercise 1.35. Prove Euler’s formula: Suppose that K is a ﬁeld and F is a homogeneous polynomial of degree d in the polynomial ring K[x0 , . . . , xn ]. Show that n ∂F xi = dF. ∂xi i=0
1.6. Primary decomposition Suppose that R is a ring. An ideal Q in R is primary if Q = R and if for x, y ∈ R, xy ∈ Q implies either x ∈ Q or y n ∈ Q for some n > 0. √ Proposition 1.36. Let Q be a primary ideal in a ring R. Then Q is the smallest prime ideal of R containing Q. √ Proof. It suﬃces √ to show that n Q is a prime ideal. Suppose x, y ∈ nR are such that xy ∈ Q. Then (xy) ∈ Q for some √ n > 0. Then √ either x ∈ Q mn ∈ Q for some m > 0. Thus either x ∈ Q or y ∈ Q. or y √ If p is a prime ideal, an ideal Q is called pprimary if Q is primary and Q = p. √ Proposition 1.37. If I is a maximal ideal m, then I is mprimary.
14
1. A Crash Course in Commutative Algebra
Proof. [13, Proposition 4.2]. Lemma 1.38. If the Qi are pprimary, then Q =
n
i=1 Qi
is pprimary.
Proof. [13, Lemma 4.3].
A primary decomposition of an ideal I in R is an expression of I as a ﬁnite intersection of primary ideals, (1.4)
I=
n
Qi .
i=1
The ideal I is called decomposable if it has a primary decomposition. If I is decomposable, then I has a minimal (or irredundant) primary decom√ Q are all distinct and position, that is, an expression (1.4) where the i j=i Qj ⊂ Qi for all i. By Lemma 1.38, every decomposable ideal I has a minimal primary decomposition. Theorem 1.39. In a Noetherian ring R, every ideal has a primary decomposition (and hence has a minimal primary decomposition). Proof. [13, Theorem 7.13].
Let M be an Rmodule. A prime ideal p is an associated prime of M if p is the annihilator Ann(x) = {r ∈ R  rx = 0} for some x ∈ M . The set of associated primes of M is denoted by Ass(M ) or AssR (M ). In the case of an ideal I of R, it is traditional to abuse notation and call the associated primes of R/I the associated primes of I. An element a in a ring R is called a zero divisor for an Rmodule M if there exists a nonzero x ∈ M such that ax = 0. Otherwise, a is M regular. Theorem 1.40. Let A be a Noetherian ring and M a nonzero Amodule. 1) Every maximal element of the family of ideals F = {Ann(x)  0 = x ∈ M } is an associated prime of M . 2) The set of zero divisors for M is the union of all the associated primes of M . Proof. 1) We must show that if Ann(x) is a maximal element of F , then it is prime. If a, b ∈ A are such that abx = 0 but bx = 0, then by maximality, Ann(bx) = Ann(x). Hence ax = 0. 2) If ax = 0 for some x = 0, then a ∈ Ann(x) ∈ F . By 1), there is an associated prime of M containing Ann(x).
1.6. Primary decomposition
15
Another important set of prime ideals associated to a module M is the support of M , which is Supp(M ) = {prime ideals p of R  Mp = 0}. Theorem 1.41. Let R be a Noetherian ring and M a ﬁnitely generated Rmodule. Then: 1) Ass(M ) is a ﬁnite set. 2) Ass(M ) ⊂ Supp(M ). 3) Any minimal element of Supp(M ) is in Ass(M ). Proof. [107, (7.G) on page 52] and [107, Theorem 9], or [106, Theorem 6.5]. The minimal elements of the set Ass(M ) are called minimal or isolated prime ideals belonging to M . The others are called embedded primes. We have that a prime ideal P of R is a minimal prime of an ideal I (a minimal prime of R/I) if I ⊂ P , and if Q is a prime ideal of R such that I ⊂ Q ⊂ P , then Q = P . Theorem 1.42. Let I be a decomposable ideal and let I = ni=1 Qi be a minimal primary decomposition of I. Then: 1) Ass(R/I) = {
Qi  1 ≤ i ≤ n}.
2) The isolated primary components (the primary components Qi corresponding to minimal prime ideals pi ) are uniquely determined by I.
Proof. [13, Theorem 4.5 and Corollary 4.11].
Proposition 1.43. Let S be a multiplicatively n closed subset of a ring R and let I be a decomposable ideal. Let I = i=1 Qi be a minimal primary √ decomposition of I. Let pi = Qi and suppose that the Qi are indexed so that S ∩ pi = ∅ for m < i ≤ n and S ∩ pi = ∅ for 1 ≤ i ≤ m. Then S −1 I =
m
S −1 Qi
i=1
is a minimal primary decomposition of S −1 I in S −1 R, with pi S −1 R. Proof. [13, Proposition 4.9].
S −1 Qi =
16
1. A Crash Course in Commutative Algebra
Exercise 1.44. Let K be a ﬁeld and R = K[x, y] be a polynomial ring. Let I = (x2 y, xy 2 ). Compute a minimal primary decomposition of I. Compute √ the set Ass(R/I). Identify the minimal and embedded √ primes. Compute I and compute a minimal primary decomposition of √I. Identify the minimal and embedded primes. Compute the set Ass(R/ I). Compute minimal primary decompositions of Ip and the set Ass(Ip ) when p = (x) and when p = (x, y). Identify the minimal and embedded primes. Exercise 1.45. Let K be a ﬁeld and R = K[x, y, z]/(z 2 − xy) = K[x, y, z]. Compute a minimal primary decomposition of the ideal (z). Exercise 1.46. Suppose that I is an ideal in a Noetherian ring R and √ I = I. Show that all elements of Ass(R/I) are minimal and the minimal primary decomposition of I is p. I= {minimal primes p of I}
Exercise 1.47. Suppose that R is a Noetherian ring and I ⊂ J are ideals of R. Show that I = J if and only if Im = Jm for all maximal ideals m of R.
1.7. Integral extensions [13, Chapter 5], [95, Chapter VII, Section 1] and [160, Sections 1–4 of Chapter V] are good references for this section. Deﬁnition 1.48. Suppose that R is a subring of a ring S. An element u ∈ S is integral over R if u satisﬁes a relation un + a1 un−1 + · · · + an−1 u + an = 0 with a1 , . . . , an ∈ R. Theorem 1.49. Suppose that R is a subring of a ring S and u ∈ S. The following are equivalent: 1) u is integral over R. 2) R[u] is a ﬁnitely generated Rmodule. 3) R[u] is contained in a subring T of S such that T is a ﬁnitely generated Rmodule. Proof. [13, Proposition 5.1].
We have the following immediate corollaries. Corollary 1.50. Let u1 , . . . , un be elements of S which are each integral over R. Then the subring R[u1 , . . . , un ] of S is a ﬁnitely generated Rmodule.
1.7. Integral extensions
17
Corollary 1.51. Suppose that R is a subring of a ring S. Let R = {u ∈ S  u is integral over R}. Then R is a ring. Proof. If x, y ∈ R, then the subring R[x, y] of S is a ﬁnitely generated Rmodule by Corollary 1.50. The elements x + y and xy are in R[x, y] so x + y and xy are integral over R by Theorem 1.49. R is called the integral closure of R in S. This construction is particularly important when R is a domain and S is the quotient ﬁeld of R. In this case, R is called the normalization of R. R is said to be normal if R = R. If R is a domain and S is a ﬁeld extension of the quotient ﬁeld of R, then the integral closure of R in S is called the normalization of R in S. We now state some theorems which will be useful. Lemma 1.52. Let A ⊂ B be rings with B integral over A and let S be a multiplicatively closed subset of A. Then S −1 B is integral over S −1 A. Proof. Suppose b ∈ B satisﬁes a relation bn + a1 bn−1 + · · · + an = 0 with a1 , a2 , . . . , an ∈ A and s ∈ S. Then n b a1 b n−1 an = 0. + + ··· + s s s s
Theorem 1.53 (Noether’s normalization lemma). Let R be a ﬁnitely generated Lalgebra, where L is a ﬁeld. Then there exist y1 , . . . , yr ∈ R such that the subring L[y1 , . . . , yr ] of R is a polynomial ring over L and R is integral over L[y1 , . . . , yr ]. Proof. We give a proof with the assumption that L is an inﬁnite ﬁeld. For a proof when L is a ﬁnite ﬁeld, we refer to [161, Theorem 25 on page 200]. Write R = L[x1 , . . . , xn ]. Suppose that R is not a polynomial ring over L, so there exists a nonzero f in the polynomial ring L[z1 , . . . , zn ] over L such that f (x1 , . . . , xn ) = 0. Let d = deg f and let fd be the homogeneous part of f of degree d, so that zn−1 z1 d ,..., ,1 . fd = z n fd zn zn By Theorem 1.4, there exist c1 , . . . , cn−1 ∈ L such that fd (c1 , . . . , cn−1 , 1) = 0. Set yi = xi − ci xn for 1 ≤ i ≤ n − 1. Then 0 = f (x1 , . . . , xn ) = f (y1 + c1 xn , . . . , yn−1 + cn−1 xn , xn ) + · · · + gd = fd (c1 , . . . , cn−1 , 1)xdn + g1 xd−1 n
18
1. A Crash Course in Commutative Algebra
with all gi ∈ L[y1 , . . . , yn−1 ], so that xn is integral over L[y1 , . . . , yn−1 ]. The theorem then follows by induction on n. Theorem 1.54. Let R be a domain which is a ﬁnitely generated algebra over a ﬁeld K. Let Q be the quotient ﬁeld of R and let L be a ﬁnite algebraic extension of Q. Then the integral closure R of R in L is a ﬁnitely generated Rmodule and is also a ﬁnitely generated Kalgebra. Lemma 1.55. Suppose that R is a Noetherian ring, M is a ﬁnitely generated Rmodule, and N is a submodule of M . Then N is a ﬁnitely generated Rmodule. Proof. This follows from (1) of the “basic criteria” for a module to be Noetherian of [95, page 413] and [95, Proposition 1.4, page 415]. Let B be a ring and A be a subring. Let P be a prime ideal of A and let Q be a prime ideal of B. We say that Q lies over P if Q ∩ A = P . Proposition 1.56. Let A be a subring of a ring B, let P be a prime ideal of A, and assume B is integral over A. Then P B = B and there exists a prime ideal Q of B lying over P . Proof. We ﬁrst show that P B = B. By Lemma 1.52, it suﬃces to show that P S = S where S = BP . Suppose that P S = S. Then there is a relation 1 = a1 b1 + · · · + an bn with ai ∈ P and bi ∈ S. Let R = AP and S0 = R[b1 , . . . , bn ]. Then P S0 = S0 and S0 is a ﬁnitely generated Rmodule by Corollary 1.50, and so S0 = 0 by Nakayama’s lemma, Lemma 1.18, a contradiction. Thus we have that P BP is contained in a maximal ideal m of BP . Then P AP ⊂ m ∩ AP . But P AP is the maximal ideal of AP so m ∩ AP = P AP . Let Q be the inverse image of m in B. We have that P ⊂ Q ∩ A. Suppose f ∈ Q ∩ A. Then f1 ∈ m ∩ AP = P AP and so f ∈ P . Thus P = Q ∩ A. We will further develop the theory of integral extensions in Section 21.2. Exercise 1.57. Suppose that R is a domain which is contained in a ﬁeld p K of characteristic p > 0. Suppose that f ∈ K is such that √ f ∈ R. Let S = R[f ]. Suppose that Q is a prime ideal in R. Show that QS is a prime ideal. Exercise 1.58. Suppose that K is a ﬁeld and A is a subring of K. Let B be the integral closure of A in K. Let S be a multiplicatively closed subset of A. Show that S −1 B is the integral closure of S −1 A in K. Exercise 1.59. Let K be a ﬁeld and R be a polynomial ring over K. Show that R is integrally closed in its quotient ﬁeld.
1.8. Dimension
19
1.8. Dimension In this section we deﬁne the height of an ideal and the dimension (Krull dimension) of a ring. Deﬁnition 1.60. The height, ht(P ), of a prime ideal P in a ring R is the supremum of all natural numbers n such that there exists a chain (1.5)
P0 ⊂ P1 ⊂ · · · ⊂ Pn = P
of distinct prime ideals. The dimension dim R of R is the supremum of the heights of all prime ideals in R. If P is a prime ideal in a ring R, then dim RP = ht(P ) by Proposition 1.43. A chain (1.5) is maximal if the chain cannot be lengthened by adding an additional prime ideal somewhere in the chain. Deﬁnition 1.61. The height of an ideal I in a ring R is ht(I) = inf{ht(P )  P is a prime ideal of R and I ⊂ P }. Theorem 1.62. Let B be a Noetherian ring and let A be a Noetherian subring over which B is integral. Then dim A = dim B. Proof. [107, Theorem 20, page 81].
Theorem 1.63. Let K be a ﬁeld and A be a ﬁnitely generated Kalgebra which is a domain. Let L be the quotient ﬁeld of A. Then dim A = trdegK L, the transcendence degree of L over K. Proof. The dimension of a polynomial ring over K in n variables is n by [107, Theorem 22]. The proof of the theorem now follows from Theorem 1.53 (Noether’s normalization lemma) and Theorem 1.62. An example of a Noetherian ring which has inﬁnite dimension is given in [121, Example 1 of Appendix A1, page 203]. Rings which are ﬁnitely generated Kalgebras have the following nice property. Theorem 1.64. Let K be a ﬁeld and A be a ﬁnitely generated Kalgebra which is a domain. For any prime ideal p in A we have that ht(p) + dim A/p = dim A. Proof. [28, Theorem A.16] or [13, Chapter 11].
There are Noetherian rings which do not satisfy the equality of Theorem 1.64 [121, Example 2, Appendix A1].
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1. A Crash Course in Commutative Algebra
The following theorem is of fundamental importance. Theorem 1.65 (Krull’s principal ideal theorem). Let A be a Noetherian ring, and let f ∈ A be an element which is neither a zero divisor nor a unit. Then every minimal prime ideal p containing f has height 1. Proof. [13, Corollary 11.17].
Proposition 1.66. A Noetherian domain A is a UFD if and only if every prime ideal of height 1 in A is principal. Proof. [106, Theorem 20.1] or [23, Chapter 7, Section 3] or [50, Proposition 3.11].
1.9. Depth Let R be a ring and M be an Rmodule. Elements x1 , . . . , xr ∈ R are said to be an M regular sequence if 1) for each 1 ≤ i ≤ r, xi is a nonzero divisor on M/(x1 , . . . , xi−1 )M (xi y = 0 for all nonzero y ∈ M/(x1 , . . . , xi−1 )M ) and 2) M = (x1 , . . . , xr )M . Deﬁnition 1.67. Let R be a Noetherian ring, I be an ideal in R, and M be a ﬁnitely generated Rmodule. We deﬁne depthI M to be the maximal length of an M regular sequence x1 , . . . , xr with all xi ∈ I. Deﬁnition 1.68. A Noetherian ring R is said to be CohenMacaulay if depthI R = ht(I) for every maximal ideal I of R. We give some examples of CohenMacaulay rings in the following theorem and proposition. Theorem 1.69. Let A be a CohenMacaulay ring. Then the polynomial ring A[x1 , . . . , xn ] is a CohenMacaulay ring. In particular, a polynomial ring over a ﬁeld is CohenMacaulay. Proof. [107, Theorem 33].
Proposition 1.70. Let A be a CohenMacaulay ring and J = (a1 , . . . , ar ) be an ideal of height r. Then A/J v is CohenMacaulay for every v > 0. Proof. [107, Proposition, page 112].
A proof of the following theorem is given in [50, Corollary 18.14] or [107, Theorem 32].
1.9. Depth
21
Theorem 1.71 (Unmixedness theorem). Let R be a CohenMacaulay ring. If I = (x1 , . . . , xn ) is an ideal such that ht(I) = n, then all associated primes of I are minimal primes of I and have height n. Lemma 1.72. Suppose that R is a ring and I, P1 , . . . , Pr are ideals in R suchthat the Pi are prime ideals. Suppose that I ⊂ Pi for each i. Then I ⊂ i Pi . Proof. We may omit the Pi which are contained in some other Pj and suppose that Pi ⊂ Pj if i = j. We prove the lemma by induction on r. Suppose r = 2 and I ⊂ P1 ∪ P2 . Choose x ∈ I \ P2 and y ∈ I \ P1 . Then x ∈ P1 so y + x ∈ P1 . Thus y and y + x ∈ P2 so x ∈ P2 , a contradiction. Now suppose r > 2. Then IP1 · · · Pr−1 ⊂ Pr since Pr is a prime ideal. Choose x ∈ IP1 · · · Pr−1 \ Pr . Let S = I \ (P1 ∪ · · · ∪ Pr−1 ). By induction, S = ∅. Suppose I ⊂ P1 ∪ · · · ∪ Pr . Then S ⊂ Pr . Suppose s ∈ S. Then s + x ∈ S and thus both s and s + x are in Pr , and so x ∈ Pr , a contradiction. Lemma 1.73. Suppose that R is a Noetherian ring, m is a maximal ideal of R, and M is a ﬁnite Rmodule. Then depthm M = 0 if and only if m ∈ AssR (M ). Proof. If m is an associated prime for M , then there exists x ∈ M such that m = Ann(x). Thus depthm M = 0. Suppose depthm M = 0. Then all elements of m are zero divisors for M . Now the set of all zero divisors for M is the union of the ﬁnitely many associated primes of R by Theorems 1.40 and 1.41. Thus m is an associated prime of M by Lemma 1.72. A proof of the following lemma is given in [50, Corollary 18.6]. Lemma 1.74. Let R be a Noetherian ring, m be a maximal ideal of R, and 0 → N → N → N → 0 be a short exact sequence of nonzero ﬁnitely generated Rmodules. Then 1) depthm N ≥ min{depthm N, depthm N − 1}, 2) depthm N ≥ min{depthm N, depthm N + 1}. Example 1.75. There exists a domain A and a nonzero element f ∈ A such that the ideal f A has an embedded prime. We now construct such an example. Let K be a ﬁeld. We will ﬁrst show that the twodimensional domain R = K[s4 , s3 t, st3 , t4 ]
22
1. A Crash Course in Commutative Algebra
which is a subring of the polynomial ring K[s, t] has depthm (R) = 1 where m = (s4 , s3 t, st3 , t4 ) (so R is not CohenMacaulay). Let S = K[s4 , s3 t, s2 t2 , st3 , t4 ]. The domain S contains R as a subring, realizing S = R + s2 t2 R as a ﬁnitely generated Rmodule. We have a short exact sequence of Rmodules 0→R→S→M →0 where M = S/R. We have that depthm S ≥ 1 since S is a domain which is not a ﬁeld. Let a be the class of s2 t2 in M . Consider the surjective Rmodule homomorphism φ : R → M deﬁned by φ(f ) = f a for f ∈ R. Since ma = 0, φ induces an isomorphism of Rmodules M ∼ = R/m. By Lemma 1.74 we have that depthm R ≤ 1, so that depthm R = 1 since R is a domain which is not a ﬁeld. Thus R has the following attribute: For every nonzero f ∈ m, depthm R/(f ) = 0, so by Lemma 1.73, m is an embedded prime for the ideal (f ); that is, a minimal primary decomposition of f Rm is (1.6)
f Rm = Q 1 ∩ · · · ∩ Q t ∩ Q 0
where the Qi are Pi primary for a height 1 prime Pi in Rm (a minimal prime of f Rm ) and Q0 is a nontrivial mm primary ideal. The following theorem will be useful. Theorem 1.76. Suppose that R is a CohenMacaulay ring and J = (g1 , . . . , gs ) is an ideal in R such that g1 , . . . , gs is an Rregular sequence. Then J i /J i+1 = R/J[g 1 , . . . , g s ] grJ (R) = i≥0
is a polynomial ring over R/J in g 1 , . . . , g s , where g i is the class of gi in J/J 2 . Proof. This follows from [107, Theorem27, page 98] and the equivalence (***) on page 98 of [107].
1.10. Normal rings and regular rings Normal and regular rings play an important role in algebraic geometry. In normal rings, the concepts of zeros and poles of a function are welldeﬁned, and regular rings correspond to nonsingular spaces. We begin this section with some properties of normal rings which we will use. A normal ring is deﬁned in Section 1.7.
1.10. Normal rings and regular rings
23
Lemma 1.77. Suppose that A is a domain with quotient ﬁeld K. Then A= AP P
where the intersection in K is over all maximal ideals P of A. Proof. Suppose x ∈ K. Let D = {a ∈ A  ax ∈ A}. The element x is in A if and only if D = A, and x is in AP if and only if D ⊂ P . Thus if x ∈ A, there exists a maximal ideal P of A such that D ⊂ P , and so x ∈ AP . Corollary 1.78. Suppose that A is a domain. Then A is normal if and only if AP is normal for all maximal ideals P of A. Proof. If A is normal, then S −1 A is normal for every multiplicatively closed subset of A not containing 0. Since A = AP by Lemma 1.77, where the intersection is over all maximal ideals P of A, the domain A is normal if and only if AP is normal for all maximal ideals P . A stronger intersection theorem holds for height 1 primes. Theorem 1.79. Let A be a Noetherian normal domain. Then: 1) All associated primes of a nonzero principal ideal have height 1. 2) A=
Ap
p
where the intersection in K is over all height 1 prime ideals p of A. Proof. [107, Theorem 38] or [106, Theorem 11.5].
We now develop some concepts to deﬁne a regular local ring. Deﬁnition 1.80. Suppose that R is a local ring with maximal ideal mR . The associated graded ring of R is grmR (R) = miR /mi+1 R . i≥0
Theorem 1.81. Suppose that R is a Noetherian local ring with maximal ideal mR . Then: 1) dim grmR (R) = dim R. 2) dimR/mR mR /m2R ≥ dim R. Proof. Equation 1) is proven in [107, Theorem 17] or [106, Theorem 13.9], using the theory of Hilbert polynomials. Equation 2) follows from [160, Theorem 30, page 240, and Theorem 31, page 241] or [107, (12.J)].
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1. A Crash Course in Commutative Algebra
If A is a local ring with maximal ideal mA and residue ﬁeld κ = A/mA , then the tangent space of A is deﬁned as (1.7)
T (A) = Homκ (mA /m2A , κ).
Deﬁnition 1.82. A Noetherian local ring R with maximal ideal mR is a regular local ring if dimR/mR mR /m2R = dim R. Since dimκ T (R) = dimκ mR /m2R , we always have that dimκ T (R) ≥ dim R and R is regular if and only if dimκ T (R) = dim R. We now state some useful properties of regular local rings and their relation to normal rings. Theorem 1.83. Let A be a ring such that for every prime ideal P of A the localization AP is regular. Then the polynomial ring A[x1 , . . . , xn ] has the same property. In particular, every local ring of a polynomial ring over a ﬁeld is a regular local ring. Proof. [107, Theorem40]
Theorem 1.84. Suppose that R is a regular local ring. Then R is a CohenMacaulay normal domain. Proof. This follows from [161, Corollary 1 on page 302] and [107, Theorem 36]. The proofs of the following theorems are through homological algebra. Theorem 1.85. A Noetherian ring A is normal if and only if it satisﬁes the following two condtions: 1) For every prime ideal p ⊂ A of height 1, Ap is regular. 2) For every prime ideal p ⊂ A of height ≥ 2, we have depth Ap ≥ 2. Proof. [107, Theorem 39, page 125].
Corollary 1.86. Suppose that R is a regular local ring and f ∈ R is nonzero and is not a unit. Then R/(f ) is normal if and only if (R/(f ))p is regular for all prime ideals p of R/(f ) of height 1. Proof. Let A = R/(f ). We must show that condition 2) of Theorem 1.85 holds. We have that R is a CohenMacaulay domain by Theorem 1.84. Since f is Rregular, we have that AP is CohenMacaulay for all prime ideals P of A by [107, Theorem 30], and so depth(AP ) = dim AP = ht(P ).
Theorem 1.87. Suppose that R is a normal Noetherian local ring of dimension 1. Then R is a regular local ring.
1.10. Normal rings and regular rings
25
Proof. This follows from Theorem 1.85.
Theorem 1.88. Suppose that R is a regular local ring and P is a prime ideal in R. Then RP is a regular local ring. Proof. [107, Corollary, page 139] or [106, Theorem 19.3].
Theorem 1.89 (Auslander and Buchsbaum). Suppose that R is a regular local ring. Then R is a UFD. Proof. [15] or [107, Theorem 48] or [106, Theorem 20.3].
Chapter 2
Aﬃne Varieties
In this chapter we deﬁne aﬃne and quasiaﬃne varieties and their regular functions and regular maps. We develop the basic properties of aﬃne varieties. Recall that throughout this book, k will be a ﬁxed algebraically closed ﬁeld. In Sections 2.1–2.4 we develop a correspondence between the commutative algebra of ﬁnitely generated kalgebras which are domains (or reduced) and the geometry of algebraic varieties (or algebraic sets) in an aﬃne space. In Section 2.5, we study the open sets in the Zariski topology on an aﬃne variety (which are the quasiaﬃne varieties) and the regular functions and regular maps on such open sets. We show in Lemma 2.83 and Proposition 2.93 that every aﬃne variety X has the basis for the Zariski topology consisting of the open sets D(f ) for f ∈ k[X] which are (isomorphic to) aﬃne varieties. In Section 2.6, we deﬁne rational maps on an aﬃne variety X. A rational map on X is determined by a regular map on a dense open subset U of X.
2.1. Aﬃne space and algebraic sets Aﬃne nspace over k is An = Ank = {(a1 , . . . , an )  a1 , . . . , an ∈ k}. An element p = (a1 , . . . , an ) ∈ An is called a point. The ring of regular functions on An is the kalgebra of polynomial mappings k[An ] = {f : An → A1  f ∈ k[x1 , . . . , xn ]}. Here k[x1 , . . . , xn ] is the polynomial ring over k in the variables x1 , . . . , xn . 27
28
2. Aﬃne Varieties
Since an algebraically closed ﬁeld is inﬁnite, the natural surjective ring homomorphism k[x1 , . . . , xn ] → k[An ] is an isomorphism by Theorem 1.4, as shown in Exercise 1.13. Thus we may identify the ring k[An ] with the polynomial ring k[x1 , . . . , xn ]. The zeros of a regular function f ∈ k[An ] are Z(f ) = {p ∈ An  f (p) = 0}. If T ⊂ k[An ] is a subset, then the set of common zeros of the elements of T is Z(T ) = {p ∈ An  f (p) = 0 for all f ∈ T }. A subset W of An is called an algebraic set if there exists a subset T of k[An ] such that W = Z(T ). If I is the ideal in k[An ] generated by T , then Z(T ) = Z(I). By Corollary 1.27 every algebraic set in An is the set of common zeros of a ﬁnite number of polynomials. Proposition 2.1. Suppose that I1 , I2 , {Iα }α∈S are ideals in k[An ] = k[x1 , . . . , xn ]. Then: 1) Z(I1 I2 ) = Z(I1 ) ∪ Z(I2 ). 2) Z( α∈S Iα ) = α∈S Z(Iα ). 3) Z(k[An ]) = ∅. 4) An = Z(0). Proof of 1). Suppose that p ∈ Z(I1 ) ∪Z(I2 ). Then p ∈ Z(I1 ) or p ∈ Z(I2 ). Thus for every f ∈ I1 we have f (p)= 0 or for every g ∈ I2 we have that r g(p) = 0. If f ∈ I1 I2 , thenf = i=1 fi gi for some f1 , . . . , fr ∈ I1 and g1 , . . . , gr ∈ I2 . Thus f (p) = fi (p)gi (p) = 0, so that p ∈ Z(I1 I2 ). Now suppose that p ∈ Z(I1 I2 ) and p ∈ Z(I1 ). Then there exists f ∈ I1 such that f (p) = 0. For any g ∈ I2 , we have f g ∈ I1 I2 so that f (p)g(p) = 0. Since f (p) = 0, we have that g(p) = 0. Thus p ∈ Z(I2 ). Proposition 2.1 tells us that: 1. The union of two algebraic sets is an algebraic set. 2. The intersection of any family of algebraic sets is an algebraic set. 3. ∅ and An are algebraic sets. We thus have a topology on An , deﬁned by taking the closed sets to be the algebraic sets. The open sets are the complements of algebraic sets in An (any union of open sets is open, any ﬁnite intersection of open sets is
2.1. Aﬃne space and algebraic sets
29
open, the empty set is open, and An is open). This topology is called the Zariski topology. If Y is a subset of An , we will denote the Zariski closure of Y in An by Y . Example 2.2. Suppose that I is a nontrivial ideal in k[A1 ] = k[x]; that is, I = (0) and I = k[x]. Then I = (f ) where f = (x − α1 ) · · · (x − αr ) for some α1 , . . . , αr ∈ k since k[x] is a PID and k is algebraically closed. Thus Z(I) = {α1 , . . . , αr }. The open sets in A1 are thus A1 , the complement of ﬁnitely many points in A1 , and ∅. We see that the Zariski topology is not Hausdorﬀ (to be Hausdorﬀ, distinct points must have disjoint neighborhoods). A nonempty subset Y of a topological space X is said to be irreducible if it cannot be expressed as a union Y = Y1 ∪ Y2 of two proper subsets, each of which is closed in Y (∅ is not irreducible). Example 2.3. A1 is irreducible as all proper closed subsets are ﬁnite and A1 is inﬁnite. Deﬁnition 2.4. An aﬃne algebraic variety is an irreducible closed subset of An . An aﬃne algebraic set is a closed subset of An . Given a subset Y of An , the ideal of Y in k[An ] is I(Y ) = {f ∈ k[An ]  f (p) = 0 for all p ∈ Y }. We now state Hilbert’s Nullstellensatz. Theorem 2.5. Let F be an algebraically closed ﬁeld, I be an ideal in the polynomial ring R = F [x1 , . . . , xn ], and f ∈ R be a polynomial which vanishes at all points of Z(I). Then f r ∈ I for some r ∈ Z+ . Our proof is based on Lang’s proof in [95, Chapter IX, Section 1]. To prove Theorem 2.5 we require some preliminary results. Proposition 2.6. Let A be a subring of a ring B and assume that B is integral over A. Let φ : A → L be a homomorphism into a ﬁeld L which is algebraically closed. Then φ has an extension to a homomorphism of B into L. Proof. Let P be the kernel of φ and S = A \ P . We have a natural commutative diagram B → S −1 B ↑ ↑ −1 A → S A = AP
30
2. Aﬃne Varieties
and φ induces a natural homomorphism φ of AP into L which factors φ, by deﬁning φ(x) x = φ y φ(y) −1 for x ∈ A and y ∈ S. Let C = S B, which is integral over AP . Let m be the maximal ideal of AP . By Proposition 1.56, there exists a maximal ideal n of C which lies over m. Then C/n is a ﬁeld which is an algebraic extension of AP /m, and AP /m is isomorphic to the subﬁeld φ(AP ) of L. Since the kernel of φ is m, φ induces a natural factorization AP → AP /m → L of φ. We can embed C/n into L since C/n is algebraic over AP /m and L is algebraically closed [95, Theorem 2.8, page 233], to make a commutative diagram C → C/n ↑ ↑ → A /m → L AP p giving a homomorphism of B into L which extends φ. Theorem 2.7. Let F be a ﬁeld and F [y1 , . . . , yn ] be a ﬁnitely generated F algebra. If F [y1 , . . . , yn ] is a ﬁeld, then F [y1 , . . . , yn ] is algebraic over F . Proof. Let L be an algebraic closure of F . Suppose that F [y] = F [y1 , . . . , yn ] is a ﬁeld which is not algebraic over F . Let t1 , . . . , tr (with r ≥ 1) be a transcendence basis of F [y] over F . The elements y1 , . . . , yn are algebraic over N = F (t1 , . . . , tr ) = F (t). Let fi (x) ∈ N [x] be the minimal polynomial of yi over N . If we multiply the fi by a suitable nonzero element of F [t] = F [t1 , . . . , tr ], we get polynomials in N [x], all of whose coeﬃcients lie in F [t]. Let a1 (t), . . . , an (t) be the leading coeﬃcients of these polynomials, and let a(t) = a1 (t) · · · an (t). Since a(t) = 0, there exist t1 , . . . , tr ∈ L such that a(t ) = a(t1 , . . . , tr ) = 0, by Theorem 1.4, so that ai (t ) = 0 for all i. Each yi is integral over the ring 1 1 F t1 , . . . , tr , ,..., . a1 (t) an (t) Consider the F algebra homomorphism Ψ : F [t1 , . . . , tr ] → L such that Ψ is the identity on F and Ψ(ti ) = ti for 1 ≤ i ≤ r. Let P be the kernel of Ψ. We have an extension of Ψ to Ψ(f ) f = Ψ : F [t]P → L deﬁned by Ψ g Ψ(g) for f ∈ F [t] and g ∈ F [t] \ P . Since ai (t) ∈ P for 1 ≤ i ≤ n, we have that y1 , . . . , yn are integral over F [t]P . By Proposition 2.6, we have an extension
2.1. Aﬃne space and algebraic sets
31
of Ψ to a homomorphism from F [t]P [y1 , . . . , yn ] into L which restricts to Ψ, giving an F algebra homomorphism F [y] → L which is an inclusion since F [y] is a ﬁeld. Thus F [y] is algebraic over F , giving a contradiction. Corollary 2.8. Let F be an algebraically closed ﬁeld and I be an ideal in the polynomial ring R = F [x1 , . . . , xn ]. Then either I = R or there exists α ∈ AnF such that f (α) = 0 for all f ∈ I. Proof. Suppose that I = R. Then I is contained in some maximal ideal m of R (as the ring R/I has a maximal ideal) and R/m is a ﬁeld, which is a ﬁnitely generated F algebra. By Theorem 2.7, this ﬁeld is algebraic over F and so is equal to F as F is algebraically closed. Thus there exist a1 , . . . , an ∈ F such that m = (x1 − a1 , . . . , xn − an ) and f (a1 , . . . , an ) = 0 for all f ∈ I since I ⊂ m. The above proof establishes the following corollary. Corollary 2.9. Suppose that F is an algebraically closed ﬁeld and I is an ideal in the polynomial ring F [x1 , . . . , xn ]. Then I is a maximal ideal if and only if there exist a1 , . . . , an ∈ F such that I = (x1 −a1 , x2 −a2 , . . . , xn −an ). We now prove Theorem 2.5. We may assume that f = 0. Let Y be a variable and let I be the ideal in R[Y ] generated by I and 1 − f Y . By Corollary 2.8, the ideal I = R[Y ], so there exist gi ∈ R[Y ] and hi ∈ I such that 1 = g0 (1 − Y f ) + g1 h1 + · · · + gr hr . Substitute f1 for Y and multiply by an appropriate positive power f m of f to clear denominators on the righthand side, to conclude that f m ∈ I. The following proposition is proven in Exercise 2.14. Proposition 2.10. The following statements hold: 1) Suppose that Y is a subset of An . Then I(Y ) is an ideal in k[An ]. 2) If T1 ⊂ T2 are subsets of k[An ], then Z(T2 ) ⊂ Z(T1 ). 3) If Y1 ⊂ Y2 are subsets of An , then I(Y2 ) ⊂ I(Y1 ). 4) For any two subsets Y1 , Y2 of An , we have I(Y1 ∪Y2 ) = I(Y1 )∩I(Y2 ). √ 5) For any ideal a of k[An ], we have I(Z(a)) = a. 6) For any subset Y of An , Z(I(Y )) = Y , the Zariski closure of Y . Theorem 2.11. A closed set W ⊂ An is irreducible if and only if I(W ) is a prime ideal.
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2. Aﬃne Varieties
Proof. Suppose that W is irreducible and f, g ∈ k[An ] are such that f g ∈ I(W ). Then W ⊂ Z(f g) = Z(f )∪Z(g). Thus W = (Z(f )∩W )∪(Z(g)∩W ) expresses W as a union of closed sets. Since W is irreducible, we have W ⊂ Z(f ) or W ⊂ Z(g). Thus f ∈ I(W ) or g ∈ I(W ). We have veriﬁed that I(W ) is a prime ideal. Now suppose that W is not irreducible. Then W = Z1 ∪ Z2 where Z1 and Z2 are proper closed subsets of W . The ideal I(Z1 ) is not a subset of I(Z2 ). If it were, then we would have Z2 = Z(I(Z2 )) ⊂ Z(I(Z1 )) = Z1 by 2) and 6) of Proposition 2.10, which is impossible. Thus there exists f1 ∈ k[An ] which vanishes on Z1 but not on Z2 . Similarly, there exists f2 ∈ k[An ] which vanishes on Z2 and not on Z1 . We have f1 f2 ∈ I(W ), but f1 , f2 ∈ I(W ). Thus I(W ) is not a prime ideal. Theorem 2.12. Every closed set in An is the union of ﬁnitely many irreducible ones. Proof. Suppose that Z is an algebraic set in An which is not the union of ﬁnitely many irreducible ones. Then Z = Z1 ∪ Z2 where Z1 and Z2 are proper closed subsets of Z and either Z1 or Z2 is not a ﬁnite union of irreducible closed sets. By induction, we can construct an inﬁnite chain of proper inclusions Z ⊃ W1 ⊃ W2 ⊃ · · · giving an inﬁnite chain of proper inclusions I(Z) ⊂ I(W1 ) ⊂ I(W2 ) ⊂ · · · of ideals in k[An ] (by 3) and 6) of Proposition 2.10), a contradiction to Corollary 1.27. Exercise 2.13. Prove 2), 3), and 4) of Proposition 2.1. Exercise 2.14. Prove Proposition 2.10. Exercise 2.15. Suppose that X ⊂ An is an aﬃne algebraic set. Show that I(X) = I(X). Exercise 2.16. Show that An is irreducible in the Zariski topology. Exercise 2.17. Suppose that k[x1 , . . . , xn ] is a polynomial ring over k. Suppose that I ⊂ k[x1 , . . . , xn ] in an ideal. Let R = k[x1 , . . . , xn ]/I. Let xi be the class of xi in R, so that R = k[x1 , . . . , xn ]. Show that an ideal m in R is a maximal ideal of R if and only if there exist a1 , . . . , an ∈ k such that m = (x1 − a1 , x2 − a2 , . . . , xn − an ).
2.2. Regular functions and regular maps of aﬃne algebraic sets
33
Exercise 2.18. Suppose that F ∈ k[x1 , . . . , xn ] is a nonzero nonunit ( ∈ k). Show that Z(F ) ⊂ An is irreducible if and only if F is a positive power of an irreducible element of k[x1 , . . . , xn ]. Warning: The conclusion of this exercise can be false if k is not algebraically closed. Exercise 2.19. Let Y = Z(x21 − x2 x3 , x1 x3 − x1 ). Show that Y is a union of three irreducible components. Describe them and ﬁnd their prime ideals. Exercise 2.20. Identify A2 with A1 × A1 in the natural way. Show that the Zariski topology on A2 is not the product topology of the Zariski topology on the two copies of A1 . Exercise 2.21. Suppose that X is an irreducible topological space. a) Suppose that U is a nonempty open subset. Show that U is irreducible. b) Suppose that U1 and U2 are nonempty open sets. Show that U1 ∩ U2 = ∅.
2.2. Regular functions and regular maps of aﬃne algebraic sets Deﬁnition 2.22. Suppose X ⊂ An is a closed set. The regular functions on X are the polynomial maps on X, k[X] = {f : X → A1  f ∈ k[An ]}, which is a subalgebra of the kalgebra Map(X, A1 ) of maps from X to A1 . We have a natural surjective kalgebra homomorphism, given by restriction, k[An ] → k[X]. An element f ∈ k[An ] is in the kernel if and only if f (q) = 0 for all q ∈ X, which holds if and only if f ∈ I(X). Thus k[X] ∼ = k[An ]/I(X). We have that k[X] is a reduced ring by Exercise 2.15. Deﬁnition 2.23. Suppose that X is an aﬃne algebraic set. If T ⊂ k[X], then ZX (T ) = {p ∈ X  f (p) = 0 for all f ∈ T }, a subset of X. Suppose that Y ⊂ X is a subset. Then IX (Y ) = {f ∈ k[X]  f (p) = 0 for all p ∈ Y }. We readily verify that IX (Y ) is an ideal in k[X]. When there is no ambiguity, we will usually write Z(T ) for ZX (T ) and I(Y ) for IX (Y ).
34
2. Aﬃne Varieties
Lemma 2.24. Suppose that X is a closed subset of An . Let res : k[An ] → k[X] be the restriction map. 1) Suppose that Y ⊂ X. Then res−1 (IX (Y )) = IAn (Y ). 2) Suppose that I is an ideal in k[X]. Then ZAn (res−1 (I)) = ZX (I). Proof. The map res : k[An ] → k[X] is surjective with kernel IAn (X). We ﬁrst prove 1). Since Y ⊂ X, f ∈ k[An ] vanishes on Y if and only if the restriction res(f ) of f to X vanishes on Y . Thus formula 1) holds. Now we prove 2). For p ∈ ZX (I) and f ∈ k[An ], f (p) = res(f )(p) since ZX (I) ⊂ X. Thus f ∈ res−1 (I) implies f (p) = 0 for all p ∈ ZX (I), so that ZX (I) ⊂ ZAn (res−1 (I)). Now 0 ∈ I since I is an ideal, so IAn (X) ⊂ res−1 (I). Suppose that p ∈ ZAn (res−1 (I)). Then p ∈ ZAn (IAn (X)) = X. Since res is surjective, p ∈ X, and f (p) = 0 for all f ∈ res−1 (I), we have that g(p) = 0 for all g ∈ I. Thus p ∈ ZX (I). We see from Lemma 2.24 that the natural isomorphism of k[X] with An (X) identiﬁes the ideal IX (Y ), for Y a subset of X, with the quotient IAn (Y )/IAn (X).
k[An ]/I
Theorem 2.25. Suppose that X is a closed subset of An . Then the conclusions 1)–4) of Proposition 2.1 hold, with An replaced with X. We thus obtain a topology on a closed subset X of An , where the closed sets are ZX (I) for ideals I ⊂ k[X]. This topology is the restriction topology of the Zariski topology on An . We call this the Zariski topology on X. If Y is a subset of X, Y will denote the Zariski closure of Y in X. A closed irreducible subset of an aﬃne variety X is called a subvariety of X. An open subset of an aﬃne variety is called a quasiaﬃne variety. An aﬃne algebraic set is a closed subset of An . A quasiaﬃne algebraic set is an open subset of a closed subset of An . Theorem 2.26. Suppose that X is a closed subset of An . Then the conclusions 1)–6) of Proposition 2.10 hold, with An replaced with X. Proof. We have already observed that the conclusion 1) holds. We will establish that 5) of Proposition 2.10 holds for algebraic sets. We ﬁrst establish that √ res−1 (a) = res−1 ( a). (2.1)
2.2. Regular functions and regular maps of aﬃne algebraic sets
35
To prove this, observe that √ f ∈ res−1 ( a) if and only if
res(f n ) = res(f )n ∈ a for some positive integer n
if and only if
f n ∈ res−1 (a) f ∈ res−1 (a).
if and only if We have that
IAn (ZAn (res−1 (a))) by 1) and 2) of Lemma 2.24 res−1 (IX (ZX (a))) = = res−1 (a) by 5) of Proposition 2.10. Thus IX (ZX (a)) = res(res−1 (IX (ZX (a)))) = res(
res−1 (a)) =
√
a
by (2.1).
Theorem 2.27. Suppose that X is a closed subset of An . A closed set W ⊂ X is irreducible if and only if IX (W ) is a prime ideal in k[X]. Deﬁnition 2.28. Suppose X ⊂ An is a closed set. A map φ : X → Am is a regular map if there exist f1 , . . . , fm ∈ k[X] such that φ = (f1 , f2 , . . . , fm ). A regular map φ = (f1 , . . . , fm ) : X → Am induces a kalgebra homomorphism φ∗ : k[Am ] → k[X] by φ∗ (g) = g ◦ φ for g ∈ k[Am ]. Writing k[Am ] = k[y1 , . . . , ym ], we see that φ∗ is determined by φ∗ (yi ) = fi for 1 ≤ i ≤ m. For g = g(y1 , . . . , ym ) ∈ k[Am ], we have φ∗ (g) = g(φ∗ (y1 ), . . . , φ∗ (ym )) = g(f1 , . . . , fm ). Example 2.29. Let C = Z(y 2 − x(x2 − 1)) ⊂ A2 . Let φ : C → A1 be the projection on the ﬁrst factor, so that φ(u, v) = u for (u, v) ∈ C. φ∗ : k[A1 ] = k[t] → k[C] = k[x, y]/(y 2 − x(x2 − 1)) = k[x, y] is the kalgebra homomorphism induced by t → x. Here x is the class of x in k[C] and y is the class of y in k[C]. Example 2.30. Let ψ : A1 → A2 be deﬁned by ψ(s) = (s2 , s3 ) for s ∈ A1 . ψ ∗ : k[A2 ] = k[x, y] → k[A1 ] = k[t] is the kalgebra homomorphism induced by x → t2 and y → t3 . Proposition 2.31. Suppose that X is a closed subset of An , Y is a closed subset of Am , and φ : X → Am is a regular map. Then φ(X) ⊂ Y if and only if I(Y ) ⊂ kernel φ∗ : k[Am ] → k[X].
36
2. Aﬃne Varieties
Proof. We have that φ(X) ⊂ Y holds if and only if h(φ(p)) = 0 for all h ∈ I(Y ) and p ∈ X, which holds if and only if φ∗ (h) = 0 for all h ∈ I(Y ), which holds if and only if I(Y ) ⊂ kernel φ∗ . Corollary 2.32. Suppose set and φ : X → Am √ that X is∗ an aﬃne algebraic ∗ is a regular map. Then kernel —φ = kernel φ , and φ(X) = Z(kernel φ∗ ), where φ(X) is the Zariski closure of φ(X) in An . √ Proof. The fact that kernel φ∗ = kernel φ∗ follows from the fact that k[Am ]/kernel φ∗ is isomorphic to a subring of the reduced ring k[X]. Let S = {closed subsets Y of Am  φ(X) ⊂ Y }. A closed set Y is in S if and only if I(Y ) ⊂ kernel φ∗ by Proposition 2.31 and W = Z(kernel φ∗ ) ∈ S since I(W ) = kernel φ∗ by 5) of Proposition 2.10. Thus I(Y ) = kernel φ∗ , Y ∈S
and
∗
Z(kernel φ ) = Z =
I(Y )
Y ∈S
Z(I(Y )) by 2) of Proposition 2.1
Y ∈S
=
Y by 6) of Proposition 2.10
Y ∈S
= φ(X).
Example 2.33. The image of a regular map may be neither closed nor open. Let φ : A2 → A2 be deﬁned by φ(u, v) = (u, uv). Then φ(A2 ) = A2 \ {(0, y)  y = 0}. Deﬁnition 2.34. Suppose that X ⊂ An and Y ⊂ Am are closed sets. A map φ : X → Y is a regular map if φ is the restriction of the range of a ˜ ⊂Y. regular map φ˜ : X → Am , such that φ(X) Suppose that φ : X → Y is a regular map as in the deﬁnition. Let π : k[Am ] = k[y1 , . . . , ym ] → k[Y ] be the restriction map, which has kernel I(Y ). ˜ We have that φ(X) ⊂ Y , so I(Y ) ⊂ kernel(φ˜∗ ) by Proposition 2.31. Thus ∗ φ˜ induces a kalgebra homomorphism φ∗ : k[Y ] ∼ = k[Am ]/I(Y ) → k[X]. Thus writing φ˜ = (f1 , . . . , fm ), where f1 , . . . , fm ∈ k[X], and k[Y ] = k[y1 , . . . , y m ], where y i = π(yi ) for 1 ≤ i ≤ m are the restrictions of yi to Y , we have that fi = φ∗ (y i ) = φ˜∗ (yi ) for 1 ≤ i ≤ m, and for g(y1 , . . . , y m ) ∈ k[Y ], we have that φ∗ (g) = g(φ∗ (y 1 ), . . . , φ∗ (y m )) = g(f1 , . . . , fm ).
2.2. Regular functions and regular maps of aﬃne algebraic sets
37
Deﬁnition 2.35. A regular map φ : X → Y is dominant if φ(X) is dense in Y . Proposition 2.36. Suppose that φ : X → Y is a regular map of aﬃne algebraic sets and Z ⊂ Y is a closed set. Then φ−1 (Z) = Z(φ∗ (I(Z))). Corollary 2.37. Suppose that X and Y are aﬃne algebraic sets and φ : X → Y is a regular map. Then φ is continuous. Proposition 2.38. Suppose that φ : X → Y is a regular map of aﬃne algebraic sets. Then φ∗ : k[Y ] → k[X] is injective if and only if φ(X) = Y . Proof. We have that φ(X) = Z(kernel φ∗ ) by Corollary 2.32 and Z(kernel φ∗ ) = Y if and only if kernel φ∗ = I(Y ) = (0).
Lemma 2.39. Suppose that φ : X → Y and ψ : Y → Z are regular maps of aﬃne algebraic sets. Then ψ ◦φ : X → Z is a regular map of aﬃne algebraic sets. Further, (ψ ◦ φ)∗ = φ∗ ◦ ψ ∗ : k[Z] → k[X]. Proposition 2.40. Suppose that X and Y are aﬃne algebraic sets and Λ : k[Y ] → k[X] is a kalgebra homomorphism. Then there is a unique regular map φ : X → Y such that φ∗ = Λ. Proof. We ﬁrst prove existence. We have that Y is a closed subset of An , giving a surjective kalgebra homomorphism π : k[An ] = k[y1 , . . . , yn ] → k[Y ]. Let y i = π(yi ) for 1 ≤ i ≤ n, so that k[Y ] = k[y1 , . . . , y n ]. Deﬁne a regular map φ˜ : X → An by φ˜ = (Λ(y 1 ), . . . , Λ(y n )). Suppose f (y1 , . . . , yn ) ∈ k[An ]. Then φ˜∗ (f (y1 , . . . , yn )) = f ◦ φ˜ = f (Λ(y 1 ), . . . , Λ(y n )) = Λ(f (y 1 , . . . , y n )) = Λ(π(f )) since Λ and π are kalgebra homomorphisms. Thus φ˜∗ = Λ ◦ π, and so I(Y ) = kernel π ⊂ kernel φ˜∗ . ˜ Thus φ(X) ⊂ Y by Proposition 2.31. Let φ : X → Y be the induced regular map. The map φ∗ is the homomorphism induced by φ˜∗ on the quotient k[y1 , . . . , yn ]/I(Y ) = k[Y ]. Thus φ∗ = Λ. We now prove uniqueness. Suppose that φ : X → Y and ψ : X → Y are regular maps such that φ∗ = ψ ∗ = Λ. Suppose that φ = ψ. Then there exists p ∈ X such that φ(p) = ψ(p). Let q1 = φ(p) and q2 = ψ(p). There exists f ∈ I(q1 ) \ I(q2 ) since I(q1 ) and I(q2 ) are distinct maximal ideals of k[Y ]. Thus f (q1 ) = 0 but f (q2 ) = 0. We have (φ∗ f )(p) = f (φ(p)) = f (q1 ) = 0
38
but
2. Aﬃne Varieties
(ψ ∗ f )(p) = f (ψ(p)) = f (q2 ) = 0.
Thus φ∗ = ψ ∗ , a contradiction, so we must have that φ = ψ, and thus φ is unique. Deﬁnition 2.41. Suppose that X and Y are aﬃne algebraic sets. We say that X and Y are isomorphic if there are regular maps φ : X → Y and ψ : Y → X such that ψ ◦ φ = idX and φ ◦ ψ = idY . Proposition 2.42. Suppose that φ : X → Y is a regular map of aﬃne algebraic sets. Then φ is an isomorphism if and only if φ∗ : k[Y ] → k[X] is an isomorphism of kalgebras. Proof. First suppose that the regular map φ : X → Y is an isomorphism. Then there exists a regular map ψ : Y → X such that ψ ◦ φ = idX and φ ◦ ψ = idY . Thus (ψ ◦ φ)∗ = idk[X] and (φ ◦ ψ)∗ = idk[Y ] . Now (ψ ◦ φ)∗ = φ∗ ◦ ψ ∗ and (φ ◦ ψ)∗ = ψ ∗ ◦ φ∗ by Lemma 2.39, so φ∗ : k[Y ] → k[X] is a kalgebra isomorphism with inverse ψ ∗ . Now assume that φ∗ : k[Y ] → k[X] is a kalgebra isomorphism. Let Λ : k[X] → k[Y ] be the kalgebra inverse of φ∗ . By Proposition 2.40, there exists a unique regular map ψ : Y → X such that ψ ∗ = Λ. Now by Lemma 2.39, (ψ ◦ φ)∗ = φ∗ ◦ ψ ∗ = φ∗ ◦ Λ = idk[X] and
(φ ◦ ψ)∗ = ψ ∗ ◦ φ∗ = Λ ◦ φ∗ = idk[Y ] .
Since (idX )∗ = idk[X] , by uniqueness in Proposition 2.40, we have that ψ ◦ φ = idX . Similarly, φ ◦ ψ = idY . Thus φ is an isomorphism. Deﬁnition 2.43. Suppose that X is an aﬃne algebraic set and t1 , . . . , tr ∈ k[X] are such that t1 , . . . , tr generate k[X] as a kalgebra. Let φ : X → Ar be the regular map deﬁned by φ = (t1 , . . . , tr ). Then t1 , . . . , tr are called coordinate functions on X and φ is called a closed embedding. That the map φ of Deﬁnition 2.43 is called a closed embedding is justiﬁed by the following proposition. Proposition 2.44. Suppose that X is an aﬃne algebraic set and t1 , . . . , tr are coordinate functions on X. Let φ : X → Ar be the associated closed embedding φ = (t1 , . . . , tr ), and let Y = φ(X). Then Y is a closed subset of Ar with ideal I(Y ) = kernel φ∗ : k[Ar ] → k[X], and regarding φ as a regular map to Y , we have that φ : X → Y is an isomorphism. Proof. Let Y be the Zariski closure of Y in Ar . We have I(Y ) = kernel φ∗ by Corollary 2.32. Thus φ∗ : k[Ar ] → k[X] is onto with kernel I(Y ), so that
2.2. Regular functions and regular maps of aﬃne algebraic sets
39
now regarding φ as a regular map from X to Y , we have that φ∗ : k[Y ] = k[Ar ]/I(Y ) → k[X] is an isomorphism. Thus φ : X → Y is an isomorphism by Proposition 2.42, and so Y = Y . Our deﬁnition of An = {p = (a1 , . . . , an )  a1 , . . . , an ∈ k} naturally gives us particular coordinate functions, namely the coordinate functions xi for 1 ≤ i ≤ n. If B = (bij ) is an invertible n × n matrix with coeﬃcients in k n and c = (c1 , . . . , cn ) is a vector in k , then yi = nj=1 bij xj + ci for 1 ≤ i ≤ n deﬁnes another choice of coordinate functions y1 , . . . , yn on An . We deduce the following from Proposition 2.36. Lemma 2.45. Suppose that φ : X → Y is a regular map of aﬃne algebraic sets and t1 , . . . , tn are coordinate functions on Y (giving a closed embedding of Y in An ). Suppose that p ∈ Y . Then I(p) = (t1 − t1 (p), . . . , tn − tn (p)) and I(φ−1 (p)) = (φ∗ (t1 ) − t1 (p), . . . , φ∗ (tn ) − tn (p)). The results of this section show that there is an equivalence of categories between the category of reduced ﬁnitely generated kalgebras and their kalgebra homomorphisms, and the category of aﬃne algebraic sets in Ank for some n and regular maps between aﬃne algebraic sets. Further, this equivalence restricts to give an equivalence of categories between the category of ﬁnitely generated kalgebras which are domains and their kalgebra homomorphisms, and the category of aﬃne varieties in Ank for some n and regular maps between aﬃne varieties. Exercise 2.46. Is formula 2) of Lemma 2.24 always true if I is replaced by a subset T of k[X]? Exercise 2.47. Prove Theorem 2.25. Exercise 2.48. Prove 4) and 6) of Theorem 2.26. Exercise 2.49. Prove Theorem 2.27. Exercise 2.50. Prove Proposition 2.36 and deduce Lemma 2.45. Exercise 2.51. Prove Lemma 2.39. Exercise 2.52. Let k[An ] = k[x1 , . . . , xn ]. a) Suppose that a, b, c, d, e, f ∈ k with ae − bd = 0. Show that the map φ : A2 → A2 deﬁned by φ = (ax1 + bx2 + c, dx1 + ex2 + f ) is an isomorphism. Give an explicit description of φ−1 .
40
2. Aﬃne Varieties
b) Suppose that φ : An → An is deﬁned by φ = (f1 , . . . , fn ) where fi =
n
aij xj + bi
j=1
with aij , bi ∈ k and Det(aij ) = 0. Show that φ is an isomorphism of An . Give an explicit description of φ−1 . Exercise 2.53. A quadratic polynomial in k[x1 , x2 ] is a polynomial all of whose terms are monomials of degree ≤ 2. a) Let X = Z(x2 − x21 ) ⊂ A2 . Show that X is a variety and that X∼ = A1 . b) Let Y = Z(x1 x2 − 1) ⊂ A2 . Show that Y is a variety and that Y ∼ = A1 . c) Let f be any irreducible quadratic polynomial in k[x1 , x2 ], and let W = Z(f ). Assume that k has characteristic = 2. Show that W is isomorphic to X or to Y . d) Can you give a proof of c) which is valid when k has characteristic 2? Exercise 2.54. Let X = Z(x22 − x31 ) ⊂ A2 . Consider the regular map φ : A1 → X deﬁned by φ(t) = (t2 , t3 ) for t ∈ A1 . a) Show that φ is a bijection. b) Show that φ is not an isomorphism.
2.3. Finite maps In this section we interpret the algebraic notion of a ring extension being ﬁnite geometrically. Deﬁnition 2.55. Suppose that f : X → Y is a regular map of aﬃne varieties. We say that f is a ﬁnite map if k[X] is integral and thus a ﬁnitely generated module over the subring f ∗ (k[Y ]). In the case when f : X → Y is dominant, so that f ∗ : k[Y ] → k[X] is injective, it may sometimes be convenient to abuse notation and identify k[Y ] with its isomorphic image f ∗ (k[Y ]). In this way, we may sometimes write t for f ∗ (t) if t ∈ k[Y ]. Theorem 2.56. Suppose that f : X → Y is a ﬁnite map of aﬃne varieties. Then f −1 (p) is a ﬁnite set for all p ∈ Y . Proof. Let t1 , . . . , tn be coordinate functions on X. It suﬃces to show that each ti assumes only ﬁnitely many values on f −1 (p). Since k[X] is integral
2.3. Finite maps
41
over f ∗ (k[Y ]), each ti satisﬁes a dependence relation m−1 ∗ tm + · · · + f ∗ (b0 ) = 0 i + f (bm−1 )ti
with m ∈ Z+ and b0 , . . . , bm−1 ∈ k[Y ]. Suppose that q ∈ f −1 (p). Then 0 = ti (q)m + f ∗ (bm−1 )(q)ti (q)m−1 + · · · + f ∗ (b0 )(q) = ti (q)m + bm−1 (p)ti (q)m−1 + · · · + b0 (p). Thus ti (q) must be one of the ≤ m roots of this equation.
Theorem 2.57. Suppose that f : X → Y is a dominant ﬁnite map of aﬃne varieties. Then f is surjective. Proof. Let q ∈ Y . Let mq = I(q) be the ideal of q in k[Y ]. Then f −1 (q) = Z(f ∗ (mq )) by Proposition 2.36. Then f −1 (q) = ∅ if and only if f ∗ (mq )k[X] = k[X]. By Proposition 1.56, f ∗ (mq )k[X] is a proper ideal of k[X] since f ∗ (k[Y ]) ∼ = k[Y ] and mq is a prime ideal of k[Y ]. Corollary 2.58. A ﬁnite map f : X → Y of aﬃne varieties is a closed map. Proof. It suﬃces to verify that if Z ⊂ X is an irreducible closed subset, then f (Z) is closed in Y . Let W = f (Z) be the closure of f (Z) in Y . Let f = f Z : Z → W . The homomorphism f ∗ : k[Y ] → k[X] induces the ∗ homomorphism f : k[W ] = k[Y ]/I(W ) → k[X]/I(Z) = k[Z] by Proposi∗ tion 2.31. The ring k[Z] is integral over f (k[W ]) since k[X] is integral over f ∗ (k[Y ]). Thus f : Z → W is a dominant ﬁnite map, which is surjective by Theorem 2.57. Thus f (Z) = W is closed in Y . Theorem 2.59. Suppose that X is an aﬃne algebraic set. Then there exists a dominant ﬁnite map φ : X → Ar for some r. Proof. There exist, by Theorem 1.53, y1 , . . . , yr ∈ k[X] such that k[y1 , . . . , yr ] is a polynomial ring and k[X] is integral over k[y1 , . . . , yr ]. Deﬁne a regular map φ : X → Ar by φ(p) = (y1 (p), y2 (p), . . . , yr (p)) for p ∈ X. Let t1 , . . . , tr be the natural coordinate functions on Ar . Then φ∗ : k[Ar ] → k[X] is the kalgebra homomorphism deﬁned by φ∗ (ti ) = yi for 1 ≤ i ≤ r. Thus φ∗ is injective and k[X] is integral over k[Ar ], and so φ : X → Ar is dominant and ﬁnite. Exercise 2.60. Suppose that φ : X → Y is a regular map of aﬃne varieties such that φ−1 (p) is a ﬁnite set for all p ∈ Y and φ∗ : k[Y ] → k[X] is injective. Is φ necessarily a ﬁnite map? Exercise 2.61. Suppose that φ : X → Y is a surjective regular map of aﬃne varieties such that φ−1 (p) is a ﬁnite set for all p ∈ Y and φ∗ : k[Y ] → k[X] is injective. Is φ necessarily a ﬁnite map?
42
2. Aﬃne Varieties
Exercise 2.62. Suppose that φ : X → Y is a dominant regular map of varieties. Can there exist a prime ideal I in k[Y ] such that Ik[X] = k[X]? Compare this exercise with the conclusions of Proposition 1.56.
2.4. Dimension of algebraic sets Theorem 2.59 gives us a geometric way to deﬁne the dimension of an aﬃne variety: an aﬃne variety X has dimension r if there is a dominant ﬁnite map from X to Ar . We will give an algebraic deﬁnition of dimension and show that it agrees with the geometric deﬁnition. An introduction to dimension theory in rings can be found in Section 1.8. An irreducible topological space is deﬁned before Example 2.3. Deﬁnition 2.63. Suppose that X is a topological space. The dimension of X, denoted dim X, is the supremum of all numbers n such that there exists a chain (2.2)
Z0 ⊂ Z1 ⊂ · · · ⊂ Zn
of distinct irreducible closed subsets of X. The dimension of an aﬃne algebraic set or quasiaﬃne algebraic set is its dimension as a topological space. This deﬁnition of dimension works well for the Zariski topology but does not agree with the usual deﬁnition of dimension of Cn with the Euclidean topology since the only irreducible subsets in Cn (in the Euclidean topology) are the single points. We will see that the dimension of a complex variety is equal to its dimension in the Euclidean topology in Theorem 10.45. Proposition 2.64. Suppose that X is an aﬃne algebraic set. Then the dimension of X is equal to the dimension of the ring k[X] of regular functions on X. Proof. By Theorems 2.26, 2.25, and 2.27, chains Z0 ⊂ Z1 ⊂ · · · ⊂ Zn of distinct irreducible closed subsets of X correspond 11 to chains IX (Zn ) ⊂ IX (Zn−1 ) ⊂ · · · ⊂ IX (Z0 ) of distinct prime ideals in k[X].
From Theorem 1.63 we obtain the following two propositions. Proposition 2.65. The dimension of an aﬃne variety is ﬁnite. This proposition follows since if X is an aﬃne variety, then the domain k[X] is a ﬁnitely generated kalgebra, so its quotient ﬁeld is a ﬁnitely generated extension ﬁeld of k, and thus it has a ﬁnite transcendence basis.
2.4. Dimension of algebraic sets
43
Proposition 2.66. The dimension of An is dim An = n. Proposition 2.67. Suppose that X is an aﬃne algebraic set and V1 , . . . , Vr are the irreducible components of X (the distinct largest irreducible sets contained in X). Then dim X = max{dim Vi }. Proof. Suppose that (2.2) is a chain of irreducible closed subsets of X. Then Zn is contained in Vi for some i since Zn is irreducible and the Vi are the irreducible components of X. A chain (2.2) is maximal if the chain cannot be lengthened by adding an additional irreducible closed set somewhere in the chain. Corollary 2.68. Suppose that X is an aﬃne variety. Then every maximal chain of distinct prime ideals in k[X] has the same ﬁnite length equal to the dimension of k[X]. Proof. The proof is by induction on the dimension of X. If dim X = 0, then k[X] = k is a ﬁeld and the corollary is trivially true. Suppose dim X = n > 0 and the corollary is true for varieties of dimension < n. Suppose that P 0 ⊂ P1 ⊂ · · · ⊂ Pm is a maximal chain of distinct prime ideals in k[X]. 1 + dim k[X]/P1 = ht(P1 ) + dim k[X]/P1 = dim k[X] = n by Theorem 1.64, and so dim k[X]/P1 = n − 1. Now P1 /P1 ⊂ P2 /P1 ⊂ · · · ⊂ Pm /P1 is a maximal chain of distinct prime ideals in k[X]/P1 , and k[X]/P1 = k[ZX (P1 )] so by induction on n, the variety ZX (P1 ) has dimension m − 1. Thus n − 1 = dim k[X]/P1 = m − 1, and so m = n = dim X.
Some examples of noncatenary Noetherian rings (rings which do not satisfy the conclusions of Corollary 2.68) are given by Nagata in [121, Appendix A1]. Corollary 2.69. Suppose that X is an aﬃne variety. Then every maximal chain of distinct irreducible closed subsets of X has the same length (equal to dim X).
44
2. Aﬃne Varieties
Proof. Suppose that (2.2) is a maximal chain of distinct irreducible closed subsets of X. Since X is irreducible, we must have that Zn = X and Z0 is a point. Taking the sequence of ideals of (2.2), we have a maximal chain (0) = IX (Zn ) ⊂ · · · ⊂ IX (Z0 ) of distinct prime ideals in k[X]. By Corollary 2.68 and Proposition 2.64, we have that n = dim k[X] = dim X. Proposition 2.70. Suppose that X is an aﬃne variety and Y is a nontrivial open subset of X. Then dim X = dim Y . Proof. Suppose that (2.3)
Z0 ⊂ Z1 ⊂ · · · ⊂ Zn
is a sequence of distinct closed irreducible subsets of Y . Let Z i be the Zariski closure of Zi in X for 0 ≤ i ≤ n. Then (2.4)
Z0 ⊂ Z1 ⊂ · · · ⊂ Zn
is a sequence of distinct closed irreducible subsets of X since Z i ∩ Y = Zi , as Zi is closed in Y and Y is open in X. Thus dim Y ≤ dim X. In particular, dim Y is ﬁnite, so we can choose a maximal such chain (2.3). Since the chain is maximal, Z0 is a point and Zn = Y . Now if W is an irreducible closed subset of X such that the open subset W ∩ Y of W is nonempty, we then have that W ∩ Y is dense in W . In particular, if A ⊂ B are irreducible closed subsets of X such that A ∩ Y = ∅ and A ∩ Y = B ∩ Y , then we have that A = A ∩ Y = B ∩ Y = B. Thus we have that (2.4) is a maximal chain in X, and hence dim Y = dim X by Corollary 2.69. Noether’s normalization lemma, Theorem 1.53, can be used to compute the dimension of any aﬃne variety (by Theorem 1.63). In fact, we see that if φ : X → Ar is a dominant ﬁnite map, then r = dim X. Suppose that R is a Noetherian ring and I ⊂ R is an ideal. Since R is Noetherian, the ideal I has only a ﬁnite number of minimal primes P1 , . . . , Pr (Section 1.6). We have that√I ⊂ P1 ∩ · · · ∩ Pr and I = P1 ∩ · · · ∩ Pr if and only if I is a radical ideal ( I = I). Suppose that X is a closed subset of An . Let V1 , . . . , Vr be the irreducible components of X; that is, V1 , . . . , Vr are the irreducible closed subsets of X such that X = V1 ∪ · · · ∪ Vr and we have Vi ⊂ Vj if i = j. Then the minimal primes of I(X) are Pi = I(Vi ) for 1 ≤ i ≤ r. The prime ideals P i = IX (Vi ) = Pi /I(X) are the minimal primes of the ring k[X], that is, the minimal primes of the zero ideal. We have that P 1 ∩ · · · ∩ P r = (0) since I(X) is a radical ideal. The following theorem follows from Krull’s principal ideal theorem (Theorem 1.65).
2.4. Dimension of algebraic sets
45
Theorem 2.71. Suppose that X is an aﬃne variety and f ∈ k[X]. Then: 1) If f is not 0 and is not a unit in k[X], then ZX (f ) is a nonempty algebraic set, all of whose irreducible components have dimension equal to dim X − 1. 2) If f is a unit in k[X], then ZX (f ) = ∅. 3) If f = 0, then ZX (f ) = X. Proposition 2.72. Suppose that X is a variety in An . Then X has dimension n − 1 if and only if I(X) = (f ) where f ∈ k[An ] = k[x1 , . . . , xn ] is an irreducible polynomial. Proof. Suppose that I(X) = (f ) where f is irreducible. Then (f ) is a prime ideal by Proposition 1.30 (since k[An ] is a unique factorization domain) which has height 1 by Theorem 1.65. Thus dim X = dim k[X] = dim k[An ] − ht(f ) = dim k[An ] − 1 = dim An − 1 by Theorem 1.64. Now suppose that X has dimension n − 1. Then the prime ideal I(X) has height 1 by Theorem 1.64. Since the polynomial ring k[An ] is a unique factorization domain, I(X) is a principal ideal generated by an irreducible element by Proposition 1.66. If Y is an aﬃne or quasiaﬃne algebraic set contained in an aﬃne variety X, then we deﬁne the codimension of Y in X to be codimX (Y ) = dim X − dim Y . If Y is a subvariety of an algebraic variety X, then we have that codimX (Y ) is the height of the prime ideal IX (Y ) in k[X]. More generally, suppose that X is an ndimensional aﬃne variety and Y ⊂ X is an algebraic set with irreducible components Y1 , . . . , Ys . We have codimX (Y ) = = = =
dim(X) − dim(Y ) dim(X) − max{dim(Yi )} (by Proposition 2.67) min{n − dim(Yi )} min{height IX (Yi )}.
We will call a onedimensional aﬃne variety a curve and a twodimensional aﬃne variety a surface. An ndimensional aﬃne variety is called an nfold. We see that if C is a curve, then the prime ideals in k[C] are just the maximal ideals (corresponding to the points of C) and the zero ideal (corresponding to the curve C). If S is a surface and k[S] is a UFD, then the prime ideals in k[S] are the maximal ideals (corresponding to the points of S), principal ideals generated by an irreducible element (corresponding to the curves lying on S), and the zero ideal (corresponding to the surface S).
46
2. Aﬃne Varieties
If X is an nfold with n ≥ 3, then the prime ideals in k[X] are much more complicated. The prime ideals in k[A3 ] are the maximal ideals (height 3), height 2 prime ideals, principal ideals generated by an irreducible element (height 1), and the zero ideal (height 0). The height 2 prime ideals p , which correspond to curves in A3 , can be extremely complicated, although many times one has the nice case where p is generated by two elements. A height 2 prime p in k[A3 ] requires at least two generators but there is no upper bound on the minimum number of generators required to generate such a prime p. Some examples showing this are given in [5]. A complete analysis of the generators of monomial space curves is given by J¨ urgen Herzog in [76]. The book [130] by Judith Sally gives an excellent introduction to the question of the number of generators of an ideal in a local ring. Exercise 2.73. This exercise gives a criterion which can be used to determine the minimal number of generators of an ideal in a nonlocal ring. Suppose that R = k[x1 , . . . , xn ] is a ﬁnitely generated kalgebra, I ⊂ R is an ideal, and m is a maximal ideal of R. Let μ(I) be the minimal number of generators of I. Using Lemmas 1.19 and 1.28, show that μ(I) ≥ μ(Im ) = dimk Im /mIm . Exercise 2.74. Deﬁne a regular map Φ : A1 → A3 by Φ(t) = (t, t2 , t3 ) for t ∈ A1 . Let X be the image of Φ. a) Show that Φ is a ﬁnite map. b) Show that X is a variety (the image of Φ is Zariski closed). c) Show that Φ is injective. d) Determine if Φ : A1 → X is an isomorphism of varieties. e) Find a minimal set of generators of the ideal I(X); that is, ﬁnd a set of generators of I(X) with the smallest possible number of elements. Exercise 2.75. Deﬁne a regular map Φ : A1 → A3 by Φ(t) = (t2 , t3 , t4 ) for t ∈ A1 . Let X be the image of Φ. a) Show that Φ is a ﬁnite map. b) Show that X is a variety (the image of Φ is Zariski closed). c) Show that Φ is injective. d) Determine if Φ : A1 → X is an isomorphism of varieties. e) Find a minimal set of generators of the ideal I(X); that is, ﬁnd a set of generators of I(X) with the smallest possible number of elements. You may ﬁnd that Exercise 2.73 and the method of the next problem (Exercise 2.76) will be useful in this problem.
2.4. Dimension of algebraic sets
47
Exercise 2.76. Deﬁne a regular map Φ : A1 → A3 by Φ(t) = (t3 , t4 , t5 ) for t ∈ A1 . Let X be the image of Φ. a) Show that Φ is a ﬁnite map. b) Show that X is a variety (the image of Φ is Zariski closed). c) Show that Φ is injective. d) Determine if Φ : A1 → X is an isomorphism of varieties. e) Find a minimal set of generators of the ideal I(X); that is, ﬁnd a set of generators of I(X) with the smallest possible number of elements. You may ﬁnd the following outline of a solution helpful: i) Deﬁne a “weighting” on the variables x1 , x2 , x3 by setting wt(x1 ) = 3, wt(x2 ) = 4, and wt(x3 ) = 5. Deﬁne the weight of n to be wt(xl xm xn ) = 3l + 4m + 5n. Say the monomial xl1 xm 2 x 3 1 2 3 n x that an element g = almn xl1 xm 2 3 is weighted homogeneous of degree d if 3l + 4m + 5n = d whenever almn = 0. Every element f ∈ k[x1 , x2 , x3 ] has a unique expression f = i Fi where Fi is weighted homogeneous of degree i. ii) Show that f ∈ I(X) if and only if Fi ∈ I(X) for all i. Conclude that I(X) is generated by weighted homogeneous elements. iii) Show that I(X) is generated by the set of “binomials” A − B where A and B are monomials which have the same weighted degree. iv) Show that I(X) is generated by the set of weighted homogeneous binomials which are of one of the following three types: n m l n n l m xl1 − xm 2 x3 , x2 − x1 x3 , x3 − x1 x2 .
v) Make an (intelligent) guess of a set of minimal generators of I(X), consisting of weighted homogeneous binomials. Let J be the ideal generated by this set. Show that J contains all weighted homogeneous binomials, by induction on the weighted degree. Conclude that I(X) = J and your set generates I(X). vi) Now use Exercise 2.73 to show that you have found a minimal generating set. Exercise 2.77. Suppose that R is a ring. Recall (Section 1.5) that an element f ∈ R is called irreducible if f is not a unit, and f = ab with a, b ∈ R implies a or b is a unit. Suppose that X is an aﬃne variety and f ∈ k[X] is irreducible. Is ZX (f ) necessarily irreducible? Either prove this or give a counterexample. Exercise 2.78. Let X be the variety which is the image of A1 in A3 by the map φ(t) = (t3 , t4 , t5 ) of Exercise 2.76. Compute the dimension of X and the height of I(X) in k[A3 ].
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2. Aﬃne Varieties
Exercise 2.79. This exercise shows that the assumption that A is a domain is necessary in the statement of Theorem 1.64. Let V1 = Z(x) and V2 = Z(y, z) be algebraic sets in A3 . The sets V1 and V2 are irreducible, with I(V1 ) = (x) and I(V2 ) = (y, z) (you do not need to show this). Compute the dimensions of V1 and V2 and the heights of the prime ideals I(V1 ) and I(V2 ) in k[A3 ]. Let X = V1 ∪ V2 . Compute I(X). Compute the heights of the prime ideals IX (V1 ) and IX (V2 ) in k[X]. Exercise 2.80. Let X be a variety, U an open subset of X, 0 = g ∈ k[X] a nonunit, and Z an irreducible component of Z(g) ∩ U . Show that dim Z = dim X − 1. Exercise 2.81. Suppose that X ⊂ An is a nonempty closed subset such that I(X) = (f1 , . . . , fr ) is generated by r elements. Show that codimAn (X) ≤ r. Exercise 2.82. Suppose that C is a onedimensional subvariety of A3 . Let k[A3 ] = k[x, y, z]. Suppose that C is not a line parallel to the zaxis. Let π : A3 → A2 be the projection π(a, b, c) = (a, b) for (a, b, c) ∈ A3 . a) Show that the Zariski closure of π(C) is a onedimensional subvariety D of A2 and I(D) = I(π(C)) is a principal ideal (g) where g is an irreducible polynomial in k[x, y]. b) Let h = g0 (x, y)z n + · · · + gn (x, y) be an element of I(C) of smallest positive degree n in z. Prove that if f ∈ I(C) has degree m as a polynomial in z, then we have an expression f g0m = hq + v(x, y) where v(x, y) is divisible by g(x, y). c) Show that the algebraic set Z(h, g) is the union of C and ﬁnitely many lines parallel to the zaxis. d) Show that C can be deﬁned by three equations by ﬁnding t ∈ k[x, y, z] such that C = Z(g, h, t).
2.5. Regular functions and regular maps of quasiaﬃne varieties In this section we consider regular functions and regular maps on open subsets of an aﬃne variety. Lemma 2.83. Suppose that X is an aﬃne algebraic set. Then the open sets D(f ) = X \ Z(f ) for f ∈ k[X] form a basis of the Zariski topology on X. We will also denote the open set D(f ) by Xf .
2.5. Regular functions and regular maps of quasiaﬃne varieties
49
Proof. We must show that given an open subset U of X and a point q ∈ U , there exists f ∈ k[X] such that q ∈ X \ Z(f ) ⊂ U . Set m = I(q). There exists an ideal I in k[X] such that U = X \ Z(I). The fact that q ∈ U implies q ∈ Z(I) which implies I ⊂ m. Thus there exists f ∈ I such that f ∈ m. Then Z(I) ⊂ Z(f ) implies X \ Z(f ) ⊂ U . Now f ∈ m implies q = Z(m) ∈ Z(f ) so that q ∈ X \ Z(f ). The process of localization is reviewed in Section 1.4. Suppose that R is a domain with quotient ﬁeld K. If 0 = f ∈ R, then Rf is the following subring of K: g 1 Rf = R =  g ∈ R and n ∈ N . f fn If p is a prime ideal in R, then Rp is the following subring of K:
f  f ∈ R and g ∈ R \ p . Rp = g Rp is a local ring: its unique maximal ideal is pRp = pp . Suppose that X is an aﬃne variety. Let k(X) be the quotient ﬁeld of k[X]. The ﬁeld k(X) is called the ﬁeld of rational functions on X, or the function ﬁeld of X. For p ∈ X, we have that the localization
f  f, g ∈ k[X] and g(p) = 0 . k[X]I(p) = g For f ∈ k[X]I(p) , we have a value f (p) ∈ k, deﬁned as follows. Write f =
g h
g(p) with g, h ∈ k[X] and h(p) = 0 and deﬁne f (p) = h(p) ∈ k. This value is independent of choice of g and h as above. We have natural isomorphisms k[X]I(p) /I(p)I(p) ∼ = k[X]/I(p) ∼ = k, identifying the value f (p) with the residue of f in k[X]I(p) /I(p)I(p) .
Suppose that U is a nonempty open subset of X. Deﬁne the regular functions on U to be k[X]I(p) , OX (U ) = p∈U
where the intersection in k(X) is over all p ∈ U . If U = ∅, we deﬁne OX (U ) = OX (∅) = 0. Suppose that U is a nonempty open subset of X. Let Map(U, A1 ) be the set of maps from U to A1 . The set Map(U, A1 ) is a kalgebra since A1 = k is a kalgebra. We have a natural kalgebra homomorphism φ : OX (U ) → Map(U, A1 ) deﬁned by φ(f )(p) = f (p) for f ∈ OX (U ) and p ∈ U . We will show that φ is injective. Suppose f ∈ Kernel φ and p ∈ U . There exists an expression
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2. Aﬃne Varieties
f = hg where g, h ∈ k[X] and h(p) = 0. For q in the nontrivial open set g(q) U \ Z(h) we have that h(q) = f (q) = 0. Thus g(q) = 0, and so g ∈ I(U \ Z(h)) = I(X) = (0) since U \ Z(h) is a dense open subset of X. Thus g = 0, and so f = hg = 0. Hence φ is injective. We may thus identify OX (U ) with the kalgebra φ(OX (U )) of maps from U to A1 . An element f of the function ﬁeld k(X) of X induces a map f : U → A1 on some open nonempty subset U of X. For p ∈ X, we deﬁne OX,p =
OX (U ),
p∈U
where the union in k(X) is over all open sets U in X containing p. An element f ∈ OX,p thus induces a map f : U → A1 on some open neighborhood U of p in X. We will see that OX,p is a local ring (Proposition 2.86). We will denote the maximal ideal of OX,p by mX,p or by mp if there is no danger of confusion. We will denote OX,p /mp by k(p). As a ﬁeld, k(p) is isomorphic to k. Also, k(p) has a natural structure as an OX,p module. Suppose that U ⊂ V are open subsets of X and p ∈ V . We then have injective restriction maps (2.5)
OX (V ) → OX (U )
and (2.6)
OX (V ) → OX,p .
Proposition 2.84. Suppose that X is an aﬃne variety and 0 = f ∈ k[X]. Then OX (D(f )) = k[X]f . Proof. We have k[X]f = { fgn  g ∈ k[X] and n ∈ N}. If fgn ∈ k[X]f , then fgn ∈ k[X]I(p) for all p ∈ D(f ) since then f (p) = 0. Thus k[X]f ⊂ OX (D(f )). Suppose that h ∈ OX (D(f )), which is a subset of k(X). Let B = {g ∈ k[X]  gh ∈ k[X]}. If we can prove that f n ∈ B for some n, then we will have that h ∈ k[X]f , and it follows that k[X]f = OX (D(f )). By assumption, if p ∈ D(f ), then h ∈ k[X]I(p) , so there exist functions a, b ∈ k[X] such that h = ab with b(p) = 0. Then bh = a ∈ k[X] so b ∈ B, and B contains an √ element not vanishing at p. Thus Z(B) ⊂ Z(f ). We have f ∈ B by the nullstellensatz 5) of Theorem 2.26. In particular, we have that for any aﬃne variety X, (2.7)
k[X] = OX (X).
2.5. Regular functions and regular maps of quasiaﬃne varieties
51
The above proposition shows that if U = D(f ) for some f ∈ k[X], then every element of OX (U ) has the form ab where a, b ∈ k[X] and b(p) = 0 for p ∈ U . The following example, from page 44 of [116], shows that the above desirable property fails in general for an open subset U of an aﬃne variety X. Example 2.85. There exists an open subset U of an aﬃne variety X such that
f OX (U ) =  f, g ∈ k[X] and g(p) = 0 for all p ∈ U . g Proof. Write k[A4 ] = k[x, y, z, w] and let X = Z(xw − yz) ⊂ A4 and U = D(y) ∪ D(w) = X \ Z(y, w). Write k[X] = k[A4 ]/I(X) = k[x, y, z, w] where x, y, z, w are the respective classes of x, y, z, w. Let h ∈ OX (U ) be deﬁned by h = xy on D(y) and h = wz on D(w). We have that xy = wz on D(y) ∩ D(w) = X − Z(yw), so that h is a welldeﬁned function on U . Now suppose that h = fg where f, g ∈ k[X] and g does not vanish on U . We will derive a contradiction. Let Z = ZX (y, w). Then Z is a plane in X (k[Z] = k[x, y, z, w]/(xw − yz, y, w) ∼ = k[x, z]). We have that U = X \ Z. Thus ZX (g) ⊂ Z. Suppose that g does not vanish on X. Then xy = fg so xg = f y. Now p = (1, 0, 0, 0) ∈ X so (xg)(p) = 0 but (f y)(p) = 0, which is a contradiction. Thus g is not a unit in k[X]. By Theorem 2.71 (X is irreducible) all irreducible components of ZX (g) have dimension 2 = dim X − 1. Since Z is irreducible of dimension 2, we have that ZX (g) = Z. Let Z = ZX (x, z), which is another plane. We have that {(0, 0, 0, 0)} = Z ∩ Z = ZX (g) ∩ Z . But (again by Theorem 2.71) a polynomial function vanishes on an algebraic set of dimension 1 on a plane, which is a contradiction since a point has dimension 0. Proposition 2.86. Suppose that X is an aﬃne variety and p ∈ X. Then OX,p = k[X]I(p) . Proof. We have that k[X]I(p) = {f ∈k[X]f (p)=0}
k[X]f =
p∈D(f )
OX (D(f )) =
OX (U ).
p∈U
The last equality is true since the open sets D(f ) are a basis for the topology of X. Suppose that X is an aﬃne variety. From the nullstellensatz, 5) of Theorem 2.26, we have that there is a 11 correspondence between the points in X and the maximal ideals in k[X]. Thus we have the following corollary.
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Corollary 2.87. Suppose that X is an aﬃne variety. Then X is separated; that is, if p, q ∈ X are distinct points, then OX,p = OX,q . We may express (2.8)
k[X] = OX (X) =
OX,p =
p∈X
k[X]I(p) =
k[X]m
p∈X
where the last intersection is over the maximal ideals m of k[X]. Lemma 2.88. Suppose that φ : X → Y is a dominant regular map of aﬃne varieties with induced kalgebra homomorphism φ∗ : k[Y ] → k[X]. Suppose that p ∈ X and q ∈ Y . Then the following are equivalent: 1) φ(p) = q. 2) The preimage (φ∗ )−1 (I(p)) = I(q). 3) φ∗ (OY,q ) ⊂ OX,p . In 3), we consider φ∗ to be its extension to an inclusion φ∗ : k(Y ) → k(X). Proof. For p ∈ X, we have that the preimage (φ∗ )−1 (I(p)) = {f ∈ k[Y ]  φ∗ (f ) ∈ I(p)} = {f ∈ k[Y ]  (f ◦ φ)(p) = 0} = I(φ(p)). Since Y is separated (or by the nullstellensatz 5) of Theorem 2.26), φ(p) = q if and only if I(φ(p)) = I(q), so we have established the equivalence of 1) and 2). The statement 2) is equivalent to 3) follows since φ∗ (OY,q ) ⊂ OX,q
if and only if if and only if if and only if
(φ∗ )−1 (I(p)) ∩ (k[Y ] \ I(q)) = ∅ (φ∗ )−1 (I(p)) ⊂ I(q) (φ∗ )−1 (I(p)) = I(q)
since (φ∗ )−1 (I(p)) is a maximal ideal of k[Y ].
Lemma 2.89. Suppose that U is a nontrivial open subset of an aﬃne variety X. Then OX,p = k(X). p∈U
Proof. By Proposition 2.86, OX,p = k[X]I(p) for p ∈ X. So
p∈U
OX,p ⊂ k(X).
2.5. Regular functions and regular maps of quasiaﬃne varieties
53
Suppose that h ∈ k(X). Write h = fg with f, g ∈ k[X] and 0 = g. Then Z(g) ∩ U = U since U is Zariski dense in X. So there exists p ∈ U \ Z(g). Thus g(p) = 0 and so g ∈ I(p), f ∈ OX,p = k[X]I(p) , g and
h ∈ OX,p ⊂ Thus k(X) ⊂
OX,p .
p∈U
p∈U
OX,p .
We deﬁne the rational functions k(U ) on a quasiaﬃne variety U to be the quotient ﬁeld of OX (U ), which is equal to k(X), where X is the aﬃne variety containing U as an open subset. We also say that k(U ) is the function ﬁeld of U . We further deﬁne OU (V ) = OX (V ) for an open subset V of U and OU (V ) OU,p = p∈V
where the union in k(U ) is over all open sets V in U containing p. We have that OU,p = OX,p for p ∈ U . A quasiaﬃne variety is separated by Proposition 2.86. Suppose that U is a quasiaﬃne variety with ﬁeld k(U ) of rational functions on U . A function f ∈ k(U ) is said to be regular at a point p ∈ U if f ∈ OU,p . Lemma 2.90. Suppose that U is a quasiaﬃne variety and f ∈ k(U ). Then V = {p ∈ U  f ∈ OU,p } is a dense open subset of U . Proof. The quasiaﬃne variety U is an open subset of an aﬃne variety X. Suppose that p ∈ V . Since f ∈ OX,p , there exist g, h ∈ k[X] with h ∈ I(p) such that f = hg . For q ∈ D(h) ∩ U we have that h ∈ IX (q), and thus f = hg ∈ OX,q = OU,q and D(h) ∩ U is an open neighborhood of p in V . V is nonempty since we can always write f = hg for some g, h ∈ k[X] with h = 0. We have that U ∩ D(h) = ∅ since X is irreducible. Thus ∅ = D(h) ∩ U ⊂ V . Lemma 2.91. Suppose that U is a quasiaﬃne variety and p ∈ U . Let IU (p) = {f ∈ OU (U )  f (p) = 0}. Then IU (p) is a maximal ideal in OU (U ) and OU (U )IU (p) = OU,p .
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2. Aﬃne Varieties
Proof. The quasiaﬃne variety U is an open subset of an aﬃne variety X, and OU (U ) = OX (U ). We have injective restriction maps k[X] → OU (U ) → OX,p = k[X]IX (p) . The ring OX,p is a local ring with maximal ideal m = IX (p)OX,p . We have that m ∩ OU (U ) = IU (p) and m ∩ k[X] = IX (p), so we have inclusions OX,p = k[X]IX (p) ⊂ OU (U )IU (p) ⊂ OX,p .
Deﬁnition 2.92. Suppose that Y is a quasiaﬃne variety. A regular map φ : Y → Ar is a map φ = (f1 , . . . , fr ) where f1 , . . . , fr ∈ OY (Y ). Suppose that φ(Y ) ⊂ Z where Z is an open subset of an irreducible closed subset of Ar (a quasiaﬃne variety). Then φ induces a regular map φ : Y → Z. A regular map φ : Y → Z of quasiaﬃne varieties is an isomorphism if there is a regular map ψ : Z → Y such that ψ ◦ φ = idY and φ ◦ ψ = idZ . We point out that an aﬃne variety is also a quasiaﬃne variety. Proposition 2.93. Suppose that X is an aﬃne variety and 0 = f ∈ k[X]. Then the quasiaﬃne variety D(f ) is isomorphic to an aﬃne variety. Proof. A choice of coordinate functions on X gives us a closed embedding X ⊂ An . We thus have a surjection k[x1 , . . . , xn ] → k[X] with kernel I(X). Let g ∈ k[x1 , . . . , xn ] be a function which restricts to f on X. Let J be the ideal in the polynomial ring k[x1 , . . . , xn , xn+1 ] generated by I(X) and 1 − gxn+1 . We will show that J is prime, and if Y is the aﬃne variety Y = Z(J) ⊂ An+1 , then the projection of An+1 onto its ﬁrst n factors induces an isomorphism of Y with D(f ). Using Exercise 1.7 of Section 1.1, we have k[x1 , . . . , xn , xn+1 ]/J ∼ = (k[x1 , . . . , xn ]/I(X)) [xn+1 ]/(xn+1 f − 1) ∼ = k[X][ 1 ] = OX (D(f )), f
which is an integral domain (it is a subring of the quotient ﬁeld k(X) of k[X]). Thus J is a prime ideal. Now projection onto the ﬁrst n factors induces a regular map φ : Y → An . We have that Y = {(a1 , . . . , an , an+1 )  (a1 , . . . , an ) ∈ X and g(a1 , . . . , an )an+1 = 1}. Thus φ(Y ) = D(f ) ⊂ X. In particular, we have a regular map φ : Y → D(f ). Now this map is injective and onto, but to show that it is an isomorphism we have to produce a regular inverse map. Let x1 , . . . , xn be the restrictions of x1 , . . . , xn to X. Then x1 , . . . , xn , f1 ∈ OX (D(f )). Thus ψ : D(f ) → An+1 deﬁned by ψ = (x1 , . . . , xn , f1 ) is a regular map. The image of ψ is Y . We thus have an induced regular map ψ : D(f ) → Y . Composing the maps, we have that ψ ◦ φ = idY and φ ◦ ψ = idD(f ) . Thus φ is an isomorphism.
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55
We will say that a quasiaﬃne variety is aﬃne if it is isomorphic to an aﬃne variety. Deﬁnition 2.94. Suppose that φ : U → V is a regular map of quasiaﬃne varieties. 1) The map φ is called a closed embedding if there exists a closed subvariety Z of V (an irreducible closed subset) such that φ(U ) ⊂ Z and the induced regular map φ : U → Z is an isomorphism. 2) The map φ is called an open embedding if there exists an open subset W of V such that φ(U ) ⊂ W and the induced regular map φ : U → W is an isomorphism. The above deﬁnition of a closed embedding generalizes Deﬁnition 2.43. In general, a map φ : X → Y of aﬃne varieties which is continuous (in the Zariski topology) is not regular. This can be seen most easily on A1 . The closed subsets of A1 are the ﬁnite subsets and all of A1 . Thus any bijection (or ﬁnitetoone map) of sets from A1 to A1 is continuous. Proposition 2.95. Suppose that U, V are quasiaﬃne varieties and φ : U → V is a continuous map. Let φ∗ : Map(V, A1 ) → Map(U, A1 ) be deﬁned by φ∗ (f ) = f ◦ φ for f : V → A1 a map. Then the following are equivalent: 1) φ∗ maps OV (V ) into OU (U ), inducing a kalgebra homomorphism φ∗ : OV (V ) → OU (U ). 2) φ∗ maps OV,φ(p) into OU,p for all p ∈ U , inducing kalgebra homomorphisms φ∗ : OV,φ(p) → OU,p . 3) φ∗ maps OV (W ) into OU (φ−1 (W )) for all open subsets W of V , inducing kalgebra homomorphisms φ∗ : OV (W ) → OU (φ−1 (W )). Proof. Suppose that 1) holds, p ∈ U , and φ(p) = q. Then (φ∗ )−1 (IU (p)) = IV (φ(p)). This follows since for f ∈ k[V ], f ∈ IV (φ(p))
if and only if if and only if if and only if
f (φ(p)) = 0 φ∗ (f )(p) = 0 f ∈ (φ∗ )−1 (IU (p)).
Thus OV (V )I V (φ(p)) → OU (U )IU (p) and so φ∗ : OV,φ(p) → OU,p by Lemma 2.91. Thus 2) holds. Suppose that 2) holds. Then 3) follows from the deﬁnition of regular functions. Suppose that 3) holds. Then 1) follows by taking W = V . Proposition 2.96. Suppose that U and V are quasiaﬃne varieties and φ : U → V is a map. Then φ is a regular map if and only if φ is continuous and φ∗ satisﬁes the equivalent conditions of Proposition 2.95.
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Proof. Suppose that φ is regular. Then V is an open subset of an aﬃne variety Y which is a closed subset of An , such that the extension φ˜ : U → An of φ has the form φ˜ = (f1 , . . . , fn ) with f1 , . . . , fn ∈ OU (U ). We will ﬁrst establish that φ˜ is continuous. Let {Ui } be a cover of U by aﬃne open ˜ i subsets Ui . Write k[An ] = k[x1 , . . . , xn ]. It suﬃces to show that φi = φU ∗ n is continuous for all i. Now φi : k[A ] → OU (U ) → k[Ui ] is a kalgebra homomorphism. Since Ui and An are aﬃne, there exists (by Proposition 2.40) a unique regular map gi : Ui → An such that gi∗ = φ∗i . Suppose that p ∈ Ui and q ∈ An . We have that (2.9) φ∗i (IAn (q)) ⊂ IUi (p) if and only if φ∗i (f )(p) = 0 for all f ∈ IAn (q) if and only if f (φi (p)) = 0 for all f ∈ IAn (q) if and only if IAn (q) ⊂ IAn (φi (p)) if and only if IAn (q) = IAn (φi (p)) since IAn (q) and IAn (φi (p)) are maximal ideals if and only if q = φi (p) since An is separated (by Corollary 2.87). We have that φi (p) = q if and only if φ∗i (IAn (q)) ⊂ IUi (p) and similarly gi (p) = q if and only if gi∗ (IAn (q)) ⊂ IUi (p). Thus φi = gi . Since a regular map of aﬃne varieties is continuous, we have that φi is continuous. Thus φ˜ and φ are continuous. Now let φˆ be the extension φˆ : U → Y of φ. Then φˆ∗ : OY (Y ) → OU (U ) is a kalgebra homomorphism since OY (Y ) = k[Y ] = k[x1 , . . . , xn ] and φˆ∗ (xi ) = fi for all i. By 1) implies 3) of Proposition 2.95, applied to φˆ : U → Y , we have that φ∗ = φˆ∗ : OY (V ) = OV (V ) → OU (φ−1 (V )) = OU (U ) is a kalgebra homomorphism. Thus φ : U → V satisﬁes condition 1) of Proposition 2.95. Now suppose that φ : U → V is continuous and φ∗ satisﬁes the equivalent conditions of Proposition 2.95. Then the extension φ˜ : U → An satisﬁes φ˜∗ : k[An ] → OU (U ) is a kalgebra homomorphism. Since φ˜ = (φ˜∗ (x1 ), . . . , φ˜∗ (xn )), we have that φ is a regular map. Suppose that R is an integral domain with quotient ﬁeld L and I is an ideal in R. The ideal transform of I in R is ∞ R :L I n = R :L I ∞ . S(I; R) = {f ∈ L  f I n ⊂ R for some n ∈ N} = i=0
We have the following algebraic interpretation of regular functions on a quasiaﬃne variety.
2.5. Regular functions and regular maps of quasiaﬃne varieties
57
Lemma 2.97. Suppose that X is an aﬃne variety and I ⊂ k[X] is an ideal. Let U = X \ Z(I). Then OU (U ) = S(I; k[X]). Proof. Write I = (g1 , . . . , gr ) with g1 , . . . , gr ∈ k[X]. Suppose that f is in the quotient ﬁeld of k[X]. We have that f ∈ S(I; k[X]) if and only if I m f ∈ k[X] for some m > 0, which holds if and only if gin f ∈ k[X] for some n > 0 and for all i with 1 ≤ i ≤ r. This condition is equivalent to the statement that f ∈ k[X]gi for 1 ≤ i ≤ r, which is equivalent to the statement that f ∈ OX (U ) since OX (U ) = 1≤i≤r k[X]gi . Example 2.98. There are examples of quasiaﬃne varieties U such that OU (U ) is not a ﬁnitely generated kalgebra. Nagata gives examples in [122] showing this and discusses when OU (U ) is a ﬁnitely generated kalgebra. The simplest example of a quasiaﬃne variety U such that OU (U ) is not a ﬁnitely generated kalgebra that he presents is constructed from an example of Rees [125]. Nagata explains Rees’s construction on page 48 of [122]. The existence of the example then follows from properties (1) and (2) of Rees’s example, given on page 49 of [122], Proposition 4 on page 39 of [122], and the above Lemma 2.97. Exercise 2.99. Suppose that U in an open subset an aﬃne variety X and f1 , . . . , fn ∈ k[X] are such that U = D(f1 ) ∪ · · · ∪ D(fn ). Show that OX (U ) = k[X]f1 ∩ · · · ∩ k[X]fn where the intersection is in the quotient ﬁeld k(X) of X. Exercise 2.100. Let U = A1 \ {0}. a) Compute OA1 (U ), the regular functions on the quasiaﬃne variety U. b) Is U (isomorphic) to an aﬃne variety? Exercise 2.101. Let U = A2 \ {(0, 0)}. a) Compute OA2 (U ), the regular functions on the quasiaﬃne variety U. b) Is U (isomorphic) to an aﬃne variety? Exercise 2.102. At what points of the subvariety X = Z(x2 + y 2 − 1) of A2 with regular functions k[X] = k[x, y]/(x2 + y 2 − 1) = k[x, y] is the rational function 1−y x regular? Assume that the characteristic of k is = 2. Exercise 2.103. Suppose that X is an aﬃne variety and U is an open subset of X (so that U is a quasiaﬃne variety). Suppose that p ∈ U . Show that OU,p = OX,p .
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2.6. Rational maps of aﬃne varieties In this section, we deﬁne a rational map of an aﬃne variety X. We begin by noting some properties of restriction and extension of regular maps. Lemma 2.104. Suppose that X and Y are aﬃne varieties and U ⊂ V ⊂ X are nonempty open subsets. Suppose that φ, ψ : V → Y are regular maps such that φU = ψU . Then φ = ψ. Proof. There exist a closed embedding Y ⊂ An and f1 , . . . , fn ∈ OX (V ), g1 , . . . , gn ∈ OX (V ) such that φ = (f1 , . . . , fn ) and ψ = (g1 , . . . , gn ). Since φU = ψU , we have that fi U = gi U for 1 ≤ i ≤ n. Thus fi = gi for 1 ≤ i ≤ n since the restriction map OX (V ) → OX (U ) is injective. Lemma 2.105. Suppose that X, Y are aﬃne varieties, U ⊂ X is a nontrivial open subset, and φ : U → Y is a regular map. Then there exists a largest open subset W (φ) of X such that there exists a regular map ψ : W (φ) → X with the property that ψU = φ. The map ψ is uniquely determined. Proof. There exist a closed embedding Y ⊂ An and f1 , . . . , fn ∈ OX (U ) such that φ = (f1 , . . . , fn ) : U → Y . Let Vi = {p ∈ X  fi ∈ OX,p }. nThe Vi are nontrivial open subsets of X by Lemma 2.90. Then W (φ) = i=1 Vi is the largest open subset of X on which φ extends to a regular map ψ. The map ψ is uniquely determined by Lemma 2.104. We can thus deﬁne a rational map φ between aﬃne varieties X and Y to be a regular map on some nonempty open subset U of X to Y . By Lemma 2.105, φ has a unique extension as a regular map to a largest open subset W (φ) of X, and if φ and ψ are rational maps from X to Y which are regular on respective nonempty open subsets U and V such that φ and ψ agree on the intersection U ∩ V , then φ = ψ. It is usual to write φ : X Y for a rational map, to emphasize the fact that φ may not be regular everywhere. The rational maps φ : X A1 can be identiﬁed with the rational functions k(X). We now formulate the concept of a rational map as a statement in algebra. Deﬁnition 2.106. Suppose that X is an aﬃne variety. A rational map φ : X Am is an mtuple φ = (f1 , . . . , fm ) with f1 , . . . , fm ∈ k(X). Such a map φ induces a kalgebra homomorphism φ∗ : k[Am ] → k(X) by φ∗ (ti ) = fi for 1 ≤ i ≤ m where k[Am ] = k[t1 , . . . , tm ]. Suppose that Y is an aﬃne variety which is a closed subvariety of an aﬃne space Am . A rational map φ : X Y is a rational map φ : X Am such that I(Y ) is contained in the kernel of φ∗ : k[Am ] → k(X). This induces a kalgebra homomorphism φ∗ : k[Y ] → k(X).
2.6. Rational maps of aﬃne varieties
59
In particular, a rational map φ : X Y of aﬃne varieties can be understood completely by the corresponding kalgebra homomorphism φ∗ : k[Y ] → k(X). Suppose that φ = (f1 , . . . , fm ) : X Y is a rational map of aﬃne varieties. The set of points of X on which f1 , . . . , fm are all regular is a nonempty open set W (φ) by Lemma 2.90. This is the set of points on which φ is regular (Lemma 2.105). If U ⊂ W (φ) is a nonempty open subset, we have that φ∗ : k[Y ] → OX (U ). A rational map φ is completely determined by its restriction to any open subset U of X. In particular, we have that if Y is a closed subvariety of Am and φ : X Am is a rational map, then φ : X Y if and only if φ(W (φ)) ⊂ Y . A rational map φ : X Y is called dominant if φ(U ) is dense in Y when U is an open subset of X on which φ is a regular map. Lemma 2.107. Suppose that φ : X Y is a dominant rational map of aﬃne varieties. Then φ induces an injective kalgebra homomorphism φ∗ : k(Y ) → k(X) of function ﬁelds. Proof. There exists a nonempty open subset W of X on which φ is a regular map. W contains an open set D(f ) for some f ∈ k[X] by Lemma 2.83. The open set D(f ) is aﬃne with k[D(f )] = k[X]f by Propositions 2.93 and 2.84. Then we have an induced kalgebra homomorphism φ∗ : k[Y ] → k[X]f which is injective by Corollary 2.32. We thus have an induced kalgebra homomorphism of quotient ﬁelds. If φ∗ induces a welldeﬁned injective homomorphism φ∗ : k(Y ) → k(X), and if U is an aﬃne open subset of X on which φ is regular, we have that φ∗ : k[Y ] → k[U ] is injective and thus φ(U ) is dense in Y by Corollary 2.32. Thus the rational map φ : X Y is dominant. Proposition 2.108. Suppose that X and Y are aﬃne varieties and Λ : k(Y ) → k(X) is an injective kalgebra homomorphism. Then there is a unique (dominant) rational map φ : X Y such that φ∗ = Λ. Proof. Let t1 , . . . , tm be coordinate functions on Y such that k[Y ] = k[t1 , . . . , tm ]. Write Λ(ti ) = fgii with fi , gi ∈ k[X] (and gi = 0) for 1 ≤ i ≤ m. Let g = g1 g2 · · · gm . Then Λ induces a kalgebra homomorphism Λ : k[Y ] → k[X]g . Now k[X]g = k[D(g)] where D(g) is the aﬃne open subset of X, by Proposition 2.84. By Proposition 2.40, there is a unique regular map φ : D(g) → Y such that φ∗ = Λ. Since a rational map of varieties is uniquely determined by the induced regular map on a nontrivial open subset, there is a unique rational map φ : X Y inducing Λ.
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Suppose that α : X Y is a dominant rational map and β : Y Z is a rational map. Then β ◦ α : X Z is a rational map. We see this as follows. Since α is dominant, α∗ induces a homomorphism k(Y ) → k(X) so the composition α∗ β ∗ : k[Z] → k(X) is welldeﬁned. There exist open sets U of X on which α is regular and V of Y on which β is regular. Since α(U ) is dense in Y , we have that α(U ) ∩ V = ∅. Thus the nonempty open set (αU )−1 (V ) is contained in the open set W (β ◦ α) where β ◦ α is regular. Deﬁnition 2.109. A dominant rational map φ : X Y of aﬃne varieties is birational if there is a dominant rational map ψ : Y X such that ψ ◦ φ = idX and φ ◦ ψ = idY (that is, φ∗ ◦ ψ ∗ : k[X] → k(X) is equal to the inclusion id∗X and ψ ∗ ◦ φ∗ : k[Y ] → k(Y ) is equal to the inclusion id∗Y ). Proposition 2.110. A rational map φ : X Y is birational if and only if φ∗ : k(Y ) → k(X) is a kalgebra isomorphism. Theorem 2.111. A dominant rational map φ : X Y of aﬃne varieties is birational if and only if there exist nonempty aﬃne open subsets U of X and V of Y such that φ : U → V is a regular map which is an isomorphism. Proof. Suppose that there exist open sets U of X and V of Y such that φ : U → V is a regular map which is an isomorphism. Then there exists a regular map ψ : V → U such that ψ ◦ φ = idU and φ ◦ ψ = idV . Since idk[V ] = (φ ◦ ψ)∗ = ψ ∗ ◦ φ∗ and idk[U ] = (ψ ◦ φ)∗ = φ∗ ◦ ψ ∗ , we have that φ∗ : k[V ] → k[U ] is an isomorphism of kalgebras, so that φ∗ induces an isomorphism of their quotient ﬁelds, which are respectively k(Y ) and k(X). Suppose that φ∗ : k(Y ) → k(X) is an isomorphism. Then there exists a unique rational map ψ : Y X such that ψ ∗ is the inverse of φ∗ by Proposition 2.108. Let t1 , . . . , tm be coordinate functions on Y and let s1 , . . . , sn be coordinate functions on X. There are functions a1 , . . . , am , f ∈ k[X] and b1 , . . . , bn , g ∈ k[Y ], with f, g = 0, such that φ∗ (ti ) = afi for 1 ≤ i ≤ m and ψ ∗ (sj ) = gj for 1 ≤ i ≤ n. Thus φ∗ (k[Y ]) ⊂ k[X]f and ψ ∗ (k[X]) ⊂ k[Y ]g . Since ψ ∗ is the inverse of φ∗ , we have that φ∗ (k[Y ]gψ∗ (f ) ) ⊂ k[X]f φ∗(g) and ψ ∗ (k[X]f φ∗ (g) ) ⊂ k[Y ]gψ∗ (f ) . Thus φ∗ : k[Y ]gψ∗ (f ) → k[X]f φ∗(g) is an isomorphism, and φ : D(f φ∗ (g)) → D(gψ ∗ (f )) is a regular map which is an isomorphism by Proposition 2.42. b
Two aﬃne varieties X and Y are said to be birationally equivalent if there exists a birational map φ : X Y . Proposition 2.112. Every aﬃne variety X is birationally equivalent to a hypersurface Z(g) ⊂ An .
2.6. Rational maps of aﬃne varieties
61
Proof. Let dim(X) = r. Then the quotient ﬁeld k(X) is a ﬁnite separable extension of a rational function ﬁeld L = k(x1 , . . . , xr ) (by Theorem 1.14). By the theorem of the primitive element (Theorem 1.16), k(X) ∼ = L[t]/(f (t)) for some irreducible monic polynomial f (t) ∈ L[t]. Multiplying f (t) by an appropriate element a of k[x1 , . . . , xr ], we obtain a primitive polynomial g = af (t) ∈ k[x1 , . . . , xr , t], which is thus irreducible. The quotient ﬁeld of k[x1 , . . . , xr , t]/(g) is isomorphic to k(X). Thus X is bira tionally equivalent to Z(g) ⊂ Ar+1 by Proposition 2.110. Exercise 2.113. Prove Proposition 2.110. Exercise 2.114. Consider the regular map φ : A2 → A2 deﬁned by φ(a1 , a2 ) = (a1 , a1 a2 )
for (a1 , a2 ) ∈ A2 .
a) Show that φ is dominant. b) Show that φ is birational. c) Show that φ is not an isomorphism. d) Show that φ is not ﬁnite. Exercise 2.115. Is a composition of rational maps always a rational map?
Chapter 3
Projective Varieties
In this chapter we deﬁne projective and quasiprojective varieties, their regular functions and regular maps. Recall that throughout this book, k will be a ﬁxed algebraically closed ﬁeld. We develop a correspondence between the commutative algebra of standard graded kalgebras which are domains (or reduced) and the geometry of projective varieties (or projective algebraic sets) in Sections 3.1–3.2. In Section 3.4, we deﬁne the regular functions on a projective variety.
3.1. Standard graded algebras In this section we discuss algebraic methods necessary for our study of projective space and projective varieties. We begin with some general deﬁnitions of graded rings and modules. Some references on graded algebras and modules are [161, Chapter VII] and [28]. A graded ring is a ring R with a decomposition R = i∈Z Ri such that Ri Rj ⊂ Ri+j for all i, j ∈ Z. A graded Rmodule is an Rmodule M together with a decomposition M = i∈Z Mi such that Ri Mj ⊂ Mi+j for all i, j ∈ Z. The elements x ∈ Mi are called homogeneous of degree i. The degree of x is denoted by deg x. Every element f ∈ M has a unique expression as a sum with ﬁnitely many nonzero terms f = F0 + F1 + · · · where the Fi ∈ Mi are homogeneous of degree i, with Fi = 0 for all i suﬃciently large. If M is a graded Rmodule and n ∈ Z, then the twisted Rmodule M (n) is M , with the grading M (n)i = Mn+i for i ∈ Z. 63
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3. Projective Varieties
Suppose that R is a graded ring and M is a graded Rmodule. Suppose that F ∈ R is homogeneous. Then the localization MF is graded by G deg = deg(G) − ndeg(F ) Fn for G ∈ M homogeneous. We deﬁne
G M(F ) = h = n ∈ MF  G is homogeneous and deg(h) = 0 . F If p is a homogeneous prime ideal in R, then we deﬁne (3.1)
M(p) = {elements of degree 0 in V −1 M }
where V is the multiplicatively closed subset of homogeneous elements of R not in p. The ring R(p) is a local ring with maximal ideal p(p) . In particular, if R is a domain, so that the zero ideal q = (0) is a prime ideal, then R(q) is a ﬁeld. Suppose that R is a graded ring and d is a positive integer. The dth Veronese ring of R is Rid . (3.2) R(d) = i∈Z (d) Ri
= Rid for i ∈ Z. It is a graded ring with Suppose that A = i∈Z Ai and B = j∈Z Bj are graded rings. A ring homomorphism φ : A → B is said to be graded of degree s if φ(Ai ) ⊂ Bis for all i. The inclusion of the Veronese ring R(d) into R is an example of a graded ring homomorphism of degree d. Suppose that M = i∈Z Mi and N = j∈Z Nj are graded Rmodules. An Rmodule homomorphism λ : M → N is graded of degree s if λ(Mi ) ⊂ Ni+s for all i. The following lemma follows from [153, Theorem on page 151]. Lemma 3.1. Suppose that R is a graded ring and I ⊂ R is an ideal. Then the following are equivalent: 1) Suppose that f ∈ I and f = Fi where Fi is homogeneous of degree i. Then Fi ∈ I for all i. 2) I = ∞ i=−∞ (I ∩ Ri ). An ideal satisfying the conditions of Lemma 3.1 is called a homogeneous ideal. An ideal I is homogeneous if and only if I has a homogeneous set of generators. Lemma 3.2. Suppose that P is a homogeneous ideal in a graded ring R. Then P is a prime ideal if and only if it has the property that whenever F, G ∈ R are homogeneous and F G ∈ P , then F or G is in P .
3.1. Standard graded algebras
65
Proof. Suppose that P is a homogeneous ideal in R and P has the property that whenever F, G ∈ R are homogeneous and F G ∈ P , then F or G is in P . Let f, g ∈ R and suppose that f ∈ P and g ∈ P . We will show that f g ∈ P . Let f = fr + fr+1 + · · · and g = gs + gs+1 + · · · be the decompositions of f and g into their homogeneous components. Let fr+a and gs+b be the ﬁrst homogeneous components of f and g, respectively, which does not belong to P . Then fr+a , gs+b ∈ P , and so [f − (fr + fr+1 + · · · + fr+a−1 )][g − (gs + · · · + gs+b−1 )] ∈ P since P is homogeneous. Since fr + fr+1 + · · · + fr+a−1 and gs + gs+1 + · · · + gs+b−1 are in P , we have that f g ∈ P . The following lemma follows from [161, Theorem on page 152] and [161, Theorem 9 on page 153]. Lemma 3.3. Suppose that R is a graded ring and I and J are homogeneous ideals in R. Then: 1) I + J is a homogeneous ideal. 2) IJ is a homogeneous ideal. 3) I ∩ J is a homogeneous ideal. √ 4) I is a homogeneous ideal. 5) I : J is homogeneous. 6) If I admits a primary decomposition, then I admits a homogeneous primary decomposition. We now consider the case of graded rings which are quotients of polynomial rings over a ﬁeld. A thorough development of this material is in [161, Section 2, Chapter VII]. Let T be the polynomial ring T = K[x0 , x1 , . . . , xn ] over a ﬁeld K. An element f ∈ T is called homogeneous of (total) degree d if it is a Klinear combination of monomials of degree d. The degree of the monomial xi00 xi11 · · · xinn is i0 + i1 + · · · + in . Let Td be the Kvector space of all homogeneous polynomials of degree d (we include 0). Every polynomial f ∈ T has a unique expression as a sum with ﬁnitely many nonzero terms f = F0 + F1 + · · · where the Fi are homogeneous of degree i, with Fi = 0∞for all i suﬃciently large. This is equivalent to the statement that T = i=0 Ti , where T0 = K and Ti is the Kvector space of homogeneous polynomials of degree i; that is, T is a graded ring. Since T = K[T1 ] = T0 [T1 ] is generated by elements of degree 1 as a T0 = Kalgebra, we say that T is a standard graded Kalgebra.
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∞ Suppose that U = i=0 Ui is a homogeneous ideal in T . Then S = ∞ ∼ T /U = i=0 Si where Si = Ti /Ui . The ring S is a standard graded Kalgebra (elements of Si have degree i and S is generated by S1 as an S0 = Kalgebra). Every f ∈ S has a unique expression as a sum with ﬁnitely many nonzero terms f = F0 + F1 + · · · where the Fi ∈ Si are homogeneous of degree i, with Fi = 0 for all i suﬃciently large. The conclusions of Lemma 3.1 hold for ideals in S, so we can speak of homogeneous ideals in S. Suppose that U is an ideal in the standard graded polynomial ring T and S = T /U . Then S = K[x0 , . . . , xn ] is the Kalgebra generated by the if and only if classes xi of the xi in S. We can extend the grading of T to S ∞ U is a homogeneous ideal. In this case, we have that S = i=0 Si where Si a is the Kvector space generated by the monomials xj j where aj = i. If U is not homogeneous, the concept of degree is not welldeﬁned on S. As an example, if U = (x − y 2 ) ⊂ K[x, y], then x = y 2 would have to have both degree 1 and degree 2 in K[x, y]/U . Suppose that I is a homogeneous ideal in S and I = Qj is a primary decomposition of I by homogeneous ideals. Let m be the homogeneous maximal ideal m = i>0 Si . The saturation of I is the homogeneous ideal I sat which is the intersection of all the primary components of I which are not mprimary. Properties of the saturation are derived in [161, Section 2, Chapter VII], especially in Lemmas 4 and 5. We have that In = (I sat )n for ∞ sat ∞ n 0 and I = I : m = i=1 I : mi . We have an expression of our standard graded Kalgebra S as S = K[x0 , . . . , xn ] where x0 , . . . , xn are homogeneous of degree 1. Suppose that I ⊂ S is a homogeneous ideal. Then x0 xn ,..., ⊂ Sxi S(xi ) = K xi xi for 0 ≤ i ≤ n and I(xi ) =
x0 xn G ,..., G∈I . xi xi
The following lemma will be useful. Lemma 3.4. Suppose that I and J are homogeneous ideals in S. Then I(xi ) = J(xi ) for 0 ≤ i ≤ n if and only if I sat = J sat . Proof. Suppose that I(xi ) = J(xi ) for 0 ≤ i ≤ n. We then have equality of localizations Ixi = I(xi ) S = J(xi ) S = Jxi
3.2. Projective varieties
67
for 0 ≤ i ≤ n. In particular, there exists an integer t such that xti J ⊂ I for all i, so that J ⊂ I sat . Similarly, I ⊂ J sat . Suppose that I sat = J sat . Since xi ∈ m, we have by Proposition 1.43 that Ixi = Jxi for all i. Thus I(xi ) = J(xi ) for all i. Exercise 3.5. Let K be a ﬁeld and S = K[x0 , . . . , xn ] be a standard graded Kalgebra. Show that if xi is nilpotent in S, then S(xi ) = 0. Exercise 3.6. Let K be a ﬁeld and suppose that A = ∞ i=0 Ai is a graded ring, which is a ﬁnitely generated A0 = Kalgebra. Show that there exists (d) d ∈ Z+ such that A(d) is a standard graded Kalgebra (generated by A1 ). ∞ Exercise 3.7. Let A = i=0 Ai be a graded ring and X be the set of all homogeneous prime ideals in A which do not contain the ideal A+ = i>0 Ai . For each subset E of homogeneous elements of A, let V (E) be the set of all elements of X which contain E. Prove that: a) If I is the homogeneous ideal generated√by a set of homogeneous elements of E, then V (E) = V (I) = V ( I). b) V (0) = X and V (1) = V (A+ ) = ∅. c) If {Es }s∈S is any family of subsets of homogeneous elements of A, then Es = V (Es ). V s∈S
s∈S
d) V (I ∩ J) = V (IJ) = V (I) ∪ V (J) for any homogeneous ideals I, J of A This exercise shows that the sets V (E) satisfy the axioms for closed sets in a topological space. We call this topology on X the Zariski topology and write Proj(A) for this topological space.
3.2. Projective varieties In this section we deﬁne projective varieties and projective algebraic sets. [161, Sections 4 and 5 of Chapter VII] is a good reference to the algebra of this section. As usual, we assume throughout this chapter that k is an algebraically closed ﬁeld. Deﬁne an equivalence relation ∼ on k n+1 \ {(0, . . . , 0)} by (a0 , a1 , . . . , an ) ∼ (b0 , b1 , . . . , bn ) if there exists 0 = λ ∈ k such that (b0 , b1 , . . . , bn ) = (λa0 , λa1 , . . . , λan ). Projective space Pnk over k is deﬁned as Pnk = Pn = k n+1 \ {(0, . . . , 0)} / ∼ .
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The equivalence class of an element (a0 , a1 , . . . , an ) in k n+1 \ {(0, . . . , 0)} is denoted by (a0 : a1 : . . . : an ). We deﬁne the homogeneous coordinate ring of Pnk to be the standard graded polynomial ring S(Pn ) = k[x0 , x1 , . . . , xn ]. We think of the xi as “homogeneous coordinates” on Pn . Suppose that F ∈ k[x0 , . . . , xn ] is homogeneous of degree d, λ ∈ k, and a0 , . . . , an ∈ k. Then F (λa0 , . . . , λan ) = λd F (a0 , . . . , an ).
(3.3)
Thus we see that if U is a set of homogeneous polynomials in S(Pn ) (possibly of diﬀerent degrees), then the set Z(U ) = {(a0 : . . . : an ) ∈ Pn  F (a0 , . . . , an ) = 0 for all F ∈ U } is a welldeﬁned subset of Pn . If I is a homogeneous ideal in S(Pn ), we deﬁne Z(I) = {(a0 : . . . : an ) ∈ Pn  F (a0 , . . . , an ) = 0 for all F ∈ U } where U is a set of homogeneous generators of I. This is a welldeﬁned set (independent of the choice of homogeneous generators U of I). Deﬁnition 3.8. A subset Y of Pn is a projective algebraic set if there exists a set U of homogeneous elements of S(Pn ) such that Y = Z(U ). Proposition 3.9. Suppose that I1 , I2 , {Iα }α∈Λ are homogeneous ideals in S(Pn ). Then: 1) Z(I1 I2 ) = Z(I1 ) ∪ Z(I2 ). 2) Z( α∈Λ Iα ) = α∈Λ Z(Iα ). 3) Z(S(Pn )) = ∅. 4) Pn = Z(0). Proposition 3.9 tells us that: 1. The union of two algebraic sets is an algebraic set. 2. The intersection of any family of algebraic sets is an algebraic set. 3. ∅ and Pn are algebraic sets. We thus have a topology on Pn , deﬁned by taking the closed sets to be the algebraic sets. The open sets are the complements of algebraic sets in Pn . This topology is called the Zariski topology. If X is an algebraic set in Pn , then the Zariski topology on X is the subspace topology.
3.2. Projective varieties
69
Deﬁnition 3.10. A projective algebraic variety is an irreducible closed subset of Pn . A quasiprojective variety is an open subset of a projective variety. A projective algebraic set is a closed subset of Pn . A quasiprojective algebraic set is an open subset of a closed subset of Pn . A subset X of a variety Y is called a subvariety of Y if X is a closed irreducible subset of Y . Given a subset Y of Pn , the ideal I(Y ) of Y in S(Pn ) is the ideal in S(Pn ) generated by the set U = {F ∈ S(Pn )  F is homogeneous and F (p) = 0 for all p ∈ Y }. Proposition 3.11. Suppose that a is a homogeneous ideal in the standard ∞ graded polynomial ring T = k[x0 , . . . , xn ] = i=0 Ti . Then the following are equivalent: 1) Z(a) = ∅. √ 2) a is either T or the ideal T+ = d>0 Td . 3) Td ⊂ a for some d > 0. 4) asat = T . Proof. We will prove the essential implication that Z(a) = ∅ implies Td ⊂ a for some d > 0. Suppose that F1 , . . . , Fr are homogeneous generators of a. Since Z(a) = ∅, we have that the polynomials Fi (1, y1 , . . . , yn ) have no common root. By the nullstellensatz in k[y1 , . . . , yn ] (Theorem 2.5), there exist polynomials Gi (y1 , . . . , yn ) such that Gi (y1 , . . . , yn )Fi (1, y1 , . . . , yn ) = 1. i
Substituting yi = and multiplying by xl00 with l0 suﬃciently large, we obtain that xl00 ∈ a. Similarly, we have xlii ∈ a for 0 ≤ i ≤ n. Let l = max{l0 , . . . , ln } and d = (l − 1)(n + 1) + 1. Then Td ⊂ a. xi x0
Theorem 3.12 (Homogeneous nullstellensatz). Let a √ be a homogeneous ideal in the polynomial ring T = k[x0 , . . . , xn ] such that a = (x0 , . . . , xn ), and let F ∈ T be a homogeneous polynomial which vanishes at all points of Z(a) in Pn . Then F r ∈ a for some r > 0. Proof. We may suppose that a = T . Let V = Z(a) ⊂ Pn and C(V ) = ZAn+1 (a) ⊂ An+1 (regarding k[x0 , . . . , xn ] as the regular functions on An+1 ). Since a is a homogeneous ideal, a point (a0 , . . . , an ) is in C(V ) if and only if (ta0 , ta1 , . . . , tan ) is in C(V ) for all t ∈ k; further, the point (a0 : a1 : . . . : an ) ∈ V if and only if (ta0 , . . . , tan ) ∈ C(V ) for all t ∈ k. Since √ a = (x0 , . . . , xn ), we have that C(V ) contains a point other than (0, . . . , 0) by the aﬃne nullstellensatz (Theorem 2.5) and thus V = ∅. Since V = ∅, I(V ) ⊂ IAn+1 (C(V )).
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3. Projective Varieties
If a polynomial f ∈ IAn+1 (C(V )), then f (ta0 , . . . , tan ) = 0 for all (a0 : . . . : an ) ∈ V and t ∈ k. Writing f = qj=0 Fj where each Fj is a homogeneous form of degree j, we have F0 (a0 , . . . , an ) + tF1 (a0 , . . . , an ) + · · · + tq Fq (a1 , . . . , aq ) = 0 for all t ∈ k which implies that Fi (a0 , . . . , an ) = 0 for all i since an algebraically closed ﬁeld is inﬁnite. Thus f ∈ I(V ) and so I(V ) = IAn+1 (C(V√)). By the aﬃne nullstellensatz (Theorem 2.5), we have that I(C(V )) = a, and the theorem follows. Proposition 3.13. The following statements hold: 1) If T1 ⊂ T2 are subsets of S(Pn ) consisting of homogeneous elements, then Z(T2 ) ⊂ Z(T1 ). 2) If Y1 ⊂ Y2 are subsets of Pn , then I(Y2 ) ⊂ I(Y1 ). 3) For any two subsets Y1 , Y2 of Pn , we have I(Y1 ∪Y2 ) = I(Y1 )∩I(Y2 ). 4) √ If a is a homogeneous ideal in S(Pn ) with Z(a) = ∅, then I(Z(a)) = a. 5) For any subset Y of Pn , Z(I(Y )) = Y , the Zariski closure of Y . The proofs of Theorem 3.14 and Proposition 3.15 are similar to those of Theorem 2.11 and Theorem 2.12 for aﬃne space, using Proposition 3.13 instead of Proposition 2.10. Theorem 3.14. A closed set W ⊂ Pn is irreducible if and only if I(W ) is a prime ideal. Proposition 3.15. Every closed set in Pn is the union of ﬁnitely many irreducible ones. Suppose that X is a closed subset of Pn . We deﬁne the coordinate ring of X to be S(X) = S(Pn )/I(X), which is a standard graded ring. Suppose that U is a set of homogeneous elements of S(X). Then we deﬁne ZX (U ) = {p ∈ X  F (p) = 0 for all F ∈ U }. If J is a homogeneous ideal in S(X), we deﬁne ZX (J) = ZX (U ) where U is a homogeneous set of generators of J. This set is independent of choice of homogeneous generating set U of J. Given a subset Y of X, we deﬁne IX (Y ) to be the ideal in S(X) generated by the homogeneous elements of S(X) which vanish at all points of Y . When there is no danger of confusion, we will sometimes write Z(J) to denote ZX (J) and I(Y ) to denote IX (Y ). The above results in this section hold with Pn replaced by X. The Zariski topology on X is the topology whose closed sets are ZX (I) for I ⊂ S(X) a homogeneous ideal (which is the subspace topology of X).
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71
Suppose that X is a projective algebraic set and F ∈ S(X) is homogeneous. We deﬁne D(F ) = X \ Z(F ), which is an open set in X. We will also denote the open set D(F ) by XF . The proof of the following lemma is like that of Lemma 2.83. Lemma 3.16. Suppose that X ⊂ Pn is a projective algebraic set. Then the open sets D(F ) for homogeneous F ∈ S(X) form a basis for the topology of X. We now construct a correspondence between elements of R = k[y1 , . . . , yn ] (a polynomial ring in n variables) and homogeneous elements of the standard graded polynomial ring T = k[x0 , . . . , xn ]. Fix i with 0 ≤ i ≤ n. To f (y1 , . . . , yn ) ∈ R we associate the homogeneous polynomial x0 xi−1 xi+1 xn h d ,..., , ,..., , f = F (x0 , . . . , xn ) = xi f xi xi xi xi where d is the degree of f . To F (x0 , . . . , xn ) ∈ T we associate F a = F (y1 , . . . , yi , 1, yi+1 , . . . , yn ). This deﬁnition is valid for all F ∈ T . One has that (f h )a = f for all f ∈ R, and for all homogeneous F ∈ T , (F a )h = x−m i F where m is the highest power of xi which divides F . We extend h to a map from ideals in k[y1 , . . . , yn ] to homogeneous ideals in k[x0 , . . . , xn ] by taking an ideal I to the ideal I h generated by the set of homogeneous elements {f h  f ∈ I}. We also extend a to a map from homogeneous ideals in k[x0 , . . . , xn ] to ideals in k[y1 , . . . , yn ]. A homogeneous ideal J is mapped to the ideal J a = {f a  f ∈ J} = {F a  F ∈ J is homogeneous}. The properties which are preserved by these correspondences of ideals are worked out in detail in [161, Section 5 of Chapter VII] (especially Theorem 17 on page 180 and Theorem 18 on page 182). The following formulas hold: (I h )a = I and a h
(J ) =
∞
J : xji
for I an ideal in R
for J a homogeneous ideal in T
j=0
where J : xji = {f ∈ T  f xji ∈ J}. In particular, (J a )h = J if J is a homogeneous prime ideal which does not contain xi . We thus deduce from
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3. Projective Varieties
these formulas, [161, Theorem on page 180] and [161, Theorem on page 182], the following propositions. Proposition 3.17. The functions a and h give a 11 correspondence between prime ideals in k[y1 , . . . , yn ] and homogeneous prime ideals in k[x0 , . . . , xn ] which do not contain xi . √ Recall that an ideal A is radical if A = A. Proposition 3.18. The functions a and h give a 11 correspondence between radical ideals in k[y1 , . . . , yn ] and homogeneous radical ideals in k[x0 , . . . , xn ] all of whose associated prime ideals do not contain xi . Theorem 3.19. Suppose that i satisﬁes 0 ≤ i ≤ n and D(xi ) is the open subset of Pn . Then the maps φ : D(xi ) → An deﬁned by a0 ai−1 ai+1 an ,..., , ,..., φ(a0 : . . . : an ) = ai ai ai ai for (a0 : . . . : an ) ∈ D(xi ) and ψ : An → D(xi ) deﬁned by ψ(a1 , a2 , . . . , an ) = (a1 : . . . : ai : 1 : ai+1 : . . . : an ) for (a1 , . . . , an ) ∈ An are inverse homeomorphisms. Proof. The maps φ and ψ are inverse bijections of sets, so we need only show that φ and ψ are continuous. Suppose that Z ⊂ An is closed. Then Z = Z(I) for some ideal I in k[y1 , . . . , yn ]. Then φ−1 (Z) = D(xi ) ∩ Z(I h ) is closed in D(xi ), and so φ is continuous. Suppose that Z ⊂ D(xi ) is closed in D(xi ). Then Z = W ∩ D(xi ) for some closed subset W of Pn . We have that W = Z(J) for some homogeneous ideal J of k[x0 , . . . , xn ] and so ψ −1 (Z) = Z(J a ) is closed in An . It follows from Theorem 3.19 and Proposition 3.17 that if X is an algebraic set (variety) in An , then ψ induces a homeomorphism of X with W ∩ D(xi ), where W is the projective algebraic set (variety) Z(I(X)h ). If W is a projective algebraic set (variety) in Pn which is not contained in Z(xi ), then ψ induces a homeomorphism of the closed algebraic set (variety) X = Z(I(W )a ) of An with W ∩ D(xi ). Exercise 3.20. Prove Proposition 3.9. Exercise 3.21. Prove Proposition 3.13. Exercise 3.22. Suppose that p = (a0 : . . . : an ) ∈ Pn . Show that I(p) = (ai xj − aj xi  0 ≤ i, j ≤ n).
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73
Exercise 3.23. In Theorem 3.19, we showed that the open set D(xi ) of Pn is homeomorphic to An . Suppose that Y is a closed subset of the open subset D(xi ) of Pn . The closure Y of Y in Pn is the intersection of all closed subsets of Pn containing Y . Show that IAn (φ(Y ))h = IPn (Y ). Exercise 3.24. Suppose that Y ⊂ Pn is a closed subvariety of D(x0 ) such that I(φ(Y )) = (f ) for some irreducible f ∈ k[An ] = k[y1 , . . . , yn ]. Suppose that d is the degree of f , so that we have an expansion f= aj1 ,...,jn y1j1 · · · ynjn j1 +···+jn ≤d
for some aj1 ,...,jn ∈ k and so that some term aj1 ,...,jn = 0 with j1 +· · ·+jn = d. Find generators of I(φ(Y ))h . Exercise 3.25. Let φ : k[x0 , x1 , x2 , x3 ] → k[s, t] be the kalgebra homomorphism of polynomial rings deﬁned by φ(x0 ) = s3 ,
φ(x1 ) = ts2 ,
φ(x2 ) = t2 s,
φ(x3 ) = t3 .
We will say that a monomial si tj has bidegree (i, j) and a monomial xa0 xb1 xc2 xd3 has bidegree (3a + 2b + c, b + 2c + 3d). A klinear combination of monomials of a common bidegree is called a bihomogeneous form. The map φ is bihomogeneous, as it takes forms of bidegree (i, j) to forms of bidegree (i, j). Let J be the kernel of φ. Use a variation of the method of Exercise 2.76 to show that J = (x1 x2 − x0 x3 , x21 − x0 x2 , x22 − x1 x3 ). Exercise 3.26. Let I = (F1 = y1 y2 − y3 , F2 = y12 − y2 ). Show that I is a prime ideal in R = k[y1 , y2 , y3 ]. Compute the ideal I h in T = k[x0 , x1 , x2 , x3 ] and show that it is not equal to (F1h , F2h ). Hint: Consider the ideal of the “twisted cubic curve” J = (x1 x2 − x0 x3 , x21 − x0 x2 , x22 − x1 x3 ) from Exercise 3.25 which is a prime ideal.
3.3. Grassmann varieties The Grassmannian Grass(a, b) is the set of adimensional linear subspaces of k b . We will show that Grass(a, b) naturally has a structure as a projective variety. An adimensional linear subspace W of k b is determined by a basis v1 , . . . , va of W and hence by the a × b matrix ⎞ ⎛ v1 ⎟ ⎜ (3.4) A = ⎝ ... ⎠ va
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3. Projective Varieties
which has rank a. Two such matrices A and A represent the same adimensional subspace W if and only if there exists an a × a matrix B with nontrivial determinant (an element of GLa (k)) such that BA = A . This determines an equivalence relation ∼ on S = {a × b matrices A of rank a}, giving us a natural identiﬁcation of the sets Grass(a, b) and S/ ∼. We now show, by means of the Pl¨ ucker embedding, that Grass(a, b) has b a natural structure as a projective variety. Deﬁne a map ρ : S → P(a)−1 by sending the matrix A of (3.4) to the equivalence class (aI ) where the aI are the minors of a × a submatrices of A (in some ﬁxed order). At each point A of S, for at least one I, aI is not zero. If A ∼ A , then (aI ) = Det(B)(aI ) a as vectors in k ( b ) . Thus ρ(A) = ρ(A ) so ρ induces a welldeﬁned map b ρ : Grass(a, b) → P(a)−1 .
Now using linear algebra, we can show that ρ is injective and its image b is a closed subvariety of P(a)−1 , whose ideal is generated by quadrics. For more details about Grassmannians, see [81, Chapter VII of Volume 1], [136, Example 1 on page 42 of Volume 1], and [62, Section 5 of Chapter 1].
3.4. Regular functions and regular maps of quasiprojective varieties Suppose that X is a projective variety, that is, a closed irreducible subset of Pn . Then the coordinate ring S(X) = S(Pn )/I(X) of X is a standard graded kalgebra. We deﬁne
F  F, G ∈ S(X) are homogeneous of the same degree d and G = 0 . k(X) = G It is readily veriﬁed that k(X) is a ﬁeld. The ﬁeld k(X) is called the ﬁeld of rational functions on X or the function ﬁeld of X. We have that k(X) = S(X)(q) , deﬁned in (3.1), where q is the prime ideal q = (0) in S(X), the elements of degree 0 in T −1 S(X) where T is the multiplicative set of nonzero F ∈ k(X) with F, G homogeneous elements of S(X). By (3.3), if f = G homogeneous of the same degree d and p ∈ X is such that G(p) = 0, then the value f (p) ∈ k is welldeﬁned. Speciﬁcally, if p = (a0 : . . . : an ) = (b0 : . . . : bn ), then there exists 0 = λ ∈ k such that ai = λbi for 0 ≤ i ≤ n and f (a0 , . . . , an ) = =
F (a0 ,...,an ) F (λb0 ,...,λbn ) G(a0 ,...,an ) = G(λb0 ,...,λbn ) λd F (b0 ,...,bn ) F (b0 ,...,bn ) = G(b = λd G(b0 ,...,bn ) 0 ,...,bn )
f (b0 , . . . , bn ).
3.4. Regular functions and regular maps of quasiprojective varieties
75
Suppose that f ∈ k(X). We say that f is regular at p ∈ X if there exists an expression f = FG , where F, G ∈ S(X) are homogeneous of the same degree d, such that G(p) = 0. Lemma 3.27. Suppose that f ∈ k(X). Then the set U = {p ∈ X  f is regular at p} is an open subset of X. Proof. Suppose that p ∈ U . Then there are F, G ∈ S(X) which are homogeneous of the same degree d such that f = FG and G(p) = 0. Then X \ ZX (G) is an open neighborhood of p which is contained in U . Thus U is open. For p ∈ X, we deﬁne OX,p = {f ∈ k(X)  f is regular at p}. An element f ∈ OX,p induces a map f : U → A1 on some open neighborhood U of p in X. The ring OX,p is a local ring. We will write its maximal ideal as mX,p , or mp if there is no danger of confusion. We have that OX,p = S(X)(I(p)) , deﬁned in (3.1), the elements of degree 0 in the localization T −1 S(X), where T is the multiplicative set of homogeneous elements of S(X) which are not in I(p). We will denote OX,p /mp by k(p). As a ﬁeld, k(p) is isomorphic to k. Also, k(p) has a natural structure as a OX,p module. Suppose that U is an open subset of X. Then we deﬁne OX,p . OX (U ) = p∈U
Here the intersection takes place in k(X) and is over all p ∈ U . The ring OX (U ) is called the ring of regular functions on U . Suppose that U is a nontrivial open subset of X. Let Map(U, A1 ) be the set of maps from U to A1 . The set Map(U, A1 ) is a kalgebra since A1 = k is a kalgebra. We have a natural kalgebra homomorphism φ : OX (U ) → Map(U, A1 ) deﬁned by φ(f )(p) = f (p) for f ∈ OX (U ) and p ∈ U . We will show that φ is injective. Suppose f ∈ Kernel φ and p ∈ U . There exists an F where F, G ∈ S(X) are homogeneous of the same degree expression f = G d and G(p) = 0. For q in the nontrivial open set U \ Z(G) we have that F (q) G(q) = f (q) = 0. Thus F (q) = 0, and so F ∈ I(U \ Z(G)) = I(X) =
(0) = (0)
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3. Projective Varieties
F by Proposition 3.13. Thus F = 0, and so f = G = 0. Hence φ is injective. We may thus identify OX (U ) with the kalgebra φ(OX (U )) of maps from U to A1 .
An element f of the function ﬁeld k(X) of X induces a map f : U → A1 on some open subset U of X. An element f ∈ OX,p induces a map f : U → A1 on some open neighborhood U of p in X. There are examples of quasiprojective varieties U such that OU (U ) is not a ﬁnitely generated kalgebra, as discussed earlier for quasiaﬃne varieties in Example 2.98. A quasiprojective example is given by Zariski [159, page 456 ], based on his example which we will expound in Theorem 20.14. We now deﬁne a regular map of a quasiprojective variety. Deﬁnition 3.28. Suppose that X is a quasiprojective variety or a quasiaﬃne variety and Y is a quasiprojective variety or a quasiaﬃne variety. A regular map φ : X → Y is a continuous map such that for every open subset U of Y , the map φ∗ : Map(U, A1 ) → Map(φ−1 (U ), A1 ) deﬁned by φ∗ (f ) = f ◦ φ for f ∈ Map(U, A1 ) gives a kalgebra homomorphism φ∗ : OY (U ) → OX (φ−1 (U )). Deﬁnition 3.28 is consistent with our earlier deﬁnitions of regular maps of aﬃne and quasiaﬃne varieties by Propositions 2.95 and 2.96. We will show that OX (X) can be identiﬁed with the ring of regular maps from X to A1 in Theorem 3.40. Deﬁnition 3.29. Suppose that X is a quasiprojective variety or a quasiaﬃne variety and Y is a quasiprojective variety or a quasiaﬃne variety. A regular map φ : X → Y is said to be an isomorphism if there exists a regular map ψ : Y → X such that ψ ◦ φ = idX and φ ◦ ψ = idY . We will say that a quasiprojective variety is aﬃne if it is isomorphic to an aﬃne variety. Theorem 3.30. Suppose that W is a projective variety which is a closed subset of Pn , with homogeneous coordinate ring S(Pn ) = k[x0 , . . . , xn ], and xi is a homogeneous coordinate on Pn such that W ∩ D(xi ) = ∅. Then W ∩ D(xi ) is an aﬃne variety. Proof. Without loss of generality, we may suppose that i = 0. Write S(W ) = k[x0 , . . . , xn ]/I(W ) = k[x0 , . . . , xn ] where xi is the class of xi . The ring S(W ) is standard graded with the xi x having degree 1. By our assumption, x0 = 0, so x0j ∈ k(W ) for all j. We
3.4. Regular functions and regular maps of quasiprojective varieties
calculate k(W ) = S(W )(q)
77
x1 xn = QF k ,..., x0 x0
where q is the zero ideal (0) of S(W ). Let φ : W ∩ D(x0 ) → X be the homeomorphism induced by the map of Theorem 3.19, as explained after the proof of Theorem 3.19, where X is the aﬃne variety X = Z(I(W )a ) ⊂ An . We have that k[X] = k[An ]/I(X) = k[y1 , . . . , yn ]/I(X) = k[y 1 , . . . , y n ], where y i is the class of yi in k[X]. For a point p = (a0 : a1 : . . . : an ) ∈ W ∩ D(x0 ), we have that φ(p) = ( aa10 , . . . , aan0 ).
Deﬁne φ∗ : k[An ] = k[y1 , . . . , yn ] → k(W ) by f h (x0 , . . . , xn ) x1 xn ,..., = (3.5) φ∗ (f ) = f ◦ φ = f h x0 x0 (x0 )deg(f ) for f ∈ k[y1 , . . . , yn ]. ideal by Proposition 3.17, so I(X) = I(W )a The ideal I(W )a is a prime since I(X) = I(Z(I(W )a )) = I(W )a . The kernel of φ∗ is Kernel(φ∗ ) = {f ∈ k[y1 , . . . , yn ]  f h (x0 , . . . , xn ) = 0} = {f ∈ k[y1 , . . . , yn ]  f h ∈ I(W )}. Suppose f ∈ k[y1 , . . . , yn ] and f h ∈ I(W ). Then f = f ha ∈ I(W )a = I(X). If f ∈ I(X), then f h ∈ I(X)h = I(W ) by Proposition 3.17. Thus the kernel of φ∗ is I(X). We thus have that φ∗ induces an injective kalgebra homomorphism φ∗ : k[X] = k[y 1 , . . . , y n ] → k(W ), giving an induced kalgebra homomorphism φ∗ : k(X) → k(W ). By (3.5) we have that φ∗ (f ) ∈ OW,p for all f ∈ k[X] and p ∈ W ∩ D(x0 ). Suppose that p = (a0 : . . . : an ) ∈ W ∩ D(x0 ) and h ∈ OX,φ(p) . Then h = fg with f, g ∈ k[X] and g(φ(p)) = g( aa01 , . . . , aan0 ) = 0. Since a0 = 0, we see from (3.5) that φ∗ (g)(p) = 0. Thus φ∗ (h) ∈ OW,p . We thus have that F (x0 ,...,xn ) where φ∗ (OX,φ(p) ) ⊂ OW,p . Suppose that h ∈ OW,p . Then h = G(x 0 ,...,xn ) F, G are homogeneous of a common degree d and G(p) = 0. We have F = G
F xd0 G xd0
=
F (1, xx10 , . . . , xxn0 ) G(1, xx10 , . . . , xxn0 )
∗
=φ
Fa Ga
.
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3. Projective Varieties
Now Ga (φ(p)) =
G(p) ad0
= 0 so
Fa Ga
∈ OX,φ(p) . We thus have that
φ∗ : OX,φ(p) → OW,p
(3.6)
is an isomorphism for all p ∈ W ∩ D(x0 ). Since φ is a homeomorphism and by the deﬁnitions of OX and OW , we have for an open subset U of X an isomorphism (3.7) φ∗ OX (U ) = OX,q = OX,φ(p) → OW,p = OW (φ−1 (U )) q∈U
p∈φ−1 (U )
p∈φ−1 (U )
by (3.6). In particular, φ is a regular map. Now we have that the inverse map ψ of φ is determined by the map deﬁned in Theorem 3.19, as explained after the proof of Theorem 3.19, and ψ induces a map ψ ∗ : k(W ) → k(X), deﬁned by ψ ∗ (f ) = f ◦ψ for f ∈ k(W ). We have that xj = yj ψ∗ x0 for 1 ≤ j ≤ n. Thus ψ ∗ is an isomorphism of function ﬁelds with inverse φ∗ . By our above calculations, we then see that ψ is a regular map which is an inverse to φ. Thus φ is an isomorphism. Corollary 3.31. Suppose that X is a quasiaﬃne variety. Then X is isomorphic to a quasiprojective variety. Proof. The quasiaﬃne variety X is an open subset of an aﬃne variety Y ⊂ An . The aﬃne variety An is isomorphic to the open subset D(x0 ) of Pn by Theorem 3.30. Let W be the Zariski closure of Y in Pn . Then W ∩ D(x0 ) ∼ = Y by Theorem 3.30. Since X is an open subset of Y and Y is isomorphic to an open subset of W , we have that X is isomorphic to an open subset of W and thus is isomorphic to a quasiprojective variety. We will call an open subset U of a projective variety an aﬃne variety if U is isomorphic to an aﬃne variety. With this identiﬁcation, we have that all quasiaﬃne varieties are quasiprojective. Corollary 3.32. Every point p in a quasiprojective variety has an open neighborhood which is isomorphic to an aﬃne variety. Proof. Suppose that V is a quasiprojective variety and p ∈ V is a point. Then V is an open subset of a projective variety W , which is itself a closed subset of a projective space Pn . Writing S(Pn ) = k[x0 , . . . , xn ], there exists a homogeneous coordinate xi on Pn such that xi (p) = 0, so D(xi ) contains p. By Theorem 3.30, D(xi ) ∩ W is isomorphic to an aﬃne variety. We have that V ∩ (D(xi ) ∩ W ) is an open subset of D(xi ) ∩ W since V is open in
3.4. Regular functions and regular maps of quasiprojective varieties
79
W . Now by Lemma 2.83 and Proposition 2.93, there exists an aﬃne open subset U of D(xi ) ∩ W which contains p and is contained in V . The proof of Theorem 3.30 gives us the following useful formula. Suppose that W ⊂ Pn is a projective variety, and suppose that W is not contained in Z(xi ). Letting S(Pn ) = k[x0 , . . . , xn ] and S(W ) = S(Pn )/I(W ) = k[x0 , . . . , xn ], we have that xn x0 xn ∼ x0 ,..., ,..., /J (3.8) OW (D(xi )) = k =k xi xi xi xi where J
= {f ( xx0i , . . . , xxni )  f ∈ I(W )} = { xFd  F ∈ I(W ) is homogeneous of some degree d}. i
With the notation introduced before Lemma 3.4, we have that OW (D(xi )) = S(W )(x ) ∼ = S(Pn )(x ) /I(W )(x ) . i
i
i
We give a proof of this formula. Without loss of generality, we may assume that i = 0. By (3.7), the homomorphism φ∗ : k(X) → k(W ) takes OX (X) = k[X] = k[y 1 , . . . , y n ] to OW (D(x0 )). In our construction of φ∗ , we saw that x φ∗ (y j ) = x0j for all j, so x1 xn ,..., . OW (D(x0 )) = k x0 x0 Further, this last ring is the quotient of the polynomial ring xn x1 ,..., k x0 x0 by φ∗ (I(X)) = φ∗ (I(W )a ) = J. Finally, we observe that by Euler’s formula, if F (x0 , . . . , xn ) is homogeneous of degree d, then x0 xn xn x0 = xdi F . F (x0 , . . . , xn ) = F xi , . . . , xi ,..., xi xi xi xi Example 3.33. Let F = x22 x0 − x1 (x21 − x20 ) ∈ S(P2 ) = k[x0 , x1 , x2 ]. The form F is irreducible, so C = Z(F ) ⊂ P2 is a projective variety (an elliptic curve). Let y1 = xx10 and y2 = xx20 , which are coordinate functions on D(x0 ) ∼ = 2 a 2 2 A . Then I(C ∩D(x0 )) = (F ) = (f ) where f = y2 −y1 (y1 −1) ∈ k[D(x0 )] = k[y1 , y2 ]. We have that C ∩ Z(x0 ) = Z(x22 x0 − x1 (x21 − x20 ), x0 ) = Z(x0 , x1 ) = {Q} where Q = (0 : 0 : 1). Thus C is the union of the aﬃne curve C ∩ D(x0 ) and the point Q ∈ Z(x0 ) ∼ = P1 . Figure 3.1 shows this curve over C. We will see in Chapter 18 that C has genus 1, and thus (Section 18.8) C is topologically a sphere with one handle.
80
3. Projective Varieties
Q
Figure 3.1. C over C (left) and the real part of C ∩ D(x0 ) (right)
We can now calculate the regular functions on Pn . Proposition 3.34. The regular functions on Pn are OPn (Pn ) = k. Proof. Since {D(xi )  0 ≤ i ≤ n} is an open cover of Pn , ⎛ ⎞ n ⎝ OPn (Pn ) = OPn ,p = OPn ,p ⎠ p∈Pn
=
n i=0
i=0
OPn (D(xi )) =
p∈D(xi ) n i=0
x0 xn k ,..., . xi xi
Thus if g ∈ OPn (Pn ), we have expressions xn xn x0 x0 ,..., ,..., g = gi ∈k . xi xi xi xi For each i, there exists a smallest di ∈ N such that xn x0 di xi g i ,..., = fi (x0 , . . . , xn ) xi xi is a polynomial. Necessarily, we have that xi  fi in the UFD k[x0 , . . . , xn ]. Further, di is the degree of fi . If d0 = 0, then we have that g = g0 ∈ k, and we have established the proposition. Suppose that d0 > 0. Since gi = gj d for all i, j, we have that fi xj j = fj xdi i for all i, j. Since the polynomial ring k[x0 , . . . , xn ] is a UFD and xi , xj are relatively prime for i = j, we have that x0  f0 , which is a contradiction. Thus d0 = 0 and so OPn (Pn ) = k. The statement of Proposition 3.34 is true for arbitrary projective varieties W (taking the intersection over the open sets W ∩ D(xi ) such that D(xi ) ∩ W = ∅) but we need to be a little careful with the proof, as can
3.4. Regular functions and regular maps of quasiprojective varieties
81
be seen from the following example. Consider the standard graded domain T = Q[x0 , x1 ] = Q[x0 , x1 ]/(x20 + x21 ). We compute L = Q[ xx01 ] ∩ Q[ xx10 ]. We have that x20 = −x21 so xx10 = − xx01 . √ x0 ∼ L=Q = Q[ −1], = Q[t]/(t2 + 1) ∼ x1 which is larger than Q. This example shows that any proof that OW (W ) = k for a projective variety W must use the assumption that k is algebraically closed. We will give a diﬀerent proof of Theorem 3.35 in Corollary 5.15. Theorem 3.35. Suppose that W is a projective variety. Then the regular functions on W are OW (W ) = k. Proof. W is a closed irreducible subset of Pn for some n. Let S(Pn ) = k[x0 , . . . , xn ] and S(W ) = k[x0 , . . . , xn ] where xi is the class of xi in S(W ). We may suppose that xi = 0 for all i, for otherwise we have that W ⊂ Z(xi ) so that W is a closed subset of Z(xi ) ∼ = Pn−1 ⊂ Pn , and W is contained in a projective space of smaller dimension. Repeating this reduction at most a ﬁnite number of times, we eventually realize W as a closed subset of a projective space such that W ⊂ Z(xi ) for all i. Suppose that n n x0 xn . f ∈ OW (W ) = OW (D(xi )) = k ,..., xi xi i=0
i=0
Then there exist Ni ∈ N and homogeneous elements Gi ∈ S(W ) of degree Ni such that Gi f = N for 0 ≤ i ≤ n. xi i Let S be the set of homogeneous forms of degree i in S(W ) (so that S(W ) ∼ = ∞ i Ni S ). We have that x f ∈ S for 0 ≤ i ≤ n. Suppose that N ≥ N i. Ni i i=0 i Since SN is spanned (as a kvector space) by monomials of degree N in x0 , . . . , xn , for each such monomial at least one xi has an exponent ≥ Ni . Thus SN f ⊂ SN . Iterating, we have that SN f q ⊂ SN for all q ∈ N. In q particular, xN 0 f ∈ S(W ) for all q > 0. Thus the subring S(W )[f ] of the quotient ﬁeld of S(W ) is contained in x−N 0 S(W ), which is a ﬁnitely generated S(W ) module. Thus f is integral over S(W ) (by Theorem 1.49), and there exist m and a1 , . . . , am ∈ S(W ) such that f m + a1 f n−1 + · · · + am = 0. Since f has degree 0, we can replace the ai with their homogeneous components of degree 0 and still have a dependence relation. But the elements of degree 0 in S(W ) consists of the ﬁeld k. Now k[f ] is a domain since it is a
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3. Projective Varieties
subring of the quotient ﬁeld of S(W ) and so k[f ] is a ﬁnite extension ﬁeld of k. Thus f ∈ k since k is algebraically closed. Proposition 3.36. Suppose that W is a projective variety. Then W is separated (distinct points of W have distinct local rings). Proof. Suppose that W is a closed subset of a projective space Pn . Write S(W ) = S(Pn )/I(W ) = k[x0 , . . . , xn ]/I(W ) = k[x0 , . . . , xn ] where xi is the class of xi in S(W ). Suppose that p ∈ W . Then p ∈ D(xi ) for some i, and since W ∩ D(xi ) is aﬃne with x0 xn , ,..., k[W ∩ D(xi )] = k xi xi OW,p is the localization of k xx0i , . . . , xxni at a maximal ideal m. Let π : x OW,p → OW,p /mOW,p ∼ = k be the residue map. Let π( xji ) = αj ∈ k for 0 ≤ j ≤ n (with αi = 1 since xxii = 1). We have that xi (p) = 0 and xj (p) = αj xi (p). Thus p = (x0 (p) : . . . : xn (p)) = (α0 : . . . : αn ) ∈ W is uniquely determined by the ring OW,p .
Proposition 3.37. Suppose that X is a projective variety and U ⊂ X is a nonempty quasiaﬃne open subset. Then the quotient ﬁeld of OX (U ) is the function ﬁeld k(X) of X. Proof. Let p ∈ U . Then OX,p = OU,p is a localization of OU (U ) at a maximal ideal (by Lemma 2.91). By the deﬁnition of OX,p , we have that OX,p ⊂ k(X), so QF(OX (U )) ⊂ k(X). Suppose that f ∈ k(X). The F where F, G ∈ S(X) are homogeneous of a common degree function f = G d and G = 0. There exists a linear form L ∈ S(X) such that L(p) = 0. F is in the quotient ﬁeld of OX,p and hence is in the Thus LFd , LGd ∈ OX,p so G quotient ﬁeld of OU (U ). It follows from Proposition 3.37 that if X is a projective variety and p ∈ X, then k(X) = QF(OX,p ). Proposition 3.37 allows us to deﬁne the ﬁeld of rational functions or the function ﬁeld k(Y ) of a quasiprojective variety Y as k(U ), where U is a nontrivial aﬃne open subset of Y . Suppose that φ : X → Y is a regular map of quasiprojective varieties. Extending our deﬁnition for quasiaﬃne varieties, we say that φ is dominant if the Zariski closure of φ(X) in Y is equal to Y .
3.4. Regular functions and regular maps of quasiprojective varieties
83
Proposition 3.38. Suppose that X and Y are quasiprojective varieties and φ : X → Y is a dominant regular map. Then the map φ∗ : Map(Y, A1 ) → Map(X, A1 ) deﬁned by φ∗ (f ) = f ◦φ for f ∈ Map(Y, A1 ) induces an injective kalgebra homomorphism φ∗ : k(Y ) → k(X). Proof. Let V be an aﬃne open subset of Y and U be an aﬃne open subset of the open subset f −1 (V ) of X. Then the restriction of φ to a map of aﬃne varieties φ : U → V is dominant, so the kalgebra homomorphism φ∗ : k[V ] → k[U ] is injective. Taking the induced map on quotient ﬁelds, we obtain by Proposition 3.37 the desired homomorphism of function ﬁelds. The following proposition gives a useful criterion for a map of quasiprojective varieties to be regular. Proposition 3.39. Suppose that X and Y are quasiprojective varieties and φ : X → Y is a map. Let {Vi } be a collection of open aﬃne subsets covering Y and {Ui } be a collection of open subsets covering X, such that 1. φ(Ui ) ⊂ Vi for all i and 2. the map φ∗ : Map(Vi , A1 ) → Map(Ui , A1 ) deﬁned by φ∗ (f ) = f ◦ φ for f ∈ Map(Vi , A1 ) maps OY (Vi ) into OX (Ui ) for all i. Then φ is a regular map. Proof. Suppose that U is an aﬃne subset of Ui . Then φ∗ induces a kalgebra homomorphism φ∗ : k[Vi ] → k[U ] since the restriction map OY (Ui ) → OY (U ) is a kalgebra homomorphism. Thus we may reﬁne our cover {Ui } to assume that the Ui are aﬃne for all i. Let φi : Ui → Vi be the restriction of φ. Consider the kalgebra homomorphism φ∗i : k[Vi ] → k[Ui ]. Suppose that p ∈ Ui and q ∈ Vi . We have that (3.9) φ∗i (IVi (q)) ⊂ IUi (p) if and only if φ∗i (f )(p) = 0 for all f ∈ IVi (q) if and only if f (φi (p)) = 0 for all f ∈ IVi (q) if and only if IVi (q) ⊂ IVi (φi (p)) if and only if IVi (q) = IVi (φi (p)) since IVi (q) and IVi (φi (p)) are maximal ideals if and only if q = φi (p) since the aﬃne variety Vi is separated by Corollary 2.87. Now there exists a regular map gi : Ui → Vi such that gi∗ = φ∗i (by Proposition 2.40). The calculation (3.9) shows that for p ∈ Ui and q ∈ Vi we have that gi (p) = q if and only if gi∗ (IVi (q)) ⊂ IUi (p). Thus φi = gi so that φi is a regular map. In particular the φi are all continuous so that φ is continuous.
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3. Projective Varieties
Suppose that q ∈ Y and p ∈ φ−1 (q). Then there exist Ui and Vi such that p ∈ Ui and q ∈ Vi . For f ∈ k[Vi ], f ∈ IVi (q) = IVi (φi (p))
if and only if if and only if if and only if
f (φi (p)) = 0 φ∗i (f )(p) = 0 f ∈ (φ∗i )−1 (IUi (p)).
Thus (φ∗i )−1 (IUi (p)) = IVi (q), and we have an induced kalgebra homomorphism φ∗i : OVi ,q = k[Vi ]IV (q) → k[Ui ]IU (p) = OUi ,p . i
i
But this is just the statement that φ∗ : OY,q → OX,p . Thus (3.10)
φ∗ : OY,q →
OX,p .
p∈φ−1 (q)
Suppose that U is an open subset of Y . Then OY,q OY (U ) = q∈U
and OX (φ−1 (U )) =
q∈U
⎛ ⎝
⎞ OX,p ⎠ .
p∈φ−1 (q)
Thus by (3.10), we have that φ∗ : OY (U ) → OX (φ−1 (U )). We have established that φ satisﬁes Deﬁnition 3.28, and so φ is a regular map. In the following theorem we show that the regular functions on a quasiprojective variety X are the regular maps from X to A1 . Theorem 3.40. Suppose that X is a quasiprojective variety. Then OX (X) is naturally isomorphic to the kalgebra of regular maps from X to A1 . Proof. Let R be the kalgebra of regular maps from X to A1 . Let t be the natural coordinate function on A1 deﬁned by t(q) = q for q ∈ A1 . Suppose that f ∈ OX (X). Then the association p → f (p) is a welldeﬁned map from X to A1 , which we will denote by f : X → A1 . Since f ∗ (t) = f , the map f ∗ : k[A1 ] = k[t] → OX (X) is a welldeﬁned kalgebra homomorphism. We have that f : X → A1 is a regular map by Proposition 3.39, taking the trivial open cover {X} of X and the trivial aﬃne open cover {A1 } of A1 . We thus have that the rule φ(f )(p) = f (p) for f ∈ OX (X) and p ∈ X determines a welldeﬁned kalgebra homomorphism φ : OX (X) → R.
3.4. Regular functions and regular maps of quasiprojective varieties
85
The map φ is injective as shown before Deﬁnition 3.28. Now suppose that g : X → A1 ∈ R. Then g ∗ : k[A1 ] → OX (X) is a kalgebra homomorphism. Let f = g ∗ (t) ∈ OX (X). Then φ(f )(p) = [g ∗ (t)](p) = t(g(p)) = g(p) for all p ∈ X. Thus φ is surjective.
Exercise 3.41. Let W be the projective variety (surface) W = Z(x0 x1 − x2 x3 ) ⊂ P3 . We can write W as a union of an aﬃne variety, W ∩D(x0 ), and the algebraic set W ∩ Z(x0 ) ⊂ Z(x0 ) ∼ = P2 . Let x1 x2 x3 y2 = , y3 = , y1 = , x0 x0 x0 3 ∼ which are coordinate functions on D(x0 ) = A . Find the ideal I(W ∩
D(x0 )) ⊂ OP3 (D(x0 )) = k[y1 , y2 , y3 ]. What is the algebraic set W ∩ Z(x0 ) viewed as a subset of P2 ?
Exercise 3.42. Let U be the quasiaﬃne variety U = An+1 \ {(0, . . . , 0)}. Consider the map π : U → Pn deﬁned by π(a0 , a1 , . . . , an ) = (a0 : . . . : an ). Is π a regular map? Prove your answer.
Chapter 4
Regular and Rational Maps of Quasiprojective Varieties
In this chapter we deﬁne rational maps, give some useful ways to represent regular and rational maps, and give some examples. We show in Corollary 4.7 that every projective variety X has the basis of open sets D(F ) for F ∈ S(X) which are (isomorphic to) aﬃne varieties.
4.1. Criteria for regular maps Lemma 4.1. A composition of regular maps of quasiprojective varieties is a regular map. Proof. This follows from the deﬁnition of a regular map, Deﬁnition 3.28, since the composition of continuous functions is continuous and a composition of kalgebra homomorphisms is a kalgebra homomorphism. Proposition 4.2. Suppose that U and V are quasiprojective varieties. 1) Suppose that U is an open subset of V . Then the inclusion i:U →V is a regular map. 87
88
4. Regular and Rational Maps of Quasiprojective Varieties
2) Suppose that U is a closed subset of V . Then the inclusion i:U →V is a regular map. Proof. Let {Vi } be a cover of V by aﬃne open sets. Let Ui = i−1 (Vi ). Then {Ui } is an open cover of U such that i(Ui ) ⊂ Vi for all i. In both cases 1) and 2), i∗ is restriction of functions, so i∗ : OV (Vi ) → OU (Ui ) for all i. By Proposition 3.39, i is a regular map. Generalizing Deﬁnition 2.94, we have the following deﬁnition. Deﬁnition 4.3. Suppose that φ : U → V is a regular map of quasiprojective varieties. 1) The map φ is called a closed embedding if there exists a closed subvariety Z of V (an irreducible closed subset) such that φ(U ) ⊂ Z and the induced regular map φ : U → Z is an isomorphism. 2) The map φ is called an open embedding if there exists an open subset W of V such that φ(U ) ⊂ W and the induced regular map φ : U → W is an isomorphism. Proposition 4.4. Suppose that X is a quasiprojective variety and φ = (f1 , . . . , fn ) : X → An is a map. Then φ is a regular map if and only if fi are regular functions on X for all i. Proof. First suppose that φ : X → An is a regular map. Let x1 , . . . , xn be the natural coordinate functions on An , so that k[An ] = k[x1 , . . . , xn ]. We have that fi = xi ◦ φ : X → A1 is a regular map for 1 ≤ i ≤ n by Lemma 4.1, so fi ∈ OX (X) by Theorem 3.40. Now suppose that f1 , . . . , fn ∈ OX (X) and φ = (f1 , . . . , fn ) : X → An . Then the map φ∗ : k[An ] → Map(X, A1 ) has image in OX (X) and so φ is a regular map by Proposition 3.39 (taking the trivial open cover {X} of X and the trivial aﬃne open cover {An } of An ). Suppose that X is a quasiprojective variety and φ : X → Pn is a regular map. Let S(Pn ) = k[y0 , . . . , yn ]. Suppose that p ∈ X. Then there exists a j such that φ(p) ∈ D(yj ). Let V = D(yj ). Now φ−1 (V ) is an open subset of X, so by Corollary 3.32, there exists an aﬃne open neighborhood U of p in φ−1 (V ). Consider the restriction φ : U → V . Then φ is a regular map of aﬃne varieties, so on U , we can represent φ : U → V ∼ = An as φ = (f1 , . . . , fn ) for some f1 , . . . , fn ∈ k[U ] = OX (U ) ⊂ k(U ) = k(X) (by
4.1. Criteria for regular maps
89
Proposition 3.37). Thus we have a representation φ = (f1 : . . . : fj : 1 : fj+1 : . . . : fn ) on the neighborhood U of p. In summary, we have shown that there exists an open neighborhood U of p in X and regular functions f0 , . . . , fn ∈ OX (U ) such that (4.1)
φ = (f0 : . . . : fn )
on U and there are no points on U where all of the fi vanish. Suppose that q ∈ X is another point and Y is an open neighborhood of q in X with regular functions g0 , . . . , gn ∈ OX (Y ) such that (4.2)
φ = (g0 : . . . : gn )
on Y and the gi have no common zeros on Y . These two representations of φ must agree on U ∩ Y , which happens if and only if (4.3)
f i g j − fj g i = 0
for 0 ≤ i, j ≤ n
on U ∩ Y (which occurs if and only if fi gj − fj gi = 0 in k(x)). We can also use this method to construct regular maps. Suppose that X is a quasiprojective variety, {Us } is an aﬃne open cover of X, and for all Us , (4.4) fs,0 , . . . , fs,n ∈ OX (Us ) are functions that have no common zeros on Us and (4.5)
fs,i ft,j − fs,j ft,i = 0
for all s, t and 0 ≤ i, j ≤ n.
Then by Proposition 3.39 (and Proposition 2.96), the collection of maps (4.4) on an open cover {Us } of X satisfying (4.5) determines a regular map φ : X → Pn . We can thus think of a regular map φ : X → Pn as an equivalence class of expressions (f0 : f1 : . . . : fn ) with f0 , . . . , fn ∈ k(X) and such that (f0 : f1 : . . . : fn ) ∼ (g0 : . . . : gn ) if and only if fi gj − fj gi = 0 for 0 ≤ i, j ≤ n. We further have the condition that for each p ∈ X there exists a representative (f0 : f1 : . . . : fn ) of φ such that fi ∈ OX,p for all i and some fi does not vanish at p. We can reinterpret this to give another useful way to represent a regular map φ : X → Pn . The quasiprojective variety X is an open subset of a projective variety W which is a closed subvariety of a projective space Pm . In the representation (4.1) of φ near p ∈ X, we can write the regular maps as Fi where Fi , Gi are homogeneous elements of a common degree ratios fi = G i di in the homogeneous coordinate ring S(W ) (which is a quotient of S(Pm ))
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4. Regular and Rational Maps of Quasiprojective Varieties
and the Gi do not vanish near p. We can thus represent φ near p by the expression (H0 : H1 : . . . : Hn ) where the Hi = ( j=i Gj )Fi are homogeneous elements of S(W ) (or of S(Pm )) of a common degree and at least one Hi does not vanish at p. We can thus think of a regular map φ : X → Pn as an equivalence class of expressions (F0 : F1 : . . . : Fn ) with F0 , . . . , Fn homogeneous elements of S(W ) (or of S(Pm )) all having the same degree and such that (F0 : F1 : . . . : Fn ) ∼ (G0 : . . . : Gn ) if and only if Fi Gj − Fj Gi = 0 for 0 ≤ i, j ≤ n in S(W ). We further have the condition that for each p ∈ X there exists a representative (F0 : F1 : . . . : Fn ) of φ such that some Fi does not vanish at p. It is not required that the common degree of the Fi be the same as the common degree of the Gj .
4.2. Linear isomorphisms of projective space Suppose that A = (aij ) is an invertible (n + 1) × (n + 1) matrix with coeﬃcients in k (indexed as 0 ≤ i, j ≤ n). Deﬁne homogeneous elements n Si by Li = nj=0 aij xj Li of degree 1 in S = k(P ) = k[x0 , . . . , xn ] = for 0 ≤ i ≤ n. Then the Li are a kbasis of S1 so that Z(L0 , . . . , Ln ) = Z(x0 , . . . , xn ) = ∅. Thus φA : Pn → Pn deﬁned by φA = (L0 : . . . : Ln ) is a regular map. If B is another invertible (n + 1) × (n + 1) matrix with coeﬃcients in k, then we have that φA ◦ φB = φAB . Thus φA is an isomorphism of Pn , with inverse map φA−1 . We will call L0 , . . . , Ln homogeneous coordinates on Pn . Proposition 4.5. Suppose that W is a projective variety which is a closed subvariety of Pn . Suppose that L ∈ S(Pn ) is a linear homogeneous form, such that D(L) ∩ W = ∅. Then D(L) ∩ W is an aﬃne variety. Proof. Write L = nj=0 a0j xj for some a0j ∈ k not all zero, and extend the vector (a00 , . . . , a0n ) to a basis of k n+1 . Arrange this basis as the rows of the (n + 1) × (n + 1) matrix A = (aij ). Here A is necessarily invertible. Now the isomorphism φA : Pn → Pn maps D(L) to D(x0 ) and W to a projective variety φA (W ) which is not contained in Z(x0 ). We have that
4.3. The Veronese embedding
91
φA (W ) ∩ D(x0 ) is an aﬃne variety by Theorem 3.30. Thus W ∩ D(L) is aﬃne since it is isomorphic to φA (W ) ∩ D(x0 ). Composing the isomorphism φ∗A : OφA (W ) (D(x0 )) ∼ = OW (D(L)) of the above proof with the representation of OφA (W ) (D(x0 )) of (3.8), letting S(Pn ) = k[x0 , . . . , xn ] and S(W ) = S(Pn )/I(W ) = k[x0 , . . . , xn ], we obtain that x0 xn ∼ x0 xn (4.6) OW (D(L)) = k ,..., ,..., /J =k L L L L where J = {f ( xL0 , . . . , xLn )  f ∈ I(W )}. With the notation introduced before Lemma 3.4, we have that OW (D(L)) = S(W ) ∼ = S(Pn )(L) /I(W )(L) . (L)
4.3. The Veronese embedding d Suppose that d is a positive integer. Let xd0 , xd−1 0 x1 , . . . , xn bethe set of n all monomials of degree d in S(P ) = k[x0 , . . . , xn ]. There are n+d such n n+d monomials. Let e = d − 1. Since these monomials are a kbasis of Sd , d we have that Z(xd0 , xd−1 0 x1 , . . . , xn ) = ∅. Thus we have a regular map
Λ : Pn → Pe d n e deﬁned by Λ = (xd0 : xd−1 0 x1 : . . . : xn ). Let W be the closure of Λ(P ) in P .
Let Pe have the homogeneous coordinates yi0 i1 ...in where i0 , . . . , in are nonnegative integers such that i0 + · · · + in = d. The map Λ is deﬁned by the equations (4.7)
yi0 i1 ...in = xi00 xi11 · · · xinn .
We will establish that Λ is an isomorphism of Pn to W . Suppose that q ∈ W . We can verify that q = Λ(p) for some p ∈ Pn (in Theorem 5.14 we will establish this generally), so that xj (p) = 0 for some j. We have that y0...0d0...0 (q) = xdj (p) = 0, where d is in the jth place of y0...0d0...0 . Thus the aﬃne open sets Wj = D(y0...0d0...0 ) of Pe cover W . Let S(W ) = S(Pe )/I(W ) = k[{yi0 ,...,in }]. In S(W ), we have the identities (4.8) y i0 ...in y j0 ...jn = y k0 ...kn y l0 ...ln if i0 + j0 = k0 + l0 , . . . , in + jn = kn + ln . Now on each open subset Wj = D(y0...0d0...0 ) ∩ W of W we deﬁne a regular map Ψj : Wj → D(xj ) ⊂ Pn by Ψj = (y 10...0(d−1)0...0 : y 010...0(d−1)0...0 : . . . : y 0...0d0...0 : . . . : y 0...0(d−1)0...01 ).
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4. Regular and Rational Maps of Quasiprojective Varieties
Now we have that : x1 xd−1 : . . . : xn xd−1 ) = (x0 : x1 : . . . : xn ) = idD(xj ) , Ψj ◦ Λ = (x0 xd−1 j j j and the identities 0 1 n y i0 ...in y d0...0(d−1)0...0 = y i10...0(d−1)0...0 y i010...0(d−1)0...0 · · · y i0...0(d−1)0...01
whenever i0 + i1 + · · · + in = d (which are special cases of (4.8)) imply that Λ ◦ Ψj = idWj . Now we check, again using the identities (4.8), that Ψj = Ψk on Wj ∩Wk . Thus the Ψj patch by Proposition 3.39 to give a regular map Ψ : W → Pn which is an inverse to Λ. The isomorphism Λ : Pn → W is called the Veronese embedding. Recalling the deﬁnition (3.2) of a Veronese ring of a graded ring, we have a graded, degreepreserving isomorphism S(Pn )id Λ∗ : S(W ) → S(Pn )(d) = i≥0 (d)
where elements of S(Pn )i = S(Pn )id have degree i. The map is the kalgebra homomorphism deﬁned by Λ∗ (y i0 ,...,in ) = xi00 · · · xinn . We obtain the following result using the Veronese embedding, composed with a linear isomorphism, which generalizes Proposition 4.5. Proposition 4.6. Suppose that W is a projective variety which is a closed subvariety of Pn . Suppose that F ∈ S(Pn ) is a homogeneous form of degree d > 0 such that D(F ) ∩ W = ∅. Then D(F ) ∩ W is an aﬃne variety. Letting S(Pn ) = k[x0 , . . . , xn ] and S(W ) = S(Pn )/I(W ) = k[x0 , . . . , xn ], we obtain that (4.9) OW (D(F )) = k[ M  M is a monomial in x0 , . . . , xn of degree d, F = F (x0 , . . . , xn )] F ∼ = k[ M F  M is a monomial in x0 , . . . , xn of degree d]/J n)  G ∈ I(W ) is a form of degree de}. With the notawhere J = { G(x0F,...,x e tion introduced before Lemma 3.4, we have that ∼ OW (D(F )) = S(W ) = S(Pn )(F ) /I(W )(F ) .
(F )
Corollary 4.7. Suppose that X is a projective variety. Then the set of aﬃne open subsets D(F ) for F ∈ S(X) is a basis of the Zariski topology on X.
4.4. Rational maps of quasiprojective varieties
93
Proof. This follows from Proposition 4.6 and Lemma 3.16.
Exercise 4.8. Suppose that k has characteristic = 2. Let W be the conic W = Z(x20 + x21 + x22 ) ⊂ P3 . Then W has the coordinate ring S(W ) = k[x0 , x1 , x2 ]/(x20 + x21 + x22 ). Compute OW (D(L)) where L is the linear form L = x0 + x1 + x2 . Express your answer as a quotient of a polynomial ring by an ideal. Exercise 4.9. A subset W of Pn is called a cone if there exists a closed subvariety Y of a linear subvariety Pr of Pn and a linear subvariety Z ∼ = Pn−r−1 of Pn such that Pr ∩ Z = ∅ and W is the union of all lines joining a point of Y to a point of Z. Show that a cone W is a closed subvariety of Pn and that there exist homogeneous coordinates x0 , . . . , xn on Pn such that x0 , . . . , xr are homogeneous coordinates on Pr , and if I(Y ) is the ideal of Y in S(Pr ) = k[x0 , . . . , xr ], then I(W ) = I(Y )k[x0 , . . . , xn ] in S(Pn ) = k[x0 , . . . , xn ].
4.4. Rational maps of quasiprojective varieties Suppose that X and Y are quasiprojective varieties. As in Section 2.6, we deﬁne a rational map φ : X Y to be a regular map on some nonempty open subset U of X to Y . As in the aﬃne case, we have that φ has a unique extension as a regular map to a largest open subset W (φ) of X, and if φ and ψ are rational maps from X to Y which are regular on respective open subsets U and V such that φ and ψ agree on the intersection U ∩ V , then φ = ψ. We now formulate the concept of a rational map of projective varieties in algebra. The following deﬁnition extends Deﬁnition 2.106. Deﬁnition 4.10. Suppose that X is a projective variety. A rational map φ : X Pn is an equivalence class of (n + 1)tuples φ = (f0 : . . . : fn ) with f0 , . . . , fn ∈ k(X) not all zero, where (g0 : . . . : gn ) is equivalent to (f0 : . . . : fn ) if fi gj − fj gi = 0 for 0 ≤ i, j ≤ n. A rational map φ : X Pn is regular at a point p ∈ X if and only if there exists a representation (f0 : . . . : fn ) of φ such that all of the fi are regular functions at p (fi ∈ OX,p for all i) and some fi (p) = 0. Let W (φ) be the open set of points of X on which φ is regular. Then φ : W (φ) → Pn is a regular map. Suppose that Y is a projective variety which is a closed subvariety of a projective space Pn . A rational map φ : X Y is a rational map φ : X Pn such that φ(W (φ)) ⊂ Y .
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4. Regular and Rational Maps of Quasiprojective Varieties
A rational map φ : X Y is called dominant if φ(U ) is dense in Y when U is a (nontrivial) open subset of X on which φ is a regular map. Deﬁnition 2.109 extends to deﬁne a birational map of quasiprojective varieties, as follows. Deﬁnition 4.11. A dominant rational map φ : X Y of quasiprojective varieties is birational if there exists a dominant rational map ψ : Y X such that ψ ◦ φ = idX and φ ◦ ψ = idY . Two quasiprojective varieties X and Y are said to be birationally equivalent if there exists a birational map φ : X Y . Suppose that φ : X Y is a rational map. Let U be an open subset of X such that φ : U → Y is a regular map. Let A be an open aﬃne subset of Y such that A ∩ φ(U ) = ∅, and let B be an open aﬃne subset of (φU )−1 (A). Then φ induces a regular map of aﬃne varieties φ : B → A (by Deﬁnition 3.28 and Propositions 2.95 and 2.96). From this reduction, we obtain that the results of Section 2.6 on rational maps of aﬃne varieties are also valid for projective varieties. We obtain the following generalization of Proposition 2.112. Proposition 4.12. Every projective variety X is birationally equivalent to a hypersurface Z(G) ⊂ Pn . It is sometimes convenient to interpret rational maps in terms of equivalence classes of (n + 1)tuples of homogeneous forms (H0 : . . . : Hn ) of a common degree. Recalling our analysis of regular maps in the previous section, we see that a rational map φ : X Pn can also be interpreted as an equivalence class of (n + 1)tuples (F0 : . . . : Fn ) with F0 , . . . , Fn forms of the same degree in S(X) which are not all zero, where (G0 : . . . : Gn ) is equivalent to (F0 : . . . : Fn ) if Fi Gj − Fj Gi = 0 for 0 ≤ i, j ≤ n. A rational map φ is regular at a point p ∈ X if and only if there exists a representation (F0 : . . . : Fn ) of φ by forms in S(X) of the same degree such that at least one of the Fi does not vanish at p. The image of a rational map φ : X Y is the Zariski closure in Y of φ(U ) where U is a nontrivial open subset of X on which φ is regular. Lemma 4.13. Suppose that X is a projective variety and φ : X Pn is a rational map. Then the image of φ has the coordinate ring k[H0 , . . . , Hn ] for any equivalence class of homogeneous forms H0 , . . . , Hn representing φ. Proof. Let W be the image of φ in Pn . Let V = X \ Z(H0 , . . . , Hn ). Then W is the Zariski closure of φ(V ). Let x0 , . . . , xn be our homogeneous coordinates on Pn , so that S(Pn ) = k[x0 , . . . , xn ]. We have a surjective
4.5. Projection from a linear subspace
95
kalgebra homomorphism Φ : k[x0 , . . . , xn ] → k[H0 , . . . , Hn ] deﬁned by Φ(xi ) = Hi for 0 ≤ i ≤ n. Let I be the kernel of this map. For 0 ≤ i ≤ n, we have regular maps φi = (φXHi ) : XHi → Pnxi of aﬃne varieties, with φ∗i
:
x0 xn → k[XHi ] =k ,..., xi xi
k[Pnxi ]
deﬁned by φ∗i
xj xi
=
Hj Hi
for 0 ≤ j ≤ n. The kernel of φ∗i is
x0 xn n ,...,  F ∈ I(W ) = I(W )(xi ) I(W ∩ Pxi ) = F xi xi Hn 0 and the image of φ∗i is k[ H Hi , . . . , Hi ]. For all i we have short exact sequences x0 xn φ∗i H0 Hn n ,..., →k ,... → 0. 0 → I(xi ) → S(P )(xi ) = k xi xi Hi Hi
Thus I(xi ) = I(W )(xi ) for 0 ≤ i ≤ n, and thus I sat = I(W )sat by Lemma 3.4. Since both I(W ) and I are prime ideals, we have that I(W ) = I.
4.5. Projection from a linear subspace A linear subspace E of a projective space Pn is the closed subset deﬁned by the vanishing of a set of linear homogeneous forms. Such a subvariety is isomorphic to a projective space Pd for some d ≤ n. The ideal I(E) is then minimally generated by a set of n − d linear forms; in fact a set of linear forms {L1 , . . . , Ln−d } is a minimal set of generators of I(E) if and only if L1 , . . . , Ln−d is a kbasis of the klinear subspace S1 of the homogeneous linear forms on Pn which vanish on E. We will say that E has dimension d. Suppose that E is a ddimensional linear subspace of Pn . Let L1 , . . . , Ln−d be linear forms in k[x0 , . . . , xn ] which deﬁne E. The rational map φ : Pn Pn−d−1 with φ = (L1 : . . . : Ln−d ) is called the projection from E. The map is regular on the open set Pn \ E. This map can be interpreted geometrically as follows: choose a linear subspace F of Pn of dimension n−d−1 which is disjoint from E (to ﬁnd such an F , just extend L1 , . . . , Ln−d to a kbasis L1 , . . . , Ln−d , M1 , . . . , Md+1 of S1 (the vector space of linear forms in S(Pn )), and let F = Z(M1 , . . . , Md+1 ). Suppose that p ∈ Pn \ E. Let Gp be the unique linear subspace of Pn of
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4. Regular and Rational Maps of Quasiprojective Varieties
dimension d + 1 which contains p and E. We have that Gp intersects F in a unique point. This intersection point can be identiﬁed with φ(p). This map depends on the choice of basis L1 , . . . , Ln−d of linear forms which deﬁne E. However, there is not a signiﬁcant diﬀerence if a diﬀerent basis L1 , . . . , Ln−d is chosen. In this case there is a linear isomorphism Λ : Pn−d−1 → Pn−d−1 such that (L1 : . . . : Ln−d ) = Λ ◦ (L1 : . . . : Ln−d ). Exercise 4.14. Suppose that φ : Pm → Pn is a regular map. Show that there exist homogeneous forms F0 , . . . , Fn of S(Pm ), all of a common degree, such that Z(F0 , . . . , Fn ) = ∅ and φ = (F0 : . . . : Fn ). (The conclusions of this exercise are not true for Pm replaced with a projective variety W as is shown in the next exercise). Exercise 4.15. Let W = Z(xw − yz) ⊂ P3 , where S(W ) = k[x, y, z, w]/(xw − yz) = k[x, y, z, w]. Consider the rational map φ : W P1 deﬁned by φ = (x : y). a) Show that (x : y) ∼ (z : w) as rational maps, so that φ is represented by both (x : y) and (z : w). Then show that φ is a regular map of W . b) Show that there do not exist forms F0 , F1 ∈ S(W ) of a common degree, such that φ is represented by (F0 : F1 ) and ZW (F0 , F1 ) = ∅. To do this, assume that such F0 , F1 do exist, and derive a contradiction. First show that p ∈ Wy implies F1 (p) = 0. Then show that p ∈ Ww implies F1 (p) = 0. Conclude that ZW (F1 ) ∩ (Wy ∪ Ww ) = ∅. Consider the aﬃne variety X = Z(xw−yz) ⊂ A4 , which has regular functions k[X] = k[x, y, z, w]. Let U = Xy ∪ Xw , an open subset of X. Show that FF01 ∈ OX (U ). Now consider Example 2.85 and explain how the conclusions of this example give a contradiction. Exercise 4.16. Let V be the vector space V = k n+1 , and let π : V \ {(0, . . . , 0)} → Pn be the map π(c0 , . . . , cn ) = (c0 : . . . : cn ) for (c0 , . . . , cn ) ∈ V \ {(0, . . . , 0)}. Writing S(Pn ) = k[x0 , . . . , xn ] = i≥0 Si , we have an identiﬁcation of the linear forms S1 on Pn with the linear forms on V ; that is, S1 is naturally isomorphic (as a kvector space) to the dual space V ∗ . a) Suppose that W is a linear subspace of V of dimension d > 0. Let X = π(W \ {(0, . . . , 0)}). Show that X is a closed subset of Pn
4.5. Projection from a linear subspace
97
which is isomorphic to Pd−1 . (We will say that X has dimension d − 1). Deﬁne W ⊥ = {L ∈ V ∗  L(w) = 0 for all w ∈ W }. Show that X = Z(W ⊥ ). b) Show that every linear subspace X of Pn is the image π(W \ {(0, . . . , 0)}) for some linear subspace W of V . c) Suppose that w0 , . . . , wr ∈ V are nonzero. Deﬁne the linear span L of π(w0 ), . . . , π(wr ) in Pn to be L = {π(c0 w0 +· · ·+cn wn )  c0 , . . . , cn ∈ k and c0 w0 +· · ·+cn wn = (0, . . . , 0)}. Show that L is a linear subspace of Pn . Exercise 4.17. Suppose that ⎛
v0,0 . . . ⎜ .. A=⎝ . vn,0 . . . ,
⎞ v0,m ⎟ .. ⎠ . vn,m
is an (n + 1) × (m + 1) matrix with coeﬃcients in our algebraically closed ﬁeld k, such that rank(A) = m + 1. Let vi = (v0,i , . . . , vn,i ) ∈ V = k n+1 for 0 ≤ i ≤ m, and let W be the span of v0 , . . . , vm in V . Let F be the linear subspace π(W \ {(0, . . . , 0)}) of Pn . Deﬁne φA : Pm → Pn by φA (c0 : . . . : cm ) = π(c 0 v0 + · · · + cm vm ) m = ( m v c : . . . : j=0 0,j j j=0 vn,j cj ) for (c0 , . . . , cm ) ∈ Pm . Show that φA is a regular map, which is an isomorphism onto the linear subspace F of Pn . Exercise 4.18. The purpose of this exercise is to prove the geometric interpretation of projection from a linear subspace stated in this section. We begin by recalling notation. Suppose that E is a ddimensional linear subspace of Pn . Let L1 , . . . , Ln−d be linear forms in k[x0 , . . . , xn ] which deﬁne E. The rational map φ : Pn → Pn−d−1 with φ = (L1 : . . . : Ln ) is the projection from E. The map is regular on the open set Pn \ E.
98
4. Regular and Rational Maps of Quasiprojective Varieties
Choose a linear subspace F of Pn of dimension n − d − 1 which is disjoint from E (to ﬁnd such an F , just extend L1 , . . . , Ln−d to a kbasis L1 , . . . , Ln−d , M1 , . . . , Md+1 of S1 , the vector space of linear forms in S(Pn )), and let F = Z(M1 , . . . , Md+1 ). Identifying V = k n+1 with V ∗∗ . Let {v1 , . . . , vn−d , w1 , . . . , wd+1 } be the ordered dual basis to the ordered basis {L1 , . . . , Ln−d , M1 , . . . , Md+1 } of S1 = V ∗ . Suppose that p ∈ Pn \ E. Let Gp be the unique linear subspace of Pn of dimension d + 1 which contains p and E. Show that Gp intersects F in a unique point and Gp = φA (φ(p)), where ⎛ ⎞t v1 ⎜ ⎟ A = ⎝ ... ⎠ vn−d (so that φA :
Pn−d−1
→
Pn
is an isomorphism onto F ).
Exercise 4.19. Suppose that φ : X Y is a dominant rational map of varieties, with induced inclusion of function ﬁelds φ∗ : k(Y ) → k(X). Suppose that p ∈ X. Show that φ is regular at p if and only if there exists q ∈ Y such that OY,q ⊂ OX,p , and when this happens, φ(p) = q. Exercise 4.20. Suppose that p = (a0 : . . . : an ) and q = (b0 : . . . : bn ) are points in Pn . Show that there is a unique projective line L (a onedimensional linear subvariety) in Pn containing p and q. Compute the ideal I(L) in S(Pn ). Suppose that pi = (a0 (i) : . . . : an (i)) for 1 ≤ i ≤ m are points in Pn such that the matrix (aj (i)) has rank m. Show that there is a unique linear subvariety M of Pn of dimension n − m + 1 containing pi for 1 ≤ i ≤ m. Compute the ideal I(M ) in S(Pn ).
Chapter 5
Products
In this chapter we deﬁne the product of two varieties and explore some of its basic properties. We deﬁne graphs of regular and rational maps.
5.1. Tensor products The tensor product of two modules is deﬁned as follows: Deﬁnition 5.1. Let R be a ring and M, N be Rmodules. A tensor product T of M and N is an Rmodule T and an Rbilinear mapping g : M × N → T with the following universal property: given an Rmodule P and an Rbilinear map f : M × N → P , there exists a unique Rlinear map f : T → P such that f = f ◦ g. Tensor products always exist and are uniquely determined by the universal property [13, Proposition 2.12]. The tensor product is denoted by M ⊗R N . In the section on “Tensor products of algebras” [13, pages 30–31], it is shown that the tensor product A ⊗R B of two Ralgebras A and B naturally has the structure of an Ralgebra. Tensor products behave well with respect to localization. Let S be a multiplicative set in a ring A and let M be an Amodule. Then S −1 M ∼ = S −1 A ⊗A M and by [128, Deﬁnition on page 100] and [128, Corollary 3.72] if M, N are two Amodules, then by [128, Lemma 3.77] S −1 (M ⊗A N ) ∼ = S −1 M ⊗S −1 A S −1 N. 99
100
5. Products
In the case when R = K is a ﬁeld and M = A, N = B are rings containing K, there is an alternate deﬁnition of the tensor product. This is developed by Zariski and Samuel [160, Section 14 of Chapter III]. Deﬁnition 5.2. Suppose that S is a ring containing a ﬁeld K. Two Ksubspaces L and L of S are said to be linearly disjoint over K if the following condition is satisﬁed: whenever x1 , . . . , xn are elements of L which are linearly independent over K and x1 , . . . , xm are elements of L which are linearly independent over K, then the mn products xi xj are also linearly independent over K. Many important applications of this deﬁnition are given in [160, Section 15 of Chapter II]. A useful equivalent formulation is: the Ksubspaces L and L are linearly disjoint over K if and only if: Whenever x1 , . . . , xn are elements of L which are linearly independent over K, these elements xi are also linearly independent over L . Theorem 5.3. Suppose that K is a ﬁeld and A, B, C are Kalgebras with Kalgebra isomorphisms φ and ψ of A, respectively B, to Ksubalgebras of C such that: 1. C is generated by φ(A) and ψ(B) as a Kalgebra. 2. φ(A) and φ(B) are linearly disjoint over K. Then C is a tensor product A ⊗K B. Theorem 5.3 follows from the observation that the construction of C in [160, Theorem 33 on page 179] is the same as the construction of the tensor product in [13, Proposition 2.12]. Lemma 5.4. Suppose that K is an algebraically closed ﬁeld and L, K are any ﬁeld extensions of K. Then L ⊗K K is an integral domain. Lemma 5.4 is proven in [160, Corollary 1 to Theorem 40, page 198]. Proposition 5.5. Suppose that R is a ring and G is an Rmodule. Let 0→L→M →N →0 be a short exact sequence of Rmodules. Then the sequence (5.1)
G ⊗R L → G ⊗R M → G ⊗R N → 0
is right exact. Proof. [95, Proposition 2.6, page 610] or [13, Proposition 2.18].
5.2. Products of varieties
101
An Rmodule G for which the sequence (5.1) is always short exact is called a ﬂat Rmodule. Locally free modules are aways ﬂat. Theorem 5.6. An Rmodule M is ﬂat if and only if the following condition holds. Suppose that ai ∈ R, xi ∈ M , for 1 ≤ i ≤ r and ri=1 ai xi = 0. Then ∈ R and yj ∈ M , for 1 ≤ j ≤ s, there exists an integer s and elements bij such that i ai bij = 0 for all j and xi = j bij yj for all i. Proof. [23, Corollary I.11.1, page 27].
5.2. Products of varieties Now we deﬁne products of quasiprojective varieties. We continue to assume that k is a ﬁxed algebraically closed ﬁeld. Suppose that X and Y are varieties. We will put a structure of a variety on the set X × Y which has the property that the projections π1 : X × Y → X and π2 : X × Y → Y are regular maps. We construct the product of Am and An as the variety Am × An = Am+n . As sets, this gives a natural identiﬁcation, and this identiﬁcation makes Am × An into an aﬃne variety. The projections π1 : Am × An → Am and π2 : Am × An → An are regular maps. In fact, we have that if k[Am ] = k[x1 , . . . , xm ] and k[An ] = k[y1 , . . . , yn ], then k[Am × An ] = k[x1 , . . . , xm , y1 , . . . , yn ]. This follows since the subrings k[Am ] and k[An ] of k[Am × An ] are linearly disjoint and generate k[Am × An ] as a kalgebra. Now suppose that X and Y are aﬃne varieties, with X a closed subset of Am and Y a closed subset of An . The product X × Y can be naturally identiﬁed with a subset of Am × An , and we have that X × Y is a closed subset of Am × An , as we have that X × Y = Z(π1∗ (I(X)) ∪ π2∗ (I(Y ))). Let R = k[Am × An ]. We identify π1∗ (I(X)) with I(X) and π2∗ (I(Y )) with I(Y ) in the following proposition. Proposition 5.7. The ideal I(X)R + I(Y )R is a prime ideal in R. Proof. We make use of properties of tensor products to prove this. We have that R/(I(X)R + I(Y )R) ∼ = k[X] ⊗k k[Y ] [160, Theorem 35, page 184]. We have that k(X) ⊗k k(Y ) is a domain since k(X) and k(Y ) are ﬁelds and k is algebraically closed, by Lemma 5.4. The subring of k(X) ⊗k k(Y ) generated by k[X] and k[Y ] is a tensor product of
102
5. Products
k[X] and k[Y ] over k by Theorem 5.3, so that k[X] ⊗k k[Y ] is naturally a subring of the domain k(X) ⊗k k(Y ). Thus X × Y is an aﬃne variety, with prime ideal I(X × Y ) = I(X)R + I(Y )R in R = k[Am × An ]. Products are much more subtle over nonalgebraically closed ﬁelds, as can be seen from the following example. Let A = Q[x]/(x2 + 1) and B = Q[y]/(y 2 + 1), which are ﬁelds. We have that A ⊗Q B ∼ = Q[x, y]/(x2 + 1, y 2 + 1) ∼ = Q[i][y]/(y 2 + 1) = Q[i][y]/(y − i)(y + i) is not a domain. We now construct a product Pm × Pn . As a set, we can write Pm × Pn = {(a0 : . . . : am ; b0 : . . . : bn )  (a0 : . . . : am ) ∈ Pm , (b0 : . . . : bn ) ∈ Pn }. Let S be a polynomial ring in two sets of variables, S = S(Pm × Pn ) = k[x0 , . . . , xm , y0 , . . . , yn ]. We put a bigrading on S by bideg(xi ) = (1, 0) for 0 ≤ i ≤ m and bideg(yj ) = (0, 1) for 0 ≤ j ≤ n. We have Sk,l S= k,l
where Sk,l is the kvector space generated by monomials xi00 · · · ximm y0j0 · · · ynjn where i0 + · · · + im = k and j0 + · · · + jn = l. Elements of Sk,l are called bihomogeneous of bidegree (k, l). The bigraded ring S(Pm × Pn ) is called the bihomogeneous coordinate ring of Pm × Pn . Suppose F ∈ S is bihomogeneous of bidegree (k, l) and (a0 : . . . : am ; b0 : . . . : bn ) ∈ Pm × Pn . Suppose that (c0 : . . . : cm ; d0 : . . . : dn ) is equal to (a0 : . . . : am ; b0 : . . . : bn ), so that there exist 0 = α ∈ k and 0 = β ∈ k such that ci = αai for 0 ≤ i ≤ m and dj = βbj for 0 ≤ j ≤ n. Then (5.2)
F (c0 , . . . , cm , d0 , . . . , dn ) = αk β l F (a0 , . . . , am , b0 , . . . , bn ).
Thus the vanishing of such a form at a point is welldeﬁned. We put a topology on the set Pm × Pn by taking the closed sets to be Z(A) = {(p, q) ∈ Pm × Pn  F (p, q) = 0 for F ∈ A} where A is a set of bihomogeneous forms. We can extend this deﬁnition to bihomogeneous ideals by considering the vanishing at a set of bihomogeneous generators.
5.2. Products of varieties
103
Given a subset Y of Pm × Pn , the ideal I(Y ) of Y in S is the ideal in S generated by the set U = {F ∈ S  F is bihomogeneous and F (p, q) = 0 for all (p, q) ∈ Y }. The ideal I(Y ) is a bihomogeneous ideal (it is naturally bigraded as an Smodule). We deﬁne biprojective varieties, quasibiprojective varieties, biprojective algebraic sets, quasibiprojective algebraic sets, and subvarieties of biprojective varieties analogously to the projective case. The ideal I(W ) of a biprojective subvariety W of Pm × Pn is a bigraded prime ideal in S = S(Pm × Pn ). The bihomogeneous coordinate ring of W is S(W ) = S/I(W ), which is a bigraded ring. The rational functions on W or the function ﬁeld of W is
k(W ) =
F  F, G ∈ S(W ) are bihomogeneous of the same bidegree and G = 0 . G
If f = FG ∈ k(W ) where F and G are bihomogeneous of the same bidegree and (p, q) ∈ W is such that G(p, q) = 0, then the value of f (p, q) ∈ k is welldeﬁned by (5.2). The regular functions OW,(p,q) at a point (p, q) ∈ W are the quotients FG where F, G ∈ S(W ) are bihomogeneous of the same bidegree and G(p, q) = 0. We construct regular functions on an open subset U of W as OW,(p,q) . OW (U ) = (p,q)∈U
We expand our deﬁnition of regular maps (Deﬁnition 3.28) to include quasibiprojective varieties (open subsets of biprojective varieties). Suppose that F ∈ S(W ) is bihomogenous. We deﬁne D(F ) = WF = W \Z(F ). The open sets D(F ) = WF where F is bihomogenous of bidegree (a, b) with a > 0 and b > 0 are a basis for the topology of W . The proof of Theorem 3.30 generalizes (working with a bigrading instead of a grading) to show that Wxi yj is an aﬃne variety, and we obtain the following explicit computation of OW (Wxi yj ) = k[Wxi yj ]. Write S(W ) = k[x0 , . . . , xm , y0 , . . . , yn ]/I(W ) = k[x0 , . . . , xm , y 0 , . . . , y n ] where I(W ) = (G1 , . . . , Gt ) with G1 , . . . , Gt bihomogeneous generators of I(W ). Then k[Wxi yj ] = k[ xx0i , . . . , xxmi , yy0 , . . . , yyn ] j j y y ∼ = [ x0 , . . . , xm , 0 , . . . , n ]/(F1 , . . . , Ft ), xi
xi
yj
yj
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5. Products
where
Fa = Ga
x0 xm y0 yn ,..., , ,..., xi xi yj yj
for 1 ≤ a ≤ t. If X is a projective variety which is a closed subset of Pm and Y is a projective variety which is a closed subset of Pn , then X × Y = Z(I(X)S + I(Y )S) is a closed subset of Pm ×Pn . We have that I(X)S +I(Y )S is a prime ideal in S by the proof of Proposition 5.7, so X ×Y is a subvariety of Pm ×Pn , with I(X × Y ) = I(X)S + I(Y )S. The biprojective variety X × Y has a n covering by open aﬃne sets (X × Y ) ∩ (Pm × Pn )xi yj ∼ = (X ∩ Pm xi ) × (Y ∩ Pyj ) for 0 ≤ i ≤ m and 0 ≤ j ≤ n. We have that the open set WF is (isomorphic to) an aﬃne variety if and only if F is bihomogeneous of bidegree (a, b) with both a > 0 and b > 0 (Exercise 5.20). Proposition 3.39 is valid for maps between quasibiprojective varieties; even the proof is valid in the larger setting of quasibiprojective varieties. We deduce from this that if X, Y, Z, W are quasiprojective varieties and φ : Z → X and ψ : Z → Y are regular maps, then (φ, ψ) : Z → X × Y is a regular map. If α : Z → X and β : W → Y are regular maps, then α × β : Z × W → X × Y is a regular map. Suppose that W is a closed subvariety of Pm × Pn with bihomogeneous coordinate ring S(W ) = S(Pm × Pn )/I(W ). We can represent a rational map φ : W Pl by equivalence classes (f0 : . . . : fl ) with f0 , . . . , fl ∈ k(W ) not all zero. We have (f0 : . . . : fl ) ∼ (g0 : . . . : gl ) if fi gj − fj gi = 0 for 0 ≤ i, j ≤ l. The rational map φ is regular at p ∈ W if φ has a representative (f0 : . . . : fl ) with f0 , . . . , fl ∈ OW,p and fi (p) = 0 for some i. Alternatively, we can represent rational maps from W to a projective space Pl by equivalence classes (F0 : . . . : Fl ) where the Fi ∈ S(W ) are bihomogeneous of the same bidegree. The rational map is regular at a point q ∈ W if there is a representative (F0 : . . . : Fl ) such that some Fi (q) = 0. We have that the projections π1 : X × Y → X and π2 : X × Y → Y are regular maps. If W is a closed subvariety of Pm × Pn and U ⊂ W is a quasiaﬃne open subset, then k(W ) is the quotient ﬁeld of OW (U ) (as in Proposition 3.37). Using the fact that the aﬃne open subsets (Pm × Pn )x y ∼ = Am × An of
Pm × Pn have regular functions
i j
x0 xm y0 yn ,..., , ,..., , k[(P × P )xi yj ] = k xi xi yj yj m
n
5.3. The Segre embedding
105
we see that Y = (Pm × Pn )xi ∼ = Am × Pn is covered by the aﬃne open sets (Pm × Pn )xi yj for 0 ≤ j ≤ n. We can associate a coordinate ring x0 xm S(Y ) = k ,..., , y0 , . . . , yn xi xi to Y . An element of k xx0i , . . . , xxmi has bidegree (0, 0) and yj has bidegree (0, 1) for 0 ≤ j ≤ n. Thus the bidegree makes S(Y ) a graded ring (graded by the natural numbers N). All of the theory that we have worked out above extends to this situation. We compute k[Yyj ] = S(Y )(yj ) , the elements of degree 0 in the localization S(Y )yj . Given an aﬃne variety X, we can realize X as a closed subvariety of m n Am ∼ = Pm xi ⊂ P , and then we have realized X × P as a closed subvariety m n of A × P . This gives us a graded coordinate ring for X × Pn as S(X × Pn ) = k[X][y0 , . . . , yn ]. Here, elements of k[X] have degree 0, and the variables yj have degree 1. All of the theory we developed above goes through for this coordinate ring, with this grading. Suppose that W is a closed subvariety of X ×Pn (where X is aﬃne), with coordinate ring S(W ) = S(X × Pn )/I(W ). We can represent rational maps from W to Pl by equivalence classes (F0 : . . . : Fl ) where the Fi ∈ S(W ) are homogeneous of the same degree (and not all zero). The rational map is regular at a point q ∈ W if there is a representative (F0 : . . . : Fl ) such that some Fi (q) = 0. If U is a quasiaﬃne open subset of W , then the quotient ﬁeld of OW (U ) is k(W ). If X is aﬃne and W is a projective variety with coordinate ring S(W ) = k[y0 , . . . , y n ], then X × W has the coordinate ring S(X × W ) = k[X] ⊗k S(W ) = k[X][y0 , . . . , y n ] which is a domain as shown by the proof of Proposition 5.7. Also, S(X) is graded by deg(f ) = 0 if f ∈ k[X] and deg(y i ) = 1 for 1 ≤ i ≤ n.
5.3. The Segre embedding We deﬁne the Segre embedding φ : Pm × Pn → PN where N = (n + 1)(m + 1) − 1 by φ(a0 : . . . : am ; b0 : . . . : bn ) = (a0 b0 : a0 b1 : . . . : ai bj . . . : am bn ).
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The map φ is a regular map (since ZPm ×Pn (x0 y0 , x0 y1 , . . . , xm xn ) = ∅), and it can be veriﬁed that its image is a closed subvariety of PN and φ is an isomorphism onto this image (φ is a closed embedding). If we take wij , with 0 ≤ i ≤ m and 0 ≤ j ≤ n, to be the natural homogeneous coordinates on PN , then the image of φ is the projective variety W whose ideal I(W ) is generated by {wij wkl − wkj wil  0 ≤ i, k ≤ m and 0 ≤ j, l ≤ n}. This is proven in [81, Section 2 of Chapter XI of Volume 2]. Thus the product X × Y of two quasiprojective varieties is actually (isomorphic to) a quasiprojective variety, by the Segre embedding. In fact, any quasibiprojective variety is (isomorphic to) a quasiprojective variety.
5.4. Graphs of regular and rational maps Suppose that X and Y are quasiprojective varieties and φ : X → Y is a regular map. Then we have a regular map ψ : X → X × Y deﬁned by ψ(p) = (p, φ(p)) for p ∈ X. Let Γφ be the image ψ(X) in X × Y . We call Γφ the graph of φ. Proposition 5.8. Γφ is Zariski closed in X × Y . Proof. We have an embedding of Y in a projective space Pn , as an open subset of a projective subvariety. The map φ : X → Y thus extends to a regular map φ˜ : X → Pn and Γφ = Γφ˜ ∩ (X × Y ). Thus it suﬃces to prove the proposition in the case when Y = Pn . Let i : Pn → Pn be the identity map. Then φ × i : X × Pn → Pn × Pn is a regular map. Let ΔPn ⊂ Pn × Pn be the “diagonal” {(q, q)  q ∈ Pn }. We have that S(Pn × Pn ) = k[u0 , . . . , un , v0 , . . . , vn ] where the ui are homogeneous coordinates on the Pn of the ﬁrst factor and the vj are homogeneous coordinates on the Pn of the second factor. The diagonal ΔPn is a closed subset of Pn × Pn with ΔPn = Z(ui vj − uj vi  0 ≤ i, j ≤ n). Since the preimage of a closed set by a regular map is closed, we have that Γφ = (φ × i)−1 (ΔPn ) is closed in X × Pn . The graph Γφ = ψ(X) is irreducible since X is, so Γφ is a closed subvariety of X × Y . We can extend this construction to give a useful method of studying rational maps. Suppose that X and Y are quasiprojective varieties and φ : X Y is a rational map. Let U be a nontrivial open subset of X on which φ is regular. The graph Γφ of φ is deﬁned to be the closure in X × Y of the image of the regular map p → (p, φ(p)) from U to X × Y . The graph Γφ does not depend on the choice of open subset U on which φ is regular (since Γφ is irreducible and (U × Y ) ∩ Γφ is a nontrivial open subset of Γφ ).
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107
Further, ΓφV = Γφ ∩ (V × Y ) = π1−1 (V )
(5.3)
for any open subset V of X, where π1 : Γφ → X is the projection. Proposition 5.9. Suppose that φ : X Y is a rational map of quasiprojective varieties. Then Γφ is a quasiprojective variety, and the projection π1 : Γφ → X is a birational map. If U is a nonempty open subset of X on which φ is regular, then the restriction π1 : ΓφU = Γφ ∩ (U × Y ) → U is an isomorphism. Proof. It suﬃces to prove this in the case when φ is itself a regular map, as Γφ is the Zariski closure in X × Y of the graph ΓφU for any dense open subset U of X. Now Γφ is the image of the regular map (i, φ) : X → X × Y where i : X → X is the identity map. Since X is irreducible, its image Γφ is irreducible. The set Γφ is closed in X × Y by Proposition 5.8 so Γφ is a variety. The map (i, φ) is an inverse to π1 . Since both maps are regular maps, π1 is an isomorphism, and π1 is thus birational. Proposition 5.10. Suppose that φ : X Y is a rational map of quasiprojective varieties and Γφ is the graph of φ, with projection π1 : Γφ → X. Suppose that p ∈ X. Then φ is a regular map at p if and only if the rational map π1−1 is a regular map at p. Proof. Let U be an open subset of X on which φ is regular. Then π1−1 = i × φ : U → Γφ is a regular map where i : X → X is the identity map. Suppose that the rational map π1−1 : X Γφ is regular at a point p ∈ X. Let V be an open neighborhood of p in which π1−1 is regular. We have that π2 π1−1 = φ as a rational map where π2 : Γφ → Y is the projection and π2 π1−1 : V → Y is regular, so φ is regular on V . Theorem 5.11 (Elimination theory). For 1 ≤ i ≤ r, let di be a positive integer, and let Fi = J=di biJ xJ be a homogeneous polynomial in the variables x0 , . . . , xn of degree di with indeterminate coeﬃcients biJ , where the sum is over J = (j0 , . . . , jn ) ∈ Nn+1 with J = j0 + · · · + jn = di and xJ = xj00 · · · xjnn . Then there exists a set of polynomials g1 , . . . , gt ∈ Z[{biJ  1 ≤ i ≤ r, J = di }]
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5. Products
which are homogeneous with respect to each set of variables biJ with ﬁxed i, with the following property: i
Let K be an algebraically closed ﬁeld and suppose that bJ ∈ K. Let i Fi = bJ xJ ∈ K[x0 , . . . , xn ] for 1 ≤ i ≤ r. J=di
Then a necessary and suﬃcient condition for the F i to have a common zero i in K n+1 , diﬀerent from the trivial solution (0, . . . , 0), is that the bJ are a common zero of the gl . Proof. [143, Section 80, page 8].
Theorem 5.12. Suppose that Z is a closed subset of Pn × Am . Then the image of the second projection π2 : Z → Am is Zariski closed. Proof. Let x0 , . . . , xn be homogeneous coordinates on Pn and y1 , . . . , ym be aﬃne coordinates on Am . Then I(Z) = (G1 , . . . , Gr ) for some G1 , . . . , Gr ∈ k[x0 , . . . , xn , y1 , . . . , ym ] where the Gi are homogeneous in the xi variables of some degrees di . Write Gi = J=di aiJ xJ , where aiJ ∈ k[y1 , . . . , ym ]. Let g1 , . . . , gt be the polynomials of the conclusions of Theorem 5.11, and let hl = gl (aiJ ) ∈ k[y1 , . . . , ym ] = k[Am ]. For P ∈ Am , we have π2−1 (P ) ∩ Z = Z(G1 (x, P ), . . . , Gr (x, P )) ⊂ Pn × {P } ∼ = Pn i where Gi (x, P ) = aJ (P )xJ . Thus π2−1 (P ) ∩ Z = ∅ if and only if the polynomials Gi (x, P ) have a common nontrivial zero in k n+1 , which holds if and only if P is a common zero of the hl (y) = gl (aiJ ) in k m , and this holds if and only if P ∈ Z(h1 , . . . , ht ). Thus π2 (Z) = Z(h1 , . . . , ht ) is Zariski closed. Theorem 5.12 does not hold for closed subsets of An × Am . A simple example is to take Z to be Z(xy − 1) ⊂ A2 ∼ = A1 × A1 . The projection of Z 1 onto the yaxis is the nonclosed subset A \ {(0)} of A1 . Corollary 5.13. Suppose that X is a projective variety and Y is a quasiprojective variety. Then the second projection π2 : X × Y → Y takes closed sets to closed sets. Proof. Suppose that Z is a closed subset of X × Y . After embedding X as a closed subset of a projective space Pn , so that Z ris a closed subset n n of P × Y , we may assume that X = P . Let Y = i=1 Vi be an aﬃne
5.4. Graphs of regular and rational maps
109
open cover of Y . We must show that π2 (Z) ∩ Vi is closed in Vi for all i. Let Zi = Z ∩ (Pn × Vi ), a closed subset of Pn × Vi . We have that π2 (Z) ∩ Vi = π2 (Zi ). Here Vi is isomorphic to a closed subset of Ami for some mi . Thus Zi is a closed subset of Pn ×Ami under the natural inclusion. We have projections π2i : Pn × Ami → Ami . By Theorem 5.12, π2i (Zi ) is a closed subset of Ami . Since π2i (Zi ) ⊂ Vi , we have that π2 (Zi ) = π2i (Zi ) is closed in Vi for all i. Theorem 5.14. Suppose that φ : X → Y is a regular map of projective varieties. Then the image of φ is a closed subset of Y . Proof. Apply the corollary to the closed subset Γφ of X × Y .
We now give another proof of Theorem 3.35. Corollary 5.15. Suppose that X is a projective variety. Then OX (X) = k. Proof. Suppose that f ∈ OX (X). Then f is a regular map f : X → A1 . After including A1 into P1 , we obtain a regular map f : X → P1 . By Theorem 5.14, we have that the image f (X) is closed in P1 . Since f cannot be onto, f (X) must be a ﬁnite union of points. Since X is irreducible, f (X) is irreducible, so f (X) is a single point. Thus f ∈ k. Corollary 5.16. Suppose that X is a projective variety and φ : X → An is a regular map. Then φ(X) is a point. Proof. Let πi : An → A1 be projection onto the ith factor. Then πi ◦ φ : X → A1 is a regular map, so πi ◦ φ is a constant map by the previous corollary for 1 ≤ i ≤ n. Thus φ(X) is a point. Exercise 5.17. Let t be an indeterminate, F be the ﬁeld Zp (t), and K, L be the ﬁelds K = F [x]/(xp − t),
L = F [y]/(y p − t).
Show that K ⊗F L is not reduced. Exercise 5.18. Let W ⊂ P3 be the image of P1 × P1 in P3 by the Segre map. i) Find the ideal I(W ) ⊂ k[P3 ]. ii) Suppose that p ∈ P1 . Find the ideal in k[P3 ] of the image of {p}×P1 in P3 . Find the ideal in k[P3 ] of the image of P1 × {p} in P3 . Exercise 5.19. Let a and b be positive integers. Consider the Veronese map φ1 on Pm deﬁned by the forms of degree a on Pm and the Veronese
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5. Products
map φ2 on Pn deﬁned by the forms of degree b on Pn . Let W1 be the image of φ1 and let W2 be the image of φ2 . i) Show that φ1 × φ2 : Pm × Pn → W1 × W2 is an isomorphism onto the biprojective variety W1 × W2 . ii) Show that S(W1 × W2 )(1,1) = S(Pm × Pn )(a,b) ; that is, the forms of bidegree (1, 1) on W1 × W2 are the forms of bidegree (a, b) on Pm × Pn . Exercise 5.20. Suppose that F ∈ S(Pm × Pn ) with m, n > 0 is a bihomogeneous form of bidegreee (a, b). i) Show that D(F ) = (Pm × Pn )F is (isomorphic to) an aﬃne variety if a > 0 and b > 0. n ii) Show that (Pm × Pn )F ∼ = Pm F × P if F has bidegree (a, 0) and m n m n ∼ P × P if F has bidegree (0, b). Conclude that (P × P )F = F
D(F ) = (Pm × Pn )F is not aﬃne if a = 0 or b = 0.
Hint: Use the construction of Exercise 5.19, followed by a Segre embedding. Exercise 5.21. Suppose that X is a quasiprojective variety and U1 , U2 ⊂ X are aﬃne open subsets. Show that U1 ∩ U2 is an aﬃne open subset of X; that is, show that U1 ∩ U2 is isomorphic to an aﬃne variety. Hint: Consider the open subset U1 × U2 ⊂ X × X and its intersection with the diagonal ΔX = {(p, p)  p ∈ X} ⊂ X × X.
Chapter 6
The Blowup of an Ideal
In this chapter we construct the blowup of an ideal in an aﬃne or projective variety. The more general construction of the blowup of an ideal sheaf will be given in Chapter 12. As usual, we will assume that k is a ﬁxed algebraically closed ﬁeld.
6.1. The blowup of an ideal in an aﬃne variety Suppose that X is an aﬃne variety and f0 , . . . , fr ∈ k[X] are not all zero. We can deﬁne a rational map Λf0 ,...,fr : X Pr by Λf0 ,...,fr = (f0 : . . . : fr ). Proposition 6.1. Suppose that g0 , . . . , gs ∈ k[X] and (f0 , . . . , fr ) = (g0 , . . . , gs ) are the same ideal J = (0) in k[X]. Then there is a commutative diagram of regular maps ΓΛf0 ,...,fr
ψ
/ ΓΛ g0 ,...,gs MMM MMM MMM MM&
X
where the vertical arrows are the projections and ψ is an isomorphism. 111
112
6. The Blowup of an Ideal
Proof. This will follow from Theorem 6.4. In this theorem it is shown that there are graded k[X]algebra isomorphisms S(ΓΛf0 ,...,fr ) = S(X × Pr )/I(ΓΛf0 ,...,fr ) ∼ Ji = i≥0
and S(ΓΛg0 ,...,gs ) = S(X × Ps )/I(ΓΛg0 ,...,gs ) ∼ =
J i,
i≥0
which thus induce a graded k[X]algebra isomorphism α : S(ΓΛg0 ,...,gs ) → S(ΓΛf0 ,...,fr ). Write S(ΓΛg0 ,...,gs ) = k[X][y0 , . . . , y s ] where the y i are the restriction to ΓΛg0 ,...,gs of the homogeneous coordinates on Ps . Then deﬁne α : ΓΛf0 ,...,fr → X × Ps by α(p, q ∗ ) = (p; α(y 0 )(p, q ∗ ) : . . . : α(y s )(p, q ∗ )) for (p, q ∗ ) ∈ ΓΛf0 ,...,fr . We will now establish that α is an isomorphism onto ΓΛg0 ,...,gs . Since α is an isomorphism, we have that α(y 0 ), . . . , α(y s ) generate [S(ΓΛf0 ,...,fr )]1 (the homogeneous forms of degree 1) as a k[X]module. Since α(y 0 ), . . . , α(y s ) ∈ S(Γf0 ,...,fr ) are homogeneous of degree 1 and Z(α(y 0 ), . . . , α(y s )) = ∅, we have that q → (α(y 0 )(q) : . . . : α(y s )(q)) is a regular map from ΓΛf0 ,...,fr to Ps (as explained before Section 5.3).
Since the ﬁrst projection π1 : ΓΛf0 ,...,fr → X is a regular map, the product α is a regular map α : ΓΛf0 ,...,fr → X × Ps . Suppose that F ∈ S(X × Ps ) is a homogeneous form and that q ∈ ΓΛf0 ,...,fr . We have that F (α(q)) = (α(F ))(q)
where F is the residue of F in S(ΓΛg0 ,...,gs ). Thus F (α(q)) = 0 for all F ∈ I(ΓΛg0 ,...,gs ), giving us that α is a regular map from ΓΛf0 ,...,fr into ΓΛg0 ,...,gs .
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113
Since α is an isomorphism, we can apply the above argument to β = to construct a regular map β : ΓΛg0 ,...,gs → ΓΛf0 ,...,fr which has the property that
(α)−1
α ◦ β = idΓΛg
0 ,...,gs
and
β ◦ α = idΓΛf
0 ,...,fr
,
so that α is an isomorphism with inverse β.
The following deﬁnition of the blowup of an ideal is welldeﬁned by Proposition 6.1. Deﬁnition 6.2. Suppose that X is an aﬃne variety and I ⊂ k[X] is a nonzero ideal. Suppose that I = (f0 , . . . , fr ). Let Λ : X Pr be the rational map deﬁned by Λ = (f0 : . . . : fr ). The blowup of I is B(I) = ΓΛ , with projection π : B(I) → X. The blowup B(0) of the zero ideal is deﬁned to be B(0) = ∅, with natural inclusion π of the empty set into X. If I = I(Y ) is the ideal of a subvariety Y of X, then B(I) → X is called the blowup of Y . The blowup π : B(J) → X has the property that π −1 is regular and is an isomorphism over the open set X \ Z(J), as follows from Proposition 5.9. If J = (0), then X \ Z(J) = ∅. Lemma 6.3. Suppose that R is a Noetherian ring and P ⊂ R is a prime ideal. Suppose that J, A are ideals in R such that J ⊂ P , A ⊂ P and the localizations AQ = PQ for Q a prime ideal in R such that J ⊂ Q. Then P = A :R J ∞ := {f ∈ R  f J n ⊂ A for some n ≥ 0}. Proof. By Proposition 1.43, A has an irredundant primary decomposition A = P ∩ I 1 ∩ · · · ∩ Ir where for 1 ≤ i ≤ r, Ii are Qi primary for prime ideals Qi such that J ⊂ Qi . Thus there exists n > 0 such that J n ⊂ Ii for all i. Thus J n P ⊂ A, so P ⊂ A : J ∞. Now suppose that f ∈ R is such that f J n ⊂ P for some n ≥ 0. Since P is a prime ideal and there exists an element of J n which is not in P , we have that f ∈ P . Thus A : J ∞ ⊂ P . Theorem 6.4. Suppose that X is an aﬃne variety and J ⊂ k[X] is a nonzero ideal. Let π : B(J) → X be the blowup of J. Suppose that J = (f0 , . . . , fn ), with the fi all nonzero, so that B(J) ⊂ X × Pn . Then the coordinate ring of B(J) (viewing B(J) as a closed subvariety of X × Pn ) is Ji S(B(J)) ∼ = i≥0
114
6. The Blowup of an Ideal
as a graded k[X]algebra (with the degree 0 elements of S(B(J)) being J 0 = k[X] and the degree i elements of S(B(J)) being J i ). Let R = k[X] and y0 , . . . , yn be homogeneous coordinates on Pn . Then f0 fn (6.1) k[B(J)yi ] = OB(J) (B(J)yi ) = R ,..., fi fi for 0 ≤ i ≤ n. We thus have that (6.2)
Jk[B(J)yi ] = fi k[B(J)yi ]
is a principal ideal for all i. Let A = (yi fj − yj fi  0 ≤ i, j ≤ n), a homogeneous ideal in S(X × Pn ) = k[X][y0 , . . . , yn ]. The ideal of B(J) in S(X × Pn ) is IX×Pn (B(J)) = A :S(X×Pn ) (JS(X × Pn ))∞ = {f ∈ S(X × Pn )  f J n ⊂ A for some n ≥ 0}. Proof. Let φ = (f0 : . . . : fn ) : X Pn . The variety B(J) is deﬁned to be the closure of Λ(X \ ZX (f0 , . . . , fn )) in X × Pn , where Λ(p) = (p; f0 (p) : . . . : fn (p)) for p ∈ X \ ZX (f0 , . . . , fn ). The coordinate ring of X × Pn is R[y0 , . . . , yn ] which is graded by deg yi = 1 for 0 ≤ i ≤ n. The ideal A = (yi fj − yj fi  0 ≤ i, j ≤ n) in R[y0 , . . . , yn ] is contained in I(Γφ ). Since fi is a unit in Rfi , φ is regular on Xfi and we have that ΓφXf ⊂ i Xfi × Pn is isomorphic to the open subset Xfi of X. Further, we calculate (using the fact that fi is a unit in Rfi ) that Afi is a prime ideal in Rfi [y0 , . . . , yn ] = S(Xfi × Pn ) = S(X × Pn )fi . Now we will show that IX×Pn (Γφ )fi = IXfi ×Pn (ΓφXf ) = Afi . i
We have that ΓφXf = {(p; f0 (p) : · · · : fn (p))  p ∈ Xfi }. Thus (as already i observed) Afi ⊂ I(ΓφXf ). Now i fj (6.3) Afi = yj − yi  0 ≤ j ≤ n . fi Suppose F ∈ S(Xfi × Pn ) = Rfi [y0 , y1 , . . . , yn ] is homogeneous of degree d. Then (6.3) implies that F = gyid + H with H ∈ Afi homogeneous of degree d and g ∈ Rfi . Suppose that F ∈ I(ΓφXf ). Then for all p ∈ Xfi , i
0 = F (p, f0 (p), . . . , fn (p)) = g(p)fi (p)d which implies that g(p) = 0 for all p ∈ Xfi , so that g = 0.
6.1. The blowup of an ideal in an aﬃne variety
115
If Q is a prime ideal in S(X × Pn ) such that fi ∈ Q, we have that AQ ∼ = (Afi )Qfi ∼ = (I(Γφ )fi )Qfi ∼ = I(Γφ )Q by Exercise 1.22. Since this is true for 0 ≤ i ≤ n, we have that AQ = I(Γφ )Q for Q a prime ideal in R[y0 , . . . , yn ] such that (f0 , . . . , fn ) ⊂ Q. We have that J ⊂ I(Γφ ), since, otherwise, Γφ ⊂ π −1 (Z(J)) implies X ⊂ Z(J) which implies that J ⊂ I(X) = (0), a contradiction to our assumption that J = (0). By Lemma 6.3, we have that A :S(X×Pn ) (JS(X × Pn ))∞ = I(Γφ ). Let t be an indeterminate, which we give degree 1, and let P be the kernel of the graded kalgebra homomorphism R[y0 , . . . , yn ] → R[tf0 , . . . , tfn ] ⊂ R[t] deﬁned by mapping yj → tfj . Here P is a prime ideal since R[t] is a domain. We have that A ⊂ P and for a prime ideal Q in R[y0 , . . . , yn ], we have that AQ = PQ if J ⊂ Q (this follows since after localizing at such a Q, some fi becomes invertible and so we can make a similar argument to the preceding paragraph). Assume that J ⊂ P . We will derive a contradiction. Then Pfi = Rfi [y0 , . . . , yn ] for all i, which implies Rfi [tf0 , . . . , tfn ] = 0, which is a contradiction since the fi are assumed nonzero and R is a domain. Thus J ⊂ P . Thus by Lemma 6.3, P = A :R[y0 ,...,yn ] J ∞ = I(Γφ ), and the coordinate ring of B(J) is R[y0 , . . . , yn ]/P ∼ = R[tf0 , . . . , tfn ] ∼ =
∞
J i.
i=0
We have OB(J) (B(J) ∩ (X × Pnyj )) = S(B(J))(yj ) ∼ = R[tf0 , . . . , tfn ](tfj ) tf0 tfn f f ∼ = R[ tfj , . . . , tfj ] = R[ f0j , . . . , fnj ] ⊂ k(X). As a graded k[X]algebra, we have Ji ∼ S(B(J)) ∼ = k[X][tf0 , . . . , tfn ] ⊂ k[X][t] = i≥0
where k[X][t] is the polynomial ring over k[X], graded by deg(t) = 1. The algebra i≥0 J i is called the Rees algebra of J. We have the following useful proposition, which allows us to easily compute the coordinate ring of a blowup in an important case. Proposition 6.5. With the notation of Theorem 6.4, suppose that the generators f0 , . . . , fn of J are a k[X]regular sequence. Then IX×Pn (B(J)) = A and S(B(J)) ∼ = k[X][y0 , . . . , yn ]/A.
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6. The Blowup of an Ideal
Proof. This follows from Theorem 1 of [108].
There are sophisticated techniques to compute coordinate rings S(B(J)) of blowups of ideals in more general situations. Some important results on this topic are given in the papers [77] by Herzog, Simis, and Vasconcelos, [82] and [83] by Huneke, and [138] by Simis, Ulrich, and Vasconcelos. Let W be a closed subset of the aﬃne variety X. Then π −1 (W \ Z(J)) ∼ = W \ Z(J) by Proposition 5.9. We will call the Zariski closure of π −1 (W \ Z(J)) ∼ = W \ Z(J) in B(J) the strict transform of W in B(J). Proposition 6.6. Suppose that X is an aﬃne variety and J ⊂ k[X] is an ideal. Let W be a closed subvariety of X, and let J = Jk[W ]. Then the strict transform of W in B(J) is isomorphic to B(J). The ideal of B(J)yi in k[B(J)yi ] is (6.4)
I(B(J)yi ) = I(W )k[B(J)yi ] :k[B(J)yi ] (Jk[B(J)yi ])∞ = I(W )k[B(J)yi ] :k[B(J)yi ] (fi k[B(J)yi ])∞ .
Proof. We have natural projections π : B(J) → X and π : B(J) → W . First suppose J = (0), so that B(J) = ∅. Then J ⊂ I(W ) so W ⊂ Z(J). Thus W \ Z(J) = ∅ and the Zariski closure of W \ Z(J) in B(J) is ∅. Now suppose that J = (0). Let f0 , . . . , fn be a set of generators of J. Let f i be the residues of the fi in k[W ]. Let φ : X X × Pn be the rational map φ = id × (f0 : . . . : fn ) and let φ : W W × Pn be the rational map φ = id × (f 0 : . . . : f n ). We have a commutative diagram of regular maps, where the vertical maps are the natural inclusions: X \ Z(J) ↑
φ
→
X × Pn ↑
φ
W \ Z(J) → W × Pn . Now B(J) is the closure of φ(X \ Z(J)) in X × Pn and B(J) is the closure of φ(W \ Z(J)) in W × Pn , so the conclusions of the ﬁrst paragraph of the proposition thus follow from the above diagram. The ﬁnal equation (6.4) follows from Lemma 6.3.
We now discuss an important example. Let p be the origin in X = A2 . Let m = (x1 , x2 ) be the ideal of p in = k[x1 , x2 ]. Let π : B = B(m) → X be the blowup of p (the blowup of m). Let E = π −1 (p). k[A2 ]
Letting P1 have homogeneous coordinates y0 and y1 , from formula (6.1), we know that B(m) ⊂ X × P1 has the aﬃne cover {B1 = By0 , B2 = By1 }
6.1. The blowup of an ideal in an aﬃne variety
where
117
x2 x2 x2 k[B1 ] = k[A ] = k x1 , x2 , = k x1 , x1 x1 x1 2
(6.5) and
x1 x1 = k x2 , k[B2 ] = k[A ] x2 x2 2
(6.6)
are polynomial rings (x1 , x2 algebraically independent over k implies x1 , xx21 are algebraically independent over k and x2 , xx12 are algebraically independent over k), so B1 ∼ = A2 and B2 ∼ = A2 . We have that B1 ∼ = A2 has coordinates u1 = x1 , u2 = x2 and B2 ∼ = A2 has coordinates v1 = x2 , v2 =
x1 x2 .
x1
On
B1 ∩ B2 = B1 \ Z(u2 ) = B2 \ Z(v2 ), we have u1 = v1 v2 and u2 = v12 . The map π : B → A2 satisﬁes πB1 = (u1 , u1 u2 ) and πB2 = (v1 v2 , v1 ). Let E = π −1 (p). We have that E = Z(x1 , x2 ), so I(E ∩ B1 ) = (u1 ) and I(E ∩ B2 ) = (v1 ). The projection π : B \ E → X \ {p} is an isomorphism. Since mk[B1 ] = x1 k[B1 ] and mk[B2 ] = x2 k[B2 ] by (6.2), which are prime ideals, we have that I(E ∩ B1 ) = x1 k[B1 ] and I(E ∩ B2 ) = x2 k[B2 ] and x2 x1 ∼ k ] = k[B ]/(x ) k and k[E ∩ B , k[E ∩ B1 ] = k[B1 ]/(x1 ) ∼ = 2 2 2 = x1 x2 from which we see that E ∩B1 ∼ = A1 and E ∩B2 ∼ = A1 . Since k[E ∩B1 ∩B2 ] =
k[ xx21 , xx21 ], the regular maps (1 : xx21 ) : E ∩ B1 → P1 and ( xx21 : 1) : E ∩ B2 → P1 agree on E ∩ B1 ∩ B2 . Thus the maps patch to give a welldeﬁned map E → P1 which is a regular map by Proposition 3.39. By a similar argument, the inverse map P1 → E is a regular map, so E ∼ = P1 . The blowup π : B → E is illustrated in Figure 6.1. In the ﬁgure, L1 = Z(x1 ), L2 = Z(x2 ), ˜ 2 = Z(y1 ) = strict transform of L2 . ˜ 1 = Z(y0 ) = strict transform of L1 , L L ˜1 L
˜2 L L2 π
p
E L1 Figure 6.1. The blowup π : B → X
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6. The Blowup of an Ideal
Suppose that C ⊂ A2 is a curve which contains p. Let f (x1 , x2 ) ∈ k[A2 ] be such that I(C) = (f ). Write f= aij xi1 xj2 i+j≥r
where aij ∈ k and some aij = 0 with i + j = r (r is the order of f ). Let C˜ be the strict transform of C in B. In B1 , we have that j x2 x2 i+j r = x 1 f1 x 1 , f= aij x1 x1 x1 where
x2 j . f1 = x1 By (6.4), since x1 does not divide f1 in k[B1 ], we have that
aij xi+j−r 1
I(C˜ ∩ B1 ) = f k[B1 ] :k[B1 ] (x1 k[B1 ])∞ = (f1 ). Similarly, f2 =
aij
x1 x2
i xi+j−r 2
and I(C˜ ∩ B2 ) = (f2 ).
By Proposition 6.6, we have that B(mk[C]) ∼ = C˜ (where B(mk[C]) is the blowup of p in C). Thus B(mk[C]) is covered by the two aﬃne open subsets C˜ ∩ B1 and C˜ ∩ B2 . We have that x2 x1 ˜ ˜ /(f1 ) and k[C ∩ B2 ] = k x2 , /(f2 ). k[C ∩ B1 ] = k x1 , x1 x2 Write k[C] = k[x1 , x2 ]/(f ) = k[x1 , x2 ], From equation (6.1), we have that x2 x2 ∼ /(f1 ) k[x1 , x2 ] = k x1 , x1 x1 and
x1 ∼ x1 /(f2 ), k[x1 , x2 ] = k x2 , x2 x2 p
E π C˜
Figure 6.2. The curve C and its strict transform C˜
C
6.1. The blowup of an ideal in an aﬃne variety
119
from which we see that an understanding of the ring of (6.1) as isomorphic to a quotient of a polynomial ring is often not trivial (see Exercise 1.8). The curve C and its strict transform C˜ are illustrated in Figure 6.2. From the above analysis, we are able to obtain all intrinsic properties of the blowup B(mk[C]) of the point p in C. The methods demonstrated above are generally the best way to understand the geometry of blowups. We will now compute the full coordinate ring S(B(m)) and indicate how we can compute k[B1 ] and k[B2 ] from the coordinate ring. With the notation of Theorem 6.4, we have that A = (y1 x1 − y0 x2 ) ⊂ S(X × P1 ) = k[x1 , x2 , y0 , y1 ] is a prime ideal, so IX×P1 (B(m)) = A : m∞ = A. This also follows from Proposition 6.5. In this particular example, we have the desirable condition that I(B) = (x1 y1 − x2 y0 ) = A, so I(B) is actually generated by the obvious relations A. The coordinate ring of B (as a subvariety of X × P1 ) is thus (6.7)
S(B) = k[x1 , x2 , y0 , y1 ]/(y1 x1 − y0 x2 ).
We also have (by Theorem 6.4) that mi ∼ (6.8) S(B) ∼ = k[A2 ][tx1 , tx2 ]. = i≥0
There is a natural isomorphism of graded k[X]algebras from (6.7) to (6.8) by mapping y0 → tx1 and y1 → tx2 . We have that {X × P1y0 , X × P1y1 } is an aﬃne cover of X × P1 . Thus {B1 = (X × P1y0 ) ∩ B, B2 = (X × P1y1 ) ∩ B} is an aﬃne cover of B. Now k[B1 ] = S(B)(tx1 )
tx2 x2 x2 = k[A ] = k x1 , x2 , = k x1 , tx1 x1 x1 2
since x2 = x1 xx21 . Also, x2 and xx21 are algebraically independent over k since x1 , x2 are. Thus k[B1 ] is a polynomial ring in these two variables, so that B1 is isomorphic to A2 . Similarly, we have that x1 k[B2 ] = S(B)(tx2 ) = k x2 , x2 since x1 = x2 xx12 and B2 is isomorphic to A2 . We thus recover our calculations in (6.5) and (6.6).
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6. The Blowup of an Ideal
E = π −1 (p) is the algebraic set ZB (x1 , x2 ) ⊂ B ⊂ X × P1 . We compute S(B)/(x1 , x2 )S(B) = (k[A2 ][tx1 , tx2 ])/(x1 , x2 ) ∼ = (k[A2 ]/(x1 , x2 ))[tx1 , tx2 ] = k[tx1 , tx2 ]. Since tx1 and tx2 are algebraically independent over k, this ring is isomorphic to a graded polynomial ring in two variables, which is the coordinate ring of P1 . Since E ⊂ {p} × P1 , we have that E = {p} × P1 , with I(E) = (x1 , x2 ). We thus recover the calculations following (6.6).
6.2. The blowup of an ideal in a projective variety Suppose that X ⊂ Pn is a projective variety, with coordinate ring S(X) = k[x0 , . . . , xn ]. Suppose that I is a homogeneous ideal in S(X). Each open set D(xi ) (for which xi = 0) is aﬃne with regular functions k[D(xi )] = x0 xn k xi , . . . , xi . In k[D(xi )] we have an ideal
x0 xn ˜ I(D(xi )) = f ,..., f ∈I . xi xi This deﬁnition is such that for p ∈ D(xi ) ∩ D(xj ) we have equality of ideals ˜ ˜ I(D(x i ))OX,p = I(D(xj ))OX,p . ˜ We write I˜p for the ideal I(D(x i ))OX,p in OX,p if p ∈ D(xi ). To every open subset U of X we can thus deﬁne an ideal ˜ )= OX,p = OX (U ). I˜p ⊂ I(U p∈U
p∈U
The method of the proof of Theorem 3.30 shows that we have
x0 xn ˜ ,..., Ip = f f ∈I , xi xi p∈D(xi )
˜ which we initially deﬁned to be I(D(x i )), so our deﬁnition is consistent. ˜ If Z(I) = ∅, we have that 0 = I(X) ⊂ OX (X) = k. We will see this construction again later as a special case of Theorem 11.25. We will call I˜ the ideal sheaf on X associated to I. Lemma 6.7. Suppose that I ⊂ S(X) is a homogeneous ideal. Then there exists some d ≥ 1 such that the ideal J which is generated by the elements ˜ Id (the elements of I which are homogeneous of degree d) satisﬁes I˜ = J. Proof. Let F0 , . . . , Fr be a homogeneous set of generators of I. Let d0 , . . . , dr be their respective degrees. Suppose that d ≥ max{di }, and let J be the ˜ It suﬃces to show that ideal generated by Id . We will show that J˜ = I.
6.2. The blowup of an ideal in a projective variety
121
d−di ˜ ˜ Fi ∈ J for all i, so J(D(x j )) = I(D(xj )) for all j. This follows since xj that xn x0 ,..., ∈ J˜(D(xj )). Fi xj xj
Deﬁnition 6.8. Suppose that X is a projective variety and I ⊂ S(X) = k[x0 , . . . , xn ] is a nonzero homogeneous ideal. Let d and J be as in the conclusions of Lemma 6.7. Suppose that J = (F0 , . . . , Fr ), where F0 , . . . , Fr are homogeneous generators of degree d. Let Λ : X Pr be the rational map deﬁned by Λ = (F0 : . . . : Fr ). The blowup of I is B(I) = ΓΛ , with projection π : B(I) → X. We have that π −1 (D(xi )) = ΓΛD(xi ) by (5.3) and ΛD(xi ) = ( Fxd0 : . . . : Fr ). xdi
i
Since ˜ ˜ I(D(x i )) = J(D(xi )) =
F0 Fr ,..., d d xi xi
,
we see that the restriction of π to π −1 (D(xi )) → D(xi ) is the blowup of the ˜ ideal I(D(x i )) in k[D(xi )] for 0 ≤ i ≤ r. We thus have that Deﬁnition 6.8 is welldeﬁned by Proposition 6.1 (applied on each aﬃne open set D(xi )), so that B(I) is independent of choice of d and choice of generators of J (of the same degree d). More generally, for any aﬃne open subset U of X, the restriction of π ˜ ) in k[U ] (this will be proven to π −1 (U ) → U is the blowup of the ideal I(U in Lemma 12.3). Theorem 6.9. Suppose that X ⊂ Pm and Y ⊂ Pn are projective varieties and φ : Y → X is a birational regular map. Then φ is the blowup of a homogeneous ideal in S(X). Proof. Let ψ : X Y be the inverse rational map to φ. Deﬁne a regular map γ : Y → X × Y by γ(q) = (φ(q), q) for q ∈ Y . The image of γ is closed in X × Y by Proposition 5.8 and the fact that the map X × Y → Y × X given by (a, b) → (b, a) is an isomorphism. The map γ is an isomorphism onto its image since the inverse map is the projection onto Y . We have that π1 (γ(q)) = φ(q) for q ∈ Y . By Theorem 2.111, there exist aﬃne open subsets V of Y and U of X such that φ : V → U is an isomorphism. For q ∈ V , γ(q) = (p, ψ(p)) where p = φ(q). Since φ : V → U is an isomorphism, γ : V → ΓψU is an isomorphism. Thus γ(Y ) ⊂ Γψ since Y is the closure of V , Y is irreducible, and Γψ is closed. The image γ(Y ) contains ΓψU . Thus Γψ = γ(Y ) since both Γψ and γ(Y ) are closed and irreducible. We thus have a commutative
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6. The Blowup of an Ideal
diagram γ
/ Γψ @@ @@ π φ @@
Y @ @
X
where γ is an isomorphism. Choose forms F0 , . . . , Fr ∈ S(X) of a common degree, so that (F0 : . . . : Fr ) is a representative of the rational map ψ. Then π : Γψ → X is the blowup of the ideal I = (F0 , . . . , Fr ). Suppose that Λ : X Y is a rational map of projective varieties, with Y ⊂ Pn . Suppose that (F0 : . . . : Fn ) represents the rational map Λ; that is, F0 , . . . , Fn ∈ S(X) have the same degree, U = X \ Z(F0 , . . . , Fn ) = ∅, and ΛU = (F0 : . . . : Fn ). The graph ΓΛ is the closure of ΓΛV in X × Y for any dense open subset V of X on which Λ is regular (ΓΛV is the image of V in V × Y of the map p → (p, Λ(p))). We thus have that ΓΛ = Γ(F0 :...:Fn ) = B(I) where I is the ideal I = (F0 , . . . , Fn ) ⊂ S(X). We obtain the statement that if (F0 : . . . : Fn ) and (G0 : . . . : Gn ) are two representations of Λ, so that F0 , . . . , Fn are homogeneous of a common degree, G0 , . . . , Gn are homogeneous of a common degree, and Fi Gj − Fj Gi = 0 for 0 ≤ i, j ≤ n, then B(I) is isomorphic to B(J), where I = (F0 , . . . , Fn )
and J = (G0 , . . . , Gn ).
˜ In general, Z(I) = Z(J) for two such representations, and I˜ = J. The reason this works out is that two very diﬀerent ideals can have the same blowup. For instance, if X is aﬃne, I ⊂ k[X] is an ideal, and 0 = f ∈ k[X], then B(I) is isomorphic to B(f I). To prove this, suppose that I = (g0 , . . . , gr ) ⊂ k[X]. The rational map (g0 : . . . : gr ) : X Pr is the same as the rational map (f g0 : . . . , : f gr ) : X Pr so the two ideals have the same blowup. In particular, X is isomorphic to B(f k[X]) for any nonzero f ∈ k[X]. As an example, consider the projective variety A = Z(xy − zw) ⊂ P3 . Let S(A) = k[x, y, z, w]/(xy − zw) = k[x, y, z, w]. Consider the regular map φ : A → P1 which has the representations φ = (x : z) = (w : y).
6.2. The blowup of an ideal in a projective variety
123
˜ Let I = (x, z) and J = (w, y). Since φ is regular, we have that B(I) ˜ and B(J) are isomorphic to A by Proposition 5.9. We can verify this by computing the blowups on the aﬃne cover {D(x), D(y), D(z), D(w)} of A. For instance, on D(x), we have y z w y z w
y z w , , =k , , / − . k[A ∩ D(x)] = k x x x x x x x xx We have z ˜ = k[A ∩ D(x)] I(D(x)) = 1, x and w y w z w w ˜ , = , = , J(D(x)) = x x x xx x and we see that both ideals are principal ideals. The blowup of a subvariety Y of a projective variety X is the blowup of a homogeneous ideal I, which has a set of generators of a common degree, ˜ ) = I such that I(U X (Y )(U ) for all aﬃne open sets U ⊂ X. We ﬁnd such an ideal by applying Lemma 6.7 to IX (Y ). If π : B(I) → X is the blowup of a homogeneous ideal in the coordinate ring S(X) of a projective variety X and W is a closed subvariety of X, then as in the case when X is aﬃne, the strict transform of W in B(I) is deﬁned to be the Zariski closure of π −1 (W \ Z(I)) ∼ = W \ Z(I) in B(I). Deﬁnition 6.10. Suppose that f : X → Y is a birational regular map of quasiprojective varieties. Let U be the largest open subset of Y on which the inverse of f is a regular map. Suppose that Z is a subvariety of Y . The proper transform of Z by f is the closure of f −1 (Z ∩ U ) in X. In determining the diﬀerence between the strict transform and proper transform of a subvariety under a blowup, the following lemma is useful. Lemma 6.11. Suppose that I is an ideal in the regular functions of an aﬃne variety X and π : B(I) → X is the blowup of I. Then the largest open subset U of X on which π −1 is regular (and thus an isomorphism) is the set U = {p ∈ X  IOX,p is a principal ideal}. Proof. Suppose that IOX,p is a principal ideal. Then there exists an aﬃne open neighborhood V of p such that Ik[V ] is principal. Suppose that Ik[V ] = (f ). Then π −1 (V ) = B(Ik[V ]) is the graph of the rational map f˜ : V P0 deﬁned by f . But P0 is a single point, so f˜ is the regular map which contracts V to this point. The blowup of Ik[V ] is the graph of the regular map f˜ : V → P0 , so that π −1 (V ) = B(Ik[V ]) → V is an isomorphism.
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6. The Blowup of an Ideal
Now suppose that V is an aﬃne open subset of X such that π −1 (V ) = B(Ik[V ]) → V is an isomorphism. We have that IOB(I),q is a principal ideal for all q ∈ B(Ik[V ]) by (6.2), so IOX,p is a principal ideal for all p ∈ V since π is an isomorphism above V .
Use the methods of the example after Proposition 6.6 (before the computation of the coordinate ring of B(m)) in doing these exercises. Describe the aﬃne rings that come up in the computations as explicit quotients of polynomial rings. Lemma 6.11 should also be useful. Exercise 6.12. Analyze the strict transform C˜ of the curve C = Z(y 2 − x3 ) under the blowup π : B(p) → A2 of the origin p, and compute the rings of ˜ (See Exercise 1.9). Let regular functions on the natural aﬃne cover of C. −1 1 ∼ ˜ E = π (p) = P . What is E ∩ C? Exercise 6.13. Analyze the strict transform C˜ of the curve C = Z(x2 y − xy 2 + x4 + y 4 ) under the blowup π : B(p) → A2 of the origin p, and compute the rings of regular functions on the natural aﬃne cover of ˜ ˜ Let E = π −1 (p) ∼ C. = P1 . What is E ∩ C? Exercise 6.14. Analyze the blowup π : B(W ) → A3 of the following subvarieties of A3 . Describe π −1 (W ). a) W = {(0, 0, 0)}. b) W = Z(x1 , x2 ). c) Compute the strict transform of the surface S = Z(x31 +x2 x23 ) under each of these blowups. d) Compute the strict transform of the surface T = Z(x31 + x22 ) under each of these blowups. e) Compute the strict transform of the curve S = Z(x3 , x21 −x32 ) under each of these blowups. Exercise 6.15. Consider the rational map φ : P2 P2 which is represented by (x0 x1 : x0 x2 : x1 x2 ). This rational map is called a quadratic transformation. Let P1 = (0 : 0 : 1), P2 = (0 : 1 : 0), P3 = (1 : 0 : 0) ∈ P2 . Let L1,2 be the projective line in P2 containing P1 and P2 (that is, the onedimensional linear subspace of P3 containing these two points), let L1,3 be the projective line containing P1 and P3 , and let L2,3 be the projective line containing P2 and P3 . a) Show that φ2 = id as a rational map. Explain why this tells us that φ is birational.
6.2. The blowup of an ideal in a projective variety
125
b) Show that the ideal I = (x0 x1 , x0 x2 , x1 x2 ) in S(P2 ) = k[x0 , x1 , x2 ] is the intersection of prime ideals I = I(P1 ) ∩ I(P2 ) ∩ I(P3 ). c) Let π : B(I) → P2 be the blowup of I. Show that B(I) is the graph of φ. d) Show that for 0 ≤ i ≤ 2, π −1 (D(xi )) → D(xi ) is isomorphic to the blowup of a point in A2 which we analyzed at the end of this section; that is, show that there is a commutative diagram of regular maps α
π −1 (D(xi )) π
D(xi )
β
/B
λ
/ A2
where λ : B → A2 is the blowup of the origin and α, β are isomorphisms. e) Determine the largest open subset of P2 on which φ is a regular map. Explain why part d) of this problem tells us the answer to this question. f) Explain the geometry of the projections π1 : Γφ → P2 and π2 : Γφ → P2 in terms of the points P1 , P2 , P3 and their preimages by π1 and π2 , and the lines L1,2 , L1,3 , L2,3 and their strict transforms by π1 and π2 . Exercise 6.16. Let X = Z(xy − zw) ⊂ A4 , an aﬃne 3fold with k[X] = k[x, y, z, w]/(xy − zw) = k[x, y, z, w]. Let S1 be the aﬃne surface S1 = Z(x, z) and S2 be the aﬃne surface S2 = Z(x, w). These surfaces are subvarieties of X. a) Let π : B(m) → X be the blowup of m = (x, y, z, w) (the blowup of the point p = (0, 0, 0, 0)). Show that π : π −1 (X \ {p}) → X \ {p} is an isomorphism and E = π −1 (p) is a surface which is isomorphic to P1 × P1 . b) Compute the strict and proper transforms of S1 in B(m). Are they the same? Compute the strict and proper transforms of S2 in B(m). Are they the same? c) Let π1 : B(I1 ) → X be the blowup of I1 = (x, z) (the blowup of S1 ). Show that π1 : π1−1 (X \ {p}) → X \ {p} is an isomorphism and F1 = π1−1 (p) is isomorphic to P1 .
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6. The Blowup of an Ideal
d) Compute the strict and proper transforms of S1 in B(I1 ). Are they the same? Compute the strict and proper transforms of S2 in B(I1 ). Are they the same? e) Answer the questions of part c) for the blowup π2 : B(I2 ) → X of the ideal I2 = (x, w) (the blowup of S2 ), letting F2 = π2−1 (p). f) Answer the questions of part d) for the blowup of I2 . g) Show that there is a natural regular map φ1 : B(m) → B(I1 ) such that (φ1 E) : E → F1 induces the projection on the ﬁrst factor P1 × P1 → P1 . h) Show that there is a natural regular map φ2 : B(m) → B(I2 ) such that (φ2 E) : E → F2 induces the projection on the second factor P1 × P1 → P1 . i) Show that the induced birational map B(I1 ) B(I2 ) is not regular.
Chapter 7
Finite Maps of Quasiprojective Varieties
In this chapter we explore properties of ﬁnite maps.
7.1. Aﬃne and ﬁnite maps Deﬁnition 7.1. Suppose that X and Y are quasiprojective varieties and φ : X → Y is a regular map. 1) The map φ is aﬃne if for every q ∈ Y there exists an aﬃne neighborhood U of q in Y such that φ−1 (U ) is an aﬃne open subset of X. 2) The map φ is ﬁnite if for every q ∈ Y there exists an aﬃne neighborhood U of q in Y such that φ−1 (U ) is an aﬃne open subset of X and φ : φ−1 (U ) → U is a ﬁnite map of aﬃne varieties (as deﬁned in Deﬁnition 2.55). It follows from the more general statement of Theorem 7.5 below that if X and Y are aﬃne varieties and φ : X → Y is a ﬁnite map of quasiprojective varieties, then φ is a ﬁnite map of aﬃne varieties (as deﬁned in Deﬁnition 2.55). Lemma 7.2. Suppose that X is an aﬃne variety. Then X is quasicompact (every open cover has a ﬁnite subcover). 127
128
7. Finite Maps of Quasiprojective Varieties
Proof. Let {Ui }i∈Λ be an open cover of X. We may reﬁne the cover by basic open sets D(f ) with f ∈ k[X] and may so assume that each Ui = D(fi ). Since i∈Λ Ui = X, we have that Z(I) = ∅, where I is the ideal I = (fi  i ∈ Λ). Thus I = k[X] and so there exist a positive integer r and i1 , . . . , ir ∈ Λ such that (fi1 , . . . , fir ) = k[X]. Thus {Ui1 , . . . , Uir } is a ﬁnite cover of X. Lemma 7.3. Suppose that A is a ring and f1 , . . . , fn ∈ A are such that the ideal (f1 , . . . , fn ) = A. Suppose that N is a positive integer. Then (f1N , . . . , fnN ) = A. Proof. Since (f1 , . . . , fn ) = A, there exist g1 , . . . , gn ∈ A such that f1 g1 + · · · + fn gn = 1. Thus 1 = (f1 g1 + · · · + fn gn )nN (nN )! i1 i2 in N N = i1 +···+in =nN i1 !i2 !···in ! (f1 g1 ) (f2 g2 ) · · · (fn gn ) ∈ (f1 , . . . , fn ). Lemma 7.4. Suppose that A is a domain which is a subring of a domain B and there exist f1 , . . . , fn ∈ A such that 1) the ideal (f1 , . . . , fn ) = A and 2) the localization Bfi is a ﬁnitely generated Afi algebra for all i. Then B is a ﬁnitely generated Aalgebra. Further suppose that Bfi is a ﬁnitely generated Afi module for all i. Then B is a ﬁnitely generated Amodule. Proof. By assumption, there exist ri ∈ Z+ for 1 ≤ i ≤ n and zi1 , . . . , ziri ∈ Bfi for 1 ≤ i ≤ n such that Bfi = Afi [zi1 , . . . , ziri ]. After possibly multiplying the zij by a positive power of fi , we may assume that zij ∈ B for all i, j. Let C = A[{zij }]. C is a ﬁnitely generated Aalgebra which is a subring of B. We will show that B = C. Suppose that b ∈ B. Then b ∈ Bfi implies there are polynomials gi ∈ Afi [x1 , . . . , xri ] such that b = gi (zi1 , . . . , zi,ri ) for 1 ≤ i ≤ n. Since the polynomials gi have only a ﬁnite number of nonzero coeﬃcients, which are
7.1. Aﬃne and ﬁnite maps
129
in Afi , there exists a positive integer N such that fiN gi ∈ A[x1 , . . . , xri ] for 1 ≤ i ≤ n. Thus fiN b = fiN gi (zi1 , . . . , ziri ) ∈ A[zi1 , . . . , ziri ] ⊂ C ci fiN = 1. Thus for all i. By Lemma 7.3, there exist ci ∈ A such that
ci fiN b ∈ C. b= ci fiN b = Now suppose that the Bfi are ﬁnitely generated Afi modules. Suppose that b ∈ B. Then b is integral over Afi for all i, so there exists N > 0 such N that f i b is integral over A for all i. By Lemma 7.3, there exist ci ∈ A such that ci fiN = 1. Thus b = i ci fiN b is integral over A. Theorem 7.5. Suppose that φ : X → Y is a regular map of quasiprojective varieties. 1) Suppose that φ is aﬃne and U is an aﬃne open subset of Y . Then V = φ−1 (U ) is an aﬃne open subset of X. 2) Suppose that φ is ﬁnite and U is an aﬃne open subset of Y . Then V = φ−1 (U ) is an aﬃne open subset of X, and the restriction of φ to a regular map from V to U is a ﬁnite map of aﬃne varieties. Proof. Let T be the Zariski closure of φ(X) in Y . Then for all aﬃne open subsets U of Y , T ∩ U is aﬃne and the inclusion of T ∩ U into U is a ﬁnite map of aﬃne varieties. Thus we may assume that φ is dominant. Suppose that φ is aﬃne and dominant and U is an aﬃne open subset of Y . We will ﬁrst show that φ−1 (U ) → U is an aﬃne map. Suppose that q ∈ U . Then there exists an aﬃne neighborhood W of q in Y such that φ−1 (W ) is aﬃne. There exists f ∈ k[W ] such that q ∈ Wf ⊂ U ∩ W since such open sets are a basis of the topology on W . Thus φ−1 (Wf ) = φ−1 (W )φ∗ (f ) is aﬃne by Proposition 2.93. We conclude that φ : φ−1 (U ) → U is an aﬃne map. Let A = k[U ] = OY (U ). Suppose that q ∈ U and let W be an aﬃne neighborhood of q in U such that φ−1 (W ) is aﬃne. Then there exists f ∈ A ⊂ k[W ] such that q ∈ Uf ⊂ W since open sets of this form are a basis for the topology on U . The inclusions Wf ⊂ Uf ⊂ W imply that Wf = Uf and so φ−1 (Uf ) = φ−1 (Wf ) = φ−1 (W )φ∗ (f ) is aﬃne by Proposition 2.93. Since U is aﬃne, it is 7.2) so there exist nquasicompact (by Lemma −1 (U ) is aﬃne for all i. U = U and φ f1 , . . . , fn ∈ A such that fi i=1 fi Thus ZU (f1 , . . . , fn ) = ∅, so (f1 , . . . , fn ) = I(ZU (f1 , . . . , fn )) = A by the nullstellensatz. Thus (7.1)
(f1 , . . . , fn ) = A.
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7. Finite Maps of Quasiprojective Varieties
Let Vi = φ−1 (Ufi ) for 1 ≤ i ≤ n. The Vi are an aﬃne cover of V = φ−1 (U ). Let B = OY (V ). The map φ gives us an injective kalgebra homomorphism φ∗ : A → B ⊂ k(X). Let Bi = OY (Vi ) = k[Vi ] for 1 ≤ i ≤ n. The restriction of φ to Vi gives 11 kalgebra homomorphisms φ∗ : k[Ufi ] = Afi → Bi = k[Vi ] ⊂ k(X), realizing the Bi as ﬁnitely generated Afi algebras (Bi are ﬁnitely generated kalgebras since the Vi are aﬃne). Now Vi ∩ Vj is precisely the open subset (Vi )φ∗ (fj ) of Vi , so for all i, j, Vi ∩ Vj is aﬃne with regular functions k[Vi ∩ Vj ] = (Bi )φ∗(fj ) = (Bj )φ∗ (fi ) ⊂ k(X) by Propositions 2.84 and 2.93. Since φ∗ (fj ) does not vanish on Vj , φ∗ (fj ) is a unit in Bj , so (Bj )φ∗ (fj ) = Bj , and Bj ⊂ (Bj )φ∗ (fi ) = (Bi )φ∗ (fj ) for all i, j. Now B = OY (V ) =
n
OY (Vi ) =
i=1
We compute
Bφ∗ (fj ) =
n
i=1
= φ∗ (fj )
Bi .
i=1
Bi
n
n
(Bi )φ∗ (fj ) = Bj .
i=1
By Lemma 7.4, B is a ﬁnitely generated Aalgebra, and since A is a ﬁnitely generated kalgebra, B is a ﬁnitely generated kalgebra. Thus there exists an aﬃne variety Z such that k[Z] = B. Let t1 , . . . , tm ∈ B generate B as a kalgebra (the ti are coordinate functions on Z). Since B = OX (V ), α = (t1 , . . . , tm ) induces a regular map α : V → Z. Now the αi = α  Vi induce isomorphisms αi : Vi → Zφ∗ (fi ) of aﬃne varieties for all i, since αi∗ induces an isomorphism of regular functions. We may thus deﬁne regular maps ψi : Zφ∗ (fi ) → Vi which are isomorphisms by requiring that (ψi )∗ = (αi∗ )−1 . The ψi patch to give a continuous map ψ : V → Z which is a regular map by Proposition 3.39. Since ψ is an inverse to α, we have that V ∼ = Z is an aﬃne variety. Now suppose that X → Y is a ﬁnite map of quasiprojective varieties. Then we may choose f1 , . . . , fn ∈ A as above with the additional property that the Bfi are ﬁnitely generated Afi modules for all i. Thus B is a ﬁnite Amodule by Lemma 7.4, and so V = φ−1 (U ) → U is a ﬁnite map of aﬃne varieties. Exercise 7.6. Suppose that X is a quasiprojective variety. Show that X is quasicompact (every open cover has a ﬁnite subcover).
7.2. Finite maps
131
7.2. Finite maps Theorem 7.7. Suppose that X and Y are quasiprojective varieties and φ : X → Y is a ﬁnite regular map. Then φ is a closed map, and if φ is dominant, then φ is surjective. Proof. There exists an aﬃne cover {Vi } of Y such that the maps φ : Ui → Vi where Ui = φ−1 (Vi ) are ﬁnite maps of aﬃne varieties. We obtain this by either choosing the Vi so that φ−1 (Vi ) are aﬃne with φ : Ui → Vi ﬁnite, which exist by the deﬁnition of a ﬁnite map, or we pick an arbitrary aﬃne cover {Vi } of Y and apply Theorem 7.5 to get this statement. We have that each φ : Ui → Vi is a closed map by Corollary 2.58 and is surjective if φ is dominant by Theorem 2.57, so φ : X → Y is a closed map, which is surjective if φ is dominant. Theorem 7.8. Suppose that X and Y are quasiprojective varieties and φ : X → Y is a dominant regular map. Then φ(X) contains a nonempty open subset of Y . Proof. It suﬃces to prove the theorem for the map to an aﬃne open subset V of Y from the restriction of φ to an aﬃne open subset U contained in the preimage of V . Thus we may assume that X and Y are aﬃne. Let u1 , . . . , ur ∈ k[X] be a transcendence basis of k(X) over k(Y ). Then k[Y ] ⊂ k[Y ][u1 , . . . , ur ] = k[Y × Ar ] ⊂ k[X]. Thus φ factors as the composition φ = g ◦ h of regular maps where h : X → Y × Ar and g : Y × Ar → Y is the projection onto the ﬁrst factor. Every element v ∈ k[X] is algebraic over k(Y × Ar ). Thus there exists a polynomial f (x) = xs + b1 xs−1 + · · · + bs with bi ∈ k(Y × Ar ) such that f (v) = 0. Write bi = cai with a, ci ∈ k[Y × Ar ]. Thus v is integral over k[Y × Ar ]a . Let v1 , . . . , vm be coordinate functions on X (so that k[X] = k[v1 , . . . , vm ]). For each vi choose ai ∈ k[Y × Ar ] such that vi is integral over k[Y × Ar ]ai . Let F = a1 · · · am . Then k[X]h∗ (F ) is integral over k[Y × Ar ]F , so that h : Xh∗ (F ) → (Y × Ar )F is ﬁnite and dominant. Thus (Y × Ar )F = h(Xh∗ (F ) ) ⊂ h(X) by Theorem 7.7. It remains to show that g((Y × Ar )F ) contains a set that is open in Y . We have an expression F = fi1 ,...,ir ui11 . . . uirr ∈ k[Y × Ar ] = k[Y ][u1 , . . . , ur ] with fi1 ,...,ir ∈ k[Y ] not all zero. If p ∈ Y and some fi1 ,...,ir (p) = 0, then there exists a point q ∈ Ar such that F (p, q) = 0 (by Theorem 1.4 or the
132
7. Finite Maps of Quasiprojective Varieties
aﬃne nullstellensatz). Thus the nonempty open set Yfi1 ,...,ir = Y \ Z({fi1 ,...,ir }) ⊂ g((Y × Ar )F ) ⊂ φ(X).
A short proof of Theorem 7.8 can be obtained by using some theorems from commutative algebra on ﬂatness. We reduce to the case where X and Y are both aﬃne. By the theorem of generic ﬂatness, [107, Theorem 52 on page 158], there is a nontrivial open subset U of Y such that φ−1 (U ) → U is ﬂat. By [107, Theorem 4 on page 33] and [107, Theorem 8 on page 48], φ = φ−1 (U ) → U is an open map, so φ(X) contains a nontrivial open subset of X. Theorem 7.9. Suppose that X is a projective variety which is a closed subvariety of a projective space Pn and suppose that X ⊂ Pn \ E where E is a ddimensional linear subspace. Then the projection π : X → Pn−d−1 from E determines a dominant ﬁnite map from X to the projective variety π(X). Proof. Let y0 , . . . , yn−d−1 be homogeneous coordinates on Pn−d−1 and let L0 , . . . , Ln−d−1 be a basis of the vector space of linear forms vanishing on E. Deﬁne π by the formula π = (L0 : . . . : Ln−d−1 ). Here π is a regular map on X since E ∩ X = ∅, so the forms L0 , . . . , Ln−d−1 do not vanish simultaneously on X. ) ∩ X = PnLi ∩ X. Then Ui is an aﬃne open subset Let Ui = π −1 (Pn−d−1 yi is a of X. We will show that for all i such that Ui = ∅, Ui → π(X) ∩ Pn−d−1 yi n−d−1 ﬁnite map. The image π(X) is a closed subset of P by Theorem 5.14. is an aﬃne Hence π(X) is a projective variety and π(X)yi = π(X) ∩ Pn−d−1 yi open subset of π(X). We will show that k[Ui ] is integral over the subring k[π(X)yi ] for all i, so that π : Ui → π(X)yi is ﬁnite for all i and thus π : X → π(X) is ﬁnite and dominant. Suppose g ∈ k[Ui ]. Then g is the restriction of a form LGm where m i is the degree of the homogeneous form G ∈ S(Pn ) by formula (4.6). Let z0 , . . . , zn−d be homogeneous coordinates on Pn−d , and deﬁne a rational m n n−d . The rational map map π1 = (Lm 0 : . . . : Ln−d−1 : G) from P to P π1 induces a regular map of X and its image π1 (X) is closed in Pn−d by Theorem 5.14. Let F1 , . . . , Fs be a set of generators of I(π1 (X)) ⊂ S(Pn−d ). As X ∩ E = ∅, the forms L0 , . . . , Ln−d−1 do not vanish simultaneously on X. Thus the point (0 : . . . : 0 : 1) is not contained in π1 (X), so that ZPn−d (z0 , . . . , zn−d−1 , F1 , . . . , Fs ) = {(0 : . . . : 0 : 1)} ∩ π1 (X) = ∅. By Proposition 3.11, we have that Tl ⊂ (z0 , . . . , zn−d−1 , F1 , . . . , Fs ) for some l > 0, where Tl is the vector space of homogeneous forms of degree l on
7.2. Finite maps
133
Pn−d . In particular, we have an expression l zn−d
=
n−d−1
zj Hj +
j=0
s
Fj P j
j=1
where Hj , Pj ∈ S(Pn−d ) = k[z0 , . . . , zn−d ] are polynomials. Denoting by H (q) the homogeneous component of H of degree q, let l Φ(z0 , . . . , zn−d ) = zn−d −
n−d−1
(l−1)
zj Hj
.
j=0
We have that Φ ∈ I(π1 (X)). The homogeneous polynomial Φ has degree l, and as a polynomial in zn−d it has the leading coeﬃcient 1, so that it has an expression j l + Al−j (z0 , . . . , zn−d−1 )zn−d Φ = zn−d where the Al−j are homogeneous of degree l −j. Substitution of the deﬁning ∗ formulas π1∗ (zi ) = Lm i for 0 ≤ i ≤ n − d − 1 and π1 (zn−d ) = G induces a kalgebra homomorphism π1∗ : S(Pn−d ) → S(Pn ). Since the Fi vanish on π1 (X), we have that π1∗ (Fi ) ∈ I(X). We thus have that m π1∗ (Φ) = Φ(Lm 0 , . . . , Ln−d−1 , G) ∈ I(X)
is a homogeneous form of degree lm in S(Pn ). Dividing this form by Lml i , we obtain a relation l m
j Ln−d−1 m l−1 L0 G G + A , . . . , 1, . . . , m l−j j=0 L Li Li Lm i
i
∈ I(X ∩ PnLi ) ⊂ k[PnLi ]. of aﬃne The rational map π induces a regular map π : PnLi → Pn−d−1 yi yj Lj ∗ n−d−1 n ∗ varieties, with π : k[Pyi ] → k[PLi ] given by π yi = Li for 0 ≤
). j ≤ n − d − 1. We have that (π ∗ )−1 (I(X ∩ PnLi )) = I(π(X) ∩ Pn−d−1 yi Since g is the residue of LGm in k[Ui ] = k[X ∩ PnLi ] = k[PnLi ]/I(X ∩ PnLi ), i we obtain the desired dependence relation, showing that g is integral over ]/I(π(X)yi ). k[π(X)yi ] = k[Pn−d−1 yi Remark 7.10. Looking back at the proof, we see that we have also proved the following theorem. Writing the coordinate ring of X as S(X) = S(Pn )/I(X) and the coordinate ring of π(X) as S(π(X)) = k[y0 , . . . , y m ] = S(Pm )/I(π(X))
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7. Finite Maps of Quasiprojective Varieties
where m = n−d−1, we showed that the 11 graded kalgebra homomorphism π ∗ : S(π(X)) → S(X) deﬁned by π ∗ (y i ) = Li for 0 ≤ i ≤ m makes S(X) an integral extension of S(π(X)). By applying this theorem to a Veronese embedding of X, or modifying the proof using formula (4.9) instead of (4.6), we obtain the following generalization. Theorem 7.11. Let F0 , . . . , Fs be linearly independent forms over k of degree m > 0 on Pn that do not vanish simultaneously on a closed subvariety X ⊂ Pn . Then φ = (F0 : . . . : Fs ) determines a dominant ﬁnite map φ : X → φ(X) of projective varieties. Corollary 7.12. Suppose that X is a projective variety which is a closed subvariety of a projective space Pn . Then there exists a surjective ﬁnite map φ : X → Pm for some m. The map φ is the restriction to X of a projection from a suitable linear subspace of Pn . Proof. If X = Pn , choose a point p ∈ Pn \ X and let π : X → Pn−1 be the projection from p. The induced regular map π : X → π(X) is a ﬁnite map and π(X) is a projective variety which is a closed subset of Pn−1 by Theorem 7.9. We continue until the image of X is the whole ambient projective space. A composition of ﬁnite maps is ﬁnite so the resulting map is ﬁnite. Corollary 7.13 (Projective Noether normalization). Suppose that R is the coordinate ring of a projective variety. Then there exist m ≥ 0 and linear forms L0 , . . . , Lm in R such that the graded kalgebra homomorphism φ∗ : k[x0 , . . . , xm ] → R is an integral extension, where k[x0 , . . . , xm ] is a polynomial ring and φ∗ (xi ) = Li for 0 ≤ i ≤ m. Proof. This statement follows from Corollary 7.12 and Remark 7.10. A purely algebraic proof is given in [28, Theorem I.5.17]. Exercise 7.14. Suppose that φ : Pn → Pm is a regular map. Show that either φ(Pn ) is a point or φ is a ﬁnite map onto a closed subvariety of Pm . (Recall the conclusions of Exercise 4.14 and the assumption that F0 , . . . , Fs are linearly independent forms in Theorem 7.11.)
7.3. Construction of the normalization
135
7.3. Construction of the normalization In this section we construct the normalization of a quasiprojective variety in a ﬁnite extension of its function ﬁeld. Deﬁnition 7.15. Suppose that X is a quasiprojective variety and p ∈ X. The point p is called a normal point of X if OX,p is integrally closed in its quotient ﬁeld k(X). The variety X is called normal if all points of X are normal points of X. Proposition 7.16. Suppose that X is a normal quasiprojective variety. Then OX (X) is integrally closed in k(X). Proof. Suppose that f ∈ k(X) is integral over OX (X). Then for all p ∈ X, f is integral over OX,p , so that f ∈ OX,p . Thus OX,p = OX (X). f∈ p∈X
The remainder of this section will be devoted to the proof of the following theorem (from the proof on page 177 of [161] and of [116, Theorem 4, Section III.8]). Theorem 7.17 (Normalization, in a ﬁnite extension). Suppose X is a quasiprojective variety and Λ : k(X) → L is a kalgebra homomorphism of ﬁelds, such that L is a ﬁnite extension of k(X). Then there is a unique normal quasiprojective variety Y with function ﬁeld k(Y ) = L, and there is a dominant ﬁnite regular map π : Y → X such that π ∗ : k(X) → k(Y ) is the homomorphism Λ. If X is aﬃne, then Y is aﬃne. If X is projective, then Y is projective. The normal variety Y is called the normalization of X in L. If L = k(X), then Y is called the normalization of X. We ﬁrst prove uniqueness. Suppose that π : Y → X and π : Y → X each satisfy the conclusions of the theorem. Suppose that p ∈ X and that U is an aﬃne neighborhood of p in X. Then V = π −1 (U ) is an aﬃne open subset of Y since π is ﬁnite. Further, k[V ] is integrally closed in L and is ﬁnite over k[U ] by Theorem 7.5. Thus k[V ] is the integral closure of k[U ] in L. We thus have that k[V ] = k[V ] where V = (π )−1 (U ), so that the identity map is an isomorphism of the aﬃne varieties V and V . Since this holds for an aﬃne cover of X, we have that Y = Y by Proposition 3.39. We now prove existence for an aﬃne variety X. Let R be the integral closure of k[X] in L. Then R is a ﬁnitely generated kalgebra (by Theorem 1.54) which is a domain, so that R = k[Y ] for some aﬃne variety Y . The inclusion Λ : k[X] → k[Y ] induces a ﬁnite regular map Y → X by Proposition 2.40.
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7. Finite Maps of Quasiprojective Varieties
We now prove existence for a projective variety X, from which existence for a quasiprojective variety follows. For a graded ring A and d ∈ Z, recall that in (3.2), we deﬁne a graded (d) (d) ring A(d) by A(d) = ∞ i=−∞ Ai , where Ai = Aid . Now suppose that X ⊂ Pn is a projective variety, with homogeneous coordinate ring R = S(X) = k[x0 , . . . , xn ]/P = k[x0 , . . . , xn ]. Let α ∈ R1 be a nonzero element, and let Σ ⊂ R be the multiplicatively closed set of nonzero homogeneous elements. Then the localization Σ−1 R is graded. Lemma 7.18. There is an isomorphism of graded rings 1 ∼ −1 ∼ Σ R = k(X) α, k(X)αn , = α n∈Z
which is the localization k(X)[α]α of the standard graded polynomial ring K(X)[α] in the variable α (which has degree 1) over k(X). Proof. We have that a homogeneous element β ∈ Σ−1 R of degree d has an expression β = fa where a ∈ Ri and f ∈ Rm and i − m = d. Thus the elements of Σ−1 R of degree 0 are exactly the elements of k(X). If β has degree d = 0, then we have a β = αd d , fα where f αa d has degree 0. Thus f αa d ∈ k(X), and we have that (Σ−1 R)d = k(X)αd . In particular, Σ−1 R = k(X)[α, α1 ].
Lemma 7.19. The integral closure S of R in L[α] is a graded ring and a ﬁnitely generated Rmodule. Proof. Since R is a ﬁnitely generated kalgebra and L(α) is a ﬁnite ﬁeld extension of k(X)(α), we have that the integral closure of S in the ﬁeld L(α) is a ﬁnite R module by Theorem 1.54. Thus the submodule S is a ﬁnitely generated Rmodule by Lemma 1.55. We observe that since L[α] is normal, S is the integral closure of R in the ﬁeld L(α). We ﬁrst proof the lemma when L = k(X). Suppose that a ∈ S. Write a = as + as+1 + · · · + at where each ai ∈ L[α] is homogeneous of degree i and as = 0. The homogeneous form as is called the initial component a. Since S ⊂ k(X)[α], every element of S can be written as a quotient of two elements of R such that the denominator is a homogeneous element. Since a is integral over R, R[a] is a ﬁnitely generated Rmodule, by Theorem 1.49. Thus there exists a homogeneous element 0 = d ∈ R such that dR[a] ⊂ R.
7.3. Construction of the normalization
137
Hence for every i ≥ 0, dai ∈ R. We have that dais is the initial component of dai , so dais ∈ R since R is graded. Thus ais ∈ d1 R for all i ∈ N, so R[as ] ⊂ d1 R is a ﬁnite Rmodule by Lemma 1.55. Thus as is integral over R, and so a − as = as+1 + · · · + at is integral over R, and continuing this way, we establish that all ai are integral over R. Thus S is a graded subring of k(X)[α]. Now assume that L is a ﬁnite extension of k(X) and S is the integral Sq , a closure of R in L[α]. Let Sq = S ∩ L[α]q for all q ∈ N, and let S = graded subring of L[α] which is contained in S. Suppose β ∈ L. There exist n ∈ Z+ and ci ∈ k(X) such that β n + c1 β n−1 + · · · + cn = 0. Thus there exists a homogeneous element h ∈ R such that hβ is integral over R. We have that hβ ∈ S since hβ is homogeneous of degree deg h. Thus the quotient ﬁeld of S is L(α). As S contains R, the integral closure of S in L[α] is S. But then S = S by the ﬁrst part of the proof. Although S is graded, it may be that S is not generated in degree 1. By Exercise 3.6, there exists d ∈ Z+ such that S (d) is generated in degree 1. We have that R(d) = R ∩ k(X)[αd ]. We will show that S (d) is the integral closure of R(d) in L[αd ]. If x ∈ L[αd ] is integral over R(d) , then as an element of L[α], it is integral over R. Thus x ∈ S ∩ L[αd ] = S (d) . If x ∈ S (d) is homogeneous, then x is integral over R and we then have a homogeneous equation of integral dependence xm + f1 xm−1 + · · · + fm = 0 with fi ∈ Rdeg(x)i . But d divides deg(x), so x is integral over R(d) . (d)
Choosing a basis y 0 , . . . , y m of S1 , we have an isomorphism S (d) ∼ = k[y0 , . . . , ym ]/P ∗ = k[y 0 , . . . , y m ] where P ∗ is a homogeneous prime ideal in the polynomial ring k[y0 , . . . , ym ] and the yi all have degree 1. Let Y ⊂ Pm be the projective variety Y = Z(P ∗ ). We have S(Y ) ∼ = S (d) . We will now show that Y is normal. The ring S (d) is integrally closed in L[αd ]. Since L[αd ] is isomorphic to a polynomial ring over a ﬁeld, it is integrally closed in its quotient ﬁeld. Hence S (d) is integrally closed in its (d) quotient ﬁeld. Thus the localization Syi is integrally closed in its quotient ﬁeld (by Exercise 1.58). By (3.8),
f (d)  m ∈ Z+ and f ∈ Sm , OY (D(yi )) = ym i
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7. Finite Maps of Quasiprojective Varieties
(d)
which is the set of elements of Syi of degree 0. Thus it is the intersection (d)
(d)
Syi ∩ L taken within L[αd ], which is integrally closed in L since Syi is integrally closed. Since the local ring of every point of Y is a localization of one of the normal local rings OY (D(yi )), all of these local rings are integrally closed, so that Y is normal. − 1, we have By the Veronese embedding φ : Pn → Pe where e = n+d n e an isomorphism of X with a closed subset of P , such that the coordinate (d) ring of φ(X) satisﬁes S(φ(X)) ∼ = R(d) . Choosing a basis z 0 , . . . , z l of R1 , we have an isomorphism R(d) ∼ = k[z0 , . . . , zl ]/P = k[z 0 , . . . , z l ] where P is a homogeneous prime ideal in the polynomial ring k[z0 , . . . , zl ] and the zi all have degree 1. Our graded inclusion R(d) ⊂ S (d) gives us an expression zi = aij y i j
with aij ∈ k for all i, j. Let Li = j aij yi for 0 ≤ i ≤ l. We will now show that Z(L1 , . . . , Ll ) ∩ Y = ∅. Suppose that p ∈ Z(L1 , . . . , Ll ) ∩ Y . Since S (d) is integral over R(d) , for 0 ≤ i ≤ m we have that y i is integral over R(d) , by a homogeneous relation. Thus we have equations yini + bi1 (L0 , . . . , Ll )yini −1 + · · · + bi,ni (L0 , . . . , Ll ) ∈ P ∗ where the bij are homogeneous polynomials of degree j. Evaluating at p, we obtain that yini (p) = 0, so that yi (p) = 0 for all i, which is impossible. Thus Z(L1 , . . . , Ll ) ∩ Y = ∅. By Theorem 7.9, the rational map π = (L0 : . . . : Ll ) from Y to φ(X) ∼ =
X is a ﬁnite regular map. By our construction, the induced map π ∗ : k(X) → k(Y ) is Λ.
Exercise 7.20. Show that an aﬃne variety X is normal if and only if k[X] is integrally closed in k(X).
Chapter 8
Dimension of Quasiprojective Algebraic Sets
8.1. Properties of dimension Suppose that X is a quasiprojective algebraic set. We deﬁne the dimension of X to be its dimension as a topological space (Deﬁnition 2.63). The following proposition is proved in the same way as Proposition 2.67. Proposition 8.1. Suppose that X is a quasiprojective algebraic set and V1 , . . . , Vn are its irreducible components. Then dim X = max{dim Vi }. Theorem 8.2. Suppose that X is a projective variety. Then: 1) dim X = trdegk k(X). 2) Any maximal chain of distinct irreducible closed subsets of X has length n = dim X. 3) Suppose that U is a dense open subset of X. Then dim U = dim X. Proof. We have a closed embedding X ⊂ Pn . Let x0 , . . . , xn be homogeneous coordinates on Pn . Suppose that (8.1)
W0 ⊂ W1 ⊂ · · · ⊂ Wm
is a chain of distinct irreducible closed subsets of X. There exists an open set D(xi ) = Pnxi such that W0 ∩D(xi ) = ∅. Then since the Wi are irreducible, (8.2)
W0 ∩ D(xi ) ⊂ · · · ⊂ Wm ∩ D(xi ) 139
140
8. Dimension of Quasiprojective Algebraic Sets
is a chain of distinct irreducible closed subsets of the aﬃne variety U = X ∩ D(xi ). Thus m ≤ dim U = trdegk k(X) by Proposition 2.64, Theorem 1.63, and Proposition 3.37. Suppose that (8.1) is a maximal chain. Then (8.2) is a maximal chain. Thus m = trdegk k(X) by Proposition 2.64, Theorem 1.63, and Corollary 2.69. Now the proof that all nontrivial open subsets of X have the same dimension as X follows from the proof of Proposition 2.70. Theorem 8.3. Suppose that X is a projective variety, with homogeneous coordinate ring S(X). Then dim X + 1 = dim S(X). ∼ k(X)[t, 1 ] as graded Proof. By Lemma 7.18, The localization Σ−1 S(X) = t rings, where Σ is the multiplicatively closed set of nonzero homogeneous elements of S(X) and t is an indeterminate (with deg(t) = 1 and the elements of k(X) have degree 0). Thus the transcendence degree of the quotient ﬁeld of S(X) over k is equal to one plus the transcendence degree of k(X) over k. Theorem 8.4. Suppose that W ⊂ Pn is a projective variety of dimension ≥ 1 and F ∈ S(Pn ) is a form which is not contained in I(W ). Then W ∩ Z(F ) = ∅ and all irreducible components of Z(F ) ∩ W have dimension dim W − 1. Proof. Suppose that X is an irreducible component of W ∩ Z(F ). Then there exists an open subset D(xi ) of Pn such that X ∩ D(xi ) = ∅. Let d be the degree of F . Here xFd ∈ OPn (D(xi )) and X ∩ D(xi ) is an irreducible i
component of ZD(xi ) ( xFd ) ∩ (W ∩ D(xi )). Since i
F xdi
does not restrict to the
zero element on W ∩D(xi ), we have that X ∩D(xi ) has dimension dim W −1 by Theorem 2.71.
Suppose that Z(F ) ∩ W = ∅. Then by Theorem 7.11, φ = (F ) induces a ﬁnite regular map from W to P0 , which is a point, so that k(P0 ) = k. Since φ is ﬁnite, k(W ) is a ﬁnite ﬁeld extension of k(P0 ), so that k(W ) = k and dim W = trdegk k(W ) = 0. Corollary 8.5. Suppose that W ⊂ Pn is a projective variety and F1 , . . . , Fr ∈ S(Pn ) are forms (of degree > 0). Then dim G ≥ dim W − r for all irreducible components G of Z(F1 , . . . , Fr ) ∩ W . If r ≤ dim W , then Z(F1 , . . . , Fr ) ∩ W = ∅. (The dimension of ∅ is −1.) Proof. If r > dim W , then the corollary is certainly true, so suppose r ≤ dim W . Inductively deﬁne subvarieties Wi1 ,...,is of Pn for 1 ≤ s ≤ r and
8.2. The theorem on dimension of ﬁbers
141
natural numbers σ(i1 , . . . , is ) by W ∩ Z(F1 ) = W1 ∪ · · · ∪ Wσ(0) and Wi1 ,...,is ∩ Z(Fs+1 ) = Wi1 ,...,is ,1 ∪ · · · ∪ Wi1 ,...,is ,iσ(i1 ,...,is ) for s ≥ 1, where Wi1 ,...,is ,1 , . . . , Wi1 ,...,is ,iσ(i1 ,...,is ) are the irreducible components of Wi1 ,...,is ∩ Z(Fs+1 ). If Fs+1 ∈ I(Wi1 ,...,is ), then Wi1 ,...,is ∩ Z(Fs+1 ) = Wi1 ,...,is , so that σ(i1 , . . . , is ) = 1, Wi1 ,...,is ,1 = Wi1 ,...,is and dim(Wi1 ,...,is ∩ Z(Fs+1 )) = dim(Wi1 ,...,is ). If Fs+1 ∈ I(Wi1 ,...,is ), then dim Wi1 ,...,is ,j = dim Wi1 ,...,is − 1 for all j by Theorem 8.4 since dim Wi1 ,...,is ≥ 1 for s < r = dim W by induction on s. Since Z(F1 , . . . , Fr ) ∩ W = Wi1 ,...,ir , we have that G = Wi1 ,...,ir for some i1 , . . . , ir , and thus dim G ≥ dim W − r. Corollary 8.6. Suppose that W is a quasiprojective variety and f1 , . . . , fr ∈ OW (W ). Suppose that ZW (f1 , . . . , fr ) = ∅. Then dim G ≥ dim W − r for all irreducible components G of ZW (f1 , . . . , fr ). If Y is a quasiprojective algebraic set, contained in a quasiprojective variety X, then we deﬁne the codimension of Y in X to be codimX (Y ) = dim X − dim Y. Exercise 8.7. Show that the m in Corollary 7.12 is m = dim X. Exercise 8.8. Show that the deﬁnition of the dimension of a linear subspace of a projective variety, deﬁned in Section 4.5, agrees with Deﬁnition 2.63. Exercise 8.9. Suppose that L is an (n − 1)dimensional linear subspace of Pn , X ⊂ L is a closed subvariety, and y ∈ Pn \ L. Let Y be the union of all lines containing y and a point of X. Recall from Exercise 4.9 that Y is a projective subvariety of Pn . Show that dim Y = dim X + 1.
8.2. The theorem on dimension of ﬁbers Lemma 8.10. Suppose that p1 , . . . , ps , q1 , . . . , qr ∈ Pn are distinct points for some s, r, and n. Then there exists a homogeneous form F ∈ S(Pn ) such that F (p1 ) = · · · = F (ps ) = 0 and F (qi ) = 0 for 1 ≤ i ≤ r.
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8. Dimension of Quasiprojective Algebraic Sets
Proof. By the homogeneous nullstellensatz (Theorem 3.12), there exist homogeneous forms Fi ∈ I({p1 , . . . , ps , q1 , . . . , qi−1 , qi+1 , . . . , qr }) for 1 ≤ i ≤ r such that Fi (qi ) = 0 for 1 ≤ i ≤ r. Let di be the degree of Fi for 1 ≤ i ≤ r, and set d = d1 d2 · · · dr . The homogeneous form r d d Fi i F = i=1
satisﬁes the conclusions of the lemma.
Proposition 8.11. Suppose that X is a quasiprojective variety of dimension m ≥ 1 and p ∈ X. Then there exist an aﬃne neighborhood U of p in X and f1 , . . . , fm ∈ OX (U ) such that ZU (f1 , . . . , fm ) = {p}. Proof. X is an open subset of a projective variety W ⊂ Pn . Choose a point q1 ∈ W \ {p}. By Lemma 8.10, there exists a form F1 ∈ S(Pn ) such that F1 (q1 ) = 0 and F1 (p) = 0. Let X1 = Z(F1 ) ∩ W . By Theorem 8.4, X1 = X1,1 ∪· · ·∪X1,r is a union of irreducible components each of dimension m − 1. At least one of the components necessarily contains p. If m > 1, we continue, choosing points q1,i ∈ X1,i for 1 ≤ i ≤ r, none of which are equal to p. By Lemma 8.10, there exists a form F2 ∈ S(Pn ) such that F2 (q1,i ) = 0 for 1 ≤ i ≤ r and F2 (p) = 0. By Theorem 8.4, for each i, Z(F2 ) ∩ X1,i = X2,i,1 ∪ · · · ∪ X2,i,si is a unionof irreducible components each of dimension m − 2. Thus Z(F1 , F2 ) ∩ W = X2,i,j is a union of irreducible components of dimension m−2, at least one of which contains p. Continuing by induction, we ﬁnd homogeneous forms F1 , . . . , Fm ∈ S(Pn ) such that Z(F1 , . . . , Fm )∩W is a zerodimensional algebraic set which contains p. Thus Z(F1 , . . . , Fm )∩W = {a0 , a1 , . . . , at } for some points a0 = p, a1 , . . . , at ∈ W . Now by Lemma 8.10, there exists a form G ∈ S(Pn ) such that G(ai ) = 0 for 1 ≤ i ≤ t and G(p) = 0. Let L be a linear form on Pn such that L(p) = 0. Let di be the degree of Fi and e be the degree of G. Then F1 Fm G f1 = d1 , . . . , fm = dm , g = e ∈ OPn (D(L)). L L L Let V be an aﬃne neighborhood of p in X such that V ⊂ X ∩ D(L). Then Z(f1 , . . . , fm ) ∩ V ⊂ {a0 , . . . , at }. Let U be an aﬃne neighborhood of p in (V \ Z(g)) ∩ X. Then ZU (f1 , . . . , fm ) = {p}. A set of elements {f1 , . . . , fd } in a local ring R with maximal ideal mR of dimension d such that the ideal (f1 , . . . , fd ) is mR primary is called a system of parameters in R. Theorem 8.12. Suppose that A is a Noetherian local ring. Then A has a system of parameters.
8.2. The theorem on dimension of ﬁbers
143
Proof. [161, Theorem 20, page 288] or [50, Corollary 10.7].
Proposition 8.11 and the nullstellensatz give a direct proof that a system of parameters exists in a local ring OX,p of a point p on an algebraic variety X. Theorem 8.13. Let φ : X → Y be a dominant regular map between quasiprojective varieties. Let dim X = n and dim Y = m. Then m ≤ n and: 1) Suppose that p ∈ Y . Then dim F ≥ n − m for all irreducible components F of φ−1 (p). 2) There exists a nonempty open subset U of Y such that all irreducible components of φ−1 (p) have dimension n − m for p ∈ U . Proof. From the inclusion k(Y ) → k(X) we see that m = dim Y = trdegk k(Y ) ≤ trdegk k(X) = dim X = n. We ﬁrst prove 1). The conclusion of 1) is local in Y , so we can replace Y with an aﬃne open neighborhood U of p in Y and X with φ−1 (U ). By Proposition 8.11, we may assume that there exist f1 , . . . , fm ∈ k[Y ] such that ZY (f1 , . . . , fm ) = {p}. Thus the equations φ∗ (f1 ) = · · · = φ∗ (fm ) = 0 deﬁne φ−1 (p) in X. By Corollary 8.6, all irreducible components F of φ−1 (p) satisfy dim F ≥ n − m. Now we prove 2). We may replace Y with an aﬃne open subset W, X by an aﬃne open subset V ⊂ φ−1 (W ), and φ with the restriction of φ to V . (The theorem follows for X if it holds for each member of a ﬁnite aﬃne open cover of φ−1 (W ).) Since φ is dominant, φ determines an inclusion φ∗ : k[W ] → k[V ], hence an inclusion k(W ) = k(Y ) ⊂ k(X) = k(V ). Let S = k[W ]. Consider the subring R of k(V ) generated by k(W ) and k[V ]. This is a domain which is a ﬁnitely generated k(W )algebra. Further, the quotient ring of R is k(V ). Now k(W ) is not algebraically closed, but Noether’s normalization lemma does not need this assumption. By Noether’s normalization lemma (Theorem 1.53) we have that there exist t1 , . . . , tr in R such that t1 , . . . , tr are algebraically independent over k(W ) and R is integral over the polynomial ring k(W )[t1 , . . . , tr ]. We may assume, after multiplying by an element of k[W ], that t1 , . . . , tr ∈ k[V ]. Since the quotient ﬁeld of R is k(V ), we have by (1.1) that r = trdegk(W ) k(V ) = trdegk k(V ) − trdegk k(W ) = dim X − dim Y = n − m. Now consider the subring S[t1 , . . . , tr ] of k[V ]. Here S[t1 , . . . , tr ] is a polynomial ring over S, so S[t1 , . . . , tr ] = k[W × Ar ], and we have a factorization of φ by π
ψ
V → W × Ar → W.
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8. Dimension of Quasiprojective Algebraic Sets
We have that k[V ] is a ﬁnitely generated kalgebra, so it is a ﬁnitely generated S[t1 , . . . , tr ]algebra, say generated by v1 , . . . , vl as an S[t1 , . . . , tr ]algebra. Since R is integral over k(W )[t1 , . . . , tr ], there exist polynomials Fi (x) in the indeterminate x, Fi (x) = xdi + Pi,1 (t1 , . . . , tr )xdi −1 + · · · + Pi,di (t1 , . . . , tr ) where the Pi,j are polynomials with coeﬃcients in k(W ), such that Fi (vi ) = 0 for 1 ≤ i ≤ l. Let g ∈ S = k[W ] be a common denominator of the polynomials Pi,j . Then Pi,j ∈ (Sg )[t1 , . . . , tr ] for all i, j. Thus k[V ]g is ﬁnite over (Sg )[t1 , . . . , tr ]. Let U = D(g) ⊂ W . We then have a factorization of φ restricted to φ−1 (U ) as ψ
φ−1 (U ) → U × Ar → U π
where U is aﬃne with regular functions k[U ] = k[W ]g = Sg and φ−1 (U ) is aﬃne with regular functions k[φ−1 (U )] = k[V ]g and π is a ﬁnite map. For y ∈ U , we have that ψ −1 (y) = {y} × Ar has dimension r. Suppose that A is an irreducible closed subset of φ−1 (U ) which maps into {y} × Ar . Then π(A) is closed in U × Ar and the restriction of φ from A to π(A) is ﬁnite, so that the extension k(A) of k(π(A)) is an algebraic extension. Thus dim A = dim π(A). Since π(A) is a subvariety of {y} × Ar ∼ = Ar , we have that dim A ≤ r = n − m. By part 1) of this theorem, dim A = n−m for all irreducible components A of φ−1 (y). A short proof of Theorem 8.13 can be obtained by using some theorems from commutative algebra, particularly on ﬂatness, as we now indicate. We reduce to the case where X and Y are both aﬃne. Conclusion 1) of the theorem follows from [107, (1) of Theorem 19 on page 79]. By the theorem of generic ﬂatness, [107, Theorem 52 on page 158], there is a nontrivial open subset U of Y such that φ−1 (U ) → U is ﬂat. By consideration of an open aﬃne cover of φ−1 (U ), we obtain conclusion 2) of Theorem 8.13 from [107, Theorem 4 on page 33] and [107, (2) of Theorem 19 on page 79]. Corollary 8.14. Suppose that φ : X → Y is a dominant regular map between quasiprojective varieties. Then the sets Yk = {p ∈ Y  dim φ−1 (p) ≥ k} are closed in Y .
8.2. The theorem on dimension of ﬁbers
145
Proof. Let dim X = n and dim Y = m. By Theorem 8.13, Yn−m = Y , and there exists a proper closed subset Y of Y such that Yk ⊂ Y if k > n − m. If Zi are the irreducible components of Y and φi : φ−1 (Zi ) → Zi the restrictions of φ, then dim Zi < dim Y , and the corollary follows by induction on dim Y , applied to the restriction of φi to the irreducible components of φ−1 (Zi ).
Chapter 9
Zariski’s Main Theorem
In this chapter we prove Zariski’s main theorem and give some applications. Proposition 9.1. Suppose that X and Y are quasiprojective varieties. 1) Suppose that φ : X → Y is a dominant regular map. Then Λ = φ∗ : k(Y ) → k(X) has the property that for all p ∈ X, Λ : OY,φ(p) → OX,p is a local homomorphism (Λ(OY,φ(p) ) ⊂ OX,p and Λ−1 (mp ) = mφ(p) ). Further, q = φ(p) is the unique point q in Y such that Λ : OY,q → OX,p is a local homomorphism. 2) Conversely, suppose that Λ : k(Y ) → k(X) is a kalgebra homomorphism. Further suppose that for all p ∈ X, there exists a unique point q ∈ Y such that Λ : OY,q → OX,p and Λ−1 (mp ) = mq . Then there exists a unique dominant regular map φ : X → Y such that φ∗ : k(Y ) → k(X) is the map Λ. Proof. We ﬁrst prove statement 1). The fact that for all p ∈ X, Λ induces a local homomorphism Λ : OY,φ(p) → OX,p 147
148
9. Zariski’s Main Theorem
follows from Deﬁnition 3.28 of a regular map. Suppose that q, q ∈ Y have the properties that Λ : OY,q → OX,p with Λ−1 (mp ) = mq and Λ : OY,q → OX,p with Λ−1 (mp ) = mq . We will show that q = q . We have that Y is an open subset of a projective variety Y . Thus there exists an aﬃne open subset U of Y such that q, q ∈ U . There exists a closed subset Z of U such that Y ∩ U = U \ Z. By the nullstellensatz, there exists f ∈ k[U ] such that f (Z) = 0 but f (q) = 0 and f (q ) = 0. Then V = Uf is an aﬃne open subset of Y containing q and q . From the homomorphism Λ : k[V ] → OX,p , we see that IV (q) = Λ−1 (mp ) = IV (q ). Thus q = q since an aﬃne variety is separated by Corollary 2.87. We now prove 2). Deﬁne a map Σ : X → Y by Σ(p) = q for p ∈ X if Λ−1 (mp ) = mq . We will show that Σ is a regular map. Fix p ∈ X. Let q = Σ(p). Let Uq be an aﬃne neighborhood of q in Y and Vp be an aﬃne neighborhood of p in X. Since OY,q is a localization of k[Uq ], we have that Λ induces a homomorphism Λ : k[Uq ] → OX,p . Let k[Uq ] = k[x1 , . . . , xn ]. Since OX,p is a localization of k[Vp ], there exists f ∈ k[Vp ] \ IVp (p) such that Λ(xi ) ∈ k[Vp ]f for 1 ≤ i ≤ n. Thus Λ induces a homomorphism Λ : k[Uq ] → k[Wp ] where Wp = DVp (f ) is an aﬃne neighborhood of p in X. Thus there exists a unique regular map ψp : Wp → Uq such that ψp∗ = Λ : k[Uq ] → k[Wp ] by Proposition 2.40. By Lemma 2.88, for a ∈ Wp , ψp (a) = b if and only if Λ−1 (ma ) = mb , so that ΣWp = ψp . Now by Proposition 3.39, Σ is a regular map. We will need the following local version of Zariski’s main theorem, which was ﬁrst proven by Zariski in [152, Theorem 14]. Theorem 9.2. Let K be an algebraic function ﬁeld over a ﬁeld κ. Suppose that R and S are local domains which have K as their common quotient ﬁeld such that S dominates R (R ⊂ S and mS ∩ R = mR where mR and mS are the respective maximal ideals of R and S) and R is normal. Suppose that R and S are localizations of ﬁnitely generated κalgebras, dim R = dim S, S/mS is a ﬁnite extension of R/mR , and mR S is mS primary. Then R = S. Theorem 9.2 is an immediate consequence of Proposition 21.54, which we will prove in Section 21.6. The following theorem was proven by Zariski in [152] and [156]. Theorem 9.3 (Zariski’s main theorem). Let φ : X → Y be a birational regular map of projective varieties. Suppose that Y is normal. Let U = {q ∈ Y  φ−1 (q) is a ﬁnite set}. Then U is a dense open subset of Y and φ : φ−1 (U ) → U is an isomorphism.
9. Zariski’s Main Theorem
149
Proof. The set U is a dense open subset of Y by Theorem 8.13 and Corollary 8.14. The regular map φ induces an isomorphism φ∗ : k(Y ) → k(X) which allows us to identify k(Y ) with k(X). Suppose q ∈ U . Since X is projective, q = φ(p) for some p ∈ φ−1 (U ) by Theorem 5.14. Thus OX,p dominates OY,q . We have that mY,q OX,p is mX,p primary since φ−1 (q) is a ﬁnite set. By Theorem 9.2, OY,q = OX,p . Suppose that p ∈ X is such that OX,p dominates OY,q . Then OX,p ⊂ OX,p and so OX,p = OX,p since the dimension of both rings is equal to dim X. Thus p = p since X is separated by Proposition 3.36. By Proposition 9.1, there exists a unique dominant regular map ψ : U → φ−1 (U ) such that ψ(q) = p if and only if φ(p) = q. Thus φ : φ−1 (U ) → U is an isomorphism with inverse ψ : U → φ−1 (U ). Theorem 9.4. Suppose that φ : X → Y is a dominant regular map of projective varieties and X is normal. Suppose that k(X) is a ﬁnite ﬁeld extension of k(Y ). Let λ : Y → Y be the normalization of Y in k(X) (Theorem 7.17). Then there exists a unique regular map ψ : X → Y making a commutative diagram ψ /Y @@ @@ λ @ φ @@
[email protected]
Y.
Proof. Let Γ = Γψ ⊂ X × Y be the graph of the natural rational map ψ : X Y (since k(Y ) = k(X)). Let π1 : Γ → X and π2 → Y be the projections. The map π1 is birational. We will show that we have a commutative diagram of regular maps (9.1)
Γ
~~ ~~ π2 ~ ~ ~~ ~
π1
[email protected]
Y
@@ @@ @ λ φ @@
Y.
By the construction of the normalization Y , we have a commutative diagram k(Y )
GG ∗ GG φ GG GG G ψ∗ # / k(X). k(Y )
λ∗
150
9. Zariski’s Main Theorem
There exists a nontrivial open subset U of X such that ψ : U → Y is a regular map. The regular maps λψ : U → Y and φ : U → Y are equal as rational maps by Proposition 2.108 (which is valid for arbitrary varieties) so λψ : U → Y and φ : U → Y are the same regular map by Lemma 2.104. Thus when restricting to the open subset ΓψU of Γ, we have a commutative diagram of regular maps ΓψU     } ψ π1
π2
/Y DD DD λ D φ DD!
UD D
Y.
Thus φπ1 = λπ2 as rational maps from Γ to Y , so that they are the same regular maps from Γ to Y by Proposition 2.108 and Lemma 2.104, establishing the commutativity of the diagram (9.1). Suppose that p ∈ X. Then π1−1 (p) ⊂ {p} × λ−1 (φ(p)), which is a ﬁnite set for all p ∈ X since λ is a ﬁnite map. Thus π1 is a birational regular map of projective varieties such that all ﬁbers are ﬁnite. Since X is normal, we have that π1 is an isomorphism by Theorem 9.3, and thus the rational map ψ : X Y is a regular map. Proposition 9.5. Suppose that φ : X → Y is a dominant regular map of projective varieties such that φ−1 (q) is a ﬁnite set for all q ∈ Y . Then φ is an aﬃne map. Proof. We have a closed embedding X ⊂ Pm , giving closed embeddings X∼ = Γφ ⊂ X × Y ⊂ Pm × Y. Let π : Pm × Y → Y be the projection. Suppose that q ∈ Y . Then φ−1 (q) ∼ = X ∩ (Pm × {q}) is a ﬁnite set of points. Let H be a hyperplane section of Y such that q ∈ H and let L be a hyperplane section of Pm such that (L × {q}) ∩ X ∩ (Pm × {q}) = ∅. Then U = X ∩ (Pm L × YH ) is a closed subset of an aﬃne space so it is aﬃne. Let Z = π(X ∩ (L × YH )). We have that Z is a closed subset of YH , by Corollary 5.13, which does not contain q. We have that φ−1 (YH \Z) ∼ = X ∩(Pm ×(YH \Z)) = X ∩(Pm ×(YH \Z)) = U ∩(π −1 (YH \Z)). L
Let f ∈ k[YH ] be such that q ∈ DYH (f ) ⊂ YH \ Z. Let V = DYH (f ). Then V is an aﬃne neighborhood of q and φ−1 (V ) ∼ = U ∩ π −1 (V ) = DU (φ∗ (f )) is −1 aﬃne with k[φ (V )] = k[U ]φ∗ (f ) . Applying this construction to all q ∈ Y , we see that φ is an aﬃne map.
9. Zariski’s Main Theorem
151
Theorem 9.6. Suppose that φ : X → Y is a dominant regular map of projective varieties such that φ−1 (q) is a ﬁnite set for all q ∈ Y . Then φ is a ﬁnite map. Proof. We have that dim X = dim Y by Theorem 8.13. Now Theorem 8.2 implies k(X) is a ﬁnite extension of k(Y ). Let α : X → X be the normalization of X in k(X) and β : Y → Y be the normalization of Y in k(X) (Theorem 7.17). Then we have a commutative diagram of regular maps of projective varieties X α
X
ψ
φ
/Y
β
/Y
by Theorem 9.4. Now α and β are ﬁnite maps so they have ﬁnite ﬁbers by Theorem 2.56. Thus ψ has ﬁnite ﬁbers and is birational so ψ is an isomorphism by Theorem 9.3. Let V ⊂ Y be an aﬃne open subset. Then (φα)−1 (V ) ∼ = β −1 (V ) is aﬃne and k[(φα)−1 (V )] is a ﬁnitely generated k[V ]algebra since β is ﬁnite. Now φ−1 (V ) is aﬃne by Proposition 9.5, and k[φ−1 (V )] is a ﬁnitely generated k[V ]module by Lemma 1.55, since it is a submodule of k[(φα)−1 (V )]. Thus φ is ﬁnite. We also have the related theorem, which was ﬁrst proven by Zariski in [156]. Theorem 9.7 (Zariski’s connectedness theorem). Let X and Y be projective varieties and φ : X → Y be a birational regular map. Suppose that Y is normal at a point q ∈ Y . Then φ−1 (q) is connected. We mention a nice generalization of Zariski’s main theorem, which is proven in [67] or [124, Chapter IV] and [109, Theorem 1.8]. Theorem 9.8 (Grothendieck). Let φ : X → X be a regular map of varieties with ﬁnite ﬁbers. Then φ is a composition of an open embedding of X into a variety Y and a ﬁnite map from Y to X. Exercise 9.9. Suppose that X and Y are projective varieties such that X is normal and φ : X Y is a rational map. Let Γφ be the graph of φ, with projections π1 : Γφ → X and π2 : Γφ → Y . Suppose that p ∈ X is such that φ is not regular at p. Show that there exists a curve C ⊂ π1−1 (p) such that π2 (C) is a curve of Y .
Chapter 10
Nonsingularity
In this chapter, we explore the concept of nonsingularity of a variety. The tangent space Tp (X) to a point p on a variety X is ﬁrst deﬁned in Deﬁnition 10.7 extrinsically for an aﬃne variety embedded in An and then given an equivalent intrinsic deﬁnition, Tp (X) = Homk (mp /m2p , k), in Deﬁnition 10.8. We show that dim X ≤ dim Tp (X) in Theorem 10.11, and in Deﬁnition 10.12 we deﬁne X to be nonsingular at p if dim X = dimk Tp (X). In Proposition 10.13, we show that X is nonsingular at p if and only if OX,p is a regular local ring. Our deﬁnition of nonsingularity extends to varieties over arbitrary ﬁelds and general schemes. We prove the Jacobian criterion of nonsingularity, Proposition 10.14, in our situation of varieties over an algebraically closed ﬁeld. The Jacobian criterion of nonsingularity is valid for varieties over a perfect ﬁeld but can fail for varieties over nonperfect ﬁelds (Exercise 10.21). In Theorem 10.16, we show that the set of nonsingular points of X is a dense open subset of X. In the remainder of the chapter, we give applications of the above concepts, including the factorization of birational regular maps of nonsingular projective surfaces, Theorem 10.32, and the fact (Theorem 10.38) that a nonsingular projective variety of dimension n can be embedded in P2n+1 . We end the chapter with a proof (Theorem 10.45) that a nonsingular complex variety is a complex analytic manifold.
10.1. Regular parameters Suppose that R is a regular local ring, with maximal ideal m and residue ﬁeld κ = R/m. Let d = dim R, so that dimκ m/m2 = d. Elements u1 , . . . , ud ∈ m 153
154
10. Nonsingularity
such that the classes of u1 , . . . , ud are a κbasis of m/m2 are called regular parameters (or a regular system of parameters) in R. Lemma 10.1. Suppose that R is a regular local ring of dimension d with maximal ideal m and that p ⊂ R is a prime ideal such that dim R/p = a. Then dimR/m (p + m2 /m2 ) ≤ d − a, with equality if and only if R/p is a regular local ring. Proof. Let R = R/p, m = mR , and κ = R/m ∼ = R /m . There is a short exact sequence of κvector spaces (10.1) 0 → p/p ∩ m2 ∼ = p + m2 /m2 → m/m2 → m/(p + m2 ) = m /(m )2 → 0. The lemma now follows since dimκ m /(m )2 ≥ a by Theorem 1.81 and R is regular if and only if dimκ m /(m )2 = a. Lemma 10.2. Suppose that R is a regular local ring, with maximal ideal m and residue ﬁeld κ = R/m. Let d = dim R, and suppose that u1 , . . . , ud are regular parameters in R. Then Ii = (u1 , . . . , ui ) is a prime ideal in R of height i for all i, R/Ii is a regular local ring of dimension d − i, and the residues of ui+1 , . . . , ud in R/Ii are regular parameters in R/Ii . Proof. Let mi = m(R/Ii ) be the maximal ideal of R/Ii . Then d − i = dimκ mi /m2i ≥ dim R/Ii by Theorem 1.81. Since none of the ui are units in R, we have by Krull’s principal ideal theorem, Theorem 1.65, that dim R/Ii = ht mi ≥ d − i, so that R/Ii is a regular local ring of dimension d − i.
Lemma 10.3. Suppose that R is a regular local ring of dimension n with maximal ideal m, p ⊂ R is a prime ideal such that R/p is regular, and dim R/p = n − r. Then there exist regular parameters u1 , . . . , un in R such that p = (u1 , . . . , ur ) and ur+1 , . . . , un map to regular parameters in R/p. Proof. Consider the exact sequence (10.1). Since R and R are regular, there exist regular parameters u1 , . . . , un in R such that ur+1 , . . . , un map to regular parameters in R and u1 , . . . , ur are in p. By Lemma 10.2, (u1 , . . . , ur ) is a prime ideal of height r contained in p. Thus p = (u1 , . . . , ur ).
10.2. Local equations
155
Proposition 10.4. Suppose that x1 , . . . , xd is a regular system of parameters in a regular local ring R. For i ≤ d, let Pi = (x1 , . . . , xi ). Then Pi is a prime ideal of height i and we have equality of ordinary and symbolic powers (n)
Pi
= Pin
for all n > 0. Proof. The ideals Pi are prime ideals of height i by Lemma 10.2. The regular system of parameters x1 , . . . , xd is an Rregular sequence by Lemma 10.2, and hence R is a CohenMacaulay local ring. Thus we have equality of ordinary and symbolic powers by Proposition 1.70 and Theorem 1.71.
10.2. Local equations Suppose that X is a quasiprojective variety and Y is a closed subvariety. Deﬁne IY by IY (U ) = IU (Y ∩ U ) ⊂ k[U ] if U is an aﬃne open subset of X, and for p ∈ X, deﬁne IY,p = IU (Y ∩ U )k[U ]I(p) if U is an aﬃne open subset of X containing p. We have that IY,p is independent of the choice of aﬃne open subset U containing p. Letting X be the closure of X in a projective space Pn and Y be the closure of Y in Pn , we can construct IY as in Section 6.2, by taking IY to be the restriction of I X (Y ) to X, where IX (Y ) is the homogeneous ideal of Y in the coordinate ring S(X). This construction will be examined in more generality in Chapter 11. Deﬁnition 10.5. Suppose that X is a quasiprojective variety, Y is a closed subvariety of X, and p ∈ X. Functions f1 , . . . , fn ∈ OX,p are called local equations of Y in X at p (we also say f1 = · · · = fn = 0 are local equations of Y in X at p) if there exists an aﬃne neighborhood U of p in X such that f1 , . . . , fn generate the ideal IY (U ) of Y ∩ U in U . Lemma 10.6. Functions f1 , . . . , fn ∈ OX,p are local equations of Y in some aﬃne neighborhood of p ∈ X if and only if IY,p = (f1 , . . . , fn ), the ideal generated by f1 , . . . , fn in OX,p . Observe that IY,p = OX,p if and only if p ∈ Y , and this holds if and only if fi (p) = 0 for some i.
156
10. Nonsingularity
Proof. Suppose that f1 , . . . , fn are local equations of Y at p. Then there exists an aﬃne neighborhood U of p such that IY (U ) = (f1 , . . . , fn ), so IY,p = (f1 , . . . , fn )k[U ]I(p) = (f1 , . . . , fn )OX,p , where I(p) is the ideal of p in U . Now suppose that IY,p = (f1 , . . . , fn ). Let U be an aﬃne neighborhood of p, and suppose that IY (U ) = (g1 , . . . , gm ). Then we have that (g1 , . . . , gm )OX,p = (f1 , . . . , fn ), so there exist expressions gi =
(10.2)
n
ai,j fj
j=1
for 1 ≤ i ≤ m, where ai,j ∈ OX,p for all i, j, and we have expressions fk =
(10.3)
m
bk,l gl
l=1
for 1 ≤ k ≤ n, where bk,l ∈ OX,p for all k, l. After possibly replacing U with a smaller aﬃne neighborhood of p, we may assume that all fi , ai,j , bk,l are in k[U ]. Then (f1 , . . . , fn )k[U ] = (g1 , . . . , gm )k[U ] = IY (U ).
10.3. The tangent space Suppose that p = (b1 , . . . , bn ) ∈ An . Let xi = xi − bi for 1 ≤ i ≤ n. Translation by p is an isomorphism of An , and we have that k[An ] = k[x1 , . . . , xn ] = k[x1 , . . . , xn ] is a polynomial ring. Suppose that f ∈ k[An ]. Then f has a unique expansion f= ai1 ,...,in (x1 − b1 )i1 · · · (xn − bn )in with ai1 ,...,in ∈ k. If f (p) = 0, we have that a0,...,0 = 0, and f ≡ Lp (f ) mod I(p)2 , where (10.4)
Lp (f ) = a1,0,...,0 (x1 − b1 ) + · · · + a0,...,0,1 (xn − bn ) ∂f ∂f (p)(x1 − b1 ) + · · · + ∂x (p)(xn − bn ). = ∂x n 1
Deﬁnition 10.7 (Extrinsic deﬁnition of the tangent space). Suppose that X is an aﬃne variety, which is a closed subvariety of An , and that p ∈ X. The tangent space to X at p is the linear subvariety Tp (X) of An , deﬁned by Tp (X) = Z(Lp (f )  f ∈ I(X)).
10.3. The tangent space
157
We have that if I = (f1 , . . . , fr ), then I(Tp (X)) = (Lp (f1 ), . . . , Lp (fr )). The ideal I(p) of the point p is I(p) = (x1 , . . . , xn ) ⊂ k[An ]. Let V be the ndimensional kvector space which is spanned by x1 , . . . , xn in k[An ]. For f ∈ I(X), we have that Lp (f ) ∈ V . Let W be the subspace of V deﬁned by W = {Lp (f )  f ∈ I(X)}. Let mp be the maximal ideal of the local ring OX,p . By Lemma 1.28, mp /m2p ∼ = I(p)/(I(p)2 + I(Tp (X))) ∼ = V /W. = I(p)/(I(p)2 + I(X)) ∼ We can naturally identify the set of points of An with the dual vector space Homk (V, k) by associating to q ∈ An the linear map L → L(q) for L ∈ V . Now Homk (V /W, k) = {φ ∈ Homk (V, k)  φ(W ) = 0} = {q ∈ An  Lp (f )(q) = 0 for all f ∈ I(X)} = Tp (X). This gives us the following alternate deﬁnition of the tangent space. Deﬁnition 10.8 (Intrinsic deﬁnition of the tangent space). Suppose that X is a quasiprojective variety and p ∈ X. The tangent space to X at p is the kvector space Tp (X) deﬁned by Tp (X) = Homk (mp /m2p , k), where mp is the maximal ideal of OX,p . The vector space Tp (X) is the tangent space T (OX,p ) of the local ring OX,p , as deﬁned in (1.7). Suppose that φ : X → Y is a regular map of quasiprojective varieties and p ∈ X. Let q = φ(p). Then φ∗ : OY,q → OX,p induces a kvector space homomorphism mq /m2q → mp /m2p , where mq and mp are the maximal ideals of OY,q and OX,p respectively, and thus we have a kvector space homomorphism dφp : Tp (X) → Tq (Y ). Suppose that φ : X → Y is a regular map of aﬃne varieties, Z ⊂ X is a subvariety, W ⊂ Y is a subvariety of Y such that φ(Z) ⊂ W , and p ∈ Z. Let q = φ(p). Let φ : Z → W be the restricted map. We have prime ideals I(W ) ⊂ I(q) ⊂ k[Y ] and I(Z) ⊂ I(p) ⊂ k[X]. Here φ∗ : k[Y ] → k[X] ∗ induces φ : k[W ] = k[Y ]/I(W ) → k[X]/I(Z) = k[Z]. We have a commutative diagram of kvector spaces → I(p)/I(p)2 I(q)/I(q)2 ↓ ↓ 2 I(q)/I(q) + I(W ) → I(p)/I(p)2 + I(Z),
158
10. Nonsingularity
where the vertical arrows are surjections. Taking the associated diagram of dual kvector spaces (applying Homk (∗, k)), we get a commutative diagram dφp
←
Tq (Y ) ↑
(10.5)
Tq (W )
Tp (X) ↑
dφp
←
Tp (Z)
where the vertical arrows are the natural inclusions. Example 10.9. Suppose that φ : Am → An is a regular map, deﬁned by φ = (f1 , . . . , fn ) for some fi ∈ k[Am ] = k[x1 , . . . , xm ]. Suppose that α ∈ Am and β = φ(α). Then dφα : Tα (Am ) → Tβ (An ) is the linear map k m → k n given by multiplication by the n × m matrix ∂fi (α)). ( ∂x j Proof. φ∗ : k[y1 , . . . , yn ] = k[An ] → k[x1 , . . . , xm ] = k[Am ] is deﬁned by φ∗ (yi ) = fi for 1 ≤ i ≤ n. Let α = (α1 , . . . , αm ) and β = (β1 , . . . , βn ). We have expressions for 1 ≤ i ≤ n, (10.6)
fi = fi (α) +
m ∂fi (α)(xj − αj ) + hi ∂xj j=1
where hi ∈ I(α)2 . We have that fi (α) = βi . Here {y1 − β1 , . . . , yn − βn } is a kbasis of I(β)/I(β)2 , and {x1 −α1 , . . . , xm −αm } is a kbasis of I(α)/I(α)2 . By (10.6) and since φ∗ (yi − βi ) = fi − fi (α), we have that the induced map φ∗ : I(β)/I(β)2 → I(α)/I(α)2 is given by (10.7)
m ∂fi (α)(xj − αj ), φ (yi − βi ) = ∂xj ∗
j=1
for 1 ≤ i ≤ n. Let {δ1 , . . . , δm } be the dual basis to {x1 − α1 , . . . , xm − αm }, and let {1 , . . . , n } be the dual basis to {y1 − β1 , . . . , yn − βn }. That is,
1 if s = t, δs (xt − αt ) = 0 if s = t and
s (yt − βt ) =
1 0
if s = t, if s =
t.
Now we compute the dual map φ∗ : Homk (I(α)/I(α)2 , k) → Homk (I(β)/I(β)2 , k).
10.4. Nonsingularity and the singular locus
159
For 1 ≤ s ≤ m, we have commutative diagrams I(β)/I(β)2
II ∗ IIφ (δs ) II II II $ / k. I(α)/I(α)2 φ∗
δs
For 1 ≤ t ≤ n and 1 ≤ s ≤ m, we have ⎞ ⎛ m ∂ft ∂f t (α)(xj − αj )⎠ = (α). δs (φ∗ (yt − βt )) = δs ⎝ ∂xj ∂xs j=1
Thus φ∗ (δs ) =
n ∂ft (α)t ∂xs t=1
for 1 ≤ s ≤ m.
Exercise 10.10. Suppose that k is an algebraically closed ﬁeld of characteristic p > 0. Show that the regular map φ : A1 → A1 deﬁned by φ(α) = αp is a homeomorphism. Show that dφq : Tq (A1 ) → Tφ(q) (A1 ) is the zero map for all q ∈ A1 .
10.4. Nonsingularity and the singular locus Theorem 10.11. Suppose that X is a quasiprojective variety and p ∈ X. Then dimk Tp (X) ≥ dim X. Proof. Let d = dim X and n = dimk Tp (X) = dimk mp /m2p , where mp is the maximal ideal of OX,p . The ideal mp is generated by n elements as an OX,p module by Nakayama’s lemma, Lemma 1.18. Let mp = (f1 , . . . , fn ). There exists an aﬃne neighborhood U of p in X such that IU (p) = (f1 , . . . , fn ) by Lemma 10.6. Without loss of generality, we may assume that X is aﬃne with this property. Let π : B(p) → X be the blowup of p. Since B(p) is the graph of the rational map (f1 : . . . : fn ) : X Pn−1 , B(p) is a closed subvariety of X × Pn−1 , and so π −1 (p) ⊂ {p} × Pn−1 has dimension ≤ n − 1. But dim B(p) = d and so dim π −1 (p) = d − 1, by Krull’s principal ideal theorem (Theorem 1.65) since I(p)OB(p),q is a principal nonzero ideal in the domain OB(p),q for all q ∈ π −1 (p) (by (6.2)). Thus d ≤ n. Suppose that X is aﬃne and let S = S(B(p)) be the coordinate ring of the blowup of p in the above proof. We will show that dim S/I(p)S = dim π −1 (p) + 1 = d.
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10. Nonsingularity
Let V1 , . . . , Vn be the irreducible components of π −1 (p). Then I(p)S = I(Vi ) i
where I(Vi ) is the homogeneous ideal of Vi in S. We have that dim S/I(p)S = maxi {dim(S/I(Vi ))} since I(Vi ) are the minimal primes of I(p)S = maxi {dim(Vi ) + 1} by Theorem 8.3 = maxi {dim(Vi )} + 1 = dim π −1 (p) + 1 by Proposition 8.1 = d by the proof of Theorem 10.11. By Theorem 6.4, S ∼ = i≥0 I(p)i , and thus I(p)i /I(p)i+1 ∼ mip /mi+1 S/I(p)S ∼ = = p i≥0
i≥0
by Lemma 1.28, where mp is the maximal ideal of OX,p . This last ring is the associated graded ring of mp , mip /mi+1 grmp (OX,p ) = p . i≥0
We obtain that dim grmp (OX,p ) = dim OX,p . From Theorem 10.11, we have that dimOX,p /mp mp /m2p ≥ dim OX,p , so we recover the statements of Theorem 1.81 in our geometric setting. Deﬁnition 10.12. A point p of a quasiprojective variety X is a nonsingular point of X if dimk Tp (X) = dim X. A quasiprojective variety X is said to be nonsingular if all points of X are nonsingular points of X. We have the following proposition, which is immediate from the deﬁnition of a regular local ring, giving us an alternate algebraic deﬁnition of a nonsingular point. Proposition 10.13. A point p of a quasiprojective variety X is a nonsingular point of X if and only if OX,p is a regular local ring. Proposition 10.14 (Jacobian criterion). Suppose that X is an aﬃne variety of dimension r, which is a closed subvariety of An , and f1 , . . . , ft ∈ k[An ] = k[x1 , . . . , xn ] are a set of generators of I(X). Suppose that p ∈ X. Then dimk Tp (X) = n − s
10.4. Nonsingularity and the singular locus
161
where s is the rank of the t × n matrix ∂fi A= (p) . ∂xj In particular, s ≤ n − r, and p is a nonsingular point of X if and only if s = n − r. Proof. Going back to our analysis of Tp (X), we have that x1 , . . . , xn is a kbasis of V and W is the subspace of V spanned by {Lp (f1 ), . . . , Lp (ft )}. This subspace has dimension equal to the rank of A by (10.4). Since Tp (X) and V /W are kvector spaces of the same dimension, we have that dimk Tp (X) = n − s. Exercise 10.21 shows that the Jacobian criterion for nonsingularity of Proposition 10.14 does not always hold for a nonsingular point on a variety over a nonalgebraically closed ﬁeld in positive characteristic. An extensive study of nonsingularity of varieties over arbitrary ﬁelds is made by Zariski in his paper [154]. Corollary 10.15. Suppose that X is a quasiprojective variety. Then the set of nonsingular points of X is an open subset of X. Proof. Since X has an open cover by aﬃne open subsets, we may assume that X is aﬃne, so that X is a subvariety of An for some n. Let I(X) = ∂fi ). Let r = dim X and (f1 , . . . , ft ) and let B be the t × n matrix B = ( ∂x j let In−r (B) be the ideal generated by the determinants of (n − r) × (n − r) submatrices of B in k[An ]. By Proposition 10.14, q ∈ X is a nonsingular point if and only if q ∈ Z(In−r (B)). Thus the set of nonsingular points of X is open in X. Theorem 10.16. Suppose that X is a quasiprojective variety. Then the set of nonsingular points of X is a dense open subset of X. Proof. By Proposition 2.112, there is a birational map from X to a hypersurface Z(f ) ⊂ An , where f is irreducible in k[An ] = k[x1 , . . . , xn ]. We will show that the nonsingular locus of Z(f ) is nontrivial. Then the conclusions of the theorem follow since the nonsingular locus is open, any nontrivial open subset of a variety is dense, and birational varieties have isomorphic open subsets (by Theorem 2.111). Thus we may assume that X = Z(f ). ∂f ∂f , . . . , ∂x ) = Z(f ), Suppose that every point of X is singular. Then Z(f, ∂x n 1 ∂f ∂f so ∂xi ∈ I(X) = (f ) for all i. Since deg( ∂xi ) < deg(f ) for all i (here the degree of a polynomial is the largest total degree of a monomial appearing ∂f = 0 for all i. If in the polynomial), the only way this is possible is if ∂x i the characteristic of k is zero, this implies that f ∈ k, which is impossible.
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10. Nonsingularity
If k has positive characteristic, then f must be a polynomial in xp1 , . . . , xpn with coeﬃcients in k. Since the pth roots of these coeﬃcients are in k (as k is algebraically closed), we have that f = g p for some g ∈ k[x1 , . . . , xn ], contradicting the fact that f is irreducible. Suppose X is a quasiprojective variety. The closed algebraic set of singular points of X is called the singular locus of X. Theorem 10.17. Suppose that X is a normal quasiprojective variety. Let Z be the singular locus of X. Then codimX Z ≥ 2. Proof. We may assume that X is aﬃne. Let Y be a codimension 1 subvariety of X with ideal I(Y ) ⊂ k[X]. The ideal I(Y ) is a height 1 prime ideal and the property of being normal localizes by Exercise 1.58, so the local ring R = k[X]I(Y ) is normal of dimension 1. Thus R is a regular local ring by Theorem 1.87. Let f be a generator of the maximal ideal of R. After possibly multiplying f by an element of k[X] which is in k[X] \ I(Y ), we may assume that f ∈ k[X]. The ideal (f ) in k[X] thus has a minimal primary decomposition (f ) = I(Y ) ∩ Q1 ∩ · · · ∩ Qr where Q1 , . . . , Qr are primary ideals in k[X] which are Ai primary for respective prime ideals A1 , . . . , Ar which are not contained in I(Y ). Thus there exists q ∈ Y such that q ∈ Z(Ai ) for 1 ≤ i ≤ r and so I(Y )mq = f OY,q . We then have that there exists an aﬃne open subset U of X such that U ∩ Y = ∅ and IU (Y ) = (f ) by Lemma 10.6. The variety Y ∩ U has a nonsingular point p ∈ Y ∩ U , so that OY,p ∼ = OX,p /(f ) is a regular local ring. Let m be the maximal ideal of OX,p and let n be the maximal ideal of OY,p . There exist v 1 , . . . , v d−1 ∈ OY,p such that n = (v 1 , . . . , v d−1 ) where d = dim X. Let v1 , . . . , vd−1 be lifts of v 1 , . . . , v d−1 to OX,p . The ideal m = (v1 , . . . , vd−1 , f ) so that dimk m/m2 ≤ d. Thus OX,p is a regular local ring by Theorem 10.11 or Theorem 1.81 and so p is a nonsingular point of X. Thus Y is not contained in the singular locus Z of X, and so codimX Z ≥ 2. Lemma 10.18. Suppose that Y is an irreducible codimension 1 subvariety of a quasiprojective variety X. Suppose that p ∈ Y is a nonsingular point of X. Then there exists an irreducible element f ∈ OX,p which is a local equation of Y at p. Proof. This follows from Lemma 10.6, Theorem 1.89, and Proposition 1.66. Theorem 10.19. Suppose that X is an ndimensional nonsingular quasiprojective variety and Y is a nonsingular subvariety of X. Let π : B → X be the blowup of Y . Then B is nonsingular, and E = π −1 (Y ) is an irreducible, nonsingular codimension 1 subvariety of B. If p ∈ Y , then π −1 (p) ∼ = Pr−1 , where r = codimX Y . Suppose that x1 , . . . , xn are regular parameters in
10.4. Nonsingularity and the singular locus
163
OX,p , such that x1 = · · · = xr = 0 are local equations of Y at p and q ∈ π −1 (p). Then there exists a j with 1 ≤ j ≤ r and there exist α1 , . . . , αr ∈ k such that OB,q has regular parameters y1 , . . . , yn such that ⎧ ⎨ yj (yi + αi ) for 1 ≤ i ≤ r and i = j, for i = j, xi = yj ⎩ yi for i > r. A local equation of E in B at q is yj = 0. With the hypotheses of Theorem 10.19, we have by Proposition 6.5 that S(B) ∼ = k[X][y1 , . . . , yr ]/J where J = (yi xj − xi yj  1 ≤ i < j ≤ r). Proof. Suppose that p ∈ Y . By Lemmas 10.3 and 10.6, there exist regular parameters x1 , . . . , xn in OX,p and an aﬃne neighborhood U of p in X such that x1 = · · · = xr = 0 are local equations of Y in U , x1 = · · · = xn = 0 are local equations of p in U , and IY (U ) = (x1 , . . . , xr ). We have by Theorem 6.4 that π −1 (U ) has the aﬃne cover {V1 , . . . , Vr } where x1 xr ,..., k[Vj ] = k[U ] xj xj for 1 ≤ j ≤ r. Suppose that q ∈ π −1 (p). Then q ∈ Vj for some j. Let n = IVj (q) and m = IU (p). Without loss of generality, we may assume that j = r. We have n ∩ k[U ] = m since π(q) = p. Now k∼ = (k[U ]/m)[t1 , . . . , tr−1 ] ∼ = k[t1 , . . . , tr−1 ] = k[Vr ]/n ∼ where t1 , . . . , tr−1 are the residues of xx1r , . . . , xxr−1 in k[Vr ]/n. Since this kr algebra is isomorphic to k, we must have that ti = αi for some αi ∈ k, so that x1 xr−1 − α1 , . . . , − αr−1 . n = mk[Vr ] + xr xr Since mk[Vr ] = (xr , xr+1 , . . . , xn ), we have that the n functions x1 xr−1 − α1 , . . . , − αr−1 (10.8) xr , xr+1 , . . . , xn , xr xr generate the maximal ideal of OB,q , so that letting a = nOB,q be the maximal ideal of OB,q , we have that dimk a/a2 ≤ n. Since π : B → X is a birational map, and by Theorem 10.11, we have that n = dim OB,q ≤ dimk a/a2 ≤ n. Thus OB,q is a regular local ring, with regular parameters (10.8).
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We have that π −1 (U ) is a closed subvariety of U × Pr−1 , and xr = 0 is a local equation in Vr of π −1 (Y ∩ U ) ⊂ (Y ∩ U ) × Pr−1 . By Krull’s principal ideal theorem, Theorem 1.65, all irreducible components of π −1 (Y ∩ U ) have codimension 1 in π −1 (U ). Thus π −1 (Y ∩ U ) = (Y ∩ U ) × Pr−1 . Since B \ E → X \ Y is an isomorphism and X is nonsingular, we have that B is nonsingular. Exercise 10.20. Consider the curve X = Z(y 2 − x3 ) ⊂ A2 , where k is an algebraically closed ﬁeld of characteristic 0. a) Show that at the point p1 = (1, 1), the tangent space Tp1 (X) is the line Z(−3(x − 1) + 2(y − 1)). b) Show that at the point p2 = (0, 0), the tangent space Tp2 (X) is the entire plane A2k . c) Show that the curve is singular only at the origin p2 . d) Show that the blowup of p2 in C is nonsingular everywhere. (This blowup was constructed in Exercise 6.12). Exercise 10.21. Let p be a prime number, and let K = Fp (t) where Fp is the ﬁnite ﬁeld with p elements and t is transcendental over Fp . Show that R = K[x, y]/(xp + y p − t) is a regular ring (all localizations at prime ideals are regular). Hint: You can use the fact that a polynomial ring over a ﬁeld is a regular ring. Let f = xp + y p − t. Show that the matrix
∂f ∂f , ∂x ∂y
= (0, 0).
Conclude that the Jacobian criterion for nonsingularity of Proposition 10.14 is not valid over ﬁelds of characteristic p > 0 which are not algebraically closed (not perfect). Exercise 10.22. Let X ⊂ Pn be a projective variety of dimension r. Let I(X) = (F1 , . . . , Ft ) ⊂ S(Pn ) = k[x0 , . . . , xn ] where F1 , . . . , Ft are homogeneous forms. Show that the singular locus of X is the algebraic set Z(I(X) + In−r (M )) ⊂ Pn where M is the t × (n + 1) matrix M=
∂Fi ∂xj
and In−r (M ) is the ideal generated by the n − r minors of M . Hint: Use Euler’s formula, Exercise 1.35.
10.5. Applications to rational maps
165
10.5. Applications to rational maps Theorem 10.23. Suppose that φ : X Y is a rational map of projective varieties and X is normal. Let Z be the closed subset of X consisting of the points where φ is not a regular map. Then Z has codimension ≥ 2 in X. Proof. The variety Y is a closed subset of a projective space Pn , so after composing φ with a closed embedding, we may suppose that Y = Pn . Let W be the set of singular points of X. The closed set W has codim ≥ 2 in X by Theorem 10.17 since X is normal. Suppose that p ∈ Z ∩ (X \ W ). We will ﬁnd an aﬃne open neighborhood Up of p in X and a representative (f0 : . . . : fn ) of φ such that f0 , . . . , fn ∈ k[Up ] and ZUp (f0 , . . . , fn ) has codimension ≥ 2 in Up . Since Z ∩ Up ⊂ ZUp (f0 , . . . , fn ) and there is an open cover of X \ W by sets of this form, we will have that Z has codimension ≥ 2 in X. We will now prove the assertion. Suppose that p ∈ X \ W . Let f0 , . . . , fn ∈ k(X) be such that (f0 : . . . : fn ) is a representative of the rational map φ. After multiplying all of the fi by a suitable element of OX,p , we may assume that f0 , . . . , fn ∈ OX,p . Since OX,p is a UFD (Theorem 1.89), the greatest common divisor of a set of elements in OX,p is deﬁned. Let g be the greatest common divisor of the elements f0 , . . . , fn in OX,p . Let f i = fgi ∈ OX,p . We will show that the ideal (f 0 , . . . , f n ) in OX,p has height ≥ 2. Lemma 10.18 shows that for every codimension 1 subvariety S of X which contains p, there exists an irreducible h ∈ OX,p which is a local equation of S at p. Let U be an aﬃne neighborhood of p in X such that f 0 , . . . , f n ∈ k[U ] and h is a local equation of S in U . If S ∩ U ⊂ ZU (f 0 , . . . , f n ), then by the nullstellensatz, some power of f j is in IU (S ∩ U ) = hk[U ] for all j, so h divides f j in OX,p since h is irreducible in OX,p , giving a contradiction. Since (f 0 : . . . : f n ) is a representative of φ, we have the desired conclusion. Corollary 10.24. Every rational map φ : X Y of projective varieties such that X is a nonsingular projective curve is regular. Corollary 10.25. Every birational map of nonsingular projective curves is an isomorphism. Corollary 10.26. Every dominant rational map φ : X Y of projective curves such that X is nonsingular is regular and ﬁnite. Proof. The map φ : X → Y is regular by Corollary 10.24. Consider the injective kalgebra homomorphism φ∗ : k(Y ) → k(X). Let Z be the normalization of Y in k(X) constructed in Theorem 7.17, with ﬁnite regular
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map π : Z → Y . The variety Z is a normal projective curve by Theorem 7.17 and by 1) of Theorem 8.2. Since Z has dimension 1 and is normal, Z is nonsingular by Theorem 1.87. By Proposition 2.108 (as commented in Section 4.4, this result is valid for rational maps of projective varieties), there is a rational map ψ : Z X such that ψ ∗ : k(Z) → k(X) is the identity map. By Proposition 2.110 (as commented in Section 4.4, this result is valid for rational maps of projective varieties), ψ is birational. Since ψ is a birational map of nonsingular projective curves, ψ is an isomorphism by Corollary 10.25. Proposition 10.27. Suppose that φ : X → Y is a birational regular map of quasiprojective varieties which is not an isomorphism. Suppose that p ∈ X with q = φ(p), ψ = φ−1 is not regular at q and OY,q is a UFD (for instance if q is a nonsingular point of Y ). Then there exists a subvariety Z of X with p ∈ Z such that codimX Z = 1 and codimY φ(Z) ≥ 2. Proof. The homomorphism φ∗ : k(Y ) → k(X) is an isomorphism, with inverse ψ ∗ . Replacing Y with an aﬃne open neighborhood U of q in Y and X with an aﬃne open neighborhood of p in the preimage of U , we may assume that X and Y are aﬃne. We have that X is a closed subset of an aﬃne space An . Let t1 , . . . , tn be the coordinate functions on X (so that k[X] = k[t1 , . . . , tn ]). We may represent the rational map ψ : Y X by ψ = (g1 , . . . , gn ) where the gi ∈ k(Y ) are rational functions on Y , so that φ∗ (gi ) = ti for 1 ≤ i ≤ n (since φ∗ : k(Y ) → k(X) is the inverse of ψ ∗ : k(X) → k(Y )). Since ψ is not regular at q, at least one of the gi is not regular at q, say g1 ∈ OY,q . We have that OY,q is a UFD by assumption. Thus we have an expression g1 = uv where u, v ∈ OY,q and u, v are relatively prime. We necessarily have that v(q) = 0. We may if necessary replace Y and X with smaller aﬃne neighborhoods of q and p to obtain that u, v ∈ k[Y ]. Now ZY (u, v) cannot contain an irreducible component D which has codimension 1 in Y and contains q, since if it did, a local equation f = 0 of D at q would satisfy f  u and f  v in OY,q , which is impossible since u and v are relatively prime. Thus replacing Y and X with possibly smaller aﬃne neighborhoods of q and p, we may assume that ZY (u, v) has codimension ≥ 2 in Y . We have that t1 = φ∗ (g1 ) = φ∗ (u)/φ∗ (v), so that (10.9)
φ∗ (v)t1 = φ∗ (u)
in k(X). We have that t1 , φ∗ (u), φ∗ (v) ∈ k[X] and φ∗ (v)(p) = 0, so that p ∈ ZX (φ∗ (v)). Set Z = ZX (φ∗ (v)). Then codimX Z = 1 since p ∈ Z. By (10.9) it follows that φ∗ (u) ∈ IX (Z). Thus u, v ∈ IY (φ(Z)), so that φ(Z) ⊂ ZY (u, v), which we have shown has codimension ≥ 2 in Y .
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167
The conclusions of Proposition 10.27 may be false if OY,q is normal but is not a UFD, as is shown by the example computed in Exercise 6.16 c). In this example we construct the blowup π1 : B(I1 ) → X where X = Z(xy − zw) ⊂ A4 and I1 = (x, z)k[X] and show that B(I1 ) \ F1 ∼ = X \ {p} −1 1 ∼ where p is the origin and F1 = π1 (p) = P . Theorem 10.28. Suppose that φ : X → Y is a birational regular map of projective varieties and Y is nonsingular. Let C = {q ∈ Y  dim φ−1 (q) > 0}. The set C is a closed subset of Y by Corollary 8.14. Let G = φ−1 (C), a closed subset of X. Then: 1. φ : X \ G → Y \ C is an isomorphism. 2. codimY C ≥ 2. 3. The closed set G is a union of codimension 1 subvarieties of X. If E is one of these components, then codimY φ(E) ≥ 2. The set G is called the exceptional locus of φ. Proof. We have that φ(X \ G) = Y \ C by Theorem 5.14 since X and Y are projective. Proposition 10.27 now implies φ : X \ G → Y \ C is an isomorphism. The second statement of the theorem follows from Theorem 10.23 applied to the rational map φ−1 . Now we prove the third statement. Suppose that E is an irreducible component of G and p ∈ E is a point which is not contained in any other irreducible component of G. Let q = φ(p). Let B be an aﬃne neighborhood of q in Y , and let A be an aﬃne neighborhood of p in X such that A does not contain points of any irreducible component of G except for E and φ(A) ⊂ B. Proposition 10.27 applied to φ : A → B tells us that there exists a codimension 1 subvariety F of A such that codimY φ(F ) ≥ 2. We must have that F ⊂ G ∩ A = E ∩ A. Since E is irreducible, we have that F = E ∩ A. Exercise 10.29. Let U = A2 with regular functions k[U ] = k[x, y]. Let π : X → U be the blowup of the origin p. Let E = π −1 (p) ∼ = P1 , and let S be the set of lines through the origin in U . In this problem, the analysis of π : X → U given at the end of Section 6.1 and a careful reading of the proof of Lemma 10.30 will be helpful. a) Show that the map (of sets) Λ : P1 → S deﬁned by Λ(α : β) = L(α:β) = Z(αx + βy) for (α : β) ∈ P1 is injective and onto. b) Suppose that L is a line through the origin in U . Let L be the strict transform of L in X. Show that the map P1 → E deﬁned by (α : β) → (L(α:β) ) ∩ E is an injective and onto map of sets.
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c) Show that the map deﬁned in part b) is an isomorphism of projective varieties. d) Using the facts that X is covered by two aﬃne open sets isomorphic to A2 and that the strict transform L of a line L through p is a line in one of these charts, show that if p is the point E ∩ L , then the map dπp : Tp (L ) → Tp (U ) is injective with image equal to Tp (L). e) Show that for p = L ∩ E as in part d), the image dπp (Tp (X)) = Tp (L).
10.6. Factorization of birational regular maps of nonsingular surfaces The proof in this section is based on the proof by Shafarevich in [136]. The ﬁrst proof of Theorem 10.32 was by Zariski [151]. Lemma 10.30. Suppose that X is a nonsingular projective surface and π : Y → X is the blowup of a point p ∈ X. Let E = π −1 (p) ∼ = P1 . Then there is a 11 correspondence between points q of E and onedimensional subspaces L of Tp (X), given by q → dπq (Tq (Y )). Proof. Let u, v be regular parameters in OX,p . Let mp be the maximal ideal of OX,p . Then the kvector space mp /m2p ∼ = ku ⊕ kv. By Theorem 10.19, the distinct points q of E have regular parameters u1 , v1 in OY,q which have the forms (10.10)
u = u1 ,
v = u1 (v1 + α)
for α ∈ k,
or (10.11)
u = u1 v1 ,
v = v1 .
Let mq be the maximal ideal of OY,q . We have a kvector space isomorphism mq /m2q ∼ = ku1 ⊕ kv1 . The klinear map π ∗ : mp /m2p → mq /m2q is deﬁned by π ∗ (u) = u1 and π ∗ (v) = αu1 if (10.10) holds and
π ∗ (u) = 0 and π ∗ (v) = v1 if (10.11) holds.
Let δu1 , δv1 be the dual basis of Tq (Y ) to u1 , v1 , and let δu , δv be the dual basis of Tp (X) to u, v. Suppose that (10.10) holds. Then dπq (δu1 ) = δu1 π ∗ = δu + αδv Thus dπq (Tq (Y )) = (δu + αδv )k.
and
dπq (δv1 ) = δv1 π ∗ = 0.
10.6. Factorization of birational regular maps of nonsingular surfaces
169
Suppose that (10.11) holds. Then dπq (δu1 ) = δu1 π ∗ = 0
and
dπq (δv1 ) = δv1 π ∗ = δv .
Thus dπq (Tq (Y )) = δv k.
Lemma 10.31. Suppose that φ : X Y is a birational map of nonsingular projective surfaces. Let Γ ⊂ X × Y be the graph of φ, with projections π1 : Γ → X and π2 : Γ → Y . Suppose that φ−1 is not regular at a point q ∈ Y . Then there exists a curve D ⊂ Γ such that π1 (D) = C is a curve of X and π2 (D) = q. Proof. By Theorem 5.14, we have that π2 (Γ) = Y . We have that π2−1 is not regular at q since φ−1 is not regular at q. By Proposition 10.27, there exists a curve D ⊂ Γ such that π2 (D) = q. If π1 (D) is not a curve, we must have that π1 (D) is a point p. But then D ⊂ π1−1 (p) ∩ π2−1 (q) = {(p, q)}, which is a point, giving a contradiction. Theorem 10.32. Suppose that φ : X → Y is a birational regular map of nonsingular projective surfaces. Then φ has a factorization X = Yn → Yn−1 → · · · → Y1 → Y0 = Y, where each Yi+1 → Yi is the blowup of a point. Proof. Suppose that φ is not an isomorphism, so that φ−1 is not regular at some point q ∈ Y . Let σ : Y → Y be the blowup of q, and let E = σ −1 (q) ∼ = P1 be the exceptional locus of σ. Let φ : X Y be the rational map φ = σ −1 φ. We will show that φ is a regular map. Suppose that φ is not a regular map. Let Γ ⊂ X × Y be the graph of φ , and suppose that p ∈ X is a point where φ is not regular. Then by Lemma 10.31, there exists a curve D ⊂ π1−1 (p) such that π2 (D) is a curve C in Y . Since σ  (Y \ E) is an isomorphism onto Y \ {q}, we have that C ⊂ E, so C = E since both C and E are irreducible curves. Let ψ : Y X be the rational map ψ = (φ )−1 . By Theorem 10.23, there exists a ﬁnite set T ⊂ Y such that ψ(Y \ T ) is a regular map. We have that ψ(E \ T ) = π1 (D \ π2−1 (T )) = p and φ(p) = q. We will show that (10.12)
dφp : Tp (X) → Tq (Y )
is an isomorphism. Suppose not. Then there exists a onedimensional subspace L ⊂ Tq (Y ) such that dφp (Tp (X)) ⊂ L. Since ψ(E \ T ) = p, we have that dσq (Tq (Y )) ⊂ L for q ∈ E \ T . But this is a contradiction to the conclusions of Lemma 10.30. Thus (10.12) is an isomorphism. Now φ−1 is not regular at q, so by Proposition 10.27, there exists a curve F ⊂ X such that p ∈ F and φ(F ) = q. But then Tp (F ) ⊂ Tp (X) has dimension ≥ 1, and
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dφp (Tp (F )) = 0, a contradiction to the fact that (10.12) is an isomorphism. This contradiction shows that φ is a regular map. The exceptional locus of φ is a union of irreducible curves by Theorem 10.28. Let r be the number of irreducible components of the exceptional locus of φ. The regular map φ maps φ−1 (q) onto σ −1 (q) = E. Thus there exists a curve G ⊂ φ−1 (q) which maps onto E, so the number of irreducible components of the exceptional locus of φ is ≤ r − 1. By induction, after enough blowups of points, we obtain the desired factorization of φ. There is a very general local form of this theorem by Abhyankar [3], from which we can also deduce Theorem 10.32. Theorem 10.33 (Abhyankar). Suppose that R and S are twodimensional regular local rings with a common quotient ﬁeld K such that S dominates R (R ⊂ S and mS ∩ R = mR , where mR and mS are the respective maximal ideals of R and S). Then R → S factors by a ﬁnite sequence of quadratic transforms R = R0 → R1 → · · · → Rn = S. A quadratic transform of a regular local ring R is a local ring of the blowup of the maximal ideal; so if x, y are regular parameters in a twodimensional regular local ring R, then a quadratic transform of R → R1 is a local ring (a localization at a maximal ideal) of R[ xy ] or R[ xy ].
10.7. Projective embedding of nonsingular varieties Lemma 10.34. Suppose that R and S are local Noetherian rings with respective maximal ideals mR and mS . Let f : R → S be a local homomorphism of local Noetherian rings such that 1) R/mR → S/mS is an isomorphism, 2) mR → mS /m2S is surjective, and 3) S is a ﬁnitely generated Rmodule. Then f is surjective. Proof. By 2), we have that mS = mR S + m2S , so by Nakayama’s lemma (Lemma 1.18), we have that mR S = mS . By 1), S = f (R) + mS = f (R) + mR S, and by 3), S is a ﬁnitely generated Rmodule, so again by Nakayama’s lemma, we have that S = f (R).
10.7. Projective embedding of nonsingular varieties
171
Lemma 10.35. Suppose that φ : X → Y is a regular map of quasiprojective varieties such that 1) φ is bijective and 2) φ∗ : OY,φ(p) → OX,p is an isomorphism for all p ∈ X. Then φ is an isomorphism. Proof. We will ﬁnd an aﬃne cover {Vi } of Y such that for all i, Ui = φ−1 (Vi ) is an aﬃne open subset of X and φ∗ : k[Vi ] → k[Ui ] is an isomorphism. This is enough to conclude that φ is an isomorphism, since each map φ : Ui → Vi is then an isomorphism, with regular inverse ψi : Vi → Ui by Proposition 2.40 and Proposition 2.42. Now the ψi patch to give a continuous map ψ : Y → X, which is an inverse to φ by Proposition 3.39. Suppose that p ∈ X. Let q = φ(p) ∈ Y and let V be an aﬃne neighborhood of q in Y . Let U ⊂ φ−1 (V ) be an aﬃne neighborhood of p in X. We then have (since φ is dominant) that φ∗ : k[V ] → k[U ] is 11. Let I(q) be the ideal of q in V , and let I(p) be the ideal of p in U . By assumption, φ∗ induces an isomorphism k[V ]I(q) → K[U ]I(p) . Suppose that t1 , . . . , tr generate k[U ] as a kalgebra. Since ti ∈ k[U ]I(p) = k[V ]I(q) , there exists h ∈ k[V ]\I(q) and f1 , . . . , fr ∈ k[U ] such that ti = fhi . Thus k[V ]h → k[U ]h is an isomorphism. Now Vh is an aﬃne neighborhood of q in Y , Uh is an aﬃne neighborhood of p in X, and φ∗ : k[Vh ] → k[Uh ] is an isomorphism. Theorem 10.36. Suppose that X and Y are projective varieties and φ : X → Y is a regular map which is injective and such that dφp : Tp (X) → Tφ(p) (Y ) is injective for all p ∈ X. Then φ is a closed embedding. Proof. We have that φ(X) is a closed subvariety of Y by Theorem 5.14 since X is projective. Further, the map dφp factors through the inclusion Tq (φ(X)) ⊂ Tq (Y ), so without loss of generality, we may assume that Y = φ(X). Here X is a closed subvariety of Pn for some n. We have a sequence of maps λ
X → Γφ ⊂ X × Y ⊂ Pn × Y and a commutative diagram / Pn × Y XH HH HH HH π2 φ HHH # Y
where λ : X → Γφ , deﬁned by λ(p) = (p, φ(p)) for p ∈ X, is an isomorphism, π2 is the projection on the second factor, and the inclusions in the top row are all closed embeddings. Suppose that p ∈ X, with q = φ(p) ∈ Y , and U is an aﬃne open subset of Y which contains q. Let Z = Γφφ−1 (U ) = Γφ ∩ π2−1 (U )
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which is a closed subset of π2−1 (U ) ∼ = Pn × U . Let x0 , . . . , xn be homogeneous coordinates on Pn . Then S = k[U ][x0 , . . . , xn ] is the homogeneous coordinate ring of Pn × U , which is a graded ring, where elements of k[U ] have degree 0 and the xi have degree 1 for 0 ≤ i ≤ n. Let I(Z) be the homogeneous ideal of Z in S. Then −1
OX (φ
n
(U )) ∼ = OZ (Z) =
(S/I(Z))(xi ) ,
i=0
where (S/I(Z))(xi ) denotes the elements of degree 0 in the localization of S/I(Z) with respect to xi . Now the proof of Theorem 3.35 (with k replaced with the ring k[U ]) or Theorem 11.47 shows that OX (φ−1 (U )) is ﬁnite over k[U ]. Further, for q ∈ Y , we can compute q∈V
OX (φ−1 (V )) =
n
T(xi ) ,
i=0
where T = OY,q [x0 , . . . , xn ]/I(Z)OY,q [x0 , . . . , xn ] and the intersection is over all aﬃne open subsets V ⊂ U which contain q (since q∈V OY (V ) = OY,q ). Again, the proof of Theorem 3.35, with k replaced with the ring OY,q , shows that q∈V OX (φ−1 (V )) is ﬁnite over OY,q , where the intersection is over the aﬃne open subsets V of U containing q. Since φ is bijective and q∈V V = {q}, so that q∈V φ−1 (V ) = {p}, we have that q∈V OX (φ−1 (V )) = OX,p . Thus OX,p is ﬁnite over OY,φ(p) for p ∈ X. Since Tp (X) → Tq (Y ) is an injective homomorphism of ﬁnitedimensional kvector spaces, we have that the natural homomorphism mq /m2q → mp /m2p is a surjection, where mp is the maximal ideal of OX,p and mq is the maximal ideal of OY,q . Thus by Lemma 10.34, we have that OX,p = OY,q for all p ∈ X. Finally, we have by Lemma 10.35 that φ is an isomorphism. Corollary 10.37. Suppose that X ⊂ Pn is a projective variety and p ∈ Pn \ X. Suppose that every line through p intersects X in at most one point and p is not contained in the (Zariski closure in Pn of the) tangent space to X at any point. Then the projection π from p is a closed embedding of X into Pn−1 . Proof. Suppose that a ∈ X. Let b = π(a). Since a = p, we can make a linear change of coordinates in Pn so that p = (0 : . . . : 0 : 1) and a = (1 : 0 : . . . : 0). Now the projection π from p is deﬁned by the rational map (x0 : . . . : xn−1 ). Thus the restriction of π to D(x0 ) is just the map π : An → An−1 which is the projection onto the ﬁrst n − 1 factors. Since a is the origin in An , dπa : Ta (Pn ) → Tb (Pn−1 ), which we can identify with π : An → An−1 , is just the projection onto the ﬁrst n − 1 factors. The kernel
10.7. Projective embedding of nonsingular varieties
173
of dπa is the line ZD(x0 ) ( xx10 , . . . , xn−1 x0 ), which is the intersection of the line containing p and a with D(x0 ). From the commutative diagram (10.5) and the assumptions of the theorem, we have that dπa : Ta (X) → Tb (Pn−1 ) is injective. Now we apply Theorem 10.36 to obtain the conclusions of the corollary. Theorem 10.38. Suppose that X is a nonsingular projective variety of dimension n. Then X is isomorphic to a subvariety of P2n+1 . Proof. It suﬃces to prove that if X ⊂ PN with N > 2n+1, then there exists p ∈ PN \ X satisfying the hypotheses of Corollary 10.37. Let U1 , U2 ⊂ PN be the respective sets of points not satisfying the respective assumptions of Corollary 10.37. Let a = (a0 : . . . : aN ), b = (b0 : . . . : bN ), c = (c0 : . . . : cN ) ∈ PN . Let ⎛ ⎞ a0 · · · aN A = ⎝ b0 · · · bN ⎠ . c0 · · · cN The coeﬃcients of the linear forms which vanish on the three points a, b, c are the elements of the kernel of the linear map A : k N +1 → k 3 . The condition that a, b, c be collinear is that there are at least N − 1 independent forms in the kernel; that is, A has rank ≤ 2. Let x0 , . . . , xN be our homogeneous coordinates on PN and let y0 , . . . , yN be the induced homogeneous coordinates on X. Let z0 , . . . , zN be the corresponding homogeneous coordinates on a copy of X. Then x0 , . . . , xN , y0 , . . . , yN , z0 , . . . , zN are trihomogeneous coordinates on Pn × X × X with trigraded coordinate ring S(PN × X × X) = k[x0 , . . . , xN , y0 , . . . , yN , z0 , . . . , zN ]. Let W = {(a, b, c)  a ∈ PN , b, c ∈ X, and a, b, c are collinear in PN }. Then W is the closed set W = Z(I) of Pn × X × X where ⎛ ⎞ x0 · · · xN I = I3 ⎝ y0 · · · yN ⎠ , z0 · · · zN the ideal generated by the determinants of 3 × 3 submatrices. PN
In PN × X × X, consider the set Γ, which is the Zariski closure in × X × X of triples (a, b, c) with a ∈ PN , b, c ∈ X such that b = c
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10. Nonsingularity
and a, b, c are collinear. Let Γ be an irreducible component of Γ. Then Γ necessarily contains a point (a, b, c) such that a, b, c are collinear and b = c. Let y = (b, c) for these values of b and c. The projections PN × X × X to PN and X × X deﬁne regular maps φ : Γ → PN and ψ : Γ → X × X. Now ψ −1 (y) is a subset of (PN × {y}) ∩ W which is the set of points (a, b, c) where a is any point of the line through b and c. Hence dim ψ −1 (y) ≤ 1, and it follows from Theorem 8.13 that dim Γ ≤ 2n + 1. In particular, dim Γ ≤ 2n + 1. By the deﬁnitions of U1 and Γ, U1 ⊂ φ(Γ), and thus dim U1 ≤ dim Γ ≤ 2n + 1. Let x0 , . . . , xN be homogeneous coordinates on PN and suppose that p = (b0 : b1 : . . . : bN ) ∈ X is such that bi = 0. Suppose that F ∈ S(PN ) = k[x0 , . . . , xN ] is a homogeneous form of degree d. We will show that the Zariski closure of the tangent space to Z(F ) ∩ D(xi ) in D(xi ) ∼ = AN at p is N the projective hyperplane in P with equation N ∂F (b)xj = 0. ∂xj j=0
Let u0 , . . . , ui−1 , ui+1 , . . . , uN be coordinates on D(xi ), deﬁned by uj = if j = i and f = F (u0 , . . . , ui−1 , 1, ui+1 , . . . , uN ). For j = i, we have
xj xi
∂F ∂f = (u0 , . . . , ui−1 , 1, ui , . . . , uN ) ∂uj ∂xj so ∂f ∂F (b0 , . . . , bN ) = bd−1 i ∂xj ∂uj
bN b0 ,..., bi bi
.
By Euler’s formula, Exercise 1.35, xi
∂F ∂F = dF − xj . ∂xi ∂xj j=i
Thus, N
∂F j=0 ∂xj (b0 , . . . , bN )xj
= = =
b ∂F , bN )(xj − bji xi ) j=i ∂xj (b0 , . . .
∂f bN b0 xi uj , . . . , bd−1 j=i ∂uj i b bi i
∂f bN b0 xi bd−1 uj j=i ∂uj i bi , . . . , bi
− −
bj i bi x
bj , bi
proving the assertion. Now let Γ1 be the subset of PN × X consisting of points (a, b) such that a is in the Zariski closure of Tb (X). Let a = (a0 : . . . : aN ) ∈ PN and b = (b0 : . . . : bN ) ∈ X. Let I(X) = (F1 , . . . , Ft ) where F1 , . . . , Ft are homogeneous. Then the condition
10.8. Complex manifolds
175
that a is in the Zariski closure of Tb (X) in PN is that N ∂Fj i=0
Thus Γ1 = Z
∂xi
(b)ai = 0
N ∂Fj (y0 , . . . , yN ) i=0
∂xi
for 1 ≤ j ≤ t.
(y0 , . . . , yN )xi  1 ≤ j ≤ t
⊂ Pn × X
is a Zariski closed subset. Let Γ1 be an irreducible component of Γ1 . We necessarily have (a, b) ∈ for some b ∈ X. We have projections ψ : Γ1 → X and φ : Γ1 → PN . For our b ∈ X, we have dim ψ −1 (b) ≤ n since X is nonsingular, and hence dim Γ1 ≤ 2n, and since U2 = φ(Γ1 ), we have dim U2 ≤ 2n.
Γ1
We have shown that dim U1 ≤ 2n + 1 and dim U2 ≤ 2n. Thus if N > 2n + 1 we have that U1 ∪ U2 = PN .
10.8. Complex manifolds We have deﬁned Ank to be k n (as a set) with the Zariski topology. If we take k = C, the complex numbers, we have a ﬁner topology, the Euclidean topology on Cn . We also have the theory of analytic functions on (Euclidean) open subsets of Cn . If U ⊂ Cn is a Zariski open subset and f is a regular function (in the Zariski topology) on U , then f is also an analytic function on U . Deﬁnition 10.39. A complex manifold of dimension n is a Hausdorﬀ topological space M such that M has a covering {Ui } by open subsets with homeomorphisms φi : Ui → Vi between Ui and open subsets Vi of Cn such that φj ◦ φ−1 i : φi (Ui ∩ Uj ) → φj (Ui ∩ Uj ) are analytic (and hence bianalytic). Complex projective space PnC has a covering by Zariski open subsets Ui = An , where An is just Cn with the Zariski topology. The Euclidean topology on Cn thus gives us the Euclidean topology on each Ui . These topologies agree on Ui ∩ Uj for i = j. This deﬁnes the Euclidean topology on PnC . We have that the Euclidean topology is ﬁner than the Zariski topology (a Zariski open subset is open in the Euclidean topology). Suppose that X is a quasiprojective algebraic set which is contained in Pn . Then the Euclidean topology on X is the subspace topology of the Euclidean topology of PnC . In particular, any quasiprojective complex variety X (the base ﬁeld k is C) has the Euclidean topology, as we can transcribe the Euclidean topology to X by an embedding φ : X → Pn by prescribing that φ−1 (W ) is open in the Euclidean topology on X whenever W is a Euclidean open subset of PnC .
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10. Nonsingularity
Suppose that X is a quasiprojective variety. As commented above, we can consider X to be an open subset of a closed subvariety of a projective space Pn and give X the induced Euclidean topology. Now Pn has an open covering by open subsets Ui ∼ = An as above. Suppose that V is a Zariski open subset of X. Then V has an aﬃne covering by Zariski open subsets Wi,j = (Ui )gj ∩ X where X is the Zariski closure of X in Pn and gj ∈ C[Ui ]. Suppose that f ∈ OX (V ) is a regular function. Then each restriction of f to each Wij is a continuous map from Wij to C in the Euclidean topology, so f : V → C is continuous in the Euclidean topology. The following lemma is [120, Exercise 13, page 100]. Lemma 10.40. Suppose that X is a topological space. Then X is Hausdorﬀ if and only if the diagonal Δ = {(p, p)  p ∈ X} is closed in X × X (where X × X has the product topology). Theorem 10.41. Suppose that X is a quasiprojective variety. Then X is Hausdorﬀ in the Euclidean topology. Proof. The diagonal Δ of X ×X is closed in the Zariski topology by Proposition 5.8. The product topology of the Euclidean topologies on X is the Euclidean topology on X × X, which is ﬁner than the Zariski topology on X × X. Thus Δ is closed in X × X in the product topology of the Euclidean topology on X, so that X is Hausdorﬀ in the Euclidean topology by Lemma 10.40. We have inclusions of rings C[x1 , . . . , xn ] ⊂ C[x1 , . . . , xn ](x1 ,...,xn ) ⊂ C{x1 , . . . , xn } ⊂ C[[x1 , . . . , xn ]] where C[x1 , . . . , xn ] is the polynomial ring in n variables, C[[x1 , . . . , xn ]] is the ring of formal power series, and C{x1 , . . . , xn } is the ring of formal power series which have a positive radius of convergence (the germs of analytic functions at the origin in Cn ). Theorem 10.42 (Analytic implicit function theorem). Suppose that f ∈ C{x1 , . . . , xn } is such that f = ni=1 ai xi + (higherorder terms) with ai ∈ C and a1 = 0. Then there exist g ∈ C{x2 , . . . , xn } and a unit series u ∈ C{x1 , . . . , xn } such that f = u(x1 − g(x2 , . . . , xn )). This is a special case of the Weierstrass preparation theorem ([102, Section C.2.4], [62, page 8], or [161, pages 142–145]).
10.8. Complex manifolds
177
Corollary 10.43. Suppose that f ∈ C{x1 , . . . , xn } is such that f=
n
ai xi + (higherorder terms)
i=1
with ai ∈ C and a1 = 0. Then C{x2 , . . . , xn } → C{x1 , . . . , xn }/(f ) is an isomorphism. Proof. With the notation of Theorem 10.42, we have that the ideal (f ) = (x1 − g(x2 , . . . , xn )), so we can eliminate x1 . Corollary 10.44. Suppose that f1 , . . . , fr ∈ C{x1 , . . . , xn } are such that fi =
r
aij xj + (higherorder terms)
j=1
with Det(aij ) = 0. Then the map C{xr+1 , . . . , xn } → C{x1 , . . . , xn }/(f1 , . . . , fr ) is an isomorphism. Proof. Let B be the inverse of the matrix (aij ). Deﬁne f1 , . . . , fr ∈ C{x1 , . . . , xn } by ⎛ ⎞ ⎛ ⎞ f1 f1 ⎜ f2 ⎟ ⎜ f ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎜ .. ⎟ = B ⎜ .. ⎟ . ⎝ . ⎠ ⎝ . ⎠ fr fr Then we have an equality of ideals (f1 , . . . , fr ) = (f1 , . . . , fr ), so we may replace the fi with the fi and assume that (aij ) is the identity matrix. By Theorem 10.42, f1 = u1 (x1 − g1 (x2 , x3 , . . . , xn )) where u1 ∈ C{x1 , . . . , xn } is a unit and by considering the linear term in f1 , we see that g1 ∈ (x2 , . . . , xn )2 C{x2 , . . . , xn }. Now the ideal (f1 , . . . , fr ) = (x1 − g1 , f2 , . . . , fr ), so C{x1 , . . . , xn }/(f1 , . . . , fr ) ∼ = C{x2 , . . . , xn }/(f2 (g1 , x2 , . . . , xn ), . . . , fr (g1 , x2 , . . . , xn )) and for 2 ≤ i ≤ n, fi (g1 , x2 , . . . , xn ) = xi + (higherorder terms in x2 , . . . , xn ). Thus we have the assumptions of the corollary with a reduction of r to r − 1 and n to n − 1 and further have that the new matrix A giving the linear part of the expression of f2 , . . . , fr in terms of the variables x2 , . . . , xn is the identity matrix. By induction on r, repeating the above argument, we obtain the conclusions of the corollary.
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10. Nonsingularity
Theorem 10.45. Suppose that X is a nonsingular quasiprojective complex variety of dimension r. Then X is a complex manifold of dimension r in the Euclidean topology. Proof. X is Hausdorﬀ in the Euclidean topology by Theorem 10.41. Suppose that p ∈ X. Then there exists an aﬃne neighborhood W of p such that C[W ] = C[x1 , . . . , xn ]/p for some prime ideal p in C[x1 , . . . , xn ]. We have a closed embedding λ = (x1 , . . . , xn ) : W → Cn which is a homeomorphism in the Euclidean topology onto its image since the restrictions of the xi to W are regular functions on W and we have seen that they are thus continuous in the Euclidean topology. We can translate p so that λ(p) is the origin in Cn , and thus I(λ(p)) = (x1 , . . . , xn ). Let p = (f1 , . . . , fm ). Since X is nonsingular at p, the matrix ∂fi (λ(p)) A= ∂xj has rank n − r. After possibly permuting the variables x1 , . . . , xn and the functions f1 , . . . , fm , we may assume that if B is the (n − r) × (n − r) submatrix of A consisting of the ﬁrst n − r rows and columns of A, then Det(B) = 0, so that the classes of f1 , . . . , fn−r are linearly independent over C in I(p)/I(p)2 . Let R = C[x1 , . . . , xn ](x1 ,...,xn ) . We can extend f1 , . . . , fn−r to regular parameters f1 , . . . , fn−r , g1 , . . . , gr in the regular local ring R. By Lemma 10.2, the ideal I = (f1 , . . . , fn−r )R is a prime ideal in R of height n − r. But I ⊂ pR and I and pR are prime ideals of the same height so pC[x1 , . . . , xn ](x1 ,...,xn ) = (f1 , . . . , fn−r )C[x1 , . . . , xn ](x1 ,...,xn ) , and thus pC{x1 , . . . , xn } = (f1 , . . . , fn−r )C{x1 , . . . , xn }. Let Lr+i
n ∂fi = (p)xj for 1 ≤ i ≤ n − r ∂xj j=1
and let L1 , . . . , Lr be linear forms in x1 , . . . , xn such that {L1 , . . . , Ln } is a basis of Cx1 + · · · + Cxn . We have that C[x1 , . . . , xn ] = C[L1 , . . . , Ln ], so we may replace the xi with Li for 1 ≤ i ≤ n. By Corollary 10.44, we have that the map C{x1 , . . . , xr } → C{x1 , . . . , xn }/pC{x1 , . . . , xn } is an isomorphism, so there exist functions hr+1 , . . . , hn ∈ C{x1 , . . . , xr } such that xi − hi ∈ pC{x1 , . . . , xn } for r + 1 ≤ i ≤ n. All of the hi converge within a polydisc V = {(a1 , . . . , ar ) ∈ Cr  ai  < for all i}
10.8. Complex manifolds
179
for some small . Let π : Cn → Cr be the projection onto the ﬁrst r factors (which is analytic), and let U = π −1 (V ) ∩ λ(W ). Then π : U → V has the analytic inverse (a1 , . . . , ar ) → (a1 , . . . , ar , hr+1 (a1 , . . . , ar ), . . . , hn (a1 , . . . , ar )), for (a1 , . . . , ar ) ∈ V . Let U = λ−1 (U ) and φ = πλ : U → V . The map φ is a homeomorphism in the Euclidean topology since it is a composition of homeomorphisms in the Euclidean topology. Repeating this for every point p ∈ X, we obtain an open covering of X by open sets {Ui } (in the Euclidean topology) with homeomorphisms φi : Ui → Vi , with Vi an open subset (in the Euclidean topology) of Cr as above. We will show that they satisfy the condition of the deﬁnition of a complex manifold. Now we introduce some notation on our construction of φi : Ui → Vi . There exists an aﬃne open subset Wi of X such that Ui is an open subset of Wi (in the Euclidean topology). We have a representation C[Wi ] = C[xi,1 , . . . , xi,ni ]/pi = C[xi,1 , . . . , xi,ni ] such that φi (p) = (xi,1 (p), . . . , xi,r (p)) for p ∈ Ui and there exist analytic functions hij on Vi for r+1 ≤ j ≤ ni such that xi,j (p) = hij (xi,1 (p), . . . , xi,r (p)). Given i = j, there exist fk ∈ C[Wi ] such that Wi ∩ Wj = k (Wi )fk . The restriction map C[Wj ] → C[(Wi )fk ] = C[Wi ]fk takes xj,l to
gl (xi,1 , . . . , xi,ni ) , fk (xi,1 , . . . , xi,ni )tl
for 1 ≤ l ≤ nj , where tl ∈ N and gl and fk are polynomials in xi,1 , . . . , xi,ni . We will show that φj φ−1 i : φi (Ui ∩ Uj ) → φj (Ui ∩ Uj ) is an analytic map, showing that X is a complex manifold. To show this, it suﬃces to show that φj φ−1 i : φi ((Wi )fk ∩ Ui ∩ Uj ) → φj ((Wi )fk ∩ Ui ∩ Uj ) is an analytic map for all k. Suppose that (a1 , . . . , ar ) ∈ φi ((Wi )fk ∩Ui ∩Uj ). Let p = φ−1 i (a1 , . . . , ar ). Then xi,1 (p) = a1 , . . . , xi,r (p) = ar and xi,r+1 (p) = hir+1 (a1 , . . . , ar ), . . . , xi,ni (p) = hini (a1 , . . . , ar ).
180
10. Nonsingularity
Thus φj φ−1 i (a1 , . . . , ar ) = φj (p) = (xj,1 (p), . . . , xj,r (p)) gr (xi,1 (p), . . . , xi,ni (p)) g1 (xi,1 (p), . . . , xi,ni (p)) , . . . , = fk (xi,1 (p), . . . , xi,ni (p))t1 fk (xi,1 (p), . . . , xi,ni (p))tr = (σ1 , . . . , σr ) where σb =
gb (a1 , . . . , ar , hir+1 (a1 , . . . , ar ), . . . , hini (a1 , . . . , ar )) fk (a1 , . . . , ar , hir+1 (a1 , . . . , ar ), . . . , hini (a1 , . . . , ar ))t1
for 1 ≤ b ≤ r, showing that the homeomorphism φj φ−1 is analytic on (Wi )k ∩ Ui ∩ Uj . Exercise 10.46. We know that if X is a variety, then the diagonal ΔX = {(p, p)  p ∈ X} is closed in X × X in the Zariski topology (proof of Proposition 5.8) and that X is not Hausdorﬀ in the Zariski topology if X has positive dimension (proved for A1 in Example 2.3). Why does this not contradict Lemma 10.40?
Chapter 11
Sheaves
In this chapter we introduce the formalism of sheaves on a topological space. The most important concepts from this section are the invertible sheaves and coherent sheaves. Further discussion of sheaves can be found in Godement [59] and Hartshorne [73].
11.1. Limits In this section we deﬁne direct and inverse limits of systems of algebraic structures. A directed set I is a set with a partial order ≤ such that for any i, j ∈ I, there exists k ∈ I such that i ≤ k and j ≤ k. A directed system of Abelian groups is a set of Abelian groups {Ai }, indexed by a directed set I, such that if i ≤ j, then there is a homomorphism φij : Ai → Aj which satisﬁes φii = idAi and φik = φjk φij if i ≤ j ≤ k. Proposition 11.1. Suppose that {Ai }i∈I is a directed system of Abelian groups. Then there exists a group lim→ Ai with homomorphisms φi : Ai → lim→ Ai such that φi = φj φij for all i ≤ j ∈ I, which satisﬁes the following universal property: suppose that B is an Abelian group with homomorphisms τi : Ai → B such that τi = τj φij for i ≤ j ∈ I. Then there exists a unique homomorphism τ : lim→ Ai → B such that τ φi = τi for i ∈ I. The group lim→ Ai with homomorphisms φi : Ai → lim→ Ai is uniquely determined up to isomorphism. It is called the direct limit of {Ai }. The direct limit can be constructed as follows: let M = i∈I Ai , and let N be the subgroup generated by elements of the form a − φij (a) such 181
182
11. Sheaves
that a ∈ Ai and i ≤ j. Let φi : Ai → M/N be the natural map. Then M/N is the direct limit of the {Ai }. We give an alternate construction of the direct limit. Let {Ai , φij } be a directed system of Abelian groups. Let B be the disjoint union of pairs (Ai , ai ) such that ai ∈ Ai . Deﬁne a relation ∼ on B by (Ai , ai ) ∼ (Aj , aj ) if there is a k ≥ i, j with φik (ai ) = φjk (aj ). The relation ∼ is an equivalence relation on B. Let C = B/ ∼ be the set of equivalence classes. Let [Ai , ai ] denote the equivalence class of (Ai , ai ). The set C is a group under the following operation: [Ai , ai ] + [Aj , aj ] = [Ak , φik (ai ) + φjk (aj )] where k is any index with k ≥ i, j. Let φi : Ai → C be the map φi (a) = [Ai , a] for a ∈ Ai . The map φi is a group homomorphism, which satisﬁes φi = φj φij if i ≤ j. Lemma 11.2. The group C with the homomorphisms φi constructed above is the direct limit lim→ Ai . Proof. We will show that C with the homomorphisms φi constructed above satisﬁes the universal property of Proposition 11.1. Suppose that G is an Abelian group with homomorphisms τi : Ai → G such that τi = τj φij for i ≤ j ∈ I. Deﬁne τ : C → G by the rule τ ([Ai , ai ]) = τi (ai ). This is welldeﬁned since if [Ai , ai ] = [Aj , aj ], then there is a k with i, j ≤ k and φik (ai ) = φjk (aj ). Thus τi (ai ) = τk (φik (ai )) = τk (φjk (aj )) = τj (aj ). The map τ is a group homomorphism satisfying τi = τ φi . If τ : C → G is a group homomorphism satisfying τi = τ φi for each i, then τ ([Ai , ai ]) = τ (φi (ai )) = τi (ai ) = τ ([Ai , ai ]). Thus τ = τ . Thus C satisﬁes the universal property of the direct limit, so C with the maps φi is the direct limit of the {Ai }. Proposition 11.3. Let lim→ Ai be the direct limit of a directed system of Abelian groups {Ai , φij }. Then: 1) Suppose that x ∈ lim→ Ai . Then x = φi (a) for some i and a ∈ Ai . 2) Suppose that a ∈ Ai satisﬁes φi (a) = 0. Then there is a j ≥ i such that φij (a) = 0. Proof. This follows easily from the construction C of lim→ Ai given above. The ﬁrst statement follows since every element of C has the form [Ai , ai ] = φi (ai ) for some i and ai ∈ Ai . The second statement follows since if [Ai , ai ] = 0, then (Ai , ai ) ∼ (Ai , 0), so by the deﬁnition of the relation, there is a j ≥ i such that φij (ai ) = φij (0) = 0.
11.1. Limits
183
The above construction and proofs carry through for commutative rings {Ai }. If Mi are modules over the Ai , then we can construct a direct limit of the modules {Mi } which is a lim→ Ai module. All rings and modules are Abelian groups, so we just apply the above construction and keep track of the fact that all of the groups and homomorphisms constructed have appropriate extra structure. An important special case is when we have a direct system of groups, rings, or modules which are all subgroups, subrings, or submodules Mi of a larger group, ring, or module M and the maps in the system are inclusion maps. Then the direct limit is just the union Mi inside M . Suppose that R, R are rings and λ : R → R is a homomorphism. Suppose M, N are Rmodules and M , N are R modules. Then λ makes M , N and M ⊗R N into Rmodules. Suppose that φ : M → M and ψ : N → N are Rmodule homomorphisms. We then have an Rbilinear map M × N → M ⊗R N deﬁned by (x, y) → φ(x) ⊗ ψ(y) for x ∈ M and y ∈ N . By the universal property of the tensor product (Deﬁnition 5.1), there exists a unique homomorphism of Rmodules φ ⊗ ψ : M ⊗R N → M ⊗R N
(11.1)
satisfying (φ ⊗ ψ)(x ⊗ y) = φ(x) ⊗ ψ(y) for x ∈ M , y ∈ N . Lemma 11.4. Suppose that {Rα }α∈I is a directed system of rings and {Mα }α∈I , {Nα }α∈I are directed systems of Rα modules. Then there is a natural isomorphism of lim→ Rα modules lim(Mα ⊗Rα Nα ) ∼ = lim Mα ⊗lim→ Rα lim Nα . →
→
→
Proof. Let λα α : Rα → Rα , φα α : Mα → Mα , ψα α : Nα → Nα for α < α be the homomorphisms in the directed systems. Let R = lim→ Rα , M = lim→ Mα , N = lim→ Nα , D = lim→ (Mα ⊗Rα Nα ), with homomorphisms λα : Rα → R, φα : Mα → M , ψα : Nα → N , and χα : Mα ⊗Rα Nα → D for α ∈ I. The homomorphisms of (11.1), φα ⊗ ψα : Mα ⊗Rα Nα → M ⊗R N , induce homomorphisms μ : D → M ⊗R N by the universal property of limits (Proposition 11.1). We will now deﬁne an Rbilinear homomorphism u : M × N → D. Suppose x ∈ M , y ∈ N . There exists an index α ∈ I and xα ∈ Mα , yα ∈ Nα such that x = φα (xα ), y = ψα (yα ) by Proposition 11.3. Deﬁne u(x, y) = χα (xα ⊗ yα ) ∈ D. Since the Rα , Mα , Nα , and Mα ⊗Rα Nα are directed systems, u(x, y) is independent of choices of α, xα , yα , so u is welldeﬁned. Further, u is Rbilinear. Thus, by the universal property of tensor
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products, Deﬁnition 5.1, there exists a unique Rmodule homomorphism ξ : M ⊗R N → D satisfying ξ(x × y) = u(x, y) for x ∈ M , y ∈ N . Here ξμ and μξ are identity maps by our construction, so μ is an isomorphism with inverse ξ. An inverse system of Abelian groups is a set of Abelian groups {Ai }, indexed by a directed set I, such that if i ≤ j, then there is a homomorphism ψij : Aj → Ai which satisﬁes ψii = idAi and ψik = ψij ψjk if i ≤ j ≤ k. Proposition 11.5. Suppose that {Ai }i∈I is an inverse system of Abelian groups. Then there exists a group lim← Ai with homomorphisms ψi : lim Ai → Ai ←
such that ψi = ψij ψj for i ≤ j in I, which satisﬁes the following universal property: suppose that B is an Abelian group with homomorphisms σi : B → Ai such that σi = ψij σj for i ≤ j in I. Then there exists a unique homomorphism σ : B → lim← Ai such that σi = σψi for i ∈ I. The group lim← Ai with homomorphisms ψi : lim← Ai → Ai is uniquely determined up to isomorphism. It is called the inverse limit of the system {Ai }. The inverse limit can be deﬁned as # % $ (11.2) lim Ai = (ai ) ∈ Ai  ψij (aj ) = ai for i ≤ j . ←
i∈I
The construction of inverse limits carries through for inverse systems of commutative rings and for inverse systems of modules over a ﬁxed commutative ring. In the case when we have an inverse system of groups, rings, or modules which are all subgroups, subrings, or submodules Mi of a larger group, ring, or module M and the maps in the system are all inclusions maps, the inverse limit is the intersection Mi inside of M . Exercise 11.6. If I is a directed set and J is a subset of I, then J is called coﬁnal in I if for every i ∈ I there exists j ∈ J such that i ≤ j. Suppose {Ai }i∈I is a directed system of Abelian groups and J is a coﬁnal subset of I. Show that there is a natural isomorphism lim Aj → lim Ai →
→
where the ﬁrst limit is over the directed set J and the second limit is over the directed set I.
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185
11.2. Presheaves and sheaves In this section, we deﬁne presheaves and sheaves on a topological space. Deﬁnition 11.7. Suppose that X is a topological space. A presheaf P F of Abelian groups on X associates to every open subset U of X an Abelian group P F (U ), with P F (∅) = 0, and to every pair of open sets U1 ⊂ U2 a restriction map resU2 U1 : P F (U2 ) → P F (U1 ) which is a group homomorphism and such that resU,U = idP F (U ) for all U and if U1 ⊂ U2 ⊂ U3 , then the diagram P F (U3 )
LLL U U LLres LLL3 1 LL% / P F (U1 ) P F (U2 )
resU3 U2
resU2 U1
is commutative. We will often write f U1 for resU2 U1 (f ) and say that f U1 is the restriction of f to U1 . We will write P F U1 for the restriction of P F to U1 . Deﬁnition 11.8. Suppose that P F1 and P F2 are presheaves of Abelian groups on a topological space X. A homomorphism of presheaves of Abelian groups φ : P F1 → P F2 is a collection of homorphisms φ(U ) : P F1 (U ) → P F2 (U ) for each open subset U of X, such that if U ⊂ V , then the diagram P F1 (V ) resV,U
P F1 (U )
φ(V )
φ(U )
/ P F2 (V )
resV,U
/ P F2 (U )
is commutative. A homomorphism φ : P F1 → P F2 is an isomorphism if there exists a homomorphism ψ : P F2 → P F1 such that ψ(U ) ◦ φ(U ) = idP F1 (U ) and φ(U ) ◦ ψ(U ) = idP F2 (U ) for all open subsets U of X. The stalk of a presheaf P F at a point p is lim→ P F (U ), where we take the direct limit over the open sets U of X which contain p. Elements of P Fp are called germs. If U ⊂ X is an open subset, t ∈ P F (U ), and p ∈ U , then tp will denote the germ which is the image of t in P Fp .
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If φ : P F1 → P F2 is a homomorphism of presheaves and p ∈ X, then for open neighborhoods V ⊂ U of p, we have a natural commutative diagram P F1 (U )
P F1 (V )
φ(U )
φ(V )
/ P F2 (U ) / P F2 (V )
(P F2 )p so we have a unique induced homomorphism of stalks φp : (P F1 )p → (P F2 )p by the universal property of direct limits. Lemma 11.9. Suppose that P F is a presheaf on a topological space X, U is an open subset of X, and p ∈ U . Then (P F U )p = P Fp . Proof. By the universal property of direct limits of Proposition 11.1, we have a homomorphism π : (P F U )p → P Fp such that for V an open subset of U , we have a commutative diagram P F (V )
JJJ JJJ JJJ J$ / P Fp . (P F U )p
Now the injectivity and surjectivity of π follow from 2) and 1) of Proposition 11.3, respectively. Alternatively, the proof follows from Exercise 11.6. Deﬁnition 11.10. A presheaf of Abelian groups F on a topological space X is a sheaf of Abelian groups on X if for every open subset U of X and collection {Ui }i∈I of open sets in X with U = Ui : 1) If x1 , x2 ∈ F (U ) and resU,Ui x1 = resU,Ui x2 for all i, then x1 = x2 . 2) If xi ∈ F (Ui ) for i ∈ I are such that resUi ,Ui ∩Uj xi = resUj ,Ui ∩Uj xj for all i and j, then there is an x ∈ F (U ) such that resU,Ui x = xi for all i. We will often call condition 1) the “ﬁrst sheaf axiom” and condition 2) the “second sheaf axiom”. It is common to denote F (U ) by Γ(U, F ). An element σ ∈ Γ(X, F ) is called a global section of F . If F and G are sheaves on a topological space X, then a homomorphism of sheaves of Abelian groups φ : F → G is just
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187
a homomorphism of presheaves of Abelian groups, and an isomorphism of sheaves is an isomorphism of presheaves. Example 11.11. Suppose that X is a quasiprojective variety. Then the association U → OX (U ) for U an open subset of X is a sheaf on X. Proof. We have that OX is a presheaf with restriction of functions. We will show that OX is a sheaf. Suppose that U is an open subset of X and {Ui } is an open cover of X. The restriction map OX (U ) → OX (Ui ) is injective, so we immediately get condition 1) of the deﬁnition of a sheaf. Now if xi ∈ OX (Ui ) and xj ∈ OX (Uj ) are such that they have the same restriction in OX (Ui ∩ Uj ), then xi = xj (as elements of k(X)). Thus if resUi ,Ui ∩Uj xi = resUj ,Ui ∩Uj xj for all i, j, then the elements xi are a common element x ∈ i OX (Ui ) = OX (U ), so condition 2) of the deﬁnition of a sheaf holds. In the above example, the stalk of OX at a point p ∈ X is the direct limit over open sets U containing p, OX (U ) = OX,p , lim OX (U ) = p∈U
p∈U
as deﬁned earlier. Proposition 11.12. Suppose that P F is a presheaf of Abelian groups on a topological space X. Then there is a sheaf of Abelian groups F on X and a homomorphism f : P F → F of presheaves such that if F is a sheaf of Abelian groups on X and g : P F → F is a homomorphism of presheaves on X, then there is a unique homomorphism of sheaves h : F → F such that g = hf . The sheaf F of Proposition 11.12 is uniquely determined up to isomorphism (since it satisﬁes the stated universal property). It is called the sheaﬁﬁcation of P F . The sheaﬁﬁcation F of P F has the property that the stalks P Fp = Fp for all p ∈ X. Proof. For an open subset U of X, deﬁne F (U ) to be the set of maps & s:U → P Fp ' where P Fp is the disjoint union of the stalks P Fp for p ∈ X such that: 1) For each p ∈ U , s(p) ∈ P Fp . 2) For each p ∈ U , there exists a neighborhood V of p contained in U and an element t ∈ P F (V ) such that for all Q ∈ V , the germ tQ = s(Q).
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We have that F is a presheaf of Abelian groups with the natural restricthat U is an open tion maps (since each stalk P Fp is a group). Suppose subset of X and Ui are open subsets of X with i Ui = U . The ﬁrst sheaf axiom holds sinceF (U ) is completely determined by its stalks (condition 1) above). Since Ui = U , elements xi ∈ F (Ui ) satisfying the assumptions ' of the second sheaf axiom induce a welldeﬁned map x : U → P Fp by prescribing x(p) = (xi )p ∈ P Fp if p ∈ Ui . Since the xi ∈ F (Ui ) satisfy condition 2) above on Ui for all i, we have that x also satisﬁes condition 2), and thus x ∈ F (U ) and the second sheaf axiom is satisﬁed. Thus F (U ) is a sheaf. We have a natural homomorphism f : P F → F of presheaves deﬁned by mapping t ∈ P F (U ) to the map p → tp for p ∈ U . Now suppose that F is a sheaf of Abelian groups on X and g : P F → F is a homomorphism of presheaves. The extension h : F → F is deﬁned as follows. Suppose that U is an open subset of X and s ∈ F (U ). By condition 2), there exists an open cover {Ui } of U and ti ∈ P F (Ui ) such that sUi = f (ti ). We necessarily have that sUi ∩ Uj = f (ti )Ui ∩ Uj = f (tj )Ui ∩ Uj for all i, j. Thus the germs (ti )p = (tj )p in P Fp = Fp for all p ∈ Ui ∩ Uj . Let ui = g(Ui )(ti ) ∈ F (Ui ). Suppose that p ∈ Ui ∩ Uj . Then the germ (ui )p = gp ((ti )p ) = gp ((tj )p ) = (uj )p . Using the second statement of Proposition 11.3, we ﬁnd that there exists an open neighborhood Vp of p in Ui ∩ Uj such that ui  Vp = uj  Vp . Thus ui Ui ∩ Uj = uj Ui ∩ Uj since F is a sheaf (the ﬁrst sheaf axiom) and thus there exists a ∈ F (U ) such that aUi = ui = g(Ui )(ti ) for all i, again since F is a sheaf (the second sheaf axiom). We deﬁne h(U )(s) = a. The element a is uniquely determined by the ﬁrst sheaf axiom since we must have that h(Ui )(sUi ) = h(Ui )(f (ti )) = g(Ui )(ti ) = ui for all i. Proposition 11.13. Let φ : F → G be a homomorphism of sheaves of Abelian groups on a topological space X. Then φ is an isomorphism if and only if the induced map on stalks φp : Fp → Gp is an isomorphism for every p ∈ X. Proof. If φ is an isomorphism, then φ(U ) is a group isomorphism for all open subsets U of X so φp is an isomorphism for all p ∈ X. Suppose that φp is an isomorphism for all p ∈ X. To show that φ is an isomorphism, we will show that φ(U ) is an isomorphism for all open subsets U of X. We can then deﬁne the inverse map ψ to φ by deﬁning ψ(U ) to be the inverse to φ(U ) for all open subsets U of X. To show that φ(U ) is an isomorphism, we must show that φ(U ) is injective and surjective. We will ﬁrst show that φ(U ) is injective. Suppose that s ∈ F (U ) and φ(U )(s) = 0. Then for all p ∈ U , 0 = φ(U )(s)p = φp (sp ) = 0 in Gp . Thus
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189
for all p ∈ U , sp = 0 in Fp since φp is injective. Thus there exists an open neighborhood Wp of p in U such that sWp = 0, by 2) of Proposition 11.3. The open set U is covered by the open sets Wp for p ∈ U , so by the ﬁrst sheaf axiom, s = 0 in F (U ). Thus φ(U ) is injective. Now we will show that φ(U ) is surjective. Suppose t ∈ G(U ). For p ∈ U , let tp ∈ Gp be the germ of t at p. Since φp is surjective, there exists sp ∈ Fp such that φp (sp ) = tp . By 1) of Proposition 11.3, there exists an open neighborhood Vp of p in U and h(p) ∈ F (Vp ) such that the germ of h(p) at p is sp . By 2) of Proposition 11.3, φp (h(p)p ) = tp implies that there exists a neighborhood Wp of p in Vp such that φ(Wp )(h(p)Wp ) = tWp . So replacing Vp with Wp , we may assume that φ(Vp )(h(p)) = tVp . If p, q ∈ U are two points, then φ(Vp ∩ Vq )(h(p)) = tVp ∩ Vq = φ(Vp ∩ Vq )(h(q)). Since φ(Vp ∩ Vq ) was shown to be injective, we have that h(p)Vp ∩ Vq = h(q)Vp ∩ Vq . Thus by the second sheaf axiom, there exists s ∈ F (U ) such that sVp = h(p) for all p ∈ U . Now φ(U )(s)Vp = φ(Vp )(h(p)) = tVp for all p ∈ X and {Vp } is an open cover of U so φ(U )(s) = t by the ﬁrst sheaf axiom. Proposition 11.14 (The constant sheaf). Suppose that G is an Abelian group and X is a topological space which has the property that if U is an open subset of X, then all connected components of U are open. Deﬁne a presheaf P GX on X by P GX (U ) = G whenever U is a nonempty subset of X. Deﬁne the restriction maps to be the identity. Let GX be the sheaf on X associated to P GX . Suppose that U is an open subset of X and {Ui }i∈I are the connected components of U . Then $ G. GX (U ) ∼ = i∈I
Proof. Let P F = P GX and F = GX . By Proposition 11.12, there is a homomorphism of presheaves λ : P F → F such that for all open subsets U of X and p ∈ U , there is a commutative diagram P F (U ) → F (U ) ↓ ↓ ∼ = → Fp . P Fp Since P F (U ) = P Fp = Fp = G, we have that P F (U ) → F (U ) is an injective homomorphism for all open subsets of X. In particular, this gives
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11. Sheaves
us natural inclusions λU : G → F (U ) for all U , which is compatible with restriction for V ⊂ U , λU / F (U ) DD DD DD λV DD"
GD
F (V ). We thus have that the restriction map F (U ) → Fp = G deﬁned by φ → φp is surjective for all open subsets U of X which contain p. Suppose that p ∈ X and U is an open subset of X containing p. Suppose that φ ∈ F (U ). Let g = φp . Then (φ − λU (g))p = 0. We have that Fp = lim F (V ), →
where the limit is over open subsets V of U containing p. Thus by the second statement of Proposition 11.3, there exists an open subset V of U containing p such that the restriction (φ − λV (g))V = 0. Thus φV = λV (g), so that φq = (λV (g))q = g for all q ∈ V . We have shown that given an open subset U of X and φ ∈ F (U ), the map U → G deﬁned by p → φp is continuous if we give G the discrete topology (a point of G is an open set). Suppose that U is a connected open subset of X. For g ∈ G and φ ∈ F (U ), Wgφ = {q ∈ U  φq = g} is an open subset of U . Further, {Wgφ  g ∈ G} is an open cover of U by disjoint open sets. Thus there exists g ∈ G such that Wgφ = U . In particular if we take any p ∈ U , φq = φp for all q ∈ U . Consider the restriction homomorphism Ψ : F (U ) → G deﬁned by Ψ(φ) = φp . Suppose that φp = 0. Then φq = 0 for all q ∈ U , and so there exists an open cover {Vj } of U such that φVj = 0 for all j (by the second statement of Proposition 11.3). Thus φ = 0 by the ﬁrst sheaf axiom. We have already established that Ψ is onto. We have shown that if U is a connected open set and p ∈ U , then the restriction map F (U ) → G is an isomorphism. Suppose that U is an open subset of X, and let {Ui }i∈I be the connected components of X, which by assumption are open. Consider the group homomorphism $ F (Ui ) Λ : F (U ) → i∈I
11.2. Presheaves and sheaves
191
deﬁned by φ → {φUi }. By the ﬁrst sheaf axiom Λ is injective. Since Ui ∩ Uj = ∅ if i = j, so that F (Ui ∩ Uj ) = (0), we have by the second sheaf axiom that Λ is onto. Thus Λ is an isomorphism. Thus $ F (U ) ∼ G. = i∈I
Suppose that f : X → Y is a continuous map of topological spaces and F is a sheaf of groups on X. We deﬁne a presheaf f∗ F on Y by f∗ F (U ) = F (f −1 (U )) for U an open subset of Y . Then f∗ F is actually a sheaf. We can now recognize that the analysis in the proof of Theorem 10.36 is actually of the sheaf φ∗ Ox . We extend our deﬁnitions of presheaves and sheaves of Abelian groups to presheaves and sheaves of rings. Deﬁnition 11.15. A locally ringed space is a pair (X, OX ) where X is a topological space and OX is a sheaf of rings on X such that for each p ∈ X the stalk OX,p is a local ring. The space X is called the underlying topological space of (X, OX ) and OX is called the structure sheaf of X. A morphism of locally ringed spaces from (X, OX ) to (Y, OY ) is a pair (f, f # ) such that f : X → Y is a continuous map and f # : OY → f∗ OX is a map of sheaves of rings on Y such that for all p ∈ X, the induced map fp# : OY,f (p) → (f∗ OX )f (p) → OX,p is a local homomorphism of local rings. A morphism of locally ringed spaces X → Y is an isomorphism if there exists a morphism of locally ringed spaces Y → X which is a twosided inverse. If X is a quasiprojective variety, then X with its sheaf of regular functions OX is a locally ringed space. If X and Y are two quasiprojective varieties, then the regular maps ϕ : X → Y are morphisms, with ϕ# induced by ϕ∗ . Deﬁnition 11.16. Suppose that X is a locally ringed space. A sheaf of OX modules (an OX module) is a sheaf F on X such that for each open subset U of X the group F (U ) is an OX (U )module, and for each inclusion of open subsets V ⊂ U of X, the restriction homomorphism F (U ) → F (V ) is compatible with the module structures by the restriction ring homomorphism OX (U ) → OX (V ). A homomorphism F → G of sheaves of OX modules is a homomorphism of sheaves such that for all open subsets U of X, F (U ) → G(U ) is a homomorphism of OX (U )modules. If f : X → Y is a morphism of locally ringed spaces and F is a sheaf of OX modules, then f∗ F is a sheaf of OY modules. A subsheaf of a sheaf F is a sheaf F such that for every open subset U of X, F (U ) is a subgroup of F (U ) and the restriction maps of the sheaf
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F are induced by those of F . If F and F are OX modules, then F is a subOX module, or an OX submodule of F if F is a subsheaf of F such that F (U ) is an OX (U )submodule of F (U ) for all open subsets U of X. Example 11.17 (Ideal sheaf, on an aﬃne variety). Suppose that X is an aﬃne variety and J ⊂ k[X] is an ideal. We associate to J a presheaf J˜ on X deﬁned by ˜ )= J(U JI(p) p∈U
for an open subset U of X, where I(p) is the ideal of p in k[X] and JI(p) = Jk[X]I(p) = JOX,p is the localization of J at the maximal ideal I(p). Using the deﬁnition of J˜ in Example 11.17, we verify that J˜ is a sheaf of OX modules. We calculate that the stalk J˜p = JI(p) for p ∈ X. We also observe that, by the deﬁnition, if U ⊂ X is an open set, then J˜(U ) = JOX (U ), and if U is an aﬃne open subset of X, then the restriction J˜U is in fact the ]. tilde on U of the ideal Jk[U ] in k[U ]; that is, J˜U = Jk[U Example 11.18 (Ideal sheaf, on a projective variety). Suppose that X ⊂ Pn is a projective variety, with homogeneous coordinates x0 , . . . , xn and homogeneous coordinate ring S(X). We may suppose that none of the xi vanish everywhere on X. Suppose that J is a homogeneous ideal in S(X). We associate to J a presheaf J˜ on X deﬁned by ˜ )= J(U J(I(p)) p∈U
for an open subset U of X, where the ideal J(I(p)) in OX,p ⊂ k(X) is deﬁned to be the elements of degree 0 in the localization T −1 J where T is the multiplicative set of homogeneous elements of S(X) which are not in I(p). Using the deﬁnition of J˜ in Example 11.18, we verify that J˜ is a sheaf of OX modules. We calculate that the stalk J˜p = J(I(p)) for p ∈ X. Looking back over the analysis we made of OX in Section 3.2, we ﬁnd that ˜ x ) = J(x ) is the dehomogenization of J for 0 ≤ i ≤ n. (11.3) J(X i i
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193
Here J(xi ) denotes the elements of degree 0 in the localization Jxi . In particular, if the ideal J has the homogeneous generators F1 , . . . , Fm , then J˜(Xxi ) is the ideal in OX (Xxi ) = k[ xx0i , . . . , xxni ] generated by xn xn x0 x0 ,..., ,..., , . . . , Fm . F1 xi xi xi xi We deduce that the restriction of J˜ to Xxi is the tilde on the aﬃne variety ˜ in k[Xx ]; that is, Xxi of the ideal Γ(Xxi , J) i ˜ J˜Xxi = Γ(X xi , J). In fact, if U ⊂ X is any aﬃne open subset, then the restriction of J˜ to U is the tilde on U of the ideal Γ(J, U ) ⊂ k[U ]; that is, ˜ = Γ(U, ˜ JU J). The ideal sheaf I˜ which we have just deﬁned has been previously encountered in Chapter 6 (before Lemma 6.7). Suppose that A is a subsheaf of a sheaf B on a topological space X. Then B/A will denote the sheaf associated to the presheaf U → B(U )/A(U ) for U an open subset of X. Lemma 11.19. Suppose that A is a subsheaf of a sheaf B on a topological space X and p ∈ X. Then (B/A)p ∼ = Bp /Ap . Proof. We have that the stalk of the sheaf B/A at p is the stalk at p of the presheaf U → B(U )/A(U ) for U an open subset X by the comment after Proposition 11.12. By the universal property of direct limits, Proposition 11.1, there is a unique homomorphism Bp → (B/A)p such that for all open subsets U of X containing p, we have a commutative diagram B(U ) → B(U )/A(U ) ↓ ↓ Bp → (B/A)p which is compatible with restriction. Now 0 → A(U ) → B(U ) → B(U )/A(U ) → 0 is exact for all open subsets U of X so a diagram chase using Proposition 11.3 shows that 0 → Ap → Bp → (B/A)p → 0 is exact.
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If α : F → G is a homomorphism of sheaves on a topological space X, then we have sheaves Kernel(α), Image(α) and Cokernel(α) associated to the respective presheaves deﬁned by U → Kernel(α(U )), U → Image(α(U )), and U → Cokernel(α(U )) for U an open subset of X. A homomorphism α : F → G of sheaves on a topological space X is called injective if αp : Fp → Gp is injective for all p ∈ X. The homomorphism α is called surjective if αp : Fp → Gp is surjective for all p ∈ X. A sequence (11.4)
0→A→B→C→0
of homomorphisms of sheaves is called short exact if for all p ∈ X, the sequence of homomorphisms of groups 0 → Ap → Bp → Cp → 0 is short exact. Proving the following proposition is Exercise 11.22. Proposition 11.20. Suppose that (11.4) is a short exact sequence of sheaves and U is an open subset of X. Then (11.5)
0 → A(U ) → B(U ) → C(U )
is exact. In particular, the presheaf U → Kernel(α(U )) is a sheaf for any homomorphism of sheaves α : F → G. Example 11.21. Suppose that A is a subsheaf of a sheaf B. Then the presheaf U → B(U )/A(U ) may not be a sheaf. Further, if α : A → B is a homomorphism of sheaves, then the presheaf U → Image(α(U )) may not be a sheaf, and if (11.4) is a short exact sequence of sheaves, then B(U ) → C(U ) may not be surjective for some open subset U of X. Proof. Let p be a nonsingular point on a projective curve X. Let Ip be the ideal sheaf of p in X. For q ∈ X, we have that
OX,q if q = p, Ip,q = the maximal ideal mp ⊂ OX,p if q = p. Let P F be the presheaf deﬁned by P F (U ) = Ip (U )/Ip (U )2 for U an open set in X. Let F be the sheaf associated to P F . We calculate, from Lemma 11.19, that the stalk
0 if q = p, Fq = mp /m2p ∼ = k if q = p. We will now establish that for an open subset U of X,
k if p ∈ U, (11.6) F (U ) = 0 if p ∈ U.
11.2. Presheaves and sheaves
195
Suppose that p ∈ U and x ∈ F (U ). Then for each q ∈ U , the stalk Fq = 0. Thus the image of x in Fq is zero, so there exists an open neighborhood Uq of q, which is contained in U , such that the restriction of x in F (Uq ) is zero (by part 2) of Proposition 11.3). Thus we have a cover {Uq } of U by open subsets of U such that the restriction of x in each F (Uq ) is zero. Now 0 ∈ F (U ) also has this property, so x = 0 by the ﬁrst sheaf axiom. We have established that F (U ) = (0) if p ∈ U . Suppose that p ∈ U . We will show that the restriction homomorphism Λ : F (U ) → Fp ∼ = k is an isomorphism. Let r ∈ k. Then there exist an open neighborhood Up of p in U and φ ∈ F (Up ) such that the restriction of φ to Fp is r (by part 1) of Proposition 11.3). Let V = U \ {p}, which is an open subset of U . The point p ∈ V , so F (V ) = F (V ∩ Up ) = (0). Since the restriction of φ to F (V ∩ Up ) is zero, which is the restriction of 0 ∈ F (V ) to F (V ∩ Up ), we have by the second sheaf axiom that there exists x ∈ F (U ) which restricts to 0 in F (V ) and restricts to φ in F (Up ) so necessarily restricts to r ∈ Fp . Thus Λ is surjective. Suppose that x ∈ F (U ) and Λ(x ) = 0. Then there exists an open neighborhood Up of p in U such that the restriction of x to F (Up ) is zero, by 2) of Proposition 11.3. Since the restriction of x to V = U \ {p} is necessarily zero, we have that x = 0 in F (U ) by the ﬁrst sheaf axiom. Thus Λ is injective and is necessarily an isomorphism. From the natural inclusions of sheaves of OX modules, Ip2 ⊂ Ip ⊂ OX , we have (by applying Proposition 11.20 to each inclusion) inclusions of modules Ip2 (X) ⊂ Ip (X) ⊂ OX (X). Now OX (X) = k (by Theorem 3.35). Further, Ip (X) cannot contain a nonzero element of k since every element of Ip (X) must restrict to an element of Ip,p = mp and hence must vanish at p. Thus Ip2 (X) = Ip (X) = 0, and P F (X) = Ip (X)/Ip2 (X) = 0. In contrast, by (11.6), F (X) ∼ = k. Thus F = P F . We have (by Proposition 11.12) a natural exact sequence of sheaves α
0 → Ip2 → Ip → F → 0. The image of α is the presheaf P F , which we have already established is not a sheaf. Further, the evaluation of the above short exact sequence at X is 0→0→0→k→0 which is not short exact. Exercise 11.22. Prove Proposition 11.20.
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Exercise 11.23. Let X be a topological space, let {Ui } be an open cover of X, and suppose that for each i we have a sheaf Fi on Ui , and for each i, j an isomorphism φij : Fi Ui ∩ Uj → Fj Ui ∩ Uj such that for each i, φii = id and for each i, j, k, φik = φjk ◦ φij on Ui ∩ Uj ∩ Uk . Show that there exists a unique sheaf F on X, with isomorphisms ψi : F Ui → Fi such that for each i, j, ψj = φij ◦ ψi on Ui ∩ Uj .
11.3. Some sheaves associated to modules Theorem 11.24 (Sheaﬁﬁcation of a module on an aﬃne variety)). Suppose that X is an aﬃne variety and M is a k[X]module. Then there is a unique ˜ of OX modules on X which has the property that sheaf M ˜ (Xf ) = Mf for f ∈ k[X], M
(11.7)
˜ (X) = M ˜ (X1 ) = M → M ˜ (Xf ) = Mf is the map and the restriction map M a a → 1 . For p ∈ X, the stalk ˜ (U ) = ˜ p = lim M M p∈U
lim
f ∈k[X]\I(p)
Mf = MI(p) ,
where I(p) is the maximal ideal in k[X] of the point p and MI(p) is the localization of M at this prime ideal. ˜ (U ) Proof. The property (11.7) and the sheaf axioms uniquely determine M for U an arbitrary open subset of X. In fact, U is a ﬁnite union of open sets U = Xf1 ∪ Xf2 ∪ · · · ∪ Xfn for some fi ∈ k[X]. By the sheaf axioms, we have an exact sequence of k[X]module homomorphisms (11.8)
α ˜ (U ) → 0→M
n
β ˜ (Xf ) → M i
i=1
˜ (Xf ∩ Xf ), M i j
1≤i<j≤n
where α(g) = {resU,Xfi (g)} ˜ (U ) and for g ∈ M β({hl }) = {resXfj ,Xfi ∩Xfj (hj ) − resXfi ,Xfi ∩Xfj (hi )} ˜ (Xf ). for {hl } ∈ ni=1 M i ˜ (U ) can be identiﬁed with Since Xfi ∩ Xfj = Xfi fj , (11.8) tells us that M the kernel of β, which has the explicit form n i=1
β
Mfi →
1≤i<j≤n
Mfi fj .
11.3. Some sheaves associated to modules
197
˜ satisfying (11.7). Let U be an open subset We now construct the sheaf M ˜ (U ) to of the aﬃne variety X, with regular functions A = k[X]. Deﬁne M ' be the set of functions s : U → p∈U MI(p) such that s(p) ∈ MI(p) for each p in U and such that for each p ∈ U there is an open neighborhood V of p in U , a ∈ M , and f ∈ A such that Z(f ) ∩ V = ∅ and for each q ∈ V , ˜ is a sheaf on X as it satisﬁes the sheaf s(q) = fa ∈ MI(q) . We have that M axioms. For U an open subset of X, OX (U ) = p∈U AI(p) ⊂ k(X). It follows that the map of sheaves OX → A˜ deﬁned by associating to t ∈ OX (U ) the ˜ ) deﬁned by s(p) = t for p ∈ U is an isomorphism of map s : U → A(U sheaves of kalgebras. We thus have that A˜ ∼ = OX . We have that property (11.7) holds for A˜ by Proposition 2.84. For A˜ is naturally a sheaf of A˜ ∼ modules M , the sheaf M = OX modules. ˜ follows from a careful analysis The fact that property (11.7) holds for M ˜ of the natural map Mf → M (D(f )) for f ∈ A, as we now verify. Deﬁne an Af module homomorphism ˜ (D(f )) ψ : Mf → M by ψ( fan ) = s where s(p) =
a fn
∈ MI(p) for p ∈ D(f ).
We ﬁrst will show that ψ is injective. Suppose that ψ( fan ) = ψ( fbm ) for some fan , fbm ∈ Mf . Then fan = fbm in MI(p) for all p ∈ D(f ), so for each p, there exists hp ∈ A with hp ∈ I(p) such that hp (f m a − f n b) = 0 in M . Let J = Ann(f m a−f n b) be the annihilator of f m a−f n b in A. Then hp ∈ J and hp ∈ I(p) so J is not contained in I(p). √ Since this is true for all p ∈ D(f ), we have Z(J) ∩ D(f ) = ∅. Thus f ∈ J so f l ∈ J for some positive power l, and so f l (f m a − f n b) = 0, implying fan = fbm in Mf , and so ψ is injective. ˜ (D(f )). We can cover Now we will show that ψ is surjective. Let s ∈ M D(f ) with open sets Vi so that there are ai ∈ M and gi ∈ A such that Z(gi ) ∩ Vi = ∅ and s(p) = agii ∈ MI(p) for all p ∈ Vi . The open sets D(h) with h ∈ A are a basis of the topology of X (Lemma 2.83) so we may assume that Vi = D(hi ) for some hi ∈ A. We have D(hi ) ⊂ D(gi ) for all i so each (hi ) ⊂ (gi ) by the nullsetellensatz. Thus hni i = ci gi for some positive power ni and ci ∈ A, so that agii = chinaii in Mhi . Replacing hi with hni i (we i
have D(hi ) = D(hni i )) and ai by ci ai , we may assume that D(f ) is covered by open sets D(hi ) such that s(p) = haii for p ∈ D(hi ). Since D(f ) ⊂ D(hi ), we have that Z({hi }) = Z(hi ) ⊂ Z(f ). Since A is Noetherian, the ideal ({hi }) is generated by a ﬁnite number of the hi , say h1 , . . . , hr , so D(f ) ⊂ D(h1 )∪· · ·∪D(hr ) and we have that f n ∈ (h1 , . . . , hr )
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for some positive power n by the nullstellensatz. Thus we have an expression f n = b1 h1 + · · · + br hr
(11.9) for some b1 , . . . , br ∈ A.
a
For p ∈ D(hi ) ∩ D(hj ) = D(hi hj ), we have that s(p) = haii = hjj ∈ MI(p) . By our proof of injectivity, ψ is injective when restricted to D(hi hj ), so we a have that haii = hjj in Mhi hj . Hence, (hi hj )n (hj ai − hi aj ) = 0 in M for some n. We may pick n suﬃciently large so that this equation and (11.9) are valid for all i, j. Rewrite the equation as (hni ai ) − hn+1 (hnj aj ) = 0. hn+1 j i Then replace hi by hn+1 and ai by hni ai (we have that D(hi ) = D(hn+1 )) i i ai so that we still have that s(p) = hi for p ∈ D(hi ), and we now have that hj ai = hi aj for all i, j. Let a = i bi ai where the bi are from (11.9). Then for each j, we have bi hj ai = bi hi aj = f n aj hj a = i
i
a
so that fan = hjj on D(hj ), and thus ψ( fan ) = s, so that ψ is surjective and hence is an isomorphism. Important special cases of the construction of Theorem 11.24 are that ) k[X] = OX and if Y is a subvariety of X, then the ideal sheaf IY = I(Y where I(Y ) is the prime ideal in k[X] of Y . More generally, we have a sheaf of OX modules I˜ (an ideal sheaf) associated to any ideal I ⊂ k[X] (Example 11.17). In the case when M is a k[X]submodule of k(X) and f ∈ k[X], we have that ˜ (Xf ) = Mf = Mp (11.10) M p∈D(f )
where the intersection takes place in k(X). This follows from Lemma 1.77. We have a corresponding construction for projective varieties. Suppose that Y is an aﬃne variety and X is a closed subvariety of Y ×Pr . The variety Y × Pr has the coordinate ring S(Y × Pr ) = A[x0 , . . . , xr ] where A = k[Y ] and the polynomial ring A[x0 , . . . , xr ] over A is graded by deg(xi ) = 1 for all i. Let p ⊂ A[x0 , . . . , xr ] be the graded prime ideal ∞ p = I(X). Then the coordinate ring of X is S(X) = A[x0 , . . . , xr ]/p = i=0 Si .
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199
Theorem 11.25 (Sheaﬁﬁcation of a graded module on a projective variety). Suppose that X is a projective variety, or, more generally, Y is an aﬃne variety and X is a closed subvariety of Y × Pr , with coordinate ring S(X), ˜ of OX and N is a graded S(X)module. Then there is a unique sheaf N modules on X which has the property that (11.11)
˜ (XF ) = N(F ) N
for homogeneous F ∈ S(X)
˜ (XG ) → and for forms F, G ∈ S(X) with XF ⊂ XG , the restriction map N ˜ (XF ) is the natural map N(G) → N(F ) induced by localization. N Recall that N(F ) denotes the set of elements of degree 0 in the localization NF . Using the sheaf axioms, we can give an explicit formula similar to (11.8) ˜ (U ), when U = XF ∪· · ·∪XF for some homogeneous for the calculation of N n 1 forms Fi ∈ S(X) We calculate that for p ∈ X, the stalk (11.12)
˜p = N(I(p)) , N
where NI(p) denotes the elements of degree 0 in the localization T −1 N where T is the multiplicative system of homogeneous elements of S(X) \ I(p). ˜ to the aﬃne open In fact, we have that the restriction of the sheaf N subset XF (where F is homogeneous of positive degree) is just the sheaf (11.13)
˜ XF = N N (F )
on the aﬃne variety XF (which has regular functions k[XF ] = S(X)(F ) ), ˜ below, the deﬁnition of as follows from a comparison of the deﬁnition of N N (F ) from Theorem 11.24, and (11.12). ˜ of Theorem 11.25 is constructed for general graded modules The sheaf N N as follows. Let U be an open subset of the projective variety X, with coordinate ring S = S(X), and suppose that N is a graded Smodule. ˜ Suppose that U is an ' open subset of X. Deﬁne N (U ) to be the set of functions s : U → p∈U N(I(p)) such that s(p) ∈ NI(p) for each p ∈ U and such that for each p ∈ U there is an open neighborhood V of p in U and homogeneous elements a ∈ N and f ∈ S of the same degree such that Z(f ) ∩ V = ∅ and s(q) = fa ∈ N(I(q)) . Here I(p) is the homogeneous ideal of the point p in S and N(I(p)) is the set of elements of degree 0 in the localization T −1 N where T is the multiplicative set of homogeneous ˜ is a sheaf on X as it elements of S which are not in I(p). We have that N satisﬁes the sheaf axioms. For U an open subset of X, OX (U ) = p∈U S(I(p)) ⊂ k(X). It follows that the map of sheaves OX → S˜ deﬁned by associating to t ∈ OX (U ) the
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11. Sheaves
˜ ) deﬁned by s(p) = t for p ∈ U is an isomorphism of map s : U → S(U sheaves of kalgebras. We thus have that S˜ ∼ = OX . We have that property (11.11) holds for S˜ by Proposition 4.6. ˜ is naturally a sheaf of S˜ ∼ For general modules N , the sheaf N = OX modules. The fact that property (11.11) holds for N follows from a careful ˜ (D(F )) for F ∈ S homogeneous, analysis of the natural map N(F ) → N generalizing the proof of Theorem 11.24. It follows from the above analysis that the deﬁnition in Theorem 11.25 is consistent with Example 11.18. An important example which we have encountered before is the OX module I˜ (ideal sheaf) associated to any homogeneous ideal I of S(X). In particular, if Y is a closed algebraic set in X, with homogeneous reduced ). If X is a quasiideal I(Y ) in S(X), the ideal sheaf of Y is IY = I(Y projective variety and Y is a closed algebraic set in X, then X is an open subset of projective variety X. Letting Y be the closed algebraic set in X which is the closure of Y in X, we have that Y ∩ X = Y . We deﬁne the ideal sheaf IY on X to be the restriction IY X. Suppose that X is a projective variety or a closed subvariety of Y × Pr where Y is an aﬃne variety, with a closed embedding i : X → Y × Pr . Let S(Y × Pr ) and S(X) = S(Y × Pr )/I(X) be the respective coordinate rings. Suppose that M is a graded S(X)module. Let M be the sheaﬁﬁcation of M as a graded S(X)module. Then (11.14) i∗ M is the sheaﬁﬁcation of M regarded as a graded S(Y × Pr )module. Exercise 11.26. Suppose that φ : X → Y is regular map of aﬃne varieties, Mk[X] is a k[X]module, and F = M k[X] is the induced sheaf on X. Let Mk[Y ] be the k[Y ]module which is Mk[X] with the k[Y ]module structure induced by the homomorphism φ∗ : k[Y ] → k[X]. Let G = M k[Y ] be the induced sheaf on Y . Show that φ∗ F = G. Exercise 11.27. Prove formula (11.14).
11.4. Quasicoherent and coherent sheaves Deﬁnition 11.28. Let X be a quasiprojective variety. A sheaf of OX modules F is quasicoherent if X can be covered by aﬃne open sets Ui such that for all i, the restriction F Ui of F to Ui is isomorphic as a sheaf of OUi ˜ i for some k[Ui ]module Mi . The sheaf F is coherent modules to a sheaf M if the Mi are all ﬁnitely generated k[Ui ]modules.
11.4. Quasicoherent and coherent sheaves
201
The conclusions of the following example follow from the remark that ˜ N XF ∼ = N (F ) for homogeneous F ∈ S(X) of positive degree in equation (11.13). Example 11.29. Suppose that Y is an aﬃne variety and X is a closed subvariety of Y × Pr . Let S(X) be the graded coordinate ring of X, and let ˜ is a quasicoherent OX module. If N be a graded S(X)module. Then N ˜ is coherent. N is a ﬁnitely generated S(X)module, then N Suppose that X is a locally ringed space and F , G are sheaves of OX modules on X. Then HomOX (F , G) denotes the OX (X)module of OX module sheaf homomorphisms from F to G. Lemma 11.30. Suppose that X is an aﬃne variety, R = Γ(X, OX ), and M , N are Rmodules. Then the natural map ˜,N ˜ ) → HomR (M, N ) HomO (M X
deﬁned by ψ → ψ(X) is an isomorphism of Rmodules. ˜,N ˜) Proof. Suppose that φ ∈ HomR (M, N ). We will construct φ˜ ∈ HomOX(M ˜,N ˜ ) → HomR (M, N ). giving an inverse to the natural map HomOX (M Let U be an open subset of X. Then U = Xf1 ∪ · · · ∪ Xfn for some f1 , . . . , fn ∈ k[X]. The homomorphism φ induces a commutative diagram
i Mfi
β1
/
φfi
Mfi fj
i<j
i Nfi
β2
/
i<j
φfi fj
Nfi fj
where the horizontal maps are deﬁned as in equation (11.8). Hence we have a natural homomorphism ˜ ):M ˜ (U ), ˜ (U ) = Kernel β1 → Kernel β2 = N φ(U which is compatible with restrictions.
Lemma 11.31. Suppose that X is an aﬃne variety and R = Γ(X, OX ) = k[X]. The sequence of Rmodules M →N →P is exact if and only if the sequence of sheaves of OX modules ˜ →N ˜ → P˜ M is exact. ˜ → N ˜ is a homomorphism of OX modules, then its kernel, Thus if M ˜ for some Rmodule K. cokernel, and image are of the form K
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11. Sheaves
Proof. The sequence of sheaves is exact if and only if it is exact at all stalks; that is, Mm → Nm → Pm is exact for all maximal ideals m of R. This is equivalent to the exactness of M → N → P by [13, Proposition 3.9]. Theorem 11.32. Suppose that F is a sheaf of OX modules on a quasiprojective variety X. Then F is quasicoherent (coherent) if and only if for every aﬃne open subset U of X, there exists a (ﬁnitely generated) k[U ]˜. module M such that F U ∼ =M If F is a quasicoherent sheaf on X and U is an aﬃne open subset of X, then the above theorem tells us that F U = Γ(U, F ), by Theorem 11.24. Proof. We will ﬁrst assume that F is quasicoherent and that U is an aﬃne open subset of X. ˜ We observe that if V is an aﬃne open subset of X such that F V ∼ =M ∼ ˜ for some k[V ]module M and h ∈ k[V ], then F Vh = Mh . Since U ∩ V is covered by aﬃne open subsets Vh , we have that F U is quasicoherent since F is quasicoherent. Let R = k[U ]. If V ⊂ U is an aﬃne open subset and p ∈ V , then there exists g ∈ R such that p ∈ Ug ⊂ V , and Ug = Vg . Thus U can be covered by a ﬁnite number of open sets Ui = Ugi such that there are k[Ui ] = Rgi modules ˜ i. Mi with F Ui ∼ =M We have
F Ugi ∩ Ugj = F Ugi gj ∼ = (M i )gj .
For every open subset W of U , the sequence $ $ Γ(W ∩ Ugi , F ) → Γ(W ∩ Ugi ∩ Ugj , F ) 0 → Γ(W, F ) → i
i<j
∗ on U by is exact by the sheaf axioms. Deﬁne new sheaves Fi∗ and Fi.j
Γ(W, Fi∗ ) = Γ(W ∩ Ugi , F ) and ∗ ) = Γ(W ∩ Ugi ∩ Ugj , F ) Γ(W, Fi,j
for an open subset W of U , so the sequence of sheaves $ $ ∗ Fi∗ → Fi,j 0→F → i
i<j
11.4. Quasicoherent and coherent sheaves
203
˜ for some Rmodule, it suﬃces is exact. So to prove that F is of the form M ∗ ∗ to prove this for Fi and Fi,j , by Lemma 11.31, as a direct sum of tildes of modules on an aﬃne variety is the tilde of the sum. Viewing Mi as an Rmodule, for all g ∈ R we have ˜ i) Γ(Ug , Fi∗ ) = Γ(Ug ∩ Ugi , F ) = Γ((Ugi )g , F Ugi ) = (Mi )g = Γ(Ug , M ˜ i . The same argument shows that F ∗ = (M so that Fi∗ = M i )gj . Thus F is i,j the tilde of an Rmodule M . The conclusions of the theorem for coherent sheaves reduces by the above arguments to the statement that if U is aﬃne, with R = k[U ], f1 , . . . , fn ∈ R are such that Uf1 , . . . , Ufn is an aﬃne cover of U , and M is an Rmodule such that Mfi is a ﬁnitely generated Rfi module for 1 ≤ i ≤ n, then M is a ﬁnitely generated Rmodule. We will now establish this statement, which is a minor extension of Lemma 7.4. For 1 ≤ i ≤ n, there exist elements σi1 , . . . , σiti in M which generate Mfi as an Rfi module. Let N be the Rsubmodule of M generated by σij for 1 ≤ i ≤ n and 1 ≤ j ≤ ti . Since U = i Ufi , we have that ZU (f1 , . . . , fn ) = ∅, so that the ideal (f1 , . . . , fn ) = R. Suppose m ∈ M . Then for 1 ≤ i ≤ n, there exist rij ∈ R and λ ∈ N such that ⎛ ⎞ rij σij ⎠ = 0 fiλ ⎝m − j
in M , so that (f1λ , . . . , fnλ )m ⊂ N . But (f1λ , . . . , fnλ ) = R by Lemma 7.3 so m ∈ N . Thus M = N is a ﬁnitely generated Rmodule. Deﬁnition 11.33. Suppose that X is a quasiprojective variety and F is a coherent sheaf of OX modules. The sheaf F is said to be invertible if there exists an open cover {Ui } of X and OUi module isomomorphisms φi : OUi → F Ui for all Ui in the cover. Exercise 11.34. Suppose that X is a locally ringed space and F is a sheaf of OX modules on X. Show that there is a natural isomorphism of OX (X)modules HomOX (OX , F ) → F (X). Exercise 11.35. Suppose that X is a quasiprojective variety, F , G are OX modules, and φ : F → G is a homomorphism of OX modules. a) Suppose that F and G are quasicoherent. Show that Kernel(φ), Image(φ), and Cokernel(φ) are quasicoherent OX modules. b) Suppose that F and G are coherent. Show that Kernel(φ), Image(φ), and Cokernel(φ) are coherent OX modules.
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11. Sheaves
Exercise 11.36. Suppose that X is a quasiprojective variety. Suppose that F is a sheaf on X. Deﬁne the support of F by Supp(F ) = {p ∈ X  Fp = 0}. Suppose that F is a coherent sheaf on X. Show that Supp(F ) is a closed set. Hint: You may use the following lemma from commutative algebra [13, Exercise 19, page 46]: suppose that Y is an aﬃne variety. Let A = k[Y ], and suppose that M is a ﬁnitely generated Amodule. Then {p ∈ Y  MIY (p) = 0} is a closed subset of Y .
11.5. Constructions of sheaves from sheaves of modules In this section we give some constructions of sheaves from sheaves of modules. Our primary interest in these sheaves is in the case of coherent or quasicoherent OX modules. In this case, the equations (11.15), (11.16), (11.17), and (11.18) should be taken as the deﬁnitions of these sheaves. These formulas, which give local realizations of the sheaves in terms of commutative algebra, are all that is needed to work eﬀectively with these sheaves. Assume that X is a quasiprojective variety and F and G are quasicoherent sheaves on X. The tensor product F ⊗OX G is a quasicoherent sheaf of OX modules which is uniquely determined by the following property: if U is an aﬃne open subset of X and L = Γ(U, F ), M = Γ(U, G), then ˜ (11.15) (F ⊗O G)U = N X
where N = L ⊗k[U ] M . We can use the sheaf axioms to show that the condition (11.15) determines a unique sheaf on X. If F and G are coherent, then F ⊗OX G is coherent. The stalk (F ⊗OX G)p ∼ = Fp ⊗OX,p Gp for p ∈ X. We sometimes denote F ⊗OX G by F ⊗G when there is no danger of confusion. Assume that X is a quasiprojective variety and F is a quasicoherent sheaf on X and G is a coherent sheaf on X. The sheaf Hom OX (F , G) is a quasicoherent sheaf of OX modules which is uniquely determined by the property that if U an aﬃne open subset of X, then ˜ (11.16) Hom O (F , G)U = G X
where G = Homk[U ] (Γ(U, F ), Γ(U, G)) is the k[U ]module of k[U ]module homomorphisms from Γ(U, F ) to Γ(U, G). We can use the sheaf axioms to show that the condition (11.16) determines a unique sheaf on X. If F is coherent, then Hom OX (F , G) is coherent. For p ∈ X, the stalk Hom OX (F , G)p = HomOX,p (Fp , Gp ), the OX,p module homomorphisms from Fp to Gp , as follows from [50, Proposition 1.10].
11.5. Constructions of sheaves from sheaves of modules
205
Suppose that φ : X → Y is a regular map of quasiprojective varieties and M is a quasicoherent sheaf of OY modules. The inverse image φ∗ M of M by φ is a quasicoherent sheaf of OX modules, which has the following deﬁning property: if U is an aﬃne open subset of X and V is an aﬃne open subset of Y such that φ(U ) ⊂ V , then (11.17)
φ∗ MU = (M ⊗ k[V ] k[U ])
where M = Γ(V, M). We can use the sheaf axioms and basic properties of tensor products to show that the condition (11.17) determines a unique sheaf φ∗ (M) on X. It follows from (11.17) that φ∗ M is coherent if M is coherent. If p ∈ X and q = φ(p), then (φ∗ M)p ∼ = Mq ⊗OY,q OX,p (by Lemma 11.4). Deﬁnition 11.37. Suppose that X is a variety. A coherent OX module I is called an ideal sheaf if I is an OX submodule of OX . If I is an ideal sheaf on an aﬃne variety X, we see from Theorem 11.32 ˜ and we will see from Theorem that I = Γ(X, I) ⊂ k[X] is such that I = I, 11.48 that if X is projective with coordinate ring S(X) and I is an ideal sheaf on X, then there exists a homogeneous ideal I of S(X) such that ˜ I = I. We will see, by Proposition 11.53, that Deﬁnition 11.37 is consistent with Examples 11.17 and 11.18. Suppose that φ : X → Y is a regular map of quasiprojective varieties and I is an ideal sheaf on Y . We deﬁne the inverse image ideal sheaf IOX to be the natural image of φ∗ I in OX induced by the inclusion I ⊂ OY , giving a map φ∗ I → φ∗ OY ∼ = OX . The sheaf IOX is coherent since it is the image of a homomorphism of coherent sheaves (Exercise 11.35). This ideal sheaf has the deﬁning property that when U an aﬃne open subset of X and V an aﬃne open subset of Y such that φ(U ) ⊂ V , (11.18)
], IOX U = Ik[U
where I = Γ(V, I). If p ∈ X and q = φ(p), then (IOX )p = Iq OX,p . If I is an ideal sheaf on Y , then there exists a natural surjection of OX modules φ∗ I → IOX , but in general this map is not injective. We do have that φ∗ I/T (φ∗ (I)) ∼ = IOX , where T (φ∗ (I)) is the OX torsion of φ∗ (I) (Exercise 11.43). The fact that φ∗ I → IOX is in general not injective can be seen in the following simple example. Let A = k[x, y] and B = k[x1 , y1 ] be polynomial rings, and
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consider the injective kalgebra homomorphism A → B deﬁned by x = x1 , y = x1 y1 . Let I = (x, y) ⊂ A. We have a short exact sequence of Amodules 0 → A → A2 → I → 0, where the ﬁrst map is deﬁned by 1 → (y, −x) and the second map is deﬁned by (1, 0) → x and (0, 1) → y. Tensoring this sequence with B, we have the right exact sequence (tensoring with a module is right exact by Proposition 5.5) B → B 2 → I ⊗A B → 0, so that I ⊗A B ∼ = B 2 /(x1 y1 , −x1 )B. The class of (y1 , −1) is nonzero in I ⊗A B, but x1 (y1 , −1) = 0. Thus I ⊗A B has Btorsion, which must be in the kernel of the surjection onto the ideal IB. The above sheaves can be deﬁned more generally for sheaves of OX modules on a locally ringed space X. We will give an outline of these constructions here for the interested reader. Suppose that F and G are OX modules on a locally ringed space X. Then we have a presheaf U → F (U ) ⊗OX (U ) G(U ). We denote by F ⊗OX G (or sometimes F ⊗ G when there is no danger of confusion) the sheaf associated to this presheaf. The stalk (F ⊗ G)p ∼ = Fp ⊗OX,p Gp for p ∈ X (by Lemma 11.4). To prove that this deﬁnition gives formula (11.15) in the case when X is quasiprojective and F and G are quasicoherent, observe that we ˜ to the presheaf V → have a natural restriction homomorphism from N F (V ) ⊗OX (V ) G(V ) for V an open subset of U . Then by Proposition 11.12, ˜ → (F ⊗O G)U . This map is there is a natural homomorphism of sheaves N X an isomorphism on stalks, so it is an isomorphism of sheaves by Proposition 11.13. The quasicoherence (or coherence) of F ⊗OX G then follows from Theorem 11.32. Suppose that F and G are OX modules on a locally ringed space X. We have a presheaf U → HomOX U (F U, GU ), for U an open subset of X, the OX (U )module of OX U module sheaf homomorphisms F U → GU . This presheaf is a sheaf (since F and G are sheaves), which we write as Hom OX (F , G). Lemma 11.38. Suppose that X is a quasiprojective variety, F is quasicoherent and G is coherent on X. Then Hom OX (F , G) is quasicoherent. Further, if F and G are both coherent, then Hom OX (F , G) is coherent.
11.5. Constructions of sheaves from sheaves of modules
207
Proof. Let H = Hom OX (F , G). It suﬃces to prove the lemma when X is aﬃne. Let R = k[X], M = F (X), and N = G(X). Then N is a ﬁnitely ˜ , and G = N ˜ . Let A = HomR (M, N ). We have generated Rmodule, F = M a natural homomorphism of sheaves A˜ → H. For f ∈ R, we have ˜ A(D(f )) = HomR (M, N )f ∼ = HomRf (Mf , Nf ) (by [50, Proposition 1.10]) ∼ ˜f, N ˜f ) by Lemma 11.30 = Hom ˜ (M Rf
∼ = H(D(f )). Since any aﬃne open subset of X is a ﬁnite union of basic open sets D(fi ), ˜ ) → H(U ) is an isomorphism for all open subsets U of we thus have that A(U X by the sheaf axioms. Thus H ∼ = A˜ is quasicoherent. Further, if F and G are both coherent, so that M and N are both ﬁnitely generated Rmodules, then A is a ﬁnitely generated Rmodule, so H ∼ = A˜ is coherent. From Lemmas 11.30, 11.32, and 11.38 we see that the formula 11.16 holds. We now give an outline of the general construction of the pullback f ∗ G of a sheaf of OY modules by a morphism of ringed spaces f : X → Y . Suppose that f : X → Y is a continuous map of topological spaces and G is a sheaf on Y . The inverse image sheaf f −1 G on X is the sheaf associated to the presheaf U → lim G(V ) f (U )⊂V
where U is an open subset of X and the limit is over the open sets V of Y which contain f (U ). Let f : X → Y be a morphism of ringed spaces and G be an OY module. Then f −1 G is an f −1 OY module and we have a natural homomorphism f −1 OY → OX of sheaves of rings on X. We deﬁne the inverse image of G by the morphism f to be f ∗ G = f −1 G ⊗f −1 OY OX . The sheaf f ∗ G is a sheaf of OX modules. In the case when f : X → Y is a regular maps of quasiprojective varieties and G is quasicoherent, we have the formula (11.17). Exercise 11.39. Suppose that X is a closed subvariety of a quasiprojective variety Y . Let i : X → Y be the inclusion, and let IX be the ideal sheaf of X in Y . Show that there is a short exact sequence of sheaves of OY modules 0 → IX → OY → i∗ OX → 0. Hint: First solve this when Y is aﬃne.
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Exercise 11.40. Suppose that X is a quasiprojective variety of dimension > 0 and p ∈ X. Deﬁne a presheaf F on X by
OX (U ) if p ∈ U, F (U ) = 0 if p ∈ U for U an open subset of X. a) Show that F is a sheaf of OX modules. b) Show that F is not coherent. c) Observe that F is a subOX module of OX which is not an ideal sheaf. Exercise 11.41. Suppose that X and Y are varieties, φ : X → Y is a regular map, and F , G are OX modules. a) Show that there is a natural homomorphism of OY modules φ∗ F ⊗OY φ∗ G → φ∗ (F ⊗OX G). b) Suppose that V is a locally free sheaf on Y (every point q ∈ Y has a neighborhood U such that VU ∼ = OUn for some n). Show that there is a natural isomorphism of OY modules φ∗ (F ⊗OX φ∗ V) ∼ = φ∗ F ⊗OY V. Exercise 11.42. Suppose that X is a quasiprojective variety and I and J are ideal sheaves on X. Suppose that p ∈ X and Ip = Jp . Show that there exists a neighborhood U of p in X such that IU = J U . Exercise 11.43. If A is an integral domain and M is an Amodule, then the Atorsion submodule of M is TA (M ) = {x ∈ M  AnnA (x) = 0}. The fact that TA (M ) is a submodule of M is shown in [13, Exercise 12, Chapter 3]. It has the property that M/TA (M ) is Atorsion free (the torsion submodule is 0). Suppose that X is a quasiprojective variety and F is a quasicoherent OX module. Suppose that σ ∈ Γ(X, F ). Then we have a coherent ideal sheaf AnnX (σ) on X deﬁned by Γ(U, AnnX (σ)) = {τ ∈ Γ(U, OX )  τ (σU ) = 0} for U an open subset of X.
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209
Deﬁne a presheaf T (F ) on X by T (F )(U ) = {σ ∈ F (U )  the sheaf AnnU (σ) = 0} for U an open subset of X. Show that T (F ) is a quasicoherent OX module which has the property that T (F )(U ) = TΓ(U,OX ) (Γ(U, F )) if U is aﬃne. The sheaf T (F ) is called the sheaf of OX torsion of F .
11.6. Some theorems about coherent sheaves The principle results of this section (through Theorem 11.51) were originally proven by Serre in [132]. In this section, we ﬁrst deﬁne the twisted sheaf OX (n). We show that, on a projective variety X with coordinate ring S(X), ˜ where N is a graded S(X)module a quasicoherent sheaf is isomorphic to N ˜ where N is a ﬁnitely generated and a coherent sheaf is isomorphic to N graded S(X)module (Theorem 11.46). We show that if φ : X → Y is a regular map of projective varieties and F is a coherent sheaf on X, then φ∗ F is a coherent sheaf on Y (Theorem 11.51). Suppose that X is a projective variety or more generally a closed subvariety of Y × Pr where Y is an aﬃne variety. We have a natural closed embedding i : X → Pr or more generally a closed embedding i : X → Y × Pr so that X has the coordinate ring S(X) = S(Y × Pr )/I(X). Here S(Y × Pr ) = k[Y ][y0 , . . . , yr ] is a polynomial ring in the variables y0 , . . . , yr , with the grading that the elements of k[Y ] have degree 0 and the yi have degree 1. The graded ideal I(X) = j≥0 I(X)j , so that (11.19)
S(X) = S(Y × Pr )/I(X) = R[x0 , . . . , xr ]
where R = k[Y ]/I(X)0 and xi is the class of yi in S(X) (which has degree 1). In the case that X is a projective variety, we have a closed embedding of X in Pr for some r. Let p be a point. Then p is an aﬃne variety with k[p] = k. Thus we have a natural closed embedding of X in p × Pr ∼ = Pr , which has the coordinate ring S(p × Pr ) = k[p] ⊗k k[y0 , . . . , yr ] ∼ = k[y0 , . . . , yr ] = S(Pr ). The following theorems, which are stated for subvarieties of Y × Pr , are thus valid for the case of a subvariety of Pr , taking Y = p and k[Y ] = k[p] = k in the proofs. We require the more general statements about subvarieties of Y × Pr with Y an aﬃne variety in the proof of Theorem 11.51 and other later applications. Suppose that N is a graded S(X)module. Recall (Section 3.1) that for n ∈ Z, we deﬁne N (n) to be the module N , but with a diﬀerent grading, N (n)i = Nn+i
for i ∈ Z.
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We have a quasicoherent sheaf of OX modules N (n) on X, which is coherent if N is a ﬁnitely generated S(X)module (Example 11.29). We deﬁne a coherent sheaf OX (n) = S(X)(n). Lemma 11.44. Suppose that R is a ring and S = R[x0 , . . . , xn ] is a graded Ralgebra with deg xi = 1 for 0 ≤ i ≤ n. Let N be a graded Smodule. Then N(xi ) xji . Nxi = j∈Z
Proof. Write the graded Sxi module Nxi as Nxi = j∈Z N j , so that N 0 = N(xi ) . Suppose that j ∈ Z. We have that N(xi ) xji ⊂ N j . Suppose that f ∈ N j . Then f = xgl with l ∈ N and g ∈ Na with a − l = j. Now i g xji ∈ N(xi ) xji . f= l+j xi Suppose that N is a graded S(X)module. We have that ˜. N (n) ∼ = F ⊗ OX (n) if F = N
(11.20)
Letting S = S(X), this formula follows from the identities N (n) ∼ = N ⊗S S(n) ∼ ∼ ˜ and N (n) = N ⊗S S(n) = N ⊗ ˜ S(n). This last equality of sheaves follows S
from the sheaf axioms, since we have, by Lemma 11.44, natural isomorphisms N(xi ) ⊗S(xi ) S(n)(xi ) ∼ = N (n)(xi ) which are compatible with localization. More generally, we deﬁne F (n) = F ⊗ OX (n) if F is a sheaf of OX modules on X. We have the following useful formula, which shows that OX (n) is an invertible sheaf (Deﬁnition 11.33): Γ(Xxi , OX (n)) = Γ(Xxi , OX )xni
(11.21)
for 0 ≤ i ≤ n and n ∈ Z. We now prove formula (11.21). By Lemma 11.44, the localization S(X)(xi ) xni S(X)xi = n∈Z
as graded rings. Thus = S(X)(x ) xn Γ(Xxi , OX (n)) = Γ(Xxi , S(X)(n)) i i n. = Γ(Xx , S(X))x i
i
We have the following formula: (11.22)
i∗ (OY ×Pr (n)) ∼ = OX (n).
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211
We now establish formula (11.22). By Lemma 11.44, we have that I(X)yi = I(X)(yi ) yin n∈Z
and = I(X)(y ) y n . Γ((Y × Pr )yi , I(X)(n)) i i We have short exact sequences of S(Y × Pr )(yi ) modules 0 → I(X)(yi ) yin → S(Y × Pr )(yi ) yin → S(X)(xi ) xni → 0 so we have natural isomorphisms Γ(Xxi , OX (n)) = ∼ = ∼ = ∼ = ∼ =
S(X)(xi ) xni = S(Y × Pr )(yi ) yin /I(X)(yi ) yin S(Y × Pr )(yi ) yin ⊗S(Y ×Pr )(y ) (S(Y × Pr )(yi ) /I(X)(yi ) ) i S(Y × Pr )(yi ) yin ⊗S(Y ×Pr )(y ) S(X)(xi ) i Γ((Y × Pr )yi , OY ×Pr (n))⊗Γ((Y ×Pr )yi ,OY ×Pr ) Γ(Xxi , OX ) Γ(Xxi , i∗ (OY ×Pr (n))).
These isomorphisms are compatible with localization, so by the sheaf axioms, we have a natural isomorphism OX (n) ∼ = i∗ (OY ×Pr (n)). Suppose that G is a sheaf of OY ×Pr modules and n ∈ Z. Then i∗ (G(n)) ∼ = (i∗ G)(n).
(11.23)
We now establish (11.23). We have that G(n) = G ⊗OY ×Pr (OY ×Pr (n)) and
(11.24) i∗ (G(n)) = G ⊗OY ×Pr OY ×Pr (n) ⊗OY ×Pr OX ∼ = G ⊗OY ×Pr OX (n) by (11.22). We have i∗ G = G ⊗OY ×Pr OX and (i∗ G)(n) = G ⊗OY ×Pr OX ⊗OX OX (n) ∼ = G ⊗OY ×Pr OX (n) ∼ = i∗ (G(n)) by equation (11.24). Suppose F is a sheaf of OX modules. For all n ∈ Z, we have natural isomorphisms (11.25)
(i∗ F )(n) ∼ = i∗ (F (n)).
We now establish formula (11.25). We have that (i∗ F )(n) is the sheaf associated to the presheaf U → Γ(U ∩ X, F ) ⊗Γ(U,OY ×Pr ) Γ(U, OY ×Pr (n)) for U an open subset of Y × Pr , and we have that i∗ (F (n)) is the sheaf associated to the presheaf
U → Γ(U ∩X, F )⊗Γ(U ∩X,OX ) Γ(U ∩ X, OX ) ⊗Γ(U,OY ×Pr ) Γ(U, OY ×Pr (n)) .
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This last module is naturally isomorphic to Γ(U ∩ X, F ) ⊗Γ(U,OY ×Pr ) Γ(U, OY ×Pr (n)). Thus (i∗ F )(n) ∼ = i∗ (F (n)). A sheaf F of OX modules on a locally ringed space X is said to be generated by a ﬁnite number of global sections if there exist σ1 , . . . , σn ∈ Γ(X, F ) such that Fp = σ1 OX,p + · · · + σn OX,p for all p ∈ X. Theorem 11.45. Let X be a projective variety, or, more generally, a closed subvariety of Y × Pr where Y is an aﬃne variety, with coordinate ring ( Let F be a coherent sheaf S = S(X), and associated sheaf OX (1) = S(1). on X. Then there exists an integer m0 such that for all n ≥ m0 , the sheaf F (n) is generated by a ﬁnite number of global sections. Proof. Write the graded coordinate ring S of X as S = R[x0 , . . . , xr ] where elements of R have degree 0 and x0 , . . . , xr have degree 1, as in (11.19). Let Bi = Γ(Xxi , OX ) = R[ xx0i , . . . , xxri ] for 0 ≤ i ≤ r. Since F is coherent, ˜ i for each there exist ﬁnitely generated Bi modules Mi such that F Xxi ∼ =M i. Let {sij } be a ﬁnite number of generators of Mi for each i. We have that n B (by (11.21)), so the OX (n)Xxi is naturally isomorphic to the module x i i n sheaf F (n)Xx is naturally isomorphic to Mi ⊗ (x Bi ). i
i
We will now show that there exists a positive integer n such that xni sij extends to tij ∈ Γ(X, F (n)) for all i, j. It suﬃces to show that if n0 ∈ M0 , then xn0 n0 extends to an element of Γ(X, F (n)) for all n 0. Let Mij = Γ(Xxi xj , F ) ∼ = (Mi ) xj xi
since F is coherent. There exists λ ∈ N and ni ∈ Mi for 0 < i ≤ r such that ni n0 = x0 λ ( xi ) in Mi0 . Since
ni x0 λ ( xi )
=
nj x0 λ ( xj )
in M0ij = (Mij ) x0 , there exists a ∈ N such that xi
) ni
x0 xj
λ
− nj
x0 xi
λ *
x0 xi
a =0
11.6. Some theorems about coherent sheaves
213
in Mij . Let σi = ni
x0 xi
a+λ xa+2λ ∈ Γ(Xxi , F (a + 2λ)) = Mi xa+2λ . i i
The diﬀerences of the restrictions σi − σj
x
= [( xx0i )a+λ ni − ( xx0i )λ+a ( xji )λ nj ]xa+2λ i x = ( xx0i )a ( xji )λ [ni ( xx0j )λ − nj ( xx0i )λ ]xa+2λ =0 i
= Γ(Xxi xj , F (a + 2λ)). Thus {σi } ∈ in Mij xa+2j i of
Mi xa+2λ is an element i
Γ(X, F (a + 2λ)), extends to an element of Γ(X, F (a + 2λ)), and so n0 xn0 and so σ0 = n0 xa+2λ 0 extends to an element of Γ(X, F (n)) whenever n ≥ a + 2λ. The sections xni sij generate Mi ⊗ (xni Bi ), and so the tij generate F (n) everywhere. Theorem 11.46. Suppose that F is a quasicoherent sheaf on a projective variety X, or, more generally, on a closed subvariety X of Y × Pr where Y is an aﬃne variety, and X has the coordinate ring S(X). Then there exists ˜ . If F is coherent, there exists a a graded S(X)module M such that F ∼ =M ˜. ﬁnitely generated graded S(X)module M such that F = M Proof. Suppose that F is quasicoherent. Let S = S(X) = R[x0 , . . . , xr ] be the coordinate ring of X, and deﬁne a graded Smodule Γ(X, F (n)). M = Γ∗ (F ) = n∈Z
˜ → We will show that there is a natural isomorphism of OX modules β : M F . It suﬃces to deﬁne isomorphisms over the aﬃne open sets Xxi which agree on the Xxi xj . We have that
˜ ) = M(x ) = Γ(Xxi , M i
m  m ∈ Γ(X, F (d)) for some d . xdi
For ﬁxed d, we have that F ∼ = F (d)⊗OX (−d) and OX (−d)Xxi = x−d i k[Xxi ], so we may deﬁne M(xi ) → F (Xxi ) ∼ = [F (d)(Xxi )] ⊗ [OX (−d)(Xxi )] by m → (mXxi ) ⊗ x−d i . xdi
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Suppose m ∈ Γ(X, F (d)) and (mXxi ) ⊗ x−d i = 0. Let mj = mXxj for 0 ≤ j ≤ r. We have that mi = 0 so mj Xxi xj = 0 in Γ(Xxi xj , F (d)) ∼ = F (d)(Xxj ) xi . There exists λ ∈ N such that for all j, ( xxji )λ mj = 0 in xj
Γ(Xxi , F (d)). Consider xλi m ∈ Γ(X, F (d + λ)). Then xi λ λ xi m  Xxj = mj xλj = 0 for all j xj so xλi m = 0. Thus F (Xxi ) is injective.
m xdi
= 0 in M(xi ) . Thus the homomorphism M(xi ) →
The proof of Theorem 11.45 shows that M(xi ) → F (Xxi ) is surjective and hence is an isomorphism. ˜ (Xx ) → F  Xx = F Thus we have natural isomorphisms βi : M (Xxi ) i i which agree on Xxi ∩Xxj = Xxi xj . Thus the βi patch to give an isomorphism of sheaves β. Now suppose that F is coherent. By Theorem 11.45, F (n) is generated by a ﬁnite number of global sections for n 0. For such an n, let M be the submodule of M generated by these sections, so that M is a ﬁnitely generated Smodule. The inclusion of Smodules M ⊂ M induces an in˜ ⊂ M ˜ = F . Tensoring with OX (n), we have an clusion of OX modules M ˜ (n) ⊂ F (n) which is an isomorphism since F (n) is generated inclusion M ˜ (n). After tensoring with OX (−n), we obtain that by global sections of M ∼ ˜ = F. M The proof of the following theorem is similar to that of Theorem 3.35. Theorem 11.47. Suppose that X is a projective variety, or, more generally, a closed subvariety of Y × Pr where Y is an aﬃne variety. Let S = S(X) be the homogeneous coordinate ring of X for this embedding, and let OX (n) be Let S = the coherent sheaf S(n). n≥0 Γ(X, OX (n)). Then S is a ﬁnite Smodule and Γ(X, OX (n)) = Sn for all n 0. Proof. Write S = R[x0 , . . . , xr ] where x0 , . . . , xr have degree 1. We have that Γ(Xxi , OX (n)) = S(n)(xi ) , the elements of degree 0 in the localization S(n)xi . Now S(n) is the ring S with a diﬀerent grading, and we can identify S(n)(xi ) with the elements (Sxi )n of degree n in the localization Sxi (by Lemma 11.44). Since OX (n) is a sheaf, r r Γ(Xxi , OX (n)) = (Sxi )n Γ(X, OX (n)) = i=0
i=0
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215
where the intersection takes place in the graded ring ri=0 Sxi . Since Sn is certainly contained in (Sxi )n for all n, we have natural graded inclusions S ⊂ S ⊂
r
Sxi .
i=0
We will show that S is integral over S. Let s ∈ S be homogeneous of degree d ≥ 0. Since s ∈ Sxi for each i, there exists an integer m such that xm i s ∈ S for all i. The monomials of degree m in x0 , . . . , xr generate Sm as an Rmodule for all m. Thus there exists an n0 such that Sn s ⊂ S for all n ≥ n0 . Thus Sn s ⊂ Sn+d for all n ≥ n0 . It follows that Sn (s )q ⊂ Sn+qd for any q ≥ 1 and all n ≥ n0 . Thus (s )q ∈ xn10 S for all q ≥ 1, and so the ring S[s ] is contained in
1
n x0 0
0
S which is a ﬁnitely generated Smodule. Thus
S[s ] is a ﬁnitely generated Smodule and s is integral over S (Theorem 1.49). Thus S is contained in the integral closure of S in its quotient ﬁeld. Since S is a ﬁnitely generated kalgebra, the integral closure of S in its quotient ﬁeld is a ﬁnitely generated Smodule, by Theorem 1.54. Thus S is a ﬁnitely generated Smodule by Lemma 1.55. Let F1 , . . . , Fr be homogeneous elements in S which generate S as an Smodule. We showed above that there exists N > 0 such that SN Fi ⊂ S for all i. Thus Sn = Sn for all n ≥ max{deg(Fi )} + N . Proposition 11.48. Suppose that Y is an aﬃne variety and X is a closed subvariety of Y × Pr . Let S(X) be the graded coordinate ring of X. Suppose that I ⊂ OX is an ideal sheaf. Then there exists a graded ideal I ⊂ S(X) such that I˜ = I. Proof. Let x0 , . . . , xr be homogeneous coordinates on Pr . For n ∈ Z and p ∈ Xxi , OX (n)p = xni OX,p (by (11.21)), so that OX (n) is an ininduces an inclusion vertible sheaf. Thus the natural inclusion I ⊂ OX I(n) = I ⊗OX OX (n) ⊂ OX (n),and so Γ∗ (I) = n∈Z Γ(X, I(n)) is a graded submodule of Γ∗ (OX ) = n∈Z Γ(X, OX (n)). There exists n0 ∈ N such that Γ(X, OX (n)) = S(X)n for n ≥ n0 by Theorem 11.47. Thus I = Γ(X, I(n)) is a graded ideal in S(X). We have that I˜ = Γ ∗ (I) = I n≥n0
by Theorem 11.46 (and its proof) and Exercise 11.57.
Theorem 11.49. Let Y be an aﬃne variety and X ⊂ Y × Pn be a closed subvariety. Let F be a coherent OX module. Then Γ(X, F ) is a ﬁnitely generated k[Y ]module. In particular, if X ⊂ Pn is a projective variety, then Γ(X, F ) is a ﬁnitely generated kvector space. Proof. Let A = k[Y ] and p = I(X) which is a graded prime ideal in the ]. The coordinate ring of X is the graded domain graded ring A[x0 , . . . , xn ∼ S = A[x0 , . . . , xn ]/p = i≥0 Si , where S0 = A/p ∩ A. By Theorem 11.46,
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11. Sheaves
˜ ∼ there exists a ﬁnitely generated Smodule M such that M = F . By [73, Theorem I.7.4], there is a ﬁnite ﬁltration 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr = M of M by graded submodules, where for each i, M i /M i−1 ∼ = (S/pi )(ni ) for some homogeneous prime ideal pi ⊂ S and some integer ni . The ﬁltra˜ by coherent OX modules, and the short exact tion gives a ﬁltration of M sequences i /M i−1 → 0 ˜ i → M ˜ i−1 → M 0→M induce exact sequences i /M i−1 ). ˜ i ) → Γ(X, M ˜ i−1 ) → Γ(X, M 0 → Γ(X, M
˜ ) is a ﬁnitely generated Amodule, it thus suﬃces To show that Γ(X, M to show that each Γ(X, S/p i (ni )) is a ﬁnitely generated Amodule, which follows from (11.14), Exercise 11.27, (11.25), and Theorem 11.47.
Theorem 11.50. Suppose that φ : X → Y is a regular map of quasiprojective varieties and F is a quasicoherent sheaf on X. Then φ∗ F is a quasicoherent sheaf on Y . If Y is an aﬃne variety, X is a closed subvariety of Y × Pr , F is coherent on X, and π1 : Y × Pr → Y is the projection with restriction π 1 to X, then (π 1 )∗ F is coherent on Y . Proof. Suppose that F is quasicoherent. We will show that φ∗ F is quasi˜ for some coherent. First assume that X and Y are aﬃne, so that F = M k[X]module M . Let A = k[Y ]. Let f ∈ A and U = Yf . Then ˜ (Xf ) = Mf . (φ∗ F )(U ) = F (φ−1 (U )) = M ˜ A )(Yf ) Thus writing MA for M considered as an Amodule, we have that (M = φ∗ F (Yf ). Since any open subset of Y is a union of basic open sets Yf , we ˜ A is quasicoherent. have by the sheaf axioms that φ∗ F = M Now we prove, more generally, that φ∗ F is quasicoherent if X and Y are quasiprojective and F is quasicoherent. By Theorem 11.32, we may assume that Y is aﬃne. Cover X with a ﬁnite number of open aﬃne sets U1 , . . . , Ur . Let Ui,j = Ui ∩ Uj which is aﬃne (Exercise 5.21). By the sheaf properties, we have exact sequences of sheaves φ∗ (λi∗ F Ui ) → φ∗ (λi,j 0 → φ∗ F → ∗ F Ui,j ) i
i<j
where λi : Ui → X and λi,j : Ui,j → X are the natural inclusions. We have i,j i that φ∗ λi∗ = (φ ◦ λi )∗ and φ∗ λi,j ∗ = (φ ◦ λ )∗ , so the sheaves φ∗ λ∗ (F Ui ) and i,j φ∗ λ∗ (F Uij ) are quasicoherent by the ﬁrst part of this proof. Thus φ∗ F is quasicoherent by Exercise 11.35 and Theorem 11.32.
11.6. Some theorems about coherent sheaves
217
Now suppose that F is coherent and X is a closed subvariety of Y × Pn where Y is aﬃne. We will show that (π 1 )∗ F is coherent. Since (π 1 )∗ F is quasicoherent, it suﬃces to show that Γ(Y, (π1 )∗ F ) is a ﬁnitely generated A = k[Y ]module. But this follows from Theorem 11.49. Theorem 11.51. Suppose that φ : X → Y is a regular map of projective varieties and F is a coherent sheaf on X. Then φ∗ F is a coherent sheaf on Y . Proof. Let Γφ ⊂ X × Y be the graph of φ, with natural isomorphism j = (id, φ) : X → Γφ . Let π2 : X × Y → Y be the projection, with induced projection π 2 : Γφ → Y . Then φ∗ F ∼ = (π 2 )∗ j∗ F , which is coherent by Theorem 11.50. Suppose that X is a variety, with function ﬁeld k(X). The constant sheaf k(X) satisﬁes Γ(U, k(X)) = k(X) for all open subsets U of X by Proposition 11.14, since all open subsets of X are connected. Proposition 11.52. Suppose that φ : X → Y is a birational regular map of projective varieties and Y is normal. Then φ∗ OX = OY . Proof. The inclusion OX → k(X) of OX modules induces an inclusion φ∗ OX → φ∗ k(X) = k(X) = k(Y ) of OY modules. Since φ∗ OX is a coherent OY module by Theorem 11.51 under the natural inclusion OY → φ∗ OX and OY is normal, we have that φ∗ OX = OY . Proposition 11.53. Suppose that X is a variety and M is a quasicoherent sheaf on X such that M is a subsheaf of OX modules of the constant sheaf k(X). Suppose that U is an open subset of X. Then Mp M(U ) = p∈U
where the intersection takes place in k(X). Proof. We have that M(U ) ⊂ Γ(U, k(X)) = k(X) for all open subsets U of X by Proposition 11.14 and Proposition 11.20. Let U be an open subset of X and {V1 , . . . , Vn } be an aﬃne open cover of U . By the sheaf axioms, M(U ) = ni=1 M(Vi ), where the intersection is in k(X). The conclusions of the proposition now hold by formula (11.10). A coherent sheaf F on a variety X is locally free if for all p ∈ X, there exists an open neighborhood U of p in X such that F U is a free OU module (isomorphic to OUr for some r).
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11. Sheaves
Lemma 11.54. Let X be a quasiprojective variety, let p ∈ X, and let F be a coherent OX module. Let λ(p) = dimk Fp /mp Fp where mp is the maximal ideal of OX,p . Then λ(p) is upper semicontinuous on X; that is, for t ∈ N, the set {q ∈ X  λ(q) ≥ t} is a closed set in X. Further, there exists an open neighborhood U of p such that F U is a free OU module (isomorphic to OUr for some r) if and only if λ is constant in a neighborhood of p. Proof. Let p ∈ X and r = λ(p). Let k(p) = OX,p /mp . There exist a1 , . . . , ar ∈ Fp such that their images generate Fp ⊗ k(p). Let U1 be a neighborhood of p such that the ai lift to elements of Γ(U1 , F ).Deﬁne an OU1 module homomorphism φ : OUr 1 → F U1 by φ(b1 , . . . , br ) = ai bi . Let K be the cokernel of φ, which is a coherent OU module (Lemma 11.31). We have exact sequences φ
r → Fp → K p → 0 OX,p
and φ
k(p)r → Fp ⊗ k(p) → Kp ⊗ k(p) → 0.
The homomorphism φ is surjective, so 0 = Kp ⊗ k(p) ∼ = Kp /mp Kp . By Nakayama’s lemma, Lemma 1.18, Kp = 0. Hence p is in the open set X \ Supp(K) (Exercise 11.36). Let U2 = U1 ∩ (X \ Supp(K)). We have that KU2 = 0, so a1 , . . . , ar generate F U2 . Hence the images of a1 , . . . , ar generate Fq ⊗ k(q) for all q ∈ U2 so that λ(q) ≤ r if q ∈ U2 . We now prove the second statement. If F is a free OX module of rank r in a neighborhood U of p, then F U ∼ = (OX U )r for some r and λ(q) = r for all q ∈ U . Conversely, suppose that U1 is a nontrivial open subset of X such that λ(p) = r for all p ∈ U1 . As in the ﬁrst part of the proof, there exists a possibly smaller open subset U2 of X, which we may take to be aﬃne, with a surjective OX module homomorphism ψ
ψ : OUr 2 → F U2 → 0. Let K be the kernel of ψ. We will show that K = 0, establishing that F U2 is free, and we will then have completed the proof. Suppose that K = 0. We will derive a contradiction. We have (since U2 is aﬃne) a short exact sequence 0 → K → Rr → M → 0
11.6. Some theorems about coherent sheaves
219
˜ where K = Γ(U2 , K), R = Γ(U2 , OX ), and M = Γ(U2 , F ). Since K = K, there exists 0 = s ∈ K. Write s = (s1 , . . . , sr ) where si ∈ R and some si = 0. Then Z(s1 , . . . , sr ) = U2 (by the nullsellensatz), so there exists q ∈ U2 \ Z(s1 , . . . , sr ), giving us that s ∈ mq OUr 2 ,q . Thus the leftmost map is nonzero in the exact sequence K ⊗ k(q) → k(q)r → Fq ⊗ k(q) → 0, giving us that λ(q) < r, a contradiction.
The following extension theorem for coherent sheaves will be useful. Theorem 11.55. Suppose that X is a variety and U is an open subset of X. Suppose that F is a coherent sheaf on U and G is a quasicoherent sheaf on X such that F is an OU submodule of GU . Then there exists a coherent sheaf F on X such that F is an OX submodule of G and F U = F . Proof. Let i : U → X be the inclusion and deﬁne ρ : G → i∗ (GU ) to be the restriction ρ(V ) : G(V ) → G(U ∩ V ) for V an open subset of X. The sheaf F is a submodule of GU , so i∗ F is a submodule of i∗ (GU ). Let H be the submodule H = ρ−1 (i∗ F ) of G. Since ρU is the identity map, we have that HU = F . Let V be an aﬃne open subset of X. Then since i∗ F and G are quasicoherent and V is aﬃne, there exist k[V ]modules A, B, and C with ˜ i∗ (GU )V ∼ ˜ and ˜ i∗ F V ∼ A a submodule of C such that GV ∼ = A, = C, = B, there exists a k[V ]module homomorphism φ : B → C which induces ρV . ˜ where M is the k[V ]module Thus HV ∼ =M M = {(a, b) ∈ A ⊕ B  φ(b) = a}, and so H is quasicoherent. Let {Mi } be the directed systemof all ﬁnitely generated OX submodules of H. We have that lim→ Mi = i Mi = H, so that i Mi = H, and F = i (Mi U ). Each Mi U is coherent and F is coherent, so restricting to a ﬁnite aﬃne open cover of X, we see that F = Mi U for some i, establishing the theorem. Exercise 11.56. Suppose that F is a coherent sheaf on a projective variety X with projective coordinate ring S. Show that F is a quotient sheaf of a ﬁnite direct sum E = ni=1 OX (qi ) for some n ∈ N and q1 , . . . , qn ∈ Z. Exercise 11.57. Suppose that X is a closed subvariety of Y × Pr where Y is an aﬃne variety and S(X) is the coordinate ring of X. Suppose that M is a graded S(X)module. Suppose that d0 ∈ Z. Let N = d≥d0 Md , which ˜ and N ˜ is a graded S(X)module. Show that the quasicoherent sheaves M are isomorphic.
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11. Sheaves
Exercise 11.58. In this problem we consider the question of coherence of the pushforward of a coherent sheaf. a) Let p be the origin in A1 , let U = A1 \ {p}, and let i : U → A1 be the inclusion. Determine if i∗ OU is a coherent OA1 module. b) Let q be the origin in A2 , let V = A2 \ {q}, and let j : V → A2 be the inclusion. Determine if j∗ OV is a coherent OA2 module. Exercise 11.59. Suppose that F is a coherent sheaf on a projective variety X with homogeneous coordinate ring S(X). Show that n≥0 Γ(X, F (n)) is a ﬁnitely generated S(X)module. Show that if X is a projective variety, p ∈ X, and F is the coherent sheaf F = OX /Ip , where Ip is the ideal sheaf of p in X, then Γ∗ (F ) = n∈Z Γ(X, F (n)) is not a ﬁnitely generated S(X)module.
Chapter 12
Applications to Regular and Rational Maps
In this chapter we deﬁne the blowup of an ideal sheaf and use this to further develop the theory of regular and rational maps.
12.1. Blowups of ideal sheaves In this section, we extend the blowup of an ideal to the blowup of an ideal sheaf. Deﬁnition 12.1. Suppose that X is an aﬃne variety and I is an ideal sheaf on X. From Theorem 11.32, we have that I ∼ = I˜ where I is the ideal Γ(X, I) in k[X]. We deﬁne the blowup of the ideal sheaf I, written as π : B(I) → X, by the construction of Deﬁnition 6.2 as B(I) = B(I). If X is an aﬃne variety with ideal sheaf I and U is an aﬃne open subset of X, then a set of generators of I = Γ(X, I) is a set of generators of Γ(U, I) (by Exercise 1.47 since they generate locally at all local rings of U ) so by the construction of Deﬁnition 6.2, there is a natural commutative diagram where the horizontal arrows are open embeddings B(IU ) πU
U
/ B(I)
πX
/X
−1 (U ). identifying B(I  U ) with πX
221
222
12. Applications to Regular and Rational Maps
Deﬁnition 12.2. Suppose that X is a projective variety and I is an ideal sheaf on X. From Proposition 11.48, we can write I = I˜ where I is a homogeneous ideal in S(X). By Lemma 6.7, we can take I to be generated by homogeneous elements F0 , . . . , Fn of some common degree d. In Deﬁnition 6.8, we deﬁned the blowup π : B(I) → X of I, where B(I) is a projective variety. We deﬁne the blowup B(I) of I to be π : B(I) → X. This deﬁnition is welldeﬁned by the following lemma, and Proposition 3.39, applied to an aﬃne open cover of X. Lemma 12.3. Suppose that X is a projective variety and π : B(I) = B(I) → X is the blowup of the ideal sheaf I. Suppose that U ⊂ X is an open aﬃne subset. Then there is a natural commutative diagram where the horizontal arrows are open embeddings B(IU ) πU
/ B(I)
U
πX
/X
−1 (U ). identifying B(IU ) with πX
Proof. It is shown after Deﬁnition 6.8 that the conclusions of the lemma are true when U = Xxi for homogeneous coordinates x0 , . . . , xn on X. For an arbitrary aﬃne open subset U of X, each Xxi is aﬃne, and Uxi is an aﬃne open subset of Xxi . Further, each Uxi ∩ Uxj = Uxi xj is an aﬃne open subset of Uxi . We thus have natural commutative diagrams for all i, j B(IUxi xj )
/ B(IUx ) i
/ B(I  Xx ) i
/ B(I)
πX
Uxi xj
/ Ux i
/ Xx i
where the horizontal maps are inclusions. Since {Uxi } is an aﬃne cover of U , B(IU ) = commutative diagram of welldeﬁned maps
/X
i B(IUxi ),
so there is a
B(IU ) → B(I) ↓ ↓ U → X which are regular by Proposition 3.39.
Deﬁnition 12.4. Suppose that X is a quasiprojective variety which is an open subset of a projective variety Y and I is an ideal sheaf on X. By Theorem 11.55, there exists an ideal sheaf J of Y such that J X = I. We
12.1. Blowups of ideal sheaves
223
deﬁne the blowup B(I) of I to be B(I) = π −1 (X) → X where π : B(J ) → Y is the blowup of J . Since an open aﬃne subset of X is also an open aﬃne subset of Y , we have that the conclusions of Lemma 12.3 hold when X is assumed to be quasiprojective, showing that B(I) is welldeﬁned in Deﬁnition 12.4. We have that B(I) is a quasiprojective variety since it is an open subset of the projective variety B(J ). If X is a variety and I is an ideal sheaf of X, then I is nonzero if and only if Ip = 0 for all p ∈ X. Theorem 12.5 (The universal property of blowing up). Suppose that Y is a quasiprojective variety and I is an ideal sheaf on Y . Let π : B(I) → Y be the blowup of I. Suppose that φ : X → Y is a regular map of quasiprojective varieties such that the IOX is a nonzero locally principal ideal sheaf (there is an aﬃne open cover {Ui } of X such that the Γ(Ui , IOX ) are nonzero principal ideals). Then there exists a unique regular map ψ : X → B(I) factoring φ. A nonzero locally principal ideal sheaf is an example of an invertible sheaf (Deﬁnition 11.33). Proof. Let p ∈ X. Let V be an aﬃne neighborhood of φ(p) in Y and U be an aﬃne neighborhood of p in X such that φ(U ) ⊂ V . Let I = Γ(V, I) ⊂ k[V ]. Since IOX is locally principal, we may assume, after possibly replacing U with a smaller aﬃne neighborhood of p, that Ik[U ] = Γ(U, IOX ) is a principal ideal. Let f0 , . . . , fr be a set of generators of the k[V ]ideal I. Then B(I) = π −1 (V ) ⊂ V × Pr is the graph of the rational map V Pr deﬁned by (f0 : . . . : fr ). The projection π : B(I) → V is a regular birational map, which has the inverse rational map π −1 : V B(I) deﬁned by π −1 = id × (f0 : . . . : fr ). Thus we have a rational map π −1 φ : U B(I). Now this rational map is exactly φ×(φ∗ (f0 ) : . . . : φ∗ (fr )). The ideal Ik[U ] which is generated by φ∗ (f0 ), . . . , φ∗ (fr ) in k[U ] is by assumption principal. We will show that, after possibly replacing U with an aﬃne neighborhood of p in U , we have that there exists an i such that φ∗ (fi ) generates the ideal Ik[U ]. By our choice of U , Ik[U ] = gk[U ] for some nonzero g∗∈ k[U ]. Thus with all ai ∈ k[U ] and φ (fi ) = di g for there exists a relation ai φ∗ (fi ) = g ai di = 1 since some di ∈ k[U ] for 0 ≤ i ≤ r. Thus ( ai di )g = g and so g is nonzero and k[U ] is an integral domain. Thus there exists an i such that di ∈ IU (p). Thus Udi is an aﬃne neighborhood of p and φ∗ (fi )k[Udi ] = Ik[Udi ]. Thus after replacing U with Udi , we have that Ik[U ] = φ∗ (fi )k[U ].
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12. Applications to Regular and Rational Maps
Without loss of generality, we may assume that i = 0. Thus there exist regular functions hi ∈ k[U ] such that φ∗ (fi ) = φ∗ (f0 )hi for i > 0. In particular, φ × (φ∗ (f0 ) : . . . : φ∗ (fr )) = φ × (1 : h1 : . . . : hr ), which is a regular map on U , since the hi are regular on U and 1 is never zero. Now the rational map π −1 φ is uniquely determined and has at most one regular extension to U . Thus there is a unique regular map ψ : U → B(I) which factors φU . We can thus construct an aﬃne open cover {Ui } of X and unique regular maps ψi : Ui → B(I) factoring φUi . By Proposition 3.39 and since a regular map of a quasiprojective variety is uniquely determined by its restriction to a nontrivial open set, the ψi patch to a unique regular map ψ : X → B(I) which factors φ. Proposition 12.6. Suppose that X ⊂ Pm is a projective variety, with homogeneous coordinate functions x0 , . . . , xm on X. Suppose that φ : X Pn is a rational map, which is represented by homogeneous elements F0 , . . . , Fn ∈ S(X) = k[x0 , . . . , xm ] of a common degree d (which are necessarily not all ˜ Suppose that p ∈ X. Then zero). Let I = (F0 , . . . , Fn ) ⊂ S(X) and I = I. φ is regular at p if and only if Ip is a principal ideal in OX,p . In particular, φ is a regular map if and only if I˜ is a locally principal ideal sheaf. Proof. Suppose that Ip is a principal ideal for some p ∈ X. Without loss of generality, we may suppose that p ∈ Xx0 . Then F0 Fn x1 xm x1 xm , . . . , d = F0 (1, , . . . , ), . . . , Fn (1, , . . . , ) I(Xx0 ) = x0 x0 x0 x0 xd0 x0 x1 xm ,..., ⊂ k[Xx0 ] = k . x0 x0 Let I(p) be the ideal of p in k[Xx0 ]. We have by assumption that Ip = I(Xx0 )I(p) is a nonzero principal ideal in the local ring OX,p = k[Xx0 ]I(p) , so that some
Fj xd0
generates Ip (by Exercise 1.12 or as in the proof of Theorem
12.5). Without loss of generality,
F0 xd0
generates this ideal. Thus there exists
an aﬃne neighborhood U of p contained in Xxi such that
F0 xd0
generates the
ideal I(U ) in k[U ] (by Exercise 11.42). In particular, there exist regular functions hj ∈ k[U ] such that Fj F0 = hj d d x0 x0 Now φ is represented by ( Fxd0 : 0
F1 xd0
. . . : hn ), so that φ is regular at p.
: ... :
for all j. Fn ) xd0
which is equivalent to (1 : h1 :
12.2. Resolution of singularities
225
Now suppose that φ is regular at p. Then there exist an aﬃne neighborhood U of p and h0 , . . . , hn ∈ k[U ] such that ZU (h0 , . . . , hn ) = ∅, and φ is represented by (h0 : h1 : . . . : hn ). Without loss of generality, we may assume that p ∈ Xx0 and that U ⊂ Xx0 . Thus Fj Fi hj = hi d d x0 x0
for all i, j.
We must have that some hi satisﬁes hi (p) = 0. We may suppose that h0 (p) = 0. After possibly replacing U with a smaller aﬃne neighborhood of p, we may assume that h0 (q) = 0 for all q ∈ U , so that h0 is a unit in k[U ]. We have that h i F0 Fi = d h0 xd0 x0
for all i, so that I(U ) = Fxd0 , . . . , Fxnd k[U ] is a principal ideal, generated by F0 . xd0
0
0
Thus the localization Ip = I(U )IU (p) is a principal ideal.
Theorem 12.7 (Resolution of indeterminancy). Suppose that X ⊂ Pm is a projective variety, with homogeneous coordinate functions x0 , . . . , xm on X. Suppose that φ : X Pn is a rational map, which is represented by homogeneous elements F0 , . . . , Fn ∈ S(X) = k[x0 , . . . , xm ] of a common ˜ Let π : B(I) → X be the degree d. Let I = (F0 , . . . , Fn ) ⊂ S(X) and I = I. blowup of I, and let φ be the rational map φ = φπ : B(I) Pn . Then φ is a regular map. Proof. This follows directly from our deﬁnition of the blowup B(I) as the graph Γφ ⊂ X × Pn of the rational map φ = (F0 , . . . , Fn ). The map φπ is the projection on the second factor Pn .
12.2. Resolution of singularities In this section we survey the main results on resolution of singularities. The interested reader is referred to the book Resolution of Singularities [35] and the article [38], which gives an accessible proof of resolution of singularities of 3folds in characteristic greater than 5, for the proofs of the main results discussed in this section. Reading through Chapter 15 on schemes in this book is adequate preparation for reading [35]. Reading [35] is a good preparation for reading [38]. Deﬁnition 12.8. Suppose that X is a quasiprojective variety. A resolution of singularities of X is a closed subvariety Y of X × Pn (for some n) such that the projection π : Y → X is birational and Y is nonsingular. Lemma 12.9. Suppose that π : Y → X is a resolution of singularities. Then Y = B(I) for some ideal sheaf I on X.
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12. Applications to Regular and Rational Maps
Proof. There exists a projective variety X such that X is an open subset of X and a closed embedding of Y into X × Pn for some n. Let Y be the Zariski closure of Y in X × Pn . We have that (X × Pn ) ∩ Y = Y since Y is closed in X × Pn . The projection Y → X is a birational regular map of projective varieties so Y = B(J ) for some ideal sheaf J on X by Theorem 6.9. Letting I = J X, we have that Y is isomorphic to the blowup of I by Lemma 12.3. Theorem 12.10. Suppose that X is a quasiprojective curve (a 1dimensional variety). Then X has a resolution of singularities. Proof. There exists a projective variety X such that X is an open subset of X. Let Y be the normalization of X in the function ﬁeld k(X) of X (by Theorem 7.17), with induced regular map φ : Y → X. The variety Y is a projective variety (by Theorem 7.17), so there is a closed embedding of Y in Pn for some n. The variety Y is nonsingular by Theorem 1.87 since Y is a curve and Y is normal. We have closed embeddings Y ∼ = Γφ ⊂ Y × X ⊂ Pn × X where Γφ is the graph of φ and the projection onto X is the map φ. Thus Y is a resolution of singularities of X, and Y = Γφ ∩ (Y × X) is a resolution of singularities of X. Theorem 12.11. Suppose that X is a quasiprojective curve. Consider the sequence (12.1)
π
π
n · · · → X1 →1 X0 = X, · · · → Xn →
where Xn+1 → Xn is obtained by blowing up the (ﬁnitely many) singular points on Xn . Then this sequence is ﬁnite (there exists an n such that Xn is nonsingular) and is thus a resolution of singularities of X. Proof. It suﬃces to prove the theorem when X is projective. Then all Xn are projective. Let φ : Y → X be the normalization of X (Theorem 7.17). The variety Y is projective since X is, and Y is a resolution of singularities of X (Theorem 12.10). We have a factorization (by Corollary 10.24) π
π
n · · · → X1 →1 X Y → Xn →
of regular maps of projective varieties for all n. Since each map in the sequence is a dominant regular map of curves, the preimage of a point by each map is a ﬁnite set of points. Thus each map in the sequence is ﬁnite by Theorem 9.6. If the sequence (12.1) is inﬁnite, then there exists an aﬃne open subset U of X such that the induced sequence of preimages above U in the sequence (12.1) · · · → Un → · · · → U1 → U
12.2. Resolution of singularities
227
is inﬁnite, and each Un is singular. Let Z be the preimage of U in Y . The sets Z and Ui are all aﬃne open sets since the maps to U are ﬁnite. Let R = R0 = k[U ] and let R be the integral closure of R in the function ﬁeld k(X) of X. Then k[Z] = R. Let Ri = k[Ui ] for all i. We have inclusions in k(X) R0 ⊂ R1 ⊂ · · · ⊂ Rn ⊂ · · · ⊂ R. We have that Ri = Ri+1 for all i since mRi+1 is a locally principal ideal if m is a maximal ideal of Ri such that (Ri )m is not regular (Ui+1 is the blowup of all singular points of Ui ) and m(Ri )m is not principal (since (Ri )m is not regular). Since R is ﬁnite over R and the sequence (12.1) is assumed inﬁnite, there exists an n such that Rn = R. But then Un is nonsingular, a contradiction. An approach to resolve surface singularities is by the following algorithm. Suppose that S is a surface. Let S1 be the normalization of S, so that S1 has only ﬁnitely many singular points (by Theorem 10.17), and let S2 → S1 be the blowup of all singular points on S2 . If S2 is nonsingular, we have obtained a resolution of singularities of S. Otherwise, we can repeat, computing the normalization S3 of S2 and blowing up all singular points of S3 to obtain S4 . We can repeat as long as Si is singular, obtaining a sequence (12.2)
· · · → S2n → S2n−1 → · · · → S2 → S1 → S.
Theorem 12.12. Suppose that S is a quasiprojective surface. Then some Si is a resolution of singularities of S. This theorem was proven when the ground ﬁeld k has characteristic 0 by Zariski in [149] and was proven in arbitrary characteristic (in fact over twodimensional excellent integral schemes) by Lipman [101]. Zariski discusses early approaches to resolution of surface singularities in his book [148]. Zariski credited Walker’s proof [144] as the ﬁrst complete proof of resolution of singularities of complex surfaces. The ﬁrst proof of resolution of singularities of surfaces in positive characteristic was given by Abhyankar in [2] (by a diﬀerent method). Hironaka proved the existence of a resolution of singularities of a variety of any dimension in characteristic 0. Theorem 12.13 (Hironaka). Suppose that X is a variety over a ﬁeld of characteristic 0. Then X has a resolution of singularities. Hironaka’s ﬁrst proof is in [79]. There have been many simpliﬁcations in the proof (some papers and books on this are [22], [24], [54], [35], [90]).
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12. Applications to Regular and Rational Maps
Abhyankar [4] proved resolution of singularities of threedimensional varieties over ﬁelds of positive characteristic greater than 5. Since then, Cossart and Piltant have established the existence of a resolution of singularities for reduced threedimensional quasiexcellent schemes in [31]. All of these proofs are extremely long. A much shorter proof of Abhyankar’s result is given in [38]. Some recent papers on resolution in higher dimensions and positive characteristic are [25], [75], [80], [94], [87], [140], [141]. Exercise 12.14. Resolve the singularities of the curve C = Z(x2 −x4 −y 4 ) ⊂ A2 by blowing up points; that is, perform a sequence of blowups of points above A2 so that the strict transform of C is nonsingular.
12.3. Valuations in algebraic geometry Suppose that K is a ﬁeld. A valuation ν of K is a map ν : K × → Γν from the multiplicative group K × of nonzero elements of K onto a totally ordered Abelian group Γν , called the value group of ν. The map ν must satisfy two properties: 1. ν(f g) = ν(f ) + ν(g) for f, g ∈ K × , 2. ν(f + g) ≥ min{ν(f ), ν(g)} for f, g ∈ K × . The valuation ν extends to K by deﬁning ν(0) = ∞, which is larger than anything in Γν . The valuation ring of ν is Vν = {f ∈ K  ν(f ) ≥ 0}. The basic theory of valuation rings is explained in the paper [152], [160, Chapter V], [161, Chapter VI], and in [6]. A quick introduction is given in the section on valuation rings in of [13, Chapter 5]. The next two theorems state a couple of basic facts about valuation rings. Theorem 12.15. A valuation ring Vν is Noetherian if and only if Γν ∼ = Z. Proof. [161, Theorem 16, page 41].
Theorem 12.16. A valuation ring Vν is a normal local ring. Proof. [13, Proposition 5.18].
If K is a function ﬁeld over a ﬁeld κ, we require a valuation of K to also satisfy the third property that 3. νκ× = 0.
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229
A valuation ν of K is called divisorial if Vν is a localization of a ﬁnitely generated algebra over κ. Since such a ring is Noetherian, Γν ∼ = Z if ν is divisorial. If K is the function ﬁeld of a curve, then all of the valuation rings of K are divisorial. In fact, by [161, Theorem 9, page 17], the valuation rings of K are exactly the local rings of the points on the nonsingular projective curve whose function ﬁeld is K, and the ﬁeld K itself (the valuation ring of the trivial valuation). If K has transcendence degree larger than 1 over its ground ﬁeld κ (so that K is the function ﬁeld of a variety of dimension larger than 1), then K admits many valuations whose valuation ring is not Noetherian. The example in [161, pages 102–104] shows that every additive subgroup of the rational numbers must appear as the valuation group of a valuation of K, so K has many valuations whose value group is not ﬁnitely generated. A theory of algebraic geometry built around valuation rings is developed in Zariski’s paper [152]. This is the paper where Zariski’s main theorem ﬁrst appears. This theory requires the introduction of more local rings on a variety than we have considered up to now. Deﬁnition 12.17. Suppose that X is a quasiprojective variety and Y is a subvariety of X. Then the local ring OX,Y is deﬁned to be the localization OX,Y = k[U ]IY (U ) where U is any aﬃne open subset of X such that Y ∩U = ∅ and IY is the ideal sheaf of Y in X. A special case is when Y is a point, in which case the above deﬁnition agrees with the deﬁnition of the stalk OX,Y of OX at the point Y . This ring is independent of the choice of U . In fact, we have that if p ∈ Y , then OX,Y is the localization OX,Y = (OX,p )IY,p . We see this as follows. Suppose that U is an aﬃne neighborhood of p in X. Then OX,p = k[U ]I(p) and IY,p = (IY (U ))I(p) where I(p) is the ideal of p in k[U ]. Thus (OX,p )IY,p ∼ = k[U ]IY (U ) by Exercise 1.22 since IY (U ) ⊂ I(p). If U and V are two aﬃne open subsets of X which intersect Y , then there exists a point p in U ∩ V ∩ Y since Y is irreducible. Thus k[U ]IY (U ) = (OX,p )IY,p = k[V ]IY (V ) . If X is aﬃne, then the local rings OX,Y of X are precisely the local rings k[X]P for P ∈ Spec(k[X]) (the spectrum of a ring is deﬁned in Exercise 1.11). If X is projective, then the local rings OX,Y of X are precisely the local rings S(X)(P ) for P ∈ Proj(S(X)) (the Proj of a graded ring is deﬁned in Exercise 3.7).
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Suppose that X is a variety and K = k(X) is the function ﬁeld of X. Let ν be a valuation of K. We will say that ν dominates a local ring S contained in K if S ⊂ Vν and the maximal ideal of Vν intersects S in its maximal ideal. Theorem 12.18 (Zariski). Suppose that X is a projective variety and ν is a valuation of k(X). Then there is a unique subvariety Y of X such that ν dominates OX,Y . Proof. Let S = k[x0 , . . . , xn ] be the coordinate ring of a projective embedx ding of X. Suppose that i is such that ν( xx0i ) ≤ ν( x0j ) for 0 ≤ j ≤ n. Then x x ν( xji ) = ν( x0j ) − ν( xx0i ) ≥ 0 for 0 ≤ j ≤ n. Thus x0 xn ,..., ⊂ Vν . OX (Xxi ) = k xi xi Let p = mν ∩ OX (Xxi ) where mν is the maximal ideal of Vν . Let Y be the Zariski closure in X of the subvariety Z(p) of Xxi . Then OX,Y = (OX (Xxi ))p is dominated by ν. Suppose that Z is another closed subvariety of X which has the property that OX,Z is dominated by ν. There exists a linear form L ∈ S such that L ∈ I(Y ) and L ∈ I(Z) (by Lemma 8.10). Then Z ∩XL = ∅ and Y ∩XL = ∅. Let R = k[XL ], p1 = I(Y ∩ XL ), and p2 = I(Z ∩ XL ). Here Rp1 = OX,Y and Rp2 = OX,Z . We have that R ⊂ Vν since Rp1 ⊂ Vν . Since Vν dominates both Rp1 and Rp2 , we have that p1 = mν ∩ R = p2 . But this is impossible since I(Y ) and I(Z) are distinct prime ideals in S. We will call the subvariety Y of the conclusions of Theorem 12.18 the center of ν on X. The following lemma gives a characterization of valuation rings of divisorial valuations. Lemma 12.19. Suppose that K is an algebraic function ﬁeld over a ﬁeld κ. Suppose ν is a divisorial valuation of K. Then there exists a normal local ring R which is of ﬁnite type over κ and a height 1 prime ideal Q in R such that RQ = Vν . Conversely, given a subring R of K with these properties and a height 1 prime ideal Q of R, there exists a divisorial valuation ν of K such that RQ = Vν . Proof. A ring A = RQ as in the statement of the lemma is a normal Noetherian local ring of dimension 1, so it is a regular local ring of dimension 1 by Theorem 1.87. Let f ∈ A be a generator of its maximal ideal. Then every nonzero element h ∈ K has a unique expression h = uf n where u ∈ A is a unit and n ∈ Z. Deﬁne ν(h) = n ∈ Z. The function ν is a valuation of K as it satisﬁes the three conditions of a valuation and A = Vν .
12.3. Valuations in algebraic geometry
231
Now suppose that ν is a divisorial valuation of K. The value group of ν is Z by Theorem 12.15 since Vν is Noetherian. Let mν be the maximal ideal of Vν . There exists f ∈ mν such that ν(f ) = 1. Suppose g ∈ mν . Then ν(g) ≥ 1. Let h = fg ∈ K. We have that ν(h) ≥ 0 so h ∈ Vν . Thus g ∈ (f ). We have that mν = (f ), so that Vν is a regular local ring of dimension 1. By assumption, there exist a domain B with quotient ﬁeld K which is ﬁnitely generated over κ and a prime ideal P in B such that BP = Vν . Let R be the integral closure of B in K, which is a ﬁnitely generated κalgebra by Theorem 1.54. Then R ⊂ BP since Vν is normal. Let Q = PP ∩ R. Then Vν ∼ = RQ . We have that Q is a height 1 prime ideal in R since dim Vν = 1. Zariski’s proof of resolution of singularities of surfaces in Theorem 12.12 ([149]) is by assuming that the algorithm described before the statement of the theorem does not terminate. Then we have an inﬁnite sequence of points pi ∈ S2i such that pi maps to pi−1 for all i and there is an inﬁnite sequence of distinct local rings R → R1 → · · · → Ri → · · · where Ri = OS2i ,pi is a normal but not regular local ring. Let V = ∞ i=1 Ri . Then V is the valuation ring Vν of a valuation ν of k(S) (this uses the fact that S has dimension 2). Zariski then shows that ν has a local uniformization (deﬁned below) and then makes a delicate argument to show that this leads to a contradiction to the assumption that the localizations of all of the Ri at the ideal of pi are not regular. Deﬁnition 12.20. Suppose that K is an algebraic function ﬁeld and ν is a valuation of K. The valuation ν has a local uniformization if there exists a variety X whose function ﬁeld is K such that the center of ν on X is a regular local ring. Zariski proved local uniformization of all valuations on characteristic 0 function ﬁelds [150] and was able to show from this result that resolution of singularities is true for varieties of dimension ≤ 3 over (algebraically closed) ﬁelds of characteristic 0 [153]. Hironaka’s proof of resolution of singularities of characteristic 0 varieties [79] does not use valuations or local uniformization. The ﬁrst proof of resolution of singularities of surfaces in characteristic p > 0 was by Abhyankar [2]. It is by proving local uniformization of all valuations on a twodimensional algebraic function ﬁeld. The proof by Lipman [101] of resolution of surface singularities does not use local uniformization.
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However, all proofs of resolution of singularities of threedimensional varieties in positive characteristic use valuations and are done by proving local uniformization of all valuations of the function ﬁeld. Exercise 12.21. This exercise gives a geometric interpretation of divisorial valuations. a) Suppose that X is a normal projective variety and ν is a divisorial valuation on k(X). Let Z be the center of ν on X, so that we have an inclusion OX,Z ⊂ Vν . Show that OX,Z = Vν if and only if codimX Z = 1. b) Give an example of a divisorial valuation ν on k(P2 ) such that Vν = OP2 ,Z , where Z is the center of ν on P2 . c) Suppose that X is a normal projective variety and ν is a divisorial valuation on k(X). Show that there exists a birational regular map φ : Y → X of normal projective varieties such that if W is the center of ν on Y , then OY,W = Vν .
12.4. Factorization of birational maps The blowups of a point and of a nonsingular curve in a nonsingular projective 3fold X (a threedimensional variety) give examples of birational regular maps of nonsingular projective 3folds (by Theorem 10.19). Further examples can be found by taking products (sequences) of blowups of points and nonsingular curves. In light of the fact that birational maps of nonsingular projective surfaces can always be factored by a product of blowups of points (Theorem 10.32), it is natural to ask if every birational regular map of nonsingular projective 3folds can be factored by a product of blowups of points and nonsingular curves. This is, however, not true. Counterexamples have been given by Hironaka [78], Shannon [137], and Sally [129], all in their PhD theses (with Zariski, Abhyankar, and Kaplansky, respectively). The examples of Shannon and Sally do not even factor locally. Here is a simple example of a birational regular map of nonsingular projective 3folds which does not factor. Example 12.22. There exists a birational regular map of projective nonsingular 3folds which cannot be factored by a product of blowups of points and nonsingular curves. The example is constructed as follows. Let X = P3 with coordinate ring S(X) = k[x0 , x1 , x2 , x3 ], where we require that k has characteristic = 3. Let C ⊂ X be the curve C = Z(I) where I is the homogeneous prime ideal I = (x0 , x1 x2 x3 + x31 + x32 ).
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233
The curve C has an isolated singularity at the point p = (0 : 0 : 0 : 1). Let ˜ The π1 : X1 → X be the blowup of the curve C (of the ideal sheaf I). −1 variety X1 \ π1 (p) is nonsingular by Theorem 10.19. Let U = Xx3 ∼ = A3 . The regular functions on U are k[U ] = k[x, y, z] where x1 x2 x0 , y= , z= . x= x3 x3 x3 3 3 ˜ We have that I(U ) = (x, yz + y + z ) and p is the origin in U . We have ˜ )) has an open cover by two aﬃne open sets U1 and that π1−1 (U ) = B(I(U U2 with x yz + y 3 + z 3 and k[U2 ] = k[U ] k[U1 ] = k[U ] yz + y 3 + z 3 x by Theorem 6.4. Since k[x, y, z, s]/(s(yz + y 3 + z 3 ) − x) ∼ = k[y, z, s] is a domain, we have that k[U1 ] ∼ = k[x, y, z, s]/(s(yz + y 3 + z 3 ) − x) ∼ = k[y, z, s] by Exercise 1.10, so U1 ∼ = A3 is nonsingular. Since k[x, y, z, t]/(tx − (yz + y 3 + z 3 )) is a domain, we have that k[U2 ] ∼ = k[x, y, z, t]/(tx − (yz + y 3 + z 3 )) by Exercise 1.10. Let f = tx − (yz + y 3 + z 3 ). All singular points of U2 are in U2 ∩ π1−1 (p) = Z(x, y, z). The only point of U2 ∩ π1−1 (p) on which ∂f ∂x = t vanishes is q := Z(x, y, z, t), which is the only singular point of U2 . Further, we have that π1−1 (C) is an irreducible surface E (yz+y 3 +z 3 = 0 is a local equation of E in U1 and x = 0 is a local equation of E in U2 ) and π1−1 (α) ∼ = P1 for all α ∈ C.
Let π2 : X2 → X1 be the blowup of the point q. The open set π2−1 (U2 ) is naturally covered by four aﬃne open sets V1 , V2 , V3 , V4 . We compute their regular functions by taking the strict transform of f = 0 in the blowup of q in A4 (with k[A4 ] = k[x, y, z, t]) to see that X2 is nonsingular and π2−1 (q) is a surface F ∼ = P1 × P1 (this is similar to the calculation of Exercise 6.16). Let E be the strict transform of E on X2 . Let π = π1 π2 : X2 → X. The map π is a birational regular map of nonsingular projective 3folds which is an isomorphism over X \ π −1 (C) and π −1 (C) = E ∪ F is a union of two irreducible surfaces. Suppose that π : X2 → X factors as a product of blowups of points and nonsingular curves. Then the ﬁrst blowup must be of a point α contained in C since C is an irreducible singular curve. Let Iα be the ideal sheaf of
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α in X. We have that π −1 (α) ∼ = P1 if α = p and π −1 (α) = F ∪ γ where γ ∼ = P1 is a curve which intersects F in a point if α = p. In each case, there exists an aﬃne open subset W of X2 which intersects π −1 (α) in a curve and such that π(W ) ⊂ T for some aﬃne open neighborhood T of α in X. Here ZW (IT (α)) = π −1 (α) ∩ W which has codimension > 1 in W so Iα OW cannot be locally principal (by Krull’s principal ideal theorem). Since Iα OX2 is not locally principal, we have that π : X2 → X cannot factor through the blowup B(α) → X of α (since Iα OB(α) is locally principal). This contradiction shows that π : X2 → X is not a product of blowups of points and nonsingular curves. Hironaka [78] and Abhyankar [7] proposed the following problem: Question 12.23. Suppose that φ : X → Y is a birational regular map of nonsingular projective varieties. Does there exist a nonsingular projective variety Z and a commutative diagram of regular maps Z X
→
Y
such that the maps Z → X and Z → Y are products of blowups of nonsingular subvarieties? This question is still open, even in dimension 3 and characteristic 0. However, we have the following theorem: Theorem 12.24 (Abramovich, Karu, Matsuki, W lodarczyk [9]). Suppose that φ : X → Y is a birational regular map of nonsingular projective varieties, over a ﬁeld of characteristic 0. Then there is a factorization (for some n) X
Yn
Yn−1
Yn−2
Yn−3
···
Y0 = Y
where each diagonal arrow is a (ﬁnite) product of blowups of nonsingular subvarieties. This theorem is not known in positive characteristic (even in dimension 3). Theorem 12.24 was proven earlier for toric and toroidal varieties (W lodarczyk [147], Abramovich, Matsuki, Rashid [10]). This is a category of varieties which is built by only allowing Laurent monomials instead of arbitrary polynomials. A regular map of nonsingular aﬃne toric (or toroidal)
12.4. Factorization of birational maps
235
varieties is then a monomial map (12.3)
φ : An → Am
with φ = (M1 , . . . , Mm ) a where the Mi are monomials Mi = nj=1 xj ij for 1 ≤ i ≤ m in the coordinate functions x1 , . . . , xn of An . A regular map of nonsingular toric (or toroidal) varieties is constructed by patching together maps of the form (12.3). Question 12.25 has been proposed by Oda [123] for the restricted case of toric (toroidal varieties). Question 12.25. Suppose that φ : X → Y is a birational regular map of nonsingular projective toric (toroidal) varieties. Does there exist a nonsingular projective toric (toroidal) variety Z and a commutative diagram of regular toric maps Z X → Y such that the maps Z → X and Z → Y are product of blowups of nonsingular toric subvarieties? This question is open even in dimension 3 and characteristic 0. However, local factorization is true in all dimensions and characteristic 0. The following theorem was conjectured by Abhyankar [8, Section 8]. Theorem 12.26 (Cutkosky [33]). Suppose that φ : X → Y is a birational regular map of nonsingular projective varieties over a ﬁeld k of characteristic 0 and ν is the valuation of the function ﬁeld k(X) = k(Y ). Then there exists a nonsingular projective variety Z and a commutative diagram of regular maps Z X → Y such that there exist aﬃne neighborhoods U , V , W of the respective centers p, q, r of ν on X, Y, Z such that W is an aﬃne open subset of a sequence of blowups of nonsingular subvarieties above U and also W is an aﬃne open subset of a sequence of blowups of nonsingular subvarieties above V . The theorem was ﬁrst proven by Christensen [29] for the case of a toric valuation in dimension 3. A toric valuation ν of An is obtained by assigning positive real numbers r1 , . . . , rn as weights to the coordinate I functions x1 , . . . , xn . The valuation of a polynomial f = I ai x with I = (i1 , . . . , in ) ∈ Nn and aI ∈ k is then ν(f ) = min{i1 r1 + · · · + in rn  aI = 0}.
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The case of a toric valuation in dimension n is solved by Karu [86] and by Cutkosky and Srinivasan [45]. The case of a general valuation is solved in [33].
12.5. Monomialization of maps We ask if it is possible to put an arbitrary regular map of varieties φ : X → Y into a particularly nice form by performing sequences of blowups of nonsingular subvarieties X1 → X and Y1 → Y to obtain a new map φ1 : X1 → Y1 which has a simpler form. The simplest such form which is possible is for X1 → Y1 to be “locally monomial”. This means that for every point p of X1 , there exist uniformizing parameters near p, such that after possibly taking ´etale covers or using formal coordinates (these concepts will be explained in Chapters 14 and 21), φ1 locally has an expression as a monomial map of the type of (12.3). (Formally, every nonsingular point looks like a point on An .) Speciﬁcally, we have the following question. Question 12.27. Suppose that φ : X → Y is a dominant regular map of characteristic 0 varieties. Does there exist a commutative diagram of regular maps φ1
(12.4)
X1 → Y1 ↓ ↓ X
φ
→
Y
such that X1 and Y1 are nonsingular, the vertical arrow are products of blowups of nonsingular subvarieties, and φ1 : X1 → Y1 is locally monomial (or toroidal)? We will call such a diagram (12.4) a monomialization (toroidalization) of φ. We prove in [34] and [37] that the answer to Question 12.27 is yes if X and Y are threedimensional varieties over an algebraically closed ﬁeld k of characteristic 0. (A simpler proof of the results of [34] is given in [39].) As a corollary to this theorem, we obtain a diﬀerent proof of Theorem 12.24 in dimension 3. This theorem (the solution to Question 12.27 in dimension 3) also shows that the HironakaAbhyankar Question 12.23 for factorization of birational maps in dimension 3 and characteristic 0 is implied by the Oda Question 12.25 for factorization of birational maps of toroidal varieties in dimension 3. Question 12.27 has a negative answer if we allow ﬁelds of positive characteristic p. A simple example is the map φ : A1 → A1 deﬁned by t → tp +tp+1 . Since A1 is a nonsingular curve, blowups do not change anything, so the
12.5. Monomialization of maps
237
question asks in this case if φ itself is locally monomial. The map is given by the expression u = xp + xp+1 = δxp where δ = 1 + x is a unit in R = k[x]x . To represent u as a monomial, we 1 1 1 would have to set x = δ p x and then obtain u = xp . However, δ p = 1 + x p is ˆ = k[[x]], so such a monomialization is not possible not in the completion R by a formal change of variables. From this argument, we also see why all maps of nonsingular curves are locally monomial in characteristic 0. We can always represent a map of nonsingular curves φ : X → Y locally at a point q ∈ X by an expression u = αxn where u is a regular parameter in R = OY,φ(q) , x is a regular parameter in R = OX,q , and α is a unit in R. In this case, taking an nth root of α gives an ´etale map locally above q on X; in fact, with our 1 assumption that k = R/mq is algebraically closed, we have that α n is in the ˆ = k[[x]], so we have an expression u = xn where x = α n1 x is completion R ˆ a regular parameter of R. We have a local valuationtheoretic version of Question 12.27. Question 12.28. Suppose that φ : X → Y is a dominant regular map of projective varieties and ν is a valuation of k(X). Does there exist a commutative diagram of regular maps φ1
X1 → Y1 ↓ ↓
(12.5)
φ
→
X
Y
such that X1 and Y1 are nonsingular, there exist aﬃne neighborhoods U , V , W , Z of the respective centers q, r, q , r of ν on X, Y , X1 , and Y1 , respectively, such that W and Z are aﬃne open subsets of sequences of blowups of nonsingular subvarieties above U and V , respectively, and there are regular parameters x1 , . . . , xn in OX1 ,q and y1 , . . . , ym in OY ,r , units δ1 , . . . , δm ∈ OX1 ,q , and a matrix A = (aij ) of natural numbers such that (12.6)
yi = δi
n $
a
xj ij
for 1 ≤ i ≤ m
j=1
and (12.7)
rank(A) = m?
This question makes sense over all ﬁelds (it is true for curves in every characteristic). In characteristic 0, if (12.6) and (12.7) hold, then we can make an ´etale or formal change of variables to represent the yi as monomials
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in regular parameters in an ´etale extension or the completion of OX1 ,q (the condition (12.7) is essential!). Theorem 12.29. Question 12.28 always has a positive answer over ﬁelds of characteristic 0. This is proven in [33] and [36]. Question 12.28 has a negative answer in positive characteristic p > 0, even in dimension 2. A counterexample is given in [40]. It is shown in [43] and most generally in [41] that when X and Y have dimension 2 and the valuation has positive residue characteristic, then Question 12.28 has a positive answer for defectless extensions of valued ﬁelds. The defect is an interesting invariant of extensions of valuations (in this case from the restriction of ν to the function ﬁeld of Y to its extension ν on k(X)) which can only occur for fairly complicated valuations in characteristic p > 0. The presence of defect really says that information about the extension that you should be able to extract from the quotient of value groups is lost. The defect can be viewed as the cause of all of the trouble in local uniformization in positive characteristic [93].
Chapter 13
Divisors
We deﬁne a divisor on a normal variety X to be a formal sum of prime divisors (codimension 1 irreducible subvarieties of X) in Section 13.1. We associate to a rational function f on X a divisor (f ) (or div(f )) which is the diﬀerence of the zeros of f and the poles of f , counting multiplicity. The divisor class group Cl(X) of X (see (13.1)) is the group of equivalence classes of divisors on X, modulo the divisors of rational functions on X. In Section 13.4 we calculate some examples of divisor class groups, and in Section 13.5, we analyze divisors in the most intuitive situation, on a nonsingular projective curve. Associated to a divisor D on a normal variety X, we have a coherent sheaf OX (D), consisting of the rational functions on X whose poles are bounded by D (Section 13.2). If X is nonsingular, then the sheaf OX (D) of a divisor D on X is an invertible sheaf (see (13.2)). Further, we can cover X with aﬃne open subsets Ui such that on each Ui , D ∩ Ui is the divisor of a rational function gi ∈ k(X) on Ui . The set {(gi , Ui )} is called a Cartier divisor. The concept of a Cartier divisor will be further explored, in the more general situation of schemes, in Section 15.1. The three concepts of divisors, invertible sheaves, and Cartier divisors are equivalent on a nonsingular variety (see (13.16)). The concepts of invertible sheaves and Cartier divisors agree on varieties, but not on schemes, while the concepts of divisors and invertible sheaves are not the same on a normal (but singular) variety. The concept of invertible sheaf is the most general and is valid on an arbitrary variety (or scheme).
239
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Associated to a divisor D on a normal variety X, we have linear systems on X, parametrizing eﬀective divisors on X which are linearly equivalent to D (Section 13.6). Rational maps from a normal variety can be interpreted through linear systems. The divisors in a linear system (without ﬁxed component) are the pullbacks of linear hyperplane sections in the image. Using this interpretation of rational map, we give criteria for a rational map from a nonsingular variety to be a regular map, injective, and a closed embedding in Section 13.7. In Sections 13.8 and 13.9, we develop the geometric theory of invertible sheaves on an arbitrary variety, generalizing the theory of divisors on a normal nonsingular variety.
13.1. Divisors and the class group Suppose that X is a normal quasiprojective variety. A prime divisor on X is an irreducible codimension 1 subvariety E of X. A divisor on X is a ﬁnite formal sum D = ai Ei where the Ei are prime divisors on X and ai ∈ Z. Let Div(X) be the group (under addition) of divisors on X. The group Div(X) is the free Abelian group on the set of prime divisors of X. The support of a divisor D = ai Ei is the algebraic set in X, Supp D = Ei . ai =0
Suppose thatD1 , D2 are divisors on X. We will say that D1 ≥ D2 if D1 − D2 = ai Ei where all ai are nonnegative. A divisor D such that D ≥ 0 is called eﬀective. Now suppose that E is a prime divisor of X. We associate to E the local ring OX,E , the local ring of a subvariety which we deﬁned in Deﬁnition 12.17. Since X is normal, the singular locus of X has codimension ≥ 2 in X (by Theorem 10.17), so there exists p ∈ E which is a nonsingular point of X. Let U be an aﬃne neighborhood of p in X. Let I be the ideal of E ∩ U in k[U ] (that is, let I = IE (U ) where IE is the ideal sheaf of E in X). The ideal I is a height 1 prime ideal in k[U ]. Let I(p) be the ideal of the point p in U , which is a maximal ideal in k[U ]. Now OX,p = k[U ]I(p) and the stalk (IE )p is the localization II(p) . Thus (IE )p = II(p) is a height 1 prime ideal in OX,p . We have that OX,E is the localization (OX,p )(IE )p . Now OX,p is a regular local ring, since p is a nonsingular point of X, so the localization OX,E is a regular local ring, by Theorem 1.88. Further, OX,E has dimension 1, so that the maximal ideal of OX,E is generated by one element t. Suppose that 0 = f ∈ k(X). Then we can (uniquely) write f = tn u where u ∈ OX,E is a unit and n ∈ Z. This is true for f ∈ OX,E , and so this is true for
13.1. Divisors and the class group
241
f ∈ k(X) since every element of k(X) is a quotient of elements of OX,E . We may thus deﬁne a map νE : k(X) \ {0} → Z by νE (f ) = n if f = tn u where u is a unit in OX,E . The map νE has the properties that νE (f g) = νE (f ) + νE (g) and νE (f + g) ≥ min{νE (f ), νE (g)} for f, g ∈ k(X) \ {0}. Using the convention that νE (0) = ∞, this tells us that νE is a valuation of k(X) with valuation ring VνE = {f ∈ k(X)  νE (f ) ≥ 0} = OX,E . In fact, νE is a divisorial valuation (Lemma 12.19). Let V be a nonsingular aﬃne open subset of X such that V ∩ E = ∅ and IE (V ) = (t). Then t = 0 is a local equation of E ∩ V in V . We see that a nonzero element f ∈ k[V ] has νE (f ) = n if and only if tn divides f and no higher power of t divides f in k[V ]. Thus n is nonnegative, and n represents the order of vanishing of f along E, that is, the order of E as a “zero” of f . Also, n represents the order of E as a “pole” of f1 . Since any element f of k(X) \ {0} is a quotient of elements of k[V ], we can interpret νE (f ) as the order of the zero of f along E if νE (f ) > 0, and −νE (f ) as the order of the pole of f along E if νE (f ) < 0. If νE (f ) = 0, then E is neither a zero nor a pole of f . This thinking in terms of zeros and poles is most intuitive when X is a curve, so that a divisor is a point, and OX,E is the local ring of a point. Lemma 13.1. Suppose that X is a normal quasiprojective variety and 0 = f ∈ k(X). Then there are at most a ﬁnite number of prime divisors E on X such that νE (f ) = 0. Proof. Since every quasiprojective variety has an open cover by a ﬁnite number of aﬃne open sets, we reduce to the case when X is aﬃne. Write f = hg where g, h ∈ k[X]. Now ZX (g) has only a ﬁnite number of irreducible components, and νE (g) = 0 unless E is an irreducible component of ZX (g). The same statement holds for h, and since νE (f ) = νE (g) − νE (h), we have the statement of the lemma. We may thus deﬁne the divisor of a function 0 = f ∈ k(X) (where X is a normal quasiprojective variety) to be νE (f )E ∈ Div(X) (f ) = (f )X = where the sum is over the prime divisors E of X. The divisor of zeros of f is νE (f )E (f )0 = EνE (f )>0
242
13. Divisors
and the divisor of poles of f is (f )∞ =
−νE (f )E,
EνE (f )<0
so that (f ) = (f )0 − (f )∞ . A divisor D is called principal if D = (f ) for some f ∈ k(X). We deﬁne an equivalence relation ∼ on Div(X), called linear equivalence, by D1 ∼ D2 if there exists 0 = f ∈ k(X) such that (f )X = D1 − D2 . Now we deﬁne the divisor class group of (a normal quasiprojective variety) X to be (13.1)
Cl(X) = Div(X)/ ∼ .
We will sometimes write div(f ) or div(f )X for (f ) = (f )X .
13.2. The sheaf associated to a divisor Suppose that X is a normal quasiprojective variety. For D = ai Ei a divisor on X, with Ei prime divisors and ai ∈ Z and U an open subset of X, we deﬁne the divisor D ∩ U on U to be ai (Ei ∩ U ). D∩U = Ei Ei ∩U =∅
We deﬁne the presheaf OX (D) of “functions whose poles are bounded by D” by Γ(U, OX (D)) = {f ∈ k(X)  (f )U + D ∩ U ≥ 0} for U an open subset of X. By the convention that νE (0) = ∞ for all prime divisors E, we have 0 ∈ Γ(U, OX (D)) for all D and U . Each Γ(U, OX (D)) is a group since for f, g ∈ k(X) and prime divisor E on X, νE (f + g) ≥ min{νE (f ), νE (g)} and νE (−f ) = νE (f ). Lemma 13.2. Suppose that X is a normal quasiprojective variety and D is a divisor on X. Then OX (D) is a sheaf of OX modules. Proof. To show that OX (D) is a sheaf, we will verify that the sheaf axioms hold. Suppose that U is an open subset of X and {Ui } is an open cover of U and we have fi ∈ Γ(Ui , OX (D)) such that fi and fj have the same restriction in Γ(Ui ∩ Uj , OX (D)) for all i, j. Now for any open subset V of X we have that Γ(V, OX (D)) is a subset of k(X), so we must have that fi = fj (as elements of k(X)). Let f be this common element. Then (f )U ∩ Ui + D ∩ Ui ≥ 0 for all i, and the fact that {Ui } is an open cover
13.2. The sheaf associated to a divisor
243
of U , so that each component of D ∩ U must intersect some Ui nontrivially, implies (f )U + D ∩ U ≥ 0. Thus f ∈ Γ(U, OX (D)). Suppose that {Ui } is an open cover of an open subset U of X and f ∈ Γ(U, OX (D)) is an element such that the restriction of f is zero in Γ(Ui , OX (D)) for all i. Then we have that f = 0 as an element of k(X), so f = 0 in Γ(U, OX (D)). For U an open subset of X and f ∈ Γ(U, OX ), we have that νE (f ) ≥ 0 for all divisors E on X such that U ∩ E = ∅. Thus Γ(U, OX (D)) is a Γ(U, OX )module. Lemma 13.3. Suppose that X is a normal quasiprojective variety. Let 0 denote the divisor 0. Then OX (0) = OX . Proof. Suppose that U is an open subset of X, f ∈ OX (U ), and E is a prime divisor such that U ∩ E = ∅. Then there exists p ∈ U ∩ E. Now f ∈ OX,p implies f ∈ OX,E (which is a localization of OX,p ), so that νE (f ) ≥ 0. Thus (f )U ≥ 0, so that f ∈ OX (0)(U ). We have established that OX ⊂ OX (0), so it suﬃces by Proposition 11.13 to show that for all p ∈ X, OX,p = OX (0)p . If f ∈ OX (0)p , then νE (f ) ≥ 0 for all prime divisors E of X which contain p. Thus f ∈ (OX,p )a for all height1 prime ideals a of OX,p . We have that OX,p is a normal local ring. Thus a (OX,p )a = OX,p , where the intersection is over the height 1 prime ideals a of OX,p by Lemma 1.79, so f ∈ OX,p . Let A be a Noetherian local domain with quotient ﬁeld K. A fractional ideal M of A is an Asubmodule of K such that there exists d = 0 in A such that dM ⊂ A. A fractional ideal is necessarily a ﬁnitely generated Amodule. If M is a fractional ideal, we have a natural inclusion of Amodules HomA (M, A) = {f ∈ K  f M ⊂ A} ⊂ K. Since HomA (M, A) is a ﬁnitely generated Amodule, we have that HomA (M, A) is a fractional ideal. Given a fractional ideal M , we deﬁne its dual to be M ∨ = HomA (M, A). Let P (A) be the set of height 1 prime ideals of A. Theorem 13.4 ([23, Theorem 1, page 157]). Suppose that M is a fractional ideal of A. Then (M ∨ )p . M∨ = p∈P (A)
Lemma 13.5. Suppose that X is a normal quasiprojective variety and D is a divisor on X. Then OX (D) is a coherent sheaf of OX modules.
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13. Divisors
Proof. We ﬁrst give a proof with the assumption that X is nonsingular. Let D = a1 E1 + · · · + an En . Suppose that p ∈ X. Since p is a nonsingular point on X, every Ei such that p ∈ Ei has a local equation fi at p (Lemma 10.18). If p ∈ Ei , then fi = 1 is a local equation of Ei at p. Let Up be an aﬃne neighborhood of p such that Γ(Up , OX (−Ei )) = Γ(Up , IEi ) = (fi ) for all i. (Here (fi ) means the ideal in k[Up ].) Let f = f1a1 · · · fnan . Then (f )Up = D ∩Up . Suppose that U ⊂ Up is an open subset and g ∈ Γ(U, OX (D)). Then (g)U ≥ −D ∩ U . We compute (f g)U = (f )U + (g)U ≥ D ∩ U − D ∩ U = 0. Thus f g ∈ Γ(U, OX (0)) = Γ(U, OX ) by Lemma 13.3, so that g ∈ f1 Γ(U, OX ). Conversely, if h ∈ Γ(U, OX ), then ( hf )U ≥ −(f )U , so that hf ∈ Γ(U, OX (D)). In summary, we have that Γ(U, OX (D)) = f1 Γ(U, OX ) for all open subsets U of Up , so that we have equality of sheaves OX (D)  Up =
1 OUp . f
We now prove the lemma for normal X. Let Z be the singular locus of X, and let U = X \ Z. Let i : U → X be the inclusion. We have that codimX Z > 1 since X is normal (Theorem 10.17). Let F = OX (D)U , a coherent sheaf on U . We have a natural inclusion of OU modules F ⊂ k(U ) = k(X), where k(X) is the constant sheaf. By Theorem 11.55, there exists a coherent sheaf H on X such that HU = F and H is an OX submodule of k(X). We will denote the sheaf Hom OX (G, OX ) (which was deﬁned in Section 11.5) by G ∗ for an OX module G. Let A = (H)∗∗ . The sheaf H is a coherent OX module by Lemma 11.38, and we have natural inclusions of OX modules H ⊂ A ⊂ k(X). Further, since F is locally isomorphic to OU as an OU module, we have that AU = F . Suppose that V ⊂ X is an aﬃne open subset. Let R = k[V ]. We let N ∨ = HomR (N, R) if N is an Rmodule. Let M = Γ(V, A). By Lemma 11.38 (and its proof) we have that M = (Γ(V, H))∨∨ . By Theorem 13.4, we have that M = Q∈P (R) MQ where P (R) is the set of height 1 prime ideals of R and the intersection is in k(X). Suppose that W ⊂ V is a closed set and codimV W ≥ 2. If Q ∈ P (R), then there exists p ∈ ZV (Q)\W (since codimV W ≥ 2 and codimV Z(Q) = 1) so Q ⊂ I(p), and hence MI(p) ⊂ MQ . Thus by Proposition 11.53, MI(p) ⊂ MQ = M. M = Γ(V, A) ⊂ Γ(V \ W, A) = p∈V \W
Q∈P (R)
Thus Γ(V \ W, A) = Γ(V, A) for any aﬃne open subset V of X and closed subset W of V of codimension ≥ 2 in V . By the sheaf axioms, we have that Γ(V \ W, A) = Γ(V, A)
13.2. The sheaf associated to a divisor
245
for any open subset V of X and closed subset W of V of codimension ≥ 2 as we can express V as a union of aﬃne open sets. Suppose that V ⊂ X is an open subset. Since U = X \ Z does not contain any prime divisor of X, we have that Γ(V, OX (D)) = Γ(V ∩ U, OX (D)) = Γ(V ∩ U, A) = Γ(V, A). Thus OX (D) = A = i∗ (OX (D)U ) is a coherent OX module.
In the case that X is nonsingular, the fact that there exists an open cover {Ui } of X and fi ∈ k(X) such that there are expressions OX (D)  Ui =
(13.2)
1 OU fi i
for all i tells us that OX (D) is an invertible sheaf of OX modules (Deﬁnition 11.33). We have that the fi are “local equations” for D in the sense that (fi )Ui = D ∩ Ui for all i. The set of pairs {(fi , Ui )} determines a Cartier divisor on X (Deﬁnition 15.4). In the appendix to [126, Section 1] it is explained that the sheaves OX (D) on a normal variety X are the reﬂexive rank 1 sheaves on X [23, Chapter 7, Section 4 ]. In particular, denoting the sheaf Hom OX (G, OX ) by G ∗ for an OX module G, as in the proof of Lemma 13.5, we have that OX (−D) ∼ = OX (D)∗ and OX (D + E) ∼ = (OX (D) ⊗OX OX (E))∗∗ . If D = ri=1 ai Ei is an eﬀective divisor on a normal variety X and U is an aﬃne open subset of X, then Γ(U, OX (−D)) = {f ∈ k[U ]  νE1 (f ) ≥ a1 } ∩ · · · ∩ {f ∈ k[U ]  νEr (f ) ≥ ar } (a1 )
= p1
r) ∩ · · · ∩ p(a r
where pi is the prime ideal pi = IEi ∩U = Γ(U, OX (−Ei )) of the codimension (a ) 1 subvariety Ei ∩ U of U and pi i is the ai th symbolic power of pi , deﬁned (a ) by pi i = k[U ] ∩ (pai i k[U ]pi ), since k[U ]pi = OX,Ei is the valuation ring of νEi , with maximal ideal pi OX,Ei . Proposition 13.6. Suppose that X is a normal quasiprojective variety and D1 and D2 are divisors on X. Then D1 ∼ D2 if and only if OX (D1 ) ∼ = OX (D2 ) as sheaves of OX modules.
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13. Divisors
Proof. We will reduce to the case that X is nonsingular by the argument used in the proof of Lemma 13.5. Let V be the nonsingular locus of X and λ:V →X be the inclusion. We have by Theorem 10.17 that codimX (X \ V ) ≥ 2 since X is normal. Thus by the deﬁnition of OX (Di ), Γ(U, OX (Di )) = Γ(U ∩ V, OX (Di )) for all open subsets U of X, and so λ∗ (OX (Di )V ) ∼ = OX (Di ). Thus we may assume that X = V is nonsingular. Suppose that D1 ∼ D2 . Then there exists g ∈ k(X) such that (g) = D1 − D2 . Multiplication by g thus gives us an OX module isomorphism OX (D1 ) → OX (D2 ) since for f ∈ k(X) and U an open subset of X, (f )U + D1 ∩ U ≥ 0 if and only if (f g)U + D2 ∩ U ≥ 0. Suppose that φ : OX (D1 ) → OX (D2 ) is an OX module isomorphism. For p ∈ X, the OX,p module isomomorphism φp : OX (D1 )p → OX (D2 )p extends uniquely to a k(X)module isomorphism ψp : k(X) → k(X), since the localizations (OX (Di )p )p = k(X) for i = 1, 2 (where p is the zero ideal in OX,p ). The map ψp is deﬁned by ψp ( fg ) = 0 = g ∈ OX,p .
φp (f ) g
for f ∈ OX (D1 )p and
Suppose that U is an aﬃne open subset of X such that OX (Di )  U = for some fi ∈ k(X) for i = 1, 2. Then
1 fi OU
(13.3)
Γ(U, OX (Di )) =
1 k[U ] for i = 1, 2. fi
For g ∈ k[U ], we have that φ(U )( fg1 ) = gu f2 for some ﬁxed unit u ∈ k[U ]. Localizing at the zero ideal of k[U ] gives us a unique extension of φ(U ) to a k(X)module isomorphism k(X) → k(X). But this extension must agree with our extensions ψp for all p ∈ U (since the OX (Di ) are coherent). Since X is connected, all of our extensions ψp agree. A nonzero k(X)module homomorphism k(X) → k(X) is multiplication by a nonzero element of k(X). Thus there exists a nonzero element g ∈ k(X) such that for all open U in X, the Γ(U, OX )module isomorphism φ(U ) : Γ(U, OX (D1 )) → Γ(U, OX (D2 )) is given by multiplication by g. This tells us that for any f ∈ k(X) and open subset U of X, (f )U + D1 ∩ U ≥ 0 if and only if (gf )U + D2 ∩ U ≥ 0. By consideration of aﬃne open subsets U on which an expression (13.3) holds, we have that (g)U = ( ff21 )U = (D2 − D1 )U . Thus (g) = D2 − D1 so that D1 ∼ D2 .
13.2. The sheaf associated to a divisor
247
In equation (13.2), we associated to a divisor D on a nonsingular variety X an open cover {Ui } of X and gi ∈ k(X) which have the property that (gi ) ∩ Ui ∩ Uj = (gj ) ∩ Ui ∩ Uj for all i, j. Conversely, given an open cover {Ui } of X and gi ∈ k(X) such that (gi ) ∩ Ui ∩ Uj = (gj ) ∩ Ui ∩ Uj for all i, j, we can associate a divisor D on X, which is deﬁned by the condition that D ∩ Ui = (gi ) ∩ Ui for all i. As in the ﬁrst part of the proof of Lemma 13.5, we have that OX (Di )Ui = g1i OUi . Theorem 13.7. Suppose that X is a nonsingular quasiprojective variety and L is an invertible sheaf on X. Then there exists a divisor D on X such that L ∼ = OX (D) as OX modules. Proof. Since X is quasicompact (Exercise 7.6), X has a ﬁnite cover {U1 , . . . , Ur } with OUi isomorphisms φi : OUi → LUi . Let σi = φi (Ui )(1) ∈ L(Ui ), so that LUi = OUi σi is generated as an OUi module by σi . Thus LUi ∩ Uj = OUi ∩Uj σi = OUi ∩Uj σj and there exists a unique unit gij ∈ OX (Ui ∩Uj ) ⊂ k(X) such that σi = gij σj . Thus σi = gij gji σi implies −1 gji = gij .
(13.4)
The equality σi = gij σj = gij gjk σk implies gik = gij gjk .
(13.5) We compute
(gi1 )Ui ∩Uj = (gij gj1 )Ui ∩Uj = (gij )Ui ∩Uj + (gj1 )Ui ∩Uj = (gj1 )Ui ∩Uj since gij is a unit in OX (Ui ∩ Uj ). Thus there exists a unique divisor D on X such that D ∩ Ui = (gi1 )Ui for 1 ≤ i ≤ r, and we have that OX (−D)Ui = gi1 OUi
for 1 ≤ i ≤ r.
We have a natural inclusion of OX modules L ⊂ k(X)σ1 . For all i, we have LUi = OUi σi = gi1 OUi σ1 = (OX (−D)Ui ) σ1 . Thus L = OX (−D)σ1 ∼ = OX (−D).
Lemma 13.8. Suppose that X is a nonsingular quasiprojective variety and I is a nonzero ideal sheaf on X. Then there exists an eﬀective divisor D on X and an ideal sheaf J on X such that the support of OX /J has codimension ≥ 2 in X and I = OX (−D)J .
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Proof. We begin with some remarks about rings. Suppose that R is a local ring of an algebraic variety and I is a nonzero ideal in R. Let d be the dimension of R. Then the following are equivalent: 1. The ring R/I has dimension d − 1. 2. I has height 1 in R. 3. There exists a height 1 prime p in R such that I ⊂ p. If we further have that R is a UFD, then a height 1 prime ideal p in R has an expression p = (g) where g is an irreducible in R, so that I ⊂ p if and only if I = gJ for some ideal J of R. Thus, assuming that R is a UFD, we have that there exists a nonzero element f ∈ R and an ideal J in R such that J has height ≥ 2 in R and I = f J. Now we prove the lemma. We have that Supp(OX /I) = E1 ∪· · ·∪Er ∪W where E1 , . . . , Er are prime divisors and codimX W ≥ 2. Let I(Ei ) be the ideal sheaf of Ei in OX . Deﬁne the localization IEi of I in OX,Ei by taking any p ∈ Ei and letting IEi = (Ip )p ⊂ (OX,p )p = OX,Ei where Ip is the stalk of I at p, p is the height 1 prime ideal p = I(Ei )p in OX,p . Since OX,Ei is a onedimensional regular local ring (a discrete valuation ring), there exists a positive integer ai such that IEi = maEii where mEi is the maximal ideal of OX,Ei . Let D be the divisor D = ri=1 ai Ei . Deﬁne a presheaf J on X by J (U ) = {f ∈ OX (U )  f OX (−D) ⊂ I}. J is an ideal sheaf on X. The ideal sheaf OX (−D) has the properties that OX (−D)Ei = maEii = IEi
for 1 ≤ i ≤ r
and OX (−D)G = OX,G = IG if G is a prime divisor on X which is not one of the Ei . Suppose that p ∈ X. For 1 ≤ i ≤ r, let fi be a local equation of Ei at p (taking fi = 1 if p ∈ Ei ). Then OX (−D)p = f1a1 · · · frar OX,p and Ip = f1a1 · · · frar (Ip : f1a1 · · · frar OX,p ) = (Ip : OX (−D)p )OX (−D)p = Jp OX (−D). Thus I = OX (−D)J . We have that Supp(OX /J ) does not contain a prime divisor, since dim(OX,p /Jp ) ≤ dim X − 2 for all p ∈ X. Thus codimX (Supp(OX /J )) ≥ 2.
13.4. Calculation of some class groups
249
13.3. Divisors associated to forms Suppose that X ⊂ Pn is a normal projective variety. Let x0 , . . . , xn be homogeneous coordinates on X, so that S(X) = k[x0 , . . . , xn ]. Suppose that F ∈ S(X) is a nonzero homogeneous form of degree d. We associate to F an eﬀective divisor Div(F ) on X as follows. Let fi =
F ∈ k(X) for 0 ≤ i ≤ n. xdi
We have that
fj = fi
xi xj
d .
Let Ui = Xxi for 0 ≤ i ≤ n, so that {U0 , . . . , Un } is an aﬃne open cover of X. We have that x0 xn fi ∈ OX (Ui ) = k ,..., xi xi for all i and
xi xj
is a unit in
x0 xn xi OX (Ui ∩ Uj ) = k ,..., , xi xi xj
for all i, j. We have that (fi )Ui ∩Uj = (fj )Ui ∩Uj for all i, j, so that there is a unique divisor E on X such that E ∩ Ui = (fi )Ui
for 1 ≤ i ≤ n.
We deﬁne Div(F ) = E. We have that Div(F ) ≥ 0 since (fi )Ui ≥ 0 for all i, as fi ∈ OX (Ui ) for all i.
13.4. Calculation of some class groups Example 13.9. Cl(An ) = (0). Proof. Suppose that E is a prime divisor on An . Then the ideal of E, I(E), is a height 1 prime ideal in the polynomial ring k[An ]. Since k[An ] is a UFD, I(E) is a principal ideal, so there exists f ∈ k[An ] such that (f ) = E.
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13. Divisors
Example 13.10. Cl(Pn ) ∼ = Z. Proof. For every irreducible form F ∈ S = S(Pn ), Z(F ) is a prime divisor on Pn . Suppose that E is a prime divisor on Pn . Let I(E) be the homogeneous prime ideal of E in S = k[x0 , . . . , xn ]. Since S is a polynomial ring and I(E) is a height 1 prime ideal in S, we have that I(E) is a principal ideal. Thus there exists an irreducible homogeneous form F ∈ S such that I(E) = (F ). Now another form G in S is a generator of I(E) if and only if F = λG where λ is a unit in S; that is, λ is a nonzero element of k. We can thus associate to E the number d = deg(F ) which is called the degree of E. Thus we can deﬁne a group homomorphism deg : Div(Pn ) → Z by deﬁning deg E = d if E is a prime divisor of degree d and
deg ai Ei = ai deg(Ei ). F Suppose that f ∈ k(Pn ) = k( xx01 , . . . , xxn0 ). Then we can write f = G where F and G are homogeneous forms of a common degree d and F , G are relatively prime in S (since S is a UFD). This expression is unique up to multiplying F and G by a common nonzero element of k. Since S is a UFD and F , G are homogeneous, there are factorizations $ $ G= Gj F = Fj ,
where Fj and Gj are irreducible forms in S. Let Dj = Z(Fj ) and Ej = Z(Gj ) (the Dj may not all be distinct and the Ej may not all be distinct). Let Ui = Xxi ∼ = An for 0 ≤ i ≤ n. Then F G − = D ∩ U − Ej ∩ Ui . (f ) ∩ Ui = ∩ U ∩ U i i j i xdi xdi Thus (f ) = and deg ((f )) =
Dj −
deg Dj −
Ej ,
deg Ej = d − d = 0.
Thus we have a welldeﬁned group homomorphism deg : Cl(Pn ) → Z. The homomorphism deg is onto since a linear form has degree 1, so the associated prime divisor has degree 1.
13.4. Calculation of some class groups
251
Suppose that D is a divisor on Pn which has degree 0. Write D= Dj − Ej where Dj and Ej are prime divisors and deg Dj = deg Ej . Let d be this common degree. Let Fj be homogeneous forms such that I(Fj ) = D j , and let Gj be Fj has degree d homogeneous forms such that I(Gj ) = Ej . Then F = F ∈ k(Pn ). We have that and G = Gj has degree d, so f = G Z(Gj ) = D. (f ) = Z(Fj ) − Thus the class of D is zero in Cl(Pn ).
Example 13.11. Cl(Pm × Pn ) ∼ = Z × Z. The method of the previous example applied to the bihomogeneous coordinate ring of Pm × Pn proves this result. Let S = k[x0 , . . . , xm , y0 , . . . , yn ] be the bihomogeneous coordinate ring of Pm ×Pn . Prime divisors on Pm ×Pn correspond to irreducible bihomogeneous forms in S. Associated to a form is a bidegree (d, e). Using the fact that S is a UFD, we argue as in the previous example that this bidegree induces the desired isomorphism. A consequence of this example is that the group Cl(Pm ×Pn ) is generated by a prime divisor L1 of bidegree (1, 0) and a prime divisor L2 of bidegree (0, 1). The divisor L1 is equal to H1 × Pn , where H1 is a linear subspace of codimension 1 in Pm , and L2 = Pm × H2 , where H2 is a linear subspace of codimension 1 in Pn . A particular example is P1 × P1 . Here L1 = {p} × P1 and L2 = P1 × {q}, where p, q are any points in P1 , are generators of Cl(P1 × P1 ). Lemma 13.12. Suppose that D is a divisor on a nonsingular quasiprojective variety X and p1 , . . . , pm ∈ X are a ﬁnite set of points. Then there exists a divisor D on X such that D ∼ D and pi ∈ Supp D for i = 1, . . . , m. Proof. By assumption, X is an open subset of a projective variety X. Thus X = X \ Z for some closed subset Z of X. Choosing a projective embedding of X, by the homogeneous nullstellensatz, Theorem 3.12, we ﬁnd homogeneous forms Fi ∈ I(Z ∪ {p1 , . . . , pi−1 , pi+1 , . . . , pm }) ⊂ S(X) for 1 ≤ i ≤ m such that Fi (pi ) = 0 for 1 ≤ i ≤ m. Let di = deg(Fi ). Taking d = d1 d2 · · · dm , set m d d Fi i , F = i=1
252
13. Divisors
a homogeneous form in S(X) of degree d. Let U = X \ Z(F ). By construction, U is an aﬃne open subset of X = X \ Z = X which contains p1 , . . . , pm . We may thus assume that X is an aﬃne variety. By induction on m, we may assume that p1 , . . . , pi ∈ Supp D and pi+1 ∈ Supp D. We must construct a divisor D such that D ∼ D and p1 , . . . , pi+1 ∈ Supp D . We will prove the inductive statement in the case that D is a prime divisor. The case of a general divisor then follows by applying the statement to each of its components as necessary. Let π be a local equation of the prime divisor D in a neighborhood kl Gl , then of pi+1 . The function π is regular at pi+1 , so if (π )∞ = pi+1 ∈ Gl for all l. Then for every l, there exists fl ∈ k[X] such that fl ∈ I(Gl ) and fl (pi+1 ) = 0. The function π = π flkl has no poles on X (π ∈ Γ(X, OX (0))) so π ∈ k[X] by Lemma 13.3. Further, π = 0 is a local equation of D at pi+1 . Since pj ∈ Supp D ∪ {p1 , . . . , pj−1 , pj+1 , . . . , pi } for 1 ≤ j ≤ i, there exists gj ∈ I(D ∪ {p1 , . . . , pj−1 , pj+1 , . . . , pi }) such that gj (pj ) = 0 (by the nullstellensatz, Theorem 2.5). π(p ) Let f = π + ij=1 αj gj2 , where αj ∈ k is chosen so that αj = − gj (pjj)2 . Then (13.6)
f (pj ) = 0
for 1 ≤ j ≤ i.
Since gj (D) = 0 for all j, π divides gj in the regular local ring OX,pi+1 and we have an expression αj gj2 = hπ 2 for some h ∈ OX,pi+1 . Thus f = π(1 + πh) and since 1 + πh is a unit in OX,pi+1 , f = 0 is a local equation of D in a neighborhood of pi+1 . Thus we have an expression (f ) = D + rs D s where none of the prime divisors Ds contain pi+1 . Let D = D − (f ). Then pi+1 ∈ Supp D . Further, pj ∈ Supp (f ) for 1 ≤ j ≤ i by (13.6). Thus D satisﬁes the conclusions of the inductive step. Suppose that φ : X → Y is a regular map of nonsingular quasiprojective varieties. We will deﬁne a natural group homomorphism φ∗ : Cl(Y ) → Cl(X). Let Z be the Zariski closure of φ(X) in Y . Suppose that D is a divisor D ∼ D on Y . Then by Lemma 13.12, there is a divisor D on Y such that and Z is not contained in the support of D . Write D = ai Ei − bj Fj , where the ai , bj are positive and the Ei , Fj are distinct prime divisors. We will deﬁne a divisor φ∗ (D ) on X. Let {Ui } be an aﬃne open cover of Y such that there exist fi ∈ k(Y ) with (fi ) ∩ Ui = D ∩ Ui for all i.
13.4. Calculation of some class groups
253
After replacing the cover {Ui } with a reﬁnement, we may write fi = hgii for all i where gi and hi are regular on Ui and (gi )Ui = aj (Ej ∩ Ui ) and bj (Fj ∩ Ui ). Since Z is not contained in any of these divisors, the (hi )Ui = restrictions g i and hi of gi and hi to Ui ∩ Z are nonzero regular functions. g Thus we have induced rational functions hi ∈ k(Z). i
Let Vi = φ−1 (Ui ), which give an open cover of X. Let φ : X → Z be the dominant regular map induced by φ, with inclusion of function ﬁelds ∗ φ : k(Z) → k(X). We deﬁne φ∗ (D ) to be the divisor on X determined by the conditions ∗ gi ∗ ∩ Vi φ (D ) ∩ Vi = φ hi over the open cover {Vi } of X. If D is another divisor whose support does not contain Z such that D ∼ D , then there exists f ∈ k(Y ) such that (f ) = D − D . Since the support of (f ) contains no component of Z, f restricts to a nonzero element ∗ f of k(Z). By our construction, we have that (φ (f )) = φ∗ (D ) − φ∗ (D ). Thus we have a welldeﬁned group homomorphism φ∗ : Cl(Y ) → Cl(X), deﬁned by taking φ∗ (D) to be the class of φ∗ (D ), for any divisor D on Y which is linearly equivalent to D and whose support does not contain φ(X). We will write φ∗ (D) ∼ φ∗ (D ), even though φ∗ (D) may only be deﬁned up to equivalence class. However, the divisor φ∗ (D ) is actually a welldeﬁned divisor on X. A particular case of this construction is the inclusion i : X → Y of a closed subvariety X of Y into Y . Then the map i∗ on class groups can be considered as a restriction map: given a divisor D on Y , ﬁnd a divisor D linearly equivalent to D whose support does not contain X, and deﬁne i∗ (D) to be the class of D ∩ X (of course a prime divisor on Y might intersect X in a sum of prime divisors). Example 13.13. Suppose that X is a nonsingular surface and p ∈ X is a point. Let π : B → X be the blowup of p, with exceptional divisor π −1 (p) = E ∼ = P1 . Let i : E → B be the inclusion. Then i∗ (E) ∼ −q where q is a point on E. Proof. Let L be a curve on X which contains p and such that L is nonsingular at p. There exists a divisor D on X such that L ∼ D and p ∈ Supp(D) (by Lemma 13.12). Let L be the strict transform of L on B. Then π ∗ (L) = L + E, and π ∗ (D) ∼ π ∗ (L). Thus i∗ (π ∗ (L)) ∼ i∗ (π ∗ (D)) ∼ 0, since Supp(π ∗ (D)) ∩ E = ∅. Thus i∗ (E) ∼ i∗ (−L ) = −q, where q is the intersection point of E and L .
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13. Divisors
Theorem 13.14 (Lefschetz). Suppose that H is a nonsingular variety which is a codimension 1 closed subvariety of Pn for some n ≥ 4. Then the restriction homomorphism Cl(Pn ) ∼ = Z → Cl(H) is an isomorphism. Proofs of this theorem can be found in [62, pages 156 and 163] over C and for general ﬁelds in [64] and [74]. Since T = Z(xy − zw) ⊂ P3 is isomorphic to P1 × P1 , which has class group Z × Z, we see that the theorem fails for n < 4. The Segre embedding of P1 × P1 into P3 is the map φ = (x0 y0 : x1 y0 : x0 y1 : x1 y1 ). This map is induced by the corresponding homomorphism φ∗ : S(P3 ) = k[x, y, z, w] → S(P1 × P1 ) = k[x0 , x1 , y0 , y1 ] of coordinate rings. Let L be a form of degree 1 on P3 . Then φ∗ (L) is a bihomogeneous form of bidegree (1, 1) in S(P1 ×P1 ). Thus the divisor φ∗ (L) has bidegree (1, 1) on P1 × P1 . Since the class of L generates Cl(P3 ), we have that the restriction map Λ : Cl(P3 ) ∼ = Z → Cl(T ) ∼ = Z2 is given by Λ(n) = (n, n) for n ∈ Z. So a divisor of degree n on P3 , whose support does not contain T , restricts to a divisor on T which is linearly equivalent to n({p} × P1 + P1 × {q}) for any points p, q ∈ P1 .
13.5. The class group of a curve Suppose thatX is a nonsingular projective curve. A divisor D on X is then a sum D = ri=1 ai pi where pi are points on X and ai ∈ Z. We deﬁne the degree of the divisor D to be r ai . deg(D) = i=1
Suppose that X and Y are nonsingular projective curves and φ : X → Y is a nonconstant regular map. Then φ is ﬁnite (by Corollary 10.26). The degree of φ is deﬁned to be the degree of the ﬁnite ﬁeld extension deg(φ) = [k(X) : k(Y )]. Lemma 13.15. Let p1 , . . . , pr ∈ X, and set r OX,pi , R= i=1
where the intersection is in k(X). Then R is a principal ideal domain. There exist elements ti ∈ R such that (13.7)
νpi (tj ) = δij
for 1 ≤ i ≤ j ≤ r.
13.5. The class group of a curve
255
If u ∈ R, then u = tl11 · · · tlrr v
(13.8)
where li = νpi (u) and v is a unit in R. Proof. For 1 ≤ i ≤ r we have that (13.9)
OX,pi = {f ∈ k(X)  νpi (f ) ≥ 0}
since X is nonsingular. Let ui be a regular parameter in the onedimensional regular local ring OX,pi . Then νpi (ui ) = 1, so that (ui ) = pi + D where D is a divisor on X such that pi does not appear in D. By Lemma 13.12, there exists a divisor D on X such that the support of D is disjoint from{p1 , . . . , pr } and D ∼ D so there exists a rational function fi ∈ k(X) such that (fi ) = D − D. Let ti = ui fi . Then (ti ) = pi + D , so that ti ∈ R by (13.9) and the equations (13.7) hold. Let u ∈ R, and set li = νpi (u). We necessarily have that li ≥ 0 for all i. −lr 1 Let v = ut−l 1 · · · tr . We have that νpi (v) = 0 for all i. By (13.9) we have −1 that v and v are in OX,pi for all i. Thus v, v −1 ∈ R, so that v is a unit in R. We thus have the expression (13.8) for u. We will now show that R is a principal ideal domain. Let I be an ideal in R. Let li = inf{νpi (u)  u ∈ I} for 1 ≤ i ≤ r. Set a = tl11 · · · tlrr , and let J = aR. For u ∈ I we have that νpi ( ua ) ≥ 0 for all i, so that ua ∈ R, and thus u ∈ J. Thus I ⊂ J. We will now show that J ⊂ I. Let J = {ua−1  u ∈ I}. Then J is an ideal in R, and inf{νpi (v)  v ∈ J } = 0 for 1 ≤ i ≤ r. Hence for 1 ≤ i ≤ r, there exists ui ∈ J such that νpi (ui ) = 0; that is, ui (pi ) = 0. Let c = rj=1 uj t1 · · · tˆj · · · tr ∈ J . Then c(pi ) = 0 for all i, so that νpi (c) = 0 for all i. Thus c is a unit in R so that J = R. Thus I = aJ = aR is a principal ideal. Since φ : X → Y is ﬁnite, it is surjective, and the preimage of every point of Y in X is a ﬁnite set (by Theorems 7.5, 2.57, and 2.56). Suppose that q ∈ Y . Let φ−1 (q) = {p1 , . . . , pr }. Let V be an aﬃne neighborhood of q. Then U = φ−1 (V ) is an aﬃne open subset of X and φ : U → V is ﬁnite (by Theorem 7.5). Lemma 13.16. Let q ∈ Y , let V be an aﬃne neighborhood of q in Y , and let U = φ−1 (V ). Let φ−1 (q) = {p1 , . . . , pr }, and R=
r i=1
OX,pi .
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13. Divisors
Then R = k[U ]OY,q =
+
, ai bi  ai ∈ k[U ], bi ∈ OY,q ,
where the product is in k(X), and we identify OY,q with its image in k(X) by φ∗ . Proof. We have that OY,q ⊂ OX,pi and k[U ] ⊂ OX,pi for all i, since pi ∈ U for all i, so that k[U ]OY,q ⊂ R. Suppose that f ∈ R. Let z1 , . . . , zs ∈ X be the poles of f on U . Let wi = φ(zi ) for 1 ≤ i ≤ s. We have that f is regular at the points φ−1 (q) = {p1 , . . . , pr } since f ∈ R. There exists a function h ∈ k[V ] such that h(q) = 0 and h(wi ) = 0 for all i by Lemma 8.10. Replacing h with a suﬃciently high power of h, we then also have that νzi (hf ) ≥ 0 for all i. Thus hf has no poles on U , so that OX,q = OX (U ) = k[U ]. hf ∈ q∈U
∈ OY,q , we have that f ∈ k[U ]OY,q . Thus R ⊂ k[U ]OY,q . Theorem 13.17. Let φ−1 (q) = {p1 , . . . , pr } and R = ri=1 OX,pi . Then R is a free OY,q module of rank n = deg(φ).
Since
1 h
Proof. Since k[U ] is a ﬁnitely generated k[V ]module and OY,q is a localization of k[V ], we have that R = k[U ]OY,q is a ﬁnitely generated OY,q module. By the classiﬁcation theorem for ﬁnitely generated modules over a principal ideal domain [84, Theorem 3.10], the OY,q module R is a direct sum of a ﬁnite rank free OY,q module and a torsion OY,q module. Since R and OY,q are contained in the ﬁeld k(X), R has no OY,q torsion, so that R is a free OY,q module of ﬁnite rank. Suppose that f1 , . . . , fm is a basis of the free OY,q module R. If m > n = deg(φ), then f1 , . . . , fm are linearly dependent over k(Y ). Clearing denominators in a dependence relation over k(Y ) gives a dependence relation over OY,q . Thus m ≤ n. Let h ∈ k(Y ). Let l be the maximum of the order of a pole of h at the pj . Let t be a local parameter at q (a generator of the maximal ideal of OY,q ). Then the function tl h has no poles in the set φ−1 (q) so tl h ∈ R. Then there exists an expression tl h = m i=1 ai fi with ai ∈ R, so h=
m ai i=1
tl
fi ,
showing that f1 , . . . , fm span k(X) as a k(Y )vector space. Thus m ≥ n.
13.5. The class group of a curve
257
Theorem 13.18. Suppose that φ : X → Y is a dominant regular map of nonsingular projective curves. Then deg φ∗ (q) = deg φ for every q ∈ Y . In particular, deg φ∗ (D) = deg φ deg D for every divisor D on Y . Proof. Let t be a local parameter at q in Y (a generatorof the maximal ideal of OY,q ), and let φ−1 (q) = {p1 , . . . , pr }. Let R = ri=1 OX,pi . Let t1 , . . . , tr ∈ R be the generators of the r distinct maximal ideals mi = (ti ) of R, found in Lemma 13.15. Then any two distinct ti , tj cannot both be contained in any ml . Thus the ideal (ti , tj ) is not contained in a maximal ideal of R, so that (ti , tj ) = R; that is, the ideals (ti ) and (tj ) are coprime in R. By Lemma 13.15, we have that t = tl11 · · · tlrr v where v is a unit in R and li = νpi (t). Thus li pi and deg φ∗ (q) = li . φ∗ (q) = Since the ideals (ti ) are pairwise coprime, we have by Theorem 1.5 that R/tR ∼ =
r
R/tlii R.
i=1
We will now show that for every i, every w ∈ R has a unique expression (13.10)
w ≡ α0 + α1 ti + · · · + αli −1 tili −1 mod (tlii )
with α0 , . . . , αli −1 ∈ k. We prove this formula by induction. Suppose that it has been established that w has a unique expression mod (tsi ) w ≡ α0 + α1 ti + · · · + αs−1 ts−1 i s−1 with α0 , . . . , αs−1 ∈ k. Then v = t−s )) ∈ R. i (w − (α0 + α1 ti + · · · + αs−1 ti Let v(pi ) = αs ∈ k. Then νpi (v − αs ) > 0, so that v ≡ αs mod (ti ). Thus
w ≡ α0 + α1 ti + · · · + αs tsi mod (ts+1 ), i establishing (13.10). From (13.10), we see that dimk R/(tlii ) = li for all i, so that dimk R/(t) =
r
li = deg φ∗ (q).
i=1
Now by Theorem 13.17, we have OY,q module isomorphisms deg(φ) ∼ deg(φ) , R/(t) ∼ =k = (OY,q /(t))
from which we conclude that dimk R/(t) = deg φ.
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13. Divisors
Corollary 13.19. The degree of a principal divisor on a nonsingular projective curve X is zero. Proof. Suppose that f ∈ k(X) is a nonconstant rational function. The inclusion of function ﬁelds k(P1 ) = k(t) → k(X) deﬁned by mapping t to f determines a nonconstant rational map φ : X → P1 , which is a regular map since X is a nonsingular curve (Corollary 10.26). We have a representation φ = (f : 1). n
For p ∈ X, write f = up zp p where up is a unit in OX,p , zp is a regular parameter in OX,p , and np = νp (f ). If np ≥ 0, then φ(p) = (f (p) : 1), and if np < 0, then −np zp 1 (p) = 1 : (p) = (1 : 0). φ(p) = 1 : f up The function t is a local equation of 0 = (0 : 1) in P1 \ {(1 : 0)}, so φ∗ (0) = νp (φ∗ (t))p = νp (f )p = (f )0 . p∈φ−1 (0)
Similarly, (f )∞ .
1 t
p∈φ−1 (0)
is a local equation of ∞ = (1 : 0) in P1 \ {(0 : 1)}, so φ∗ (∞) =
By Theorem 13.18, deg(f ) = deg(f )0 − deg(f )∞ = deg φ∗ (0) − deg φ∗ (∞) = deg φ − deg φ = 0. Corollary 13.20. Suppose that X is a nonsingular projective curve. Then the surjective group homomorphism deg : Div(X) → Z induces a surjective group homomorphism deg : Cl(X) → Z. Let X be a nonsingular projective curve and let Cl0 (X) be the subgroup of Cl(X) of classes of divisors of degree 0. By the above corollary, there is a short exact sequence of groups (13.11)
deg
0 → Cl0 (X) → Cl(X) → Z → 0.
Corollary 13.21. Suppose that X is a nonsingular projective curve. Then Cl0 (X) = (0) if and only if X ∼ = P1 . In fact, if X is a nonsingular projective curve and p, q ∈ X are distinct points such that p ∼ q, then X ∼ = P1 . Proof. We proved in Example 13.10 that Cl0 (P1 ) = (0). Suppose that X is a nonsingular projective curve and p, q ∈ X are distinct points such that p ∼ q. Then there exists f ∈ k(X) such that (f ) = p − q. The inclusion of function ﬁelds k(P1 ) = k(t) → k(X) deﬁned by t → f determines a regular map φ : X → P1 , represented by (f : 1). Now φ∗ (0) = (f )0 = p, so deg φ = 1
13.6. Divisors, rational maps, and linear systems
259
by Theorem 13.18. Thus φ∗ : k(t) → k(X) is an isomorphism, and so φ is birational. Thus φ is an isomorphism by Corollary 10.25, since X and P1 are nonsingular projective curves. By Corollary 13.19, the degree, deg(L), of an invertible sheaf L on a nonsingular projective curve X is welldeﬁned, since L ∼ = OX (D) for some divisor D on X by Theorem 13.7. Exercise 13.22. Let k be an algebraically closed ﬁeld of characteristic not equal to 2, and let C be an irreducible quadric curve in P2 (C = Z(F ) where F ∈ k[x0 , x1 , x2 ] is an irreducible form of degree 2). Let p ∈ C be a point and let π : P2 P1 be the projection from p. Let φ : C P1 be the induced rational map. Show that φ is a regular map and an isomorphism.
13.6. Divisors, rational maps, and linear systems Let X be a normal quasiprojective variety and let φ : X Pn be a rational map. We may represent φ by an expression φ = (f0 : . . . : fn ) with f0 , . . . , fn ∈ k(X). We assume that none of the fi are zero. Let Di = (fi ) =
m
kij Cj
j=1
where the Cj are distinct prime divisors on X. Deﬁne the divisor D = gcd(D0 , . . . , Dn ) =
kj Cj , where kj = min kij . i
The divisor D has the properties that Di − D ≥ 0 for all i and (13.12)
if F is a divisor such that Di ≥ F for all i, then D ≥ F .
Lemma 13.23. Suppose that X is a normal variety and p ∈ X. Then φ is regular at p if and only if p ∈ ni=0 Supp(Di − D). Proof. The rational map φ is regular at p if and only if there exists α ∈ k(X) such that αfi ∈ OX,p for all i and there exists j such that αfj (p) = 0. Suppose that such an α ∈ k(X) exists. Then νE (αfi ) ≥ 0 for all i and prime divisors E which contain p, and νE (αfj ) = 0 for all prime divisors E which contain p. Thus there exists an open neighborhood U of p such that (αfi )U ≥ 0 for all i and (αfj )U = 0. Thus (α)U = −(fj )U = −Dj ∩ U . U ≥ Dj ∩ U for all i. Further, (Di − Dj ) ∩ U = (αfi )U ≥ 0, so that Di ∩ This is only possible if Dj ∩ U = D ∩ U , so that p ∈ ni=0 Supp(Di − D).
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13. Divisors
Now suppose that p ∈ ni=0 Supp(Di − D). Then there exists Ds such that ksi = ki for all i such that p ∈ Ci . Thus there exists a neighborhood U of p such that (fs ) ∩ U = D ∩ U . Let α = f1s . Then α is a local equation for the divisor −D on U . Suppose that E is a prime divisor on X such that p ∈ E. Then
0 if E ∩ U = Cj ∩ U for any Cj , νE (αfi ) = νE (α) + νE (fi ) = kij − kj if E ∩ U = Cj ∩ U for some Cj . In particular, αfi ∈ OX (0)p = OX,p for all i, and αfs does not vanish at p, so that φ is regular at p. Suppose that X is a normal projective variety and D is a divisor on X. Then Γ(X, OX (D)) = {f ∈ k(X)  (f ) + D ≥ 0} is a ﬁnitedimensional vector space over k, by Theorem 11.50, since OX (D) is coherent (Lemma 13.5). Suppose that f0 , . . . , fr ∈ Γ(X, OX (D)) are linearly independent over k. For t0 , . . . , tr ∈ k which are not all zero, we have that t0 f0 +· · ·+tr fr ∈ Γ(X, OX (D)). We thus have an associated eﬀective divisor (t0 f0 + · · · + tr fr ) + D on X, which is linearly equivalent to D. If 0 = α ∈ k, then the divisor (αt0 f0 + · · · + αtr fr ) = (t0 f0 + · · · + tr fr ). We deﬁne a linear system L ⊂ D by L = {(t0 f0 + · · · + tr fr ) + D  (t0 : . . . : tr ) ∈ Pr }. The linear system L is a family, parameterized by Pr , of eﬀective divisors on X which are linearly equivalent to D. If we take f0 , . . . , fr to be a kbasis of Γ(X, OX (D)), then we write L = D and say that L is a complete linear system. The linear system D is complete in the sense that if G is an eﬀective divisor on X which is linearly equivalent to D, then G ∈ D. Observe that the linear system L determines the subspace of Γ(X, OX (D)) spanned by f0 , . . . , fr , but we cannot recover our speciﬁc basis. Linear systems give us another way to understand rational maps. Suppose that D is a divisor on X, V is a linear subspace of Γ(X, OX (D)), and L ⊂ D is the associated linear system. Then we associate to L a rational map φL = (f0 : . . . : fn ) : X Pn , by choosing a basis f0 , . . . , fn of V . We will also denote this rational map by φV . A change of basis of V induces a linear automorphism of Pn , so maps obtained from diﬀerent choices of bases of V are the same, up to a change of homogeneous coordinates on Pn . Suppose that φ : X Pn is a rational map. Then we have a representation φ = (f0 : . . . : fn ) with fi ∈ k(X). Let Di = (fi ) and D = gcd(D0 , . . . , Dn ). Then fi ∈ Γ(X, OX (−D)) for all i. If the fi are
13.6. Divisors, rational maps, and linear systems
261
linearly independent, then φ is the rational map associated to the linear system L ⊂  − D obtained from the span V ⊂ Γ(X, OX (−D)) of f0 , . . . , fn . If some of the fi are linearly dependent, then the projection Pn Pm onto an appropriate subspace of Pn is an isomorphism on the image of φ, and the composed rational map X Pm is given by φL . Suppose that L is a linear system on a normal projective variety X. The base locus of the linear system L is the closed subset of X: Supp(F ). Base(L) = F ∈L
We will say that a linear system L is base point free if Base(L) = ∅. Example 13.24. Let A ∈ k[x0 , . . . , xn ] be a linear form on Pn . Let H = Div(A) (a hyperplane of Pn ). For r > 0, a kbasis of Γ(Pn , OX (rH)) is % # xi00 · · · xinn  i0 + · · · + in = r Ar so the complete linear system
, + n+r rH = Div ti0 ...in xi00 · · · xinn  i0 + · · · + in = r and (ti0 ...in ) ∈ P( n )−1 is base point free. Lemma 13.25. Suppose that X is a normal projective variety, D is a divisor on X, and p ∈ X. If OX (D)p is not invertible (not isomorphic to OX,p as an OX,p module), then p ∈ Base(D). Proof. Suppose that p ∈ X and p ∈ Base(D). We will show that OX (D)p is invertible. There exists f ∈ Γ(X, OX (D)) such that p ∈ Supp((f ) + D). 1 f
Let E = (f )+D. Then OX (D) → OX (E) is an isomorphism of OX modules. Since p ∈ Supp(E), OX (E)p = OX,p so OX (D)p is invertible. We point out that for a divisor D on a normal projective variety X and p ∈ X, OX (D)p is invertible if and only if there exists a local equation of D at p; in fact, if h generates OX (D)p as an OX,p module, then h1 is a local equation of D at p, as we now show. Let U be an aﬃne open neighborhood of p in X and g ∈ k(X) be such that D ∩ U = (g) ∩ U . Then, as shown in the ﬁrst part of the proof of Lemma 13.5, OX (D)U = g1 OU . Further, if V is an open subset of U , then OX (D)V = g1 OV . Thus OX (D)p = 1g OX,p , so that h = u g1 where u is a unit in OX,p , and so h1 is a local equation of D at p.
262
13. Divisors
Lemma 13.26. Suppose that X is a normal projective variety and L ⊂ D is the linear system associated to a subspace V of Γ(X, OX (D)). Then 1) The rational map φV : X Pn associated to V is independent of the divisor G such that V ⊂ Γ(X, OX (G)). 2) Suppose that E is a codimension 1 subvariety of X such that E ⊂ Base(L). Then V ⊂ Γ(X, OX (D − E)), and the linear system associated to V , regarded as a subspace of Γ(X, OX (D − E)), is {F − E  F ∈ L}. 3) Suppose that Base(L) has codimension ≥ 2 in X. Then the locus where the rational map φL is not a regular map is the closed subset Base(L). A consequence of Lemma 13.26 is that the rational map φν : X Pn is deﬁned by a linear system whose base locus has codim ≥ 2 in X. Proof. The ﬁrst statement follows since the rational map φV depends only on V . If E ⊂ Base(L) and f ∈ V , then (f ) + D ≥ E. Thus V ⊂ Γ(X, OX (D − E)). Let f0 , . . . , fn be a basis of V . Deﬁne A = −gcd((f0 ), . . . , (fn )) as before Lemma 13.23. We have that V ⊂ Γ(X, OX (A)) and the base locus of the linear system {(f ) + A  f ∈ V } has codimension ≥ 2 in X by Lemma 13.23. By (13.12) we have that A ≤ D. It remains to prove the third statement of the lemma. Since Base(L) has codimension ≥ 2 in X, we nhave that D = A. By Lemma 13.23, it suﬃces to show that Base(L) = i=0 (Supp((fi ) + A)). By Lemma 13.25, we need to show that if p ∈ X is such that OX (A)p is invertible, p ∈ ni=0 (Supp((fi ) + A)), and G ∈ L, then p ∈ Supp(G). With these assumptions, let g be a local equation of A at p. Then f i g ∈ OX,p and f i g(p) = 0 for all i. There exist t0 , . . . , tn ∈ k f ) + A so ti fi g is a local equation of G at p. Now such that G = ( t i i ti (fi g)(p) = 0 so p ∈ G. ( ti fi gi )(p) = Deﬁnition 13.27. A divisor D on a normal projective variety X is called very ample if BaseD = ∅ and the induced regular map φD : X → Pn is a closed embedding. The divisor D is called ample if OX (D) is invertible and some positive multiple of D is very ample. If D is very ample, then OX (D) is invertible by Lemma 13.25. In Example 13.24, rH is very ample if r ≥ 1 as φrH is a Veronese embedding.
13.6. Divisors, rational maps, and linear systems
263
Lemma 13.28. Suppose that φL : X → Pn is a regular map which is given by a base point free linear system L. Then L = {φ∗L (H)  H is a linear hyperplane on Pn such that φL (X) ⊂ H}. Further, if p ∈ X, then the ﬁber φ−1 L (φL (p)) =
Supp(F )
F
where the intersection is over F ∈ L such that p ∈ F . The proof of Lemma 13.28 is Exercise 13.40. Proposition 13.29. Suppose that D is a divisor on a normal projective variety X and V is a subspace of Γ(X, OX (D)) such that the associated linear system L is base point free. Let t be an indeterminate, and let R be the graded kalgebra R = k[V t]⊂ k(X)[t], where we set t to have degree Γ(X, OX (nD)) is a ﬁnitely generated 1. Then the graded kalgebra n≥0 Rmodule, where we regard n≥0 Γ(X, OX (nD)) as a graded Rmodule by the isomorphism Γ(X, OX (nD)) ∼ Γ(X, OX (nD))tn ⊂ k(X)[t]. = n≥0
n≥0
Proof. The rational map φL : X Pm (with m = dimk V − 1) is a regular map since L is base point free, and φ∗L OPm (1) ∼ = OX (D) (by Exercise 13.61). Let F = (φL )∗ OX . The sheaf F is a coherent OPm module by Theorem 11.51. We have that F (n) ∼ = (φL )∗ (φ∗L OPm (n)) ∼ = (φL )∗ OX (nD) = F ⊗ OPm (n) ∼ for n ∈ Z by Exercise 11.41. Further, Γ(X, OX (nD)) ∼ Γ(Pm , F (n)) = n≥0
n≥0
is a ﬁnitely generated S = S(Pm )module by Exercise 11.59. Thus Γ(X, OX (nD)) n≥0
is a ﬁnitely generated Rmodule since R is the image of S in k(X)[t], by the natural graded map which takes S1 onto V . Corollary 13.30. Suppose that X is a normal projective variety and D is an ample divisor on X. Then there exists m0 > 0 such that mD is very ample for m > m0 , and there exists m1 ≥ m0 such that if m > m1 and W is
264
13. Divisors
the image of X in a projective space P by the closed embedding φmD , and S is the coordinate ring of W ∼ = X by this embedding, then Γ(X, OX (nmD)) S∼ = n≥0
as graded rings. However, if D is not base point free, then it is possible that the graded kalgebra n≥0 Γ(X, OX (nD)) is not a ﬁnitely generated kalgebra, even if D is an eﬀective divisor on a nonsingular projective surface X. An example is given by Zariski in [159, Part 1, Section 2]. We will present this example in Theorem 20.14.
13.7. Criteria for closed embeddings Suppose that D is a divisor on a nonsingular projective variety X and V ⊂ Γ(X, OX (D)) is a linear subspace, with associated linear system L, such that Base(L) has codimension ≥ 2 in X. Let f0 , . . . , fn be a kbasis of V , so that the rational map φ = φV : X Pn is represented by φ = (f0 : . . . : fn ). Then we have seen that for p ∈ X, if α ∈ k(X) is a local equation of D at p, then αfi ∈ OX,p for all i, and φ = (αf0 : . . . : αfn ) is regular at p if and only if αfi (p) = 0 for some i. In this case, (αf0 : . . . : αfn ) represents φ as a regular map in a neighborhood of p. Let Ip be the ideal sheaf of p in X. Consider the commutative diagram: Γ(X, OX (D))
restriction
→
α
OX (D)p = α1 OX,p → OX,p ↓ ↓ α OX (D)p /Ip OX (D)p → OX,p /(Ip )p ∼ =k
where the isomorphism with k is the evaluation map g → g(p). The horizontal maps, given by multiplication by α, are OX,p module isomorphisms. ∼ k is The composed map Λ : Γ(X, OX (D)) → OX (D)p /Ip OX (D)p = given by f → (αf )(p) for f ∈ Γ(X, OX (D)). We see that (13.13)
φ is regular at p if and only if the natural map of V to OX (D)p /Ip OX (D)p ∼ = k is surjective.
Now suppose that φ is a regular map on X and p, q are distinct points of X. Let α be a local equation of D at p and let β be a local equation of D at q. Then we have an evaluation map Λ : Γ(X, OX (D)) → OX (D)p /Ip OX (D)p ⊕ OX (D)q /Iq OX (D)q ∼ = k2,
13.7. Criteria for closed embeddings
265
deﬁned by Λ (f ) = ((αf )(p), (βf )(q)) for f ∈ Γ(X, OX (D)). We have that φ(p) = φ(q) if and only if the vectors (αf0 (p), . . . , αfn (p)) and
(βf0 (q), . . . , βfn (q))
in k n+1 are linearly independent over k. Thus, since the row rank and column rank of the matrix αf0 (p) · · · αfn (p) βf0 (q) · · · βfn (q) are equal, (13.14) φ is regular and injective if and only if the natural map of V to OX (D)p /Ip OX (D)p ⊕ OX (D)q /Iq OX (D)q is surjective for all p = q ∈ X. Suppose that φ is an injective regular map. Then φ is a closed embedding if and only if dφp : Tp (X) → Tq (Pn ) is injective for all p ∈ X (with q = φ(p)) by Theorem 10.36. Suppose that α = 0 is a local equation of D at p. Then (αfi )(p) = 0 for some i, say (αf0 )(p) = 0. Then φ(p) is in U = Pnx0 ∼ = An , which has the regular functions xn x1 ,..., . k[U ] = k x0 x0 Let λi = k[U ] is
(αfi )(p) (αf0 )(p)
for 1 ≤ i ≤ n. Then the maximal ideal mq = Iq (U ) of q in mq =
x1 xn − λ1 , . . . , − λn . x0 x0
Now dφp is injective if and only if the dual map φ∗ : mq /m2q → mp /m2p is onto (where mp = (Ip )p ). This map is onto if and only if the classes of αf1 αfn − λ1 , . . . , − λn αf0 αf0 span (Ip )p /(Ip )2p as a kvector space. Since (αf0 )(p) = 0, this holds if and only if the classes of αf1 − λ1 (αf0 ), . . . , αfn − λn (αf0 ) span (Ip )p /(Ip )2p as a kvector space. Now {f ∈ V  (αf )(p) = 0} has the basis {f1 − λ1 f0 , . . . , fn − λn f0 }.
266
13. Divisors
Thus:
(13.15)
Suppose that φ is an injective regular map. Let Kp be the kernel of the natural map V → OX (D)p /Ip OX (D)p . Then φ is a closed embedding if and only if the natural map of Kp to Ip OX (D)p /Ip2 OX (D)p is surjective for all p ∈ X.
Lemma 13.31. Suppose that X is a nonsingular quasiprojective variety, D is a divisor on X such that OX (D) is invertible, and p1 , . . . , pn ∈ X are distinct points. Then 1) The natural OX module homomorphism OX (D) ⊗ Ip1 ⊗ · · · ⊗ Ipn → OX (D) is injective with image the coherent OX submodule Ip1 · · · Ipn OX (D) of OX (D). 2) If U is an open subset of X, then Γ(U, OX (D) ⊗ Ip1 ⊗ · · · ⊗ Ipn ) = {f ∈ Γ(U, OX (D))  pi ∈ (f )U + D ∩ U for all pi such that pi ∈ U }. Proof. The ﬁrst statement follows since Ipi ,q ∼ = OX,q if p = q and OX (D)q ∼ = OX,q for all q ∈ X and since OX,q is a ﬂat OX,q module for all q ∈ X. To prove the second statement, by the sheaf axioms, it suﬃces to prove the second statement for aﬃne open sets U such that OX (D)U is isomorphic to OU . Let U be such an open set. Let g be a local equation of D in U . Then OX (D)U = g1 OU . We may suppose that p1 , . . . , ps ∈ U and ps+1 , . . . , pn ∈ U . We compute Γ(U, OX (D) ⊗ Ip1 ⊗ · · · ⊗ Ipn ) = g1 k[U ] ⊗k[U ] IU (p1 ) ⊗k[U ] IU (p2 ) ⊗ · · · ⊗k[U ] IU (pn ) ∼ = g1 IU (p1 )IU (p2 ) · · · IU (ps ). Now f ∈ Γ(U, OX (D) ⊗ Ip1 ⊗ · · · ⊗ Ips ) if and only if gf ∈ IU (p1 ) · · · IU (ps ) which holds if and only if (gf )U ≥ 0 and (gf )(pi ) = 0 for 1 ≤ i ≤ s. These last conditions holds if and only if (f )U + D ∩ U ≥ 0 and pi ∈ (f )U + D ∩ U . Theorem 13.32. Suppose that X is a nonsingular projective variety and A and B are divisors on X. 1) Suppose that A is very ample and B is base point free. Then A+B is very ample. 2) Suppose that A is ample and B is an arbitrary divisor on X. Then there exists a positive integer n0 such that nA + B is very ample for all n ≥ n0 .
13.7. Criteria for closed embeddings
267
Proof. We ﬁrst establish 1). Since A is very ample, V = Γ(X, OX (A)) satisﬁes the conditions of (13.13), (13.14), and (13.15). Since B is base point free, Γ(X, OX (B)) generates OX (B)p as an OX,p module for all p ∈ X. Thus we have natural surjections Λ : Γ(X, OX (A)) ⊗ Γ(X, OX (B)) → (OX (A)p /Ip OX (A)p ) ⊗OXp OX (B)p ∼ OX (A + B)p /Ip OX (A + B)p = for all p ∈ X, Λ : Γ(X, OX (A)) ⊗ Γ(X, OX (B)) → (OX (A)p /Ip OX (A)p ⊕ OX (A)q /Iq OX (A)q ) ⊗OX,p OX (B)p ∼ = (OX (A + B)p /Ip OX (A + B)p ) ⊕ (OX (A + B)q Iq OX (A + B)q ) for all p, q ∈ X with p = q, and Λ : Γ(X, OX (A) ⊗ Ip ) ⊗ Γ(X,OX (B)) → Ip OX (A)p /Ip2 OX (A)p ⊗ OX (B)p ∼ = Ip OX (A + B)p /Ip2 OX (A + B)p . We have a natural kvector space homomorphism Γ(X, OX (A)) ⊗ Γ(X, OX (B)) → Γ(X, OX (A + B)) which factors Λ and Λ and a natural kvector space homomorphism Γ(X, OX (A) ⊗ Ip ) ⊗ Γ(X, OX (B)) → Γ(X, OX (A + B) ⊗ Ip ) which factors Λ . Thus A + B satisﬁes the conditions of (13.13), (13.14), and (13.15), so that A + B is very ample. We now establish 2). There exists a positive integer n1 such that n1 A is very ample, so there exists a closed embedding φ : X → Pr such that OX (n1 A) ∼ = φ∗ OPr (1) = OX (1) by Exercise 13.61. By Theorem 11.45, there exists n2 ≥ n1 such that for 0 ≤ t < n1 , OX (tA + nn1 A) is generated by global sections if n ≥ n2 and OX (B+nn1 A) is generated by global sections if n ≥ n2 . Suppose n > 3n1 n2 . Write n − n1 n2 = mn1 + t
with 0 ≤ t < n1 .
We then have m > n2 . Thus nA + B = (m − n2 )n1 A + [(n2 n1 A + tA) + (n2 n1 A + B)] is the sum of a very ample divisor and a divisor D such that D is base point free. Thus nA + B is very ample by 1) of this theorem.
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13. Divisors
Theorem 13.33. Suppose that X is a nonsingular projective variety and D is a divisor on X such that Base(D) has codimension ≥ 2 in X. Let φD be the rational map associated to Γ(X, OX (D)). Then 1) φD is a regular map if and only if for all p ∈ X, dimk Γ(X, OX (D) ⊗ Ip ) = dimk Γ(X, OX (D)) − 1. 2) φD is an injective regular map if and only if for all distinct points p and q in X, dimk Γ(X, OX (D) ⊗ Ip ⊗ Iq ) = dimk Γ(X, OX (D)) − 2. 3) φD is a closed embedding if and only if 2) holds and for all p ∈ X, dimk Γ(X, OX (D) ⊗ Ip2 ) = dimk Γ(X, OX (D)) − (1 + dim X). Proof. Consider the short exact sequence for p ∈ X, 0 → Ip → OX → OX /Ip → 0. Since OX (D) is locally free, tensoring the sequence with OX (D) gives a short exact sequence 0 → OX (D) ⊗ Ip → OX (D) → (OX /Ip ) ⊗ OX (D) → 0. Now (OX /Ip ) ⊗ OX (D) ∼ = OX /Ip , so taking global sections, we get an exact sequence 0 → Γ(X, OX (D) ⊗ Ip ) → Γ(X, OX (D)) → k, so that the conclusion of 1) follows from (13.13). Consider the short exact sequence for p = q ∈ X, which follows from Theorem 1.5 applied to an aﬃne open subset of X containing p and q, 0 → Ip Iq → OX → OX /Ip Iq ∼ = OX /Ip ⊕ OX /Iq → 0. Since the locus of points where Ip is not locally free and the locus of points where Iq is not locally free are disjoint, we have that Ip ⊗ Iq ∼ = Ip Iq . Since OX (D) is locally free, tensoring the sequence with OX (D) gives a short exact sequence 0 → OX (D)⊗Ip Iq → OX (D) → (OX /Ip )⊗OX (D)⊕(OX /Iq )⊗OX (D) → 0. Now (OX /Ip Iq ) ⊗ OX (D) ∼ = OX /Ip Iq ∼ = OX /Ip ⊕ OX /Iq , so taking global sections, we get an exact sequence 0 → Γ(X, OX (D) ⊗ Ip Iq ) → Γ(X, OX (D)) → k 2 , so that the conclusion of 2) follows from 1) of this theorem and (13.14).
13.8. Invertible sheaves
269
Finally, 3) of the theorem follows from tensoring the short exact sequence 0 → Ip2 → Ip → Ip /Ip2 → 0 with OX (D) and taking global sections, applying (13.15), and since dimk Ip /Ip2 = dim X (as p is a nonsingular point on X).
Corollary 13.34. Suppose that X is a nonsingular projective curve and D is a divisor on X. Then 1) D is base point free if and only if dimk Γ(X, OX (D − p)) = dimk Γ(X, OX (D)) − 1 for all p ∈ X. 2) Suppose that D is base point free. Then φD is injective if and only if dimk Γ(X, OX (D − p − q)) = dimk Γ(X, OX (D)) − 2 for all p = q ∈ X. 3) Suppose that D is base point free. Then φD is a closed embedding (and D is very ample) if and only if dimk Γ(X, OX (D − p − q)) = dimk Γ(X, OX (D)) − 2 for all p, q ∈ X.
13.8. Invertible sheaves The constructions of ⊗ and Hom in Section 11.5 are fairly simple for invertible sheaves. Suppose that F and G are invertible sheaves on a quasiprojective variety X. Every point p ∈ X has an aﬃne neighborhood U such that F U = σOU where σ ∈ Γ(U, F ) is a local generator of F and GU = τ OU where τ ∈ Γ(U, G) is a local generator of G. Thus F ⊗ G is invertible. In fact, if F U = σOU and GU = τ OU , then F ⊗OX GU = σ⊗τ OU . We usually write στ for σ ⊗ τ . Suppose that L is an invertible sheaf on X. Then Hom OX (L, OX ) is ˆ OU , an invertible sheaf. In fact, if LU = σOU , then Hom OX (L, OX )U = σ 1 where σ ˆ (σ) = 1. We can write Hom OX (L, OX )U = σ OU . The map α⊗β → β(α) determines an OX module isomorphism L ⊗OX Hom OX (L, OX ) ∼ = OX . We write L−1 = Hom OX (L, OX ), since then L ⊗ L−1 ∼ = OX . = L−1 ⊗ L ∼
270
13. Divisors
Thus the tensor product and inverse operation make Pic(X) = {LL is invertible}/ ∼ into an Abelian group, where two invertible sheaves are equivalent if they are isomorphic as OX modules. Pic(X) is called the Picard group of X. If X is nonsingular and D, E are divisors on X, then we have an aﬃne cover {Ui } of X and fi , gi ∈ k(X) such that OX (D)Ui =
1 OU fi i
OX (E)Ui =
1 OU . gi i
and
Then OX (D) ⊗ OX (E)Ui =
1 OU = OX (D + E)Ui , fi g i i
so that OX (D + E) ∼ = OX (D) ⊗ OX (E). We also have that OX (D)−1 Ui = fi OX Ui = OX (−D)Ui so that OX (−D) ∼ = OX (D)−1 . It follows from Proposition 13.6 and Theorem 13.7 that (with the assumption that X is nonsingular) the map (13.16)
Cl(X) → Pic(X)
deﬁned by D → OX (D) is a group isomorphism. Suppose that L is an invertible sheaf on a quasiprojective variety X and σ0 , . . . , σn ∈ Γ(X, L). We have an associated rational map φ : X Pn , which is deﬁned as follows. Suppose that U is an open subset of X such that LU is trivial; that is, there exists τ ∈ Γ(U, L) such that LU = τ OU . Write σi = τ fi for 0 ≤ i ≤ n where fi ∈ Γ(U, OX ). We deﬁne φ = (f0 : f1 : . . . : fn ) on U . This gives a welldeﬁned rational map, which we can write as φ = (σ0 : . . . : σn ). The rational map φ is a regular map if and only if σ0 , . . . , σn generate L; that is, for all p ∈ X, the restrictions of σ0 , . . . , σn to Lp generate Lp as an OX,p module. Every regular map from X to a projective space Pn can be represented in this way by an appropriate invertible sheaf L and σ0 , . . . , σn ∈ Γ(X, L) (take L = φ∗ OPn (1) and σ0 , . . . , σn to be the images of x0 , . . . , xn ∈ Γ(Pn , OPn (1)) in Γ(X, φ∗ OPn (1)). Deﬁnition 13.35. An invertible sheaf L on a projective variety X is called very ample if the global sections Γ(X, L) generate L as an OX module and the regular map φ = (σ0 : · · · : σn ) : X → Pn , where σ0 , . . . , σn is a kbasis of Γ(X, L), is a closed embedding. An invertible sheaf L on a projective variety X is called ample if some positive tensor product Ln = L⊗n is very ample.
13.9. Transition functions
271
Deﬁnition 13.35 is consistent with Deﬁnition 13.27 of very ample and ample divisors on a normal variety X. A divisor D on a normal projective variety X is very ample (ample) if and only if OX (D) is invertible and very ample (ample). Suppose that φ : X → Y is a regular map of quasiprojective varieties and L is an invertible sheaf on Y . Then φ∗ L is a coherent sheaf on X (Section 11.5). If V ⊂ Y is an open subset on which LV is trivial, then LV = τ OV for some τ ∈ Γ(V, OY ). We have that φ∗ Lφ−1 (V ) = τ OV ⊗OV Oφ−1 (V ) = (τ ⊗ 1)Oφ−1 (V ) = τ Oφ−1 (V ) . Thus φ∗ L is an invertible sheaf on X. If X and Y are nonsingular and D is a divisor on Y , then φ∗ (OY (D)) ∼ = OX (φ∗ (D)). This can be readily seen in the case that φ is dominant. We then have an injection φ∗ : k(Y ) → k(X). We cover Y with open subsets Vi such that OY (D)Vi = f1i OVi where fi is a 1 Oφ−1 (Vi ), where local equation of D on Vi . Then φ∗ OY (D)φ−1 (Vi ) = φ∗ (f i) ∗ ∗ −1 φ (fi ) is a local equation of φ (D) on φ (Vi ). In the case where φ is not dominant we must ﬁrst shift the support of D as explained after Lemma 13.12.
13.9. Transition functions Suppose that L is an invertible sheaf on a variety X. Then there exists an ∼ = open cover {Ui } of X and OUi module isomorphisms φi : OX Ui → LUi for ∼ = all i. Consider the OX Ui ∩Uj module isomorphisms φ−1 j ◦φi : OX Ui ∩Uj → OX Ui ∩ Uj . In the notation of the proof of Theorem 13.7, we have that −1 φ−1 j ◦ φi is multiplication by gij . We may thus identify φj ◦ φi with gij , which is a unit in Γ(Ui ∩ Uj , OX ) satisfying the relations (13.4), −1 , gij = gji
and (13.5), (13.17)
gik = gij gjk .
We call the gij transition functions on Ui ∩ Uj for L. Suppose that M is another invertible sheaf on X and there exist OUi module isomorphisms ψi : OX Ui → MUi . Let hij = ψj−1 ◦ ψi be transition functions on Ui ∩Uj for M. Then we have that gij hij are transition functions for L ⊗ M on Ui ∩ Uj and g1ij are transition functions for L−1 on Ui ∩ Uj . ∗ be the presheaf of (multiplicative) groups on X, deﬁned for Let OX U ⊂ X an open subset by ∗ ) = {f ∈ Γ(U, OX )  f is a unit}. Γ(U, OX ∗ is a sheaf of groups on X, and g ∈ Γ(U ∩ U , O ∗ ). Then OX ij i j X
272
13. Divisors
Lemma 13.36. Suppose that L is an invertible sheaf on a variety X and {Ui }i∈I is an open cover of X such that there exist OUi module isomorphisms ∼ ∼ = = φi : OX Ui → LUi for all i. Let φ−1 j ◦ φi : OX Ui ∩ Uj → OX Ui ∩ Uj be the respective associated transition functions, which we identify with elements ∗ ). Then L is isomorphic to O as an O module if and gij of Γ(Ui ∩ Uj , OX X X ∗ ) for all i ∈ I such that g = f f −1 for all only if there exist fi ∈ Γ(Ui , OX ij j i i, j. ∗ ) such that g = f f −1 Proof. First suppose that there exist fi ∈ Γ(Ui , OX ij j i for all i, j. Then LUi = τi OUi where τi = φi (fi ) = fi φi (1). For all i, j, on Ui ∩ Uj , we have φi = gij φj . Thus φi (1) = gij φj (1) = fj fi−1 φj (1) so that τi = τj on Ui ∩ Uj . By the second sheaf axiom, there exists τ ∈ Γ(X, L) such that τ Ui = τi for all i. Deﬁne a homomorphism of sheaves of OX modules Λ : OX → L by Λ(f ) = τ f . Here Λ is an isomorphism at all stalks at points of X, so that Λ is an isomorphism by Proposition 11.13.
Now suppose that L is isomorphic to OX as an OX module. Then there exists an isomorphism of OX modules Λ : OX → L. Let τ = Λ(1). ∗ ) such that Then LUi = τ OUi for all i. Thus there exist fi ∈ Γ(Ui , OX τ = fi φi (1) = φi (fi ). Thus on Ui ∩ Uj , we have that φi (fi ) = φj (fj ), so that fi φi (1) = fj φj (1). Since φi (1) = gij φj (1), we have that gij = fj fi−1 . Lemma 13.37. Suppose that L and M are invertible sheaves on a variety X and {Ui }i∈I is an open cover of X such that there exist OUi module ∼ ∼ = = isomorphisms φi : OX Ui → LUi and ψi : OX Ui → MUi for all i. Let ∼ ∼ = = −1 φ−1 j ◦ φi : OX Ui ∩ Uj → OX Ui ∩ Uj and ψj ◦ ψi : OX Ui ∩ Uj → OX Ui ∩ Uj be the respective associated transition functions, which we identify with ∗ ). Then L is isomorphic to M as an elements gij and hij of Γ(Ui ∩ Uj , OX ∗ ) for all i ∈ I such that OX module if and only if there exist fi ∈ Γ(Ui , OX gij = fj fi−1 hij for all i, j. ∼ OX . Proof. We have that L is isomorphic to M if and only if L ⊗ M−1 = −1 As commented above Lemma 13.36, gij hij are the transition functions of L ⊗ M−1 on Ui ∩ Uj . The conclusions of this lemma now follow from Lemma 13.36. A situation where this criterion for isomorphism is very useful is in understanding the pullback φ∗ L of an invertible sheaf under a regular map φ : Y → X of varieties. Suppose that {Ui }i∈I is an open cover of X and we have ∗ trivializations φi : OUi → LUi for all i. Let φ−1 j ◦ φi = gij ∈ Γ(Ui ∩ Uj , OX ) be the transition functions on Ui ∩ Uj for L. Let Vi = φ−1 (Ui ). For all i, we have an OY Vi module homomorphism φi ⊗1
OY Vi = OX ⊗OX OY Vi → (L ⊗OX OY ) Vi = φ∗ LVi ,
13.9. Transition functions
273
which is an isomorphism of sheaves, since it is an isomorphism at stalks of points of Vi . Computing the transition functions of φi ⊗ 1, we see that (φj ⊗ 1)−1 ◦ (φi ⊗ 1) = g ij ∈ Γ(Vi ∩ Vj , OY∗ ), where g ij = φ∗ (gij ) under ∗ ) → Γ(Vi ∩ Vj , OY∗ ). φ∗ : Γ(Ui ∩ Uj , OX
We now rework Example 13.13 using this technique. Example 13.38. Suppose that X is a nonsingular surface and p ∈ X is a point. Let π : B → X be the blowup of p. Let E = π −1 (p) ∼ = P1 be the exceptional divisor of π. Let i : E → B be the inclusion. Then i∗ (OB (E)) ∼ = OE (−q), where q is a point on E. Proof. Let x, y be regular parameters in the regular local ring OX,p . Let U be an aﬃne neighborhood of p such that x = y = 0 are local equations of p in U . Then V = π −1 (U ) is covered by two aﬃne charts U1 and U2 which satisfy k[U1 ] = k[U ][ xy ] and k[U2 ] = k[U ][ xy ]. In U1 , y = 0 is a local equation of E. Now (x, y)k[U1 ] = yk[U1 ], so that x . k[U1 ∩ E] = k[U1 ]/(x, y)k[U1 ] = k y In U2 , x = 0 is a local equation of E, so k[U2 ∩ E] = k
y
. x Now OB (E)U1 = y1 OB U1 and OB (E)U2 = x1 OB U2 . Thus g12 = xy is a transition function for OB (E) on U1 ∩ U2 , so that g 12 = i∗ ( xy ), which we can identify with xy , is the transition function for i∗ OB (E) on U1 ∩ U2 ∩ E. Now xy is the local equation of a point q in E ∩U1 , which is not contained in U2 , so we have that OE (−q)U1 ∩ E = xy OE U1 ∩ E and OE (−q)U2 ∩ E = OE U2 ∩ E. The associated transition function on U1 ∩ U2 ∩ E is h12 = xy . Since h12 = g 12 , we have that i∗ OB (E) ∼ = OE (−q) by Lemma 13.37. Suppose that L is an invertible sheaf on a nonsingular variety X and σ ∈ Γ(X, L) is nonzero. We associate an eﬀective divisor to σ in the following way. Let {Ui } be an open cover of X such that there exist σi ∈ Γ(Ui , L) such that LUi = σi OUi . We have expressions σUi = fi σi where fi ∈ Γ(Ui , OX ). We deﬁne the divisor (σ) (or div(σ)) of σ to be the divisor D on X deﬁned by D ∩ Ui = (fi )Ui . This divisor is welldeﬁned and independent of choice of local trivialization of L. The pairs {(fi , Ui )} determine a Cartier divisor on X (Deﬁnition 15.4) and the following discussion.
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Proposition 13.39. Suppose that φ : X → Y is a birational regular map of projective varieties such that Y is normal. Suppose that L is an invertible sheaf on Y . Then φ∗ φ∗ L ∼ = L. Proof. This follows from Proposition 11.52 and Exercise 11.41 since L is locally isomorphic to OY . Exercise 13.40. Prove Lemma 13.28. Exercise 13.41. Let x0 , . . . , xn be homogeneous coordinates on X = Pn , so that k[x0 , . . . , xn ] = S(Pn ). Let ti = xx0i for 1 ≤ i ≤ n, so that k(Pn ) = k(t1 , . . . , tn ). Let E = Z(x0 ), a prime divisor on X. Suppose that f (t1 , . . . , tn ) ∈ k[t1 , . . . , tn ] = k[Xx0 ] has degree m. Compute r = −νE (f ), and show that (f ) + rE ≥ 0. Exercise 13.42. Let the notation be as in Exercise 13.41. Suppose that Y is a nonsingular closed subvariety of X such that Y ⊂ Z(x0 ). Let i : Y → X be the inclusion. We have a natural surjection i∗ : k[Pnx0 ] = k[t1 , . . . , tn ] → k[Yx0 ]. Suppose that f ∈ k[t1 , . . . , tn ] is such that i∗ (f ) = 0. Show that (i∗ (f )) + ri∗ (E) is an eﬀective divisor on Y . Exercise 13.43. Suppose that m, n are positive integers. Show that Pm ×Pn is not isomorphic to Pm+n . Show that there is, however, a birational (but not regular) map Pm × Pn Pm+n . Exercise 13.44. Suppose that X is a projective variety, with graded coordinate ring S = S(X) and homogeneous coordinates x0 , . . . , xm . Assume that none of the xi vanish everywhere on X. Recall that for n ∈ Z, S(n) is S with the grading S(n)t = Sn+t . Recall from (11.21) that OX (n) := S(n) satisﬁes the following property: xm x0 n ,..., = xni Γ(Xxi , OX ). Γ(Xxi , OX (n)) = xi k xi xi a) Show that OX (n)Xxi = xni OX Xxi . Conclude that OX (n) is an invertible sheaf on X. b) Show that Sn ⊂ Γ(X, OX (n)) and that this inclusion is an equality if X = Pm . c) The following example shows that it is possible for Sn to be strictly smaller than Γ(X, OX (n)). Let s, t be algebraically independent over k and let R = k[s4 , s3 t, st3 , t4 ] be the ring of Example 1.75. Then R is a standard graded domain with the grading deg s4 = deg s3 t = deg st3 = deg t4 = 1.
13.9. Transition functions
275
Thus R is the homogeneous coordinate ring of a projective curve C in P3 (which is isomorphic to P1 ). Show that s2 t2 ∈ Γ(C, OC (1)). Show that Γ(C, OC (n)) = k[s4 , s3 t, s2 t2 , st3 , t4 ] n≥0
where k[s4 , s3 t, s2 t2 , st3 , t4 ] is graded by deg s4 = deg s3 t = deg s2 t2 = deg st3 = deg t4 = 1. d) Suppose that X is nonsingular. Let Di = Div(xi ) be the divisor of xi on X (Section 13.3). Show that OX (n) is isomorphic as an OX module to OX (nDi ). e) Suppose that X is nonsingular. Let L be a linear form on X (which does not vanish everywhere on X), and let D = Div(L) be the divisor associated to L on X. Show that OX (n) is isomorphic as an OX module to OX (nD). Exercise 13.45. Let X be a projective variety. a) Suppose that I ⊂ OX is an ideal sheaf on X, which is not equal to OX . Show that Γ(X, I) = (0). b) Suppose that X is nonsingular, and let D = ri=1 ai Ei be an eﬀective divisor on X, with ai > 0, Ei prime divisors, so that OX (−D) is the ideal sheaf OX (−D) = I(E1 )a1 · · · I(Er )ar . Show that the presheaf on X deﬁned by P (U ) = Γ(U, OX (D)) ⊗Γ(U,OX ) Γ(U, OX (−D)) for U an open subset of X is not a sheaf on X; thus this presheaf diﬀers from the sheaf OX (D) ⊗OX OX (−D). Exercise 13.46. Let X be a nonsingular variety of dimension 3, and let p ∈ X be a point. Let π : B → X be the blowup of p. Let E = π −1 (p). We know that E ∼ = Z is generated by OE (1). Let = P2 and that Pic(E) ∼ i : E → B be the inclusion. Compute i∗ OB (E) in Pic(E) in terms of the generator OE (1). Exercise 13.47. Show that the regular isomorphisms of Pn are the linear isomorphisms (Section 4.2). Conclude that the group of automorphisms of Pn is the variety 2 PGL(n, k) = Pn +2n \ Z(Det). Hint: Use Exercise 4.14 and Example 13.10. Exercise 13.48. Suppose that Y is a nonsingular projective surface and p ∈ Y is a point. Let π : X → Y be the blowup of p with exceptional
276
13. Divisors
divisor E = π −1 (p). Show that the group homomorphism Cl(Y ) ⊕ Z → Cl(X) deﬁned by ([D], n) → [π ∗ D + nE] is a group isomorphism. Exercise 13.49. Let H be a hyperplane on P2 as in Example 13.24. Let p = (0 : 0 : 1), and deﬁne V = Γ(P2 , OX (H) ⊗ Ip ). Find a basis of V and compute the linear system L = {(f ) + H  f ∈ V } as a subsystem of the complete linear system H described as the divisors of forms of degree 1 in Example 13.24. Compute the base locus of L. Describe the rational map φV and the geometry of this map. Exercise 13.50. Let H be a hyperplane on P2 as in Example 13.24. Let P1 = (0 : 0 : 1), P2 = (0 : 1 : 0), P3 = (1 : 0 : 0). Let V = Γ(P2 , OX (2H) ⊗ IP1 ⊗ IP2 ⊗ IP3 ). Find a basis of V and compute the linear system L = {(f ) + 2H  f ∈ V } as a subsystem of the complete linear system 2H described as the divisors of forms of degree 2 in Example 13.24. Compute the base locus of L. Describe the rational map φV and the geometry of this map. Exercise 13.51. Consider the projections π1 and π2 of P1 ×P1 onto the ﬁrst and second factors. Represent each map as φD for an appropriate complete linear system D on P1 × P1 . Exercise 13.52. Suppose that Y is a normal variety of dimension ≥ 2. a) Suppose that q is a nonsingular point of Y and φ : X → Y is the blowup of q with exceptional divisor E. Show that
if n ≥ 0, OX φ∗ OX (nE) = Iq−n if n < 0. b) Suppose that Y is a normal variety, D is a divisor on Y , and q1 , . . . , qr are distinct nonsingular points on Y . Let φ : X → Y be the blowup of q1 , . . . , qr with exceptional divisors E1 , . . . , Er . Show that Γ(X, OX (φ∗ (D) − E1 − · · · − Er )) = Γ(Y, OY (D) ⊗ Iq1 ⊗ · · · ⊗ Iqr ). Exercise 13.53. Suppose that X is a normal projective variety and D is a divisor on X. Show that the divisor D on X is very ample (Deﬁnition 13.27) if and only if OX (D) is a very ample invertible sheaf (Deﬁnition 13.35).
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277
Exercise 13.54. Suppose that L is an invertible sheaf on a projective variety X. Show that L is very ample if and only if there exists a closed embedding ψ : X → Pn such that ψ ∗ OPn (1) ∼ = L. Exercise 13.55. Prove Corollary 13.30. Exercise 13.56. Extend (and prove) Lemma 13.31 and Theorems 13.33 and 13.32 to invertible sheaves on an arbitrary projective variety. Exercise 13.57. Let x0 , x1 be homogeneous coordinates on X = P1 , so that k(X) = k( xx10 ). Let p0 = (1 : 0), p1 = (0 : 1), p2 = (1 : 1) ∈ P1 . Suppose that n ∈ Z. Find a kbasis of each of the following vector spaces: a) Γ(X, OX (np0 )), b) Γ(X, OX (np2 )), c) Γ(X, OX (np0 − p1 )), d) Γ(X, OX (np0 − p1 − p2 )). Exercise 13.58. Let the notation be as in Exercise 13.57, and let V be the subspace of Γ(X, OX (3p0 )) with basis {1, ( xx10 )2 , ( xx01 )3 }. Let L be the linear system L = {(f ) + 3p0  f ∈ V }, with associated rational map φ = φL : X P2 . a) Show that L is base point free. b) Compute the homogeneous coordinate ring of the image of φ in P2 . c) Is φ a closed embedding? Why or why not? Exercise 13.59. Give an example of a linear system L on a projective variety X such that L is not base point free, but the rational map φL is a closed embedding of X. Exercise 13.60. Let X ⊂ A3 be the aﬃne surface with regular functions k[X] = k[x, y, z] = k[x, y, z]/(xy − z 2 ). Deﬁne prime divisors on X by E1 = Z(x, z) and E2 = Z(y, z). a) Show that OX (−E1 ) and OX (−E2 ) are not invertible sheaves. Hint: Use a method from the proof of Theorem 10.17. b) Show that E1 ∼ −E2 . Exercise 13.61. Suppose that D is a divisor on a normal projective variety and V is a subspace of Γ(X, OX (D)) such that the associated linear system L is base point free. Let φL : X → Pm be the associated regular map. Show that φ∗L OPm (n) ∼ = OX (nD) for n ∈ Z.
Chapter 14
Diﬀerential Forms and the Canonical Divisor
In Section 14.1, we discuss the algebraic theory of derivations and diﬀerentials, which generalizes the notion of 1forms on a manifold to rings. The results of this section are used extensively in Chapters 21 and 22. In Section 14.2, we deﬁne the sheaf of 1forms on a variety X, and in Section 14.3 we deﬁne the sheaf of nforms, canonical divisors, and the divisor of a rational nform on an ndimensional nonsingular variety. We prove the useful adjunction formula (Theorem 14.21) for computing canonical divisors.
14.1. Derivations and K¨ ahler diﬀerentials Deﬁnition 14.1. Suppose that A and B are rings and λ : A → B is a ring homomorphism, making B into an Amodule by cx = λ(c)x for x ∈ B and c ∈ A. Suppose that M is a Bmodule. Then a map D : B → M is an Aderivation from B to M if D satisﬁes the following three conditions: 1) D(f + g) = D(f ) + D(g) for f, g ∈ B, 2) D(cf ) = cD(f ) if c ∈ A and f ∈ B, 3) D(f g) = f D(g) + gD(f ) for f, g ∈ B. Let DerA (B, M ) be the set of all Aderivations from B to M . It is a Bmodule. We observe that if x ∈ B and n ∈ N, then 3) implies that D(xn ) = nxn−1 D(x). Thus D(1B ) = nD(1B ) for all n > 0, and so (14.1)
D(1B ) = 0. 279
280
14. Diﬀerential Forms and the Canonical Divisor
Thus 2) implies that D(c1B ) = 0 for all c ∈ A. Conversely, if 3) holds and D(c1B ) = 0 for all c ∈ A, then D(cf ) = cD(f ) for all c ∈ A and f ∈ B. We thus see that 2) can be replaced with the condition that D(c1B ) = 0 for all c ∈ A. Aderivations are often deﬁned in this way ([161, page 120] or [107, pages 180–181]). Deﬁnition 14.2. Let F be the free Bmodule on the symbols {dbb ∈ B}, and let G be the submodule generated by the relations 1), 2), and 3) in the deﬁnition of a derivation. We deﬁne the Bmodule of K¨ahler diﬀerentials of B over A by ΩB/A = F/G. The map d = dB/A : B → ΩB/A deﬁned by letting d(f ) be the class of df is an Aderivation of B. Lemma 14.3. Suppose that M is a Bmodule. Then the map Φ : HomB (ΩB/A , M ) → DerA (B, M ) deﬁned by Φ(τ )(f ) = τ (df ) for f ∈ B and τ ∈ HomB (ΩB/A , M ) is a Bmodule isomorphism. The lemma shows that d : B → ΩB/A is a universal derivation: if M is a Bmodule and D ∈ DerA (B, M ), then there is a unique Bmodule homomorphism σ : ΩB/A → M such that D = σdB/A . The inverse Ψ to Φ (in Lemma 14.3) is deﬁned as follows. Suppose that D : B → M is an Aderivation. Deﬁne Ψ(D) = τ by
τ αi dbi = αi D(bi ). A detailed proof of Lemma 14.3 is given in [50, page 384]. Lemma 14.4. Deﬁne a Bmodule homomorphism δ : B ⊗A B → B by δ(a1 ⊗ a2 ) = a1 a2 , and let I be the kernel of δ. The quotient I/I 2 is a Bmodule. Then the map Λ : ΩB/A → I/I 2 deﬁned by taking Λ(df ) to be the class of 1 ⊗ f − f ⊗ 1 for f ∈ B is a Bmodule isomorphism. This lemma is established in [50, Theorem 16.24]. The module ΩB/A is often deﬁned to be I/I 2 as constructed in Lemma 14.3, with the Aderivation B → I/I 2 deﬁned by mapping f to the class of 1 ⊗ f − f ⊗ 1 for f ∈ B. Example 14.5. Let κ be a ring, and let B = κ[x1 , . . . , xn ] be a polynomial ring over κ. Then ΩB/κ is the free Bmodule with generators dx1 , . . . , dxn , ΩB/κ = Bdx1 ⊕ · · · ⊕ Bdxn .
14.1. Derivations and K¨ahler diﬀerentials
281
The map d : B → ΩB/κ is d(f ) =
∂f ∂f dx1 + · · · + dxn ∂x1 ∂xn
for f ∈ B. Proof. We ﬁrst use the three properties of a derivation to prove that ∂f ∂f (14.2) df = dx1 + · · · + dxn ∂x1 ∂xn for f ∈ B. We have that d(1) = 0 by (14.1). The properties of a derivation show that (14.2) holds for an arbitrary monomial. Then the formula holds ∂ are κlinear. for all f since d and the ∂x i From equation (14.2) we conclude that ΩB/κ is generated by dx1 , . . . , dxn as a Bmodule. It remains to show that dx1 , . . . , dxn are a free basis of ΩB/κ . Suppose that we have a relation (14.3)
g1 dx1 + · · · + gn dxn = 0
for some g1 , . . . , gn ∈ B. We will show that g1 = · · · = gn = 0. We have that ∂ ∂xi for 1 ≤ i ≤ n are κderivations on B. By Lemma 14.3, there exist B
∂f module homomorphisms τi : ΩB/κ → B for 1 ≤ i ≤ n such that τi (df ) = ∂x i for f ∈ B. Thus τi (dxj ) = δij (the Kronecker delta). Applying τi to (14.3), we obtain that gi = τi (g1 dx1 + · · · + gn dxn ) = 0 for 1 ≤ i ≤ n.
The following theorem is proven in [107, Theorem 58] or [50, Proposition 16.3]. Theorem 14.6. Suppose that κ is a ring, A is a κalgebra, J is an ideal of A, and B = A/J. Then there is an exact sequence of Bmodules δ
v
J/J 2 → ΩA/κ ⊗A B → ΩB/κ → 0 where δ : J → ΩA/κ ⊗A B is deﬁned by x → dA/κ (x)⊗1 and v : ΩA/κ ⊗A B → ΩB/κ is deﬁned by dA/κ (a) ⊗ b → bdB/κ (a). Example 14.7. Suppose that R is a ring and A = R[x1 , . . . , xn ] is a polynomial ring over R. Let I = (f1 , . . . , fr ) ⊂ R[x1 , . . . , xn ] be an ideal. Let B = R[x1 , . . . , xn ]/I. Let M = Bdx1 ⊕ · · · ⊕ Bdxn . We have that ΩB/R = M/df1 B + · · · + dfr B. Proof. This follows from Theorem 14.6.
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Lemma 14.8. Suppose that B is an Aalgebra and S is a multiplicative set in B. Then we have a natural isomorphism of S −1 Bmodules ΩS −1 B/A ∼ = S −1 ΩB/A . This follows from [50, Proposition 16.9]. Theorem 14.9. Suppose that f : A → B and g : B → C are ring homomorphisms. Then there is an exact sequence β
α
ΩB/A ⊗B C → ΩC/A → ΩC/B → 0 of Cmodules where α is deﬁned by α(dB/A b⊗c) = cdC/A g(b) and β(dC/A c) = dC/B c for b ∈ B and c ∈ C. Proof. [107, Theorem 57] or [106, Theorem 25.1] or [50, Proposition 16.2]. Exercise 14.10. Suppose that φ
A → A ↑ ↑ λ
κ →
κ
is a commutative diagram of rings and ring homomorphisms. The A module ΩA /κ is naturally an Amodule by ax := φ(a)x for a ∈ A and x ∈ ΩA /κ . Show that there is a natural Amodule homomorphism δφ giving a commutative diagram δφ
ΩA/κ → ΩA /κ dA/κ ↑ ↑ dA /κ A
φ
→
A .
Exercise 14.11. If A = A ⊗κ κ in Exercise 14.10, show that ΩA /κ = ΩA/κ ⊗κ κ = ΩA/κ ⊗A A . Exercise 14.12. Let k be an (algebraically closed) ﬁeld of characteristic not equal to 2 or 3. Let X be the aﬃne variety X = Z(y 2 − x3 ) ⊂ A2 . Let R = k[X]. a) Compute ΩR/k . b) Show that ΩR/k has Rtorsion (there exists a nonzero element ω of ΩR/k and a nonzero element a of R such that aω = 0).
14.2. Diﬀerentials on varieties
283
14.2. Diﬀerentials on varieties Suppose that X is a quasiprojective variety. If X is aﬃne, then we deﬁne the coherent sheaf ΩX/k on X by ΩX/k = Ω k[X]/k . We will show that there is a natural isomorphism ΩX/k (V ) ∼ = Ωk[V ]/k if V is an aﬃne open subset of X. If V = Xf for some f ∈ k[X], then ΩX/k (Xf ) = (Ωk[X]/k )f ∼ = Ωk[X]f /k ∼ = Ωk[Xf ]/k by Lemma 14.8. Suppose that V ⊂ X is an aﬃne open subset. There exist f1 , . . . , fn ∈ k[X] such that Xfi ⊂ V for all i and V = ni=1 Xfi . We then have that Xfi = Vfi for all i, so that k[X]fi = k[V ]fi . Thus Ωk[X]/k f ∼ = Ωk[V ]/k f i
i
for all i, and
Ωk[X]/k
fi fj
∼ = Ωk[V ]/k fi fj
for all i, j. We have a commutative diagram 0 → ΩX/k (V ) → 0 →
Ωk[V ]/k
→
i ΩX/k (Xfi )
↓ i Ωk[V ]/k f
i
→
ΩX/k (Xfi fj ) ↓ → i<j Ωk[V ]/k f f i<j
i j
where the horizontal arrows are exact and the vertical arrows are isomorphisms, inducing a natural isomorphism ΩX/k (V ) ∼ = Ωk[V ]/k . To deﬁne ΩX/k for arbitrary quasiprojective varieties, we cover X with open aﬃne subsets {Ui } and deﬁne ΩX/k Ui = ΩUi /k . By the universal property of K¨ahler diﬀerentials, ΩUi /k Ui ∩Uj and ΩUj /k Ui ∩Uj are naturally isomorphic (Ui ∩ Uj is aﬃne by Exercise 5.21). Thus they patch (Exercise 11.23) to give a coherent sheaf of OX modules ΩX/k . If V is an aﬃne open subset of X, we have that ΩX/k (V ) ∼ = Ωk[V ]/k . Proposition 14.13. Suppose that X is a variety and p ∈ X. Let mp be the maximal ideal of OX,p . Let k(p) = OX,p /mp ∼ = k which is naturally an OX,p module. Then the following are naturally isomorphic kvector spaces: 1) the tangent space Tp (X) = Homk (mp /m2p , k), 2) the derivations Derk (OX,p , k(p)), 3) the module of OX,p module homomorphisms HomOX,p ((ΩX/k )p , k(p)).
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14. Diﬀerential Forms and the Canonical Divisor
Proof. The vector spaces 1) and 2) are isomorphic since every kderivation D : OX,p → k(p) must vanish on m2p and thus induces a klinear map mp /m2p → k, and further, given a klinear map : mp /m2p → k, we get a kderivation D : OX,p → k(p) deﬁned by D(f ) = (f − f (p)). The vector spaces of 2) and 3) are isomorphic since HomOX,p ((ΩX/k )p , k(p)) ∼ = HomOX,p (ΩOX,p /k , k(p)) ∼ = Derk (OX,p , k(p))
by Lemmas 14.8 and 14.3.
Theorem 14.14. Suppose that X is a quasiprojective variety. Then the (nontrivial open) subset U of nonsingular points of X is the largest open subset of X on which ΩX/k is locally free. The sheaf ΩX/k U is locally free of rank equal to the dimension of X. Proof. Suppose that p ∈ X. Let λ : (ΩX/k )p → k(p) = OX,p /mp be an OX,p module homomorphism. Then for a ∈ mp and t ∈ (ΩX/k )p we have that λ(at) = aλ(t) = 0. Thus we have a natural kvector space isomorphism ∼ =
HomOX,p ((ΩX/k )p , OX,p /mp ) → Homk ((ΩX/k )p /mp (ΩX/k )p , k). This last kvector space is (noncanonically) isomorphic to (ΩX/k )p/mp (ΩX/k )p , and the ﬁrst kvector space is isomorphic to Tp (X) by Proposition 14.13. Thus dimk (ΩX/k )p /mp (ΩX/k )p = dimk Tp (X) ≥ dim X with equality if and only if p is a nonsingular point of X, by Proposition 10.14. The conclusions of the theorem now follow from Lemma 11.54. Proposition 14.15. Suppose that X is a nonsingular quasiprojective variety, p ∈ X, and x1 , . . . , xn are regular parameters in OX,p . Then there exists an open neighborhood U of p such that dx1 , . . . , dxn is a free basis of ΩU/k . This proposition is not true on varieties over nonperfect ﬁelds (see Exercise 21.78). The generalization of this proposition to arbitrary ﬁelds is given in [154], where a thorough study of the concept of singularity over arbitrary ﬁelds is made. Proof. Let A = X,p with maximal ideal m = mp . Suppose that f ∈ A. O n a x with a ∈ A and c ∈ k. Thus df = ai dxi + Then f = c + i=1 i i i ai dxi ∈ mΩA/k . By Nakayama’s lemma xi dai ∈ ΩA/k , so that df − (Lemma 1.18) we have that dx1 , . . . , dxn generate ΩA/k = (ΩX/k )p as an Amodule. Since dim X = n and p is a nonsingular point of X, by Theorem 14.14, (ΩX/k )p is a free OX,p module of rank n. Let N be the kernel of
14.2. Diﬀerentials on varieties
285
the surjection (dx1 , . . . , dxn ) : An → (ΩX/k )p . Tensoring the short exact sequence 0 → N → An → (ΩX/k )p → 0 with k(X), we see that the localization N ⊗ k(X) = 0 (the sequence is split exact since (ΩX/k )p is a free Amodule) and so N is a torsion submodule of An , and thus N = 0. Thus dx1 , . . . , dxn is a free basis of (ΩX/k )p , and so there exists an open neighborhood U of p on which dx1 , . . . , dxn is a free basis of ΩU/k . Deﬁnition 14.16. Suppose that U is an open subset of a nonsingular variety X. Elements f1 , . . . , fn ∈ Γ(U, OX ) are called uniformizing parameters on U if df1 , . . . , dfn is a free basis of ΩX/k U . Proposition 14.17. Suppose that U is an open subset of a nonsingular variety X and f1 , . . . , fn ∈ Γ(U, OX ) are uniformizing parameters on U . Suppose that p ∈ U . Then f1 − f1 (p), . . . , fn − fn (p) are regular parameters in OX,p . Proof. Let mp be the maximal ideal of OX,p . Let x1 , . . . , xn be regular parameters in OX,p . We have that f1 − f1 (p), . . . , fn − fn (p) ∈ mp . Thus there exist aij ∈ OX,p such that aij xj , fi − fi (p) = j
so that dfi ≡
aij dxj
mod mp (ΩX/k )p
for all i. Since {dfi } and {dxj } are free bases of (ΩX/k )p , we have that Det(aij ) is a unit in OX,p and the matrix (aij ) is invertible over OX,p , so that f1 − f1 (p), . . . , fn − fn (p) generate mp . Since OX,p has dimension n and dimk mp /m2p = n, we have that f1 − f1 (p), . . . , fn − fn (p) is a regular system of parameters in OX,p . Exercise 14.18. Suppose that f : X → Y is a regular map of varieties over an algebraically closed ﬁeld k. Show that there are a coherent OX module ΩX/Y and a natural exact sequence of coherent OX modules (14.4)
f ∗ ΩY /k → ΩX/k → ΩX/Y → 0
which becomes the sequence of Theorem 14.9 with A = k, B = Γ(U, OY ), and C = Γ(V, OX ) when evaluated at an open aﬃne subset V of Y which maps into an open aﬃne subset U of X.
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14.3. nforms and canonical divisors In this section, suppose that X is a nonsingular variety of dimension n. If A is a ring and M is an Amodule, then ∧n M is the quotient of the tensor product M ⊗ · · · ⊗ M with itself n times, by the Asubmodule generated by all elements x1 ⊗ · · · ⊗ xn with all xi ∈ M and where xi = xj for some i = j [95, Section 1 of Chapter XIX]. We deﬁne ΩnX/k to be the coherent sheaf on X associated to the presheaf U → ∧n ΩX/k (U ) for U an open subset of X. If U is an aﬃne open subset of X, then ΩnX/k (U ) = ∧n Ωk[U ]/k . The sheaf ΩnX/k is an invertible OX module. In fact, if U is an open subset of X such that ΩX/k U = OU dx1 ⊕ · · · ⊕ OU dxn , then ΩnX/k U = OU dx1 ∧ · · · ∧ dxn is a free OU module. The rational diﬀerential nforms on X are Ωnk(X)/k = ∧n Ωk(X)/k = k(X)df1 ∧ · · · ∧ dfn where f1 , . . . , fn ∈ k(X) are any elements such that df1 ∧ · · · ∧ dfn = 0. Let {Ui } be an aﬃne cover of X, such that there exist ωi ∈ Γ(Ui , ΩnX/k ) satisfying ΩnX/k Ui = ωi OUi . Let φ be a nonzero rational diﬀerential nform on X. Then there exist gi ∈ k(X) for all i such that φ = gi ωi . We have that (gi )Ui ∩Uj = (gj )Ui ∩Uj for all i, j since there exist units γij ∈ Γ(Ui ∩ Uj , OX ) such that ωi = γij ωj as ωi and ωj are both generators of ΩnX/k Ui ∩ Uj . Thus there exists a divisor D on X such that (14.5)
D ∩ Ui = (gi )Ui
for all i.
The divisor D is independent of our choice of {Ui } and {ωi }. We deﬁne D to be the divisor of φ. We will denote the divisor D of φ by (φ), (φ)X , div(φ), or div(φ)X . Proposition 14.19. Suppose that φ1 , φ2 are nonzero rational diﬀerential nforms on X. Then (φ1 ) ∼ (φ2 ). Let KX = (φ) be the divisor of a rational diﬀerential nform. Then ΩnX/k ∼ = OX (KX ). We call the divisor KX of a rational diﬀerential nform on X a canonical divisor of X. If X is a normal variety, we can deﬁne a canonical divisor on X by deﬁning a canonical divisor on the nonsingular locus U of X (whose
14.3. nforms and canonical divisors
287
complement has codimension ≥ 2 in X) and extending it to a divisor KX on X. We then have that i∗ ΩnU/k ∼ = OX (KX ) where i : U → X is the inclusion (as follows from the proof of Lemma 13.5). Proof. Let φ be a nonzero rational diﬀerential nform on X. Let D = (φ) and L = φOX , which is a free OX module. We have that Ωn ∼ = OX (D) = OX (D) ⊗O L ∼ X/k
X
by equation (14.5). In particular, if φ1 , φ2 are rational diﬀerential nforms, then (φ1 ) ∼ (φ2 ) (by Proposition 13.6). The following example is established in Exercise 14.23. Example 14.20. The canonical divisor KPn = −(n + 1)L where L is a linear hyperplane on Pn . Theorem 14.21 (Adjunction). Suppose that V is a nonsingular codimension 1 closed subvariety of a nonsingular variety W , so that V is a prime divisor on W , and let i : V → W be the inclusion. Then KV = i∗ (KW + V ); that is, OV (KV ) ∼ = OW (KW + V ) ⊗OW OV . Proof. Let n = dim(W ). There exist an aﬃne open cover {Ui } of a neighborhood of V in W and uniformizing parameters x1 (i), . . . , xn (i) ∈ Γ(Ui , OW ), such that x1 (i) = 0 is a local equation of V in Ui . Thus there ∗ ) such that are units gij ∈ Γ(Ui ∩ Uj , OW x1 (i) = gij x1 (j)
(14.6)
for all i, j. Since {dx1 (i), . . . , dxn (i)} and {dx1 (j), . . . , dxn (j)} are free bases of Γ(Ui ∩ Uj , ΩnW/k ), there exists al,m (i, j) ∈ Γ(Ui ∩ Uj , OW ) such that al,m (i, j)dxm (j) d(xl (i)) = ∗ ). Now we have that and hij = Det(al,m (i, j)) is in Γ(Ui ∩ Uj , OW
dx1 (i) ∧ · · · ∧ dxn (i) = hij dx1 (j) ∧ · · · ∧ dxn (j). ⎞ a22 (i, j) · · · a2n (i, j) ⎟ ⎜ .. .. cij = Det ⎝ ⎠. . . an2 (i, j) · · · ann (i, j) ∼ The ideal of V in W is IV = OW (−V ). Taking d of (14.6), we have that for all i, j, hij ≡ gij cij mod Γ(Ui ∩ Uj , OW (−V )). Thus the transition functions of ΩnW/k ⊗OV on Ui ∩Uj are g ij cij , where g ij is the image of gij in Γ(Ui ∩Uj , OV∗ ) and cij is the image of cij in Γ(Ui ∩Uj , OV∗ ).
Let
⎛
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14. Diﬀerential Forms and the Canonical Divisor
We have that the g ij are the transition functions of OW (−V ) ⊗ OV on Ui ∩ Uj . The images x2 (i), . . . , xn (i) of x2 (i), . . . , xn (i) in Γ(Ui , OV ) are uniformizing parameters on Ui ∩V since x1 (i) = 0 is a local equation of V on Ui and so dx2 (i), . . . , dxn (i) is a free basis of Γ(Ui , ΩV /k ) by Proposition 14.15. Thus the cij are transition functions for Ωn−1 V /k on Ui ∩ Uj . Since ΩnW/k ⊗OV and Ωn−1 V /k ⊗OW (−V ) have the same transition functions on Ui ∩Uj , ΩnW/k ⊗OV and Ωn−1 V /k ⊗OW (−V ) are isomorphic by Lemma 13.37. n−1 ∼ n ⊗ OW (V ) ⊗ OV . Thus Ω =Ω V /k
W/k
Corollary 14.22. Suppose that C is a nonsingular cubic curve in P2 . Then KC = 0 (so that OC (KC ) ∼ = OC ). Proof. This follows from adjunction since C ∼ 3L and KP2 = −3L, where L is a linear hyperplane on P2 . Exercise 14.23. Prove Example 14.20. Exercise 14.24. Suppose that S is a nonsingular surface and π : X → S is the blowup of a point. Let E be the exceptional divisor of π. Show that KX ∼ π ∗ (KS ) + E.
Chapter 15
Schemes
In this chapter we discuss some generalizations of quasiprojective varieties, beginning with subschemes of quasiprojective varieties. This concept will be important in later chapters. Then we will give a quick survey of some more general spaces. All of our constructions will be locally ringed spaces, which were deﬁned in Deﬁnition 11.15.
15.1. Subschemes of varieties, schemes, and Cartier divisors Deﬁnition 15.1. A closed subscheme Z of a quasiprojective variety X is a locally ringed space, consisting of the pair of a closed subset Y of X and a sheaf of rings OZ = OX /I on Y where I is an ideal sheaf on X such that Supp(OX /I) = Y . We call Y the associated topological space of Z. The subscheme Zred of X ideal sheaf of Z in X is IZ = I. We deﬁne the closed √ to be the subscheme associated to Y with ideal sheaf I. An open subscheme of a closed subscheme Z is an open subset U of a closed subsheme Z, with the sheaf OU = OZ U . We will sometimes write “subscheme” or “scheme” to mean an open or closed subscheme. A subscheme is called aﬃne if it is a closed subscheme of an aﬃne variety, projective if it is a closed subscheme of a projective variety, and quasiprojective if it is an open subscheme of a projective subscheme. Deﬁnition 15.2. A scheme X is a locally ringed space such that every point p ∈ X has an open neighborhood U such that the open subset U with the sheaf OX U is an aﬃne scheme. 289
290
15. Schemes
If Y is a closed subvariety of a quasiprojective variety X, then we can regard Y as the scheme with associated topological space Y and sheaf of rings OY ∼ = OX /IY ; in particular, we have Y = Yred . More generally, if Y is a (closed) algebraic set in a quasiprojective variety X, then we can regard Y as a closed subscheme of X, with the structure such that Y = Yred . √ If Z is a subscheme, then the natural inclusion IZ ⊂ IZ induces a surjection OZ → OZred , and so we have an inclusion of subschemes Zred ⊂ Z. We say that a scheme Z is irreducible if the associated topological space of Z is irreducible. A scheme Z is said to be reduced if OZ is reduced; that is, Z = Zred so that IZ is a reduced ideal sheaf. A scheme Z is said to be integral if Z is both reduced and irreducible. Thus a quasiprojective scheme Z is integral if and only if Z is a variety and Z is reduced if and only if Z is an algebraic set. Example 15.3. Let X = A2 with regular functions k[X] = k[x, y] and let Y be the closed subset Y = Z(y) ⊂ X. For n a positive integer, let In be the ideal In = (y n ) in k[X], and let In be the ideal sheaf In = I˜n on X. Let Yn be the locally ringed space Yn = (Y, OA2 /In ). The schemes Yn are diﬀerent (nonisomorphic) closed subschemes of X with the same underlying topological space Y . We have that (Yn )red = Y1 = Y for all n. The subscheme Y1 = Y is a subvariety of X. We consider some situations where schemes appear naturally. Suppose that φ : X → Y is a regular map of quasiprojective varieties and Z is a closed subscheme of Y . We deﬁne the schemetheoretic ﬁber XZ over Z to be the closed subscheme of X with associated topological space the subset φ−1 (Z) of X with the ideal sheaf IXZ = IZ OX , where IZ is the ideal sheaf of Z in Y . A particularly important case is of the ﬁber Xp over a point p ∈ Y . We will continue to write φ−1 (Z) for the algebraic set (XZ )red . Suppose that Z1 and Z2 are closed subschemes of a quasiprojective variety X, with respective ideal sheaves IZ1 and IZ2 . Then the schemetheoretic intersection Z1 ∩ Z2 is the closed subscheme of X with ideal sheaf IZ1 + IZ2 . Earlier, we encountered the settheoretic intersection Z1 ∩ Z2 , which is the algebraic set with ideal sheaf IZ1 + IZ2 . An eﬀective divisor D on a normal variety X can be regarded as a closed subscheme of X. The sheaf OD = OX /OX (−D). We deﬁne sheaves, quasicoherent sheaves, and coherent sheaves on a scheme as in Chapter 11. The deﬁnition of invertible sheaf and analysis in Section 13.9 are valid on an arbitrary scheme. If L is an invertible sheaf on a scheme X, then
15.1. Subschemes of varieties, schemes, and Cartier divisors
291
there exists an aﬃne cover {Ui } of X with OUi module isomorphisms φi : ∗ ), the units in OX Ui → LUi and transition functions gij ∈ Γ(Ui ∩ Uj , OX Γ(Ui , ∩Uj , OX ). The total quotient ring of a ring A is deﬁned to be QR(A) = S −1 A where S is the multiplicative set of nonzero divisors in A. Let X be a scheme. There is a sheaf K on X which has the property that K(U ) = QR(OX (U )) whenever U is aﬃne. The sheaf K is called the sheaf of total quotient rings of OX . The sheaf of multiplicative groups K∗ ∗ is the sheaf of is deﬁned to be the group of invertible elements of K and OX ∗ ⊂ K∗ . invertible elements of OX . We have inclusions OX ⊂ K and OX Deﬁnition 15.4. A Cartier divisor D on a scheme X is a collection {(Ui , fi )} where {Ui } is an open cover of X and fi ∈ Γ(Ui , K∗ ) are such that for each ∗ ). i, j, ffji ∈ Γ(Ui ∩ Uj , OX We have encountered Cartier divisors on varieties earlier, in (13.2). Cartier divisors are developed in more detail in [118, Section 9] and [73, Section II.6]. Let X be a scheme. Then for D a Cartier divisor on X, OX (D) is the invertible sheaf on X deﬁned by OX (D)Ui =
1 OX Ui . fi
We have that OX (D1 −D2 ) ∼ = OX (D1 )⊗OX (D2 )−1 if D1 and D2 are Cartier divisors. We say that D1 is linearly equivalent to D2 if OX (D1 ) ∼ = OX (D2 ). A Cartier divisor on a scheme X is eﬀective if it can be represented by {(Ui , fi )} where each fi ∈ OX (Ui ). In this case, we have an associated subscheme D of X deﬁned by ID Ui = fi OUi . Suppose that L is an invertible sheaf on a scheme X. A global section s ∈ Γ(X, L) is called a nonzero divisor if the annihilator of the stalk sp by OX,p is zero for all p ∈ X. This is equivalent to the annihilator of sUi as a Γ(Ui , OX )module being zero for all Ui in an aﬃne cover of X. Suppose that s ∈ Γ(X, L) is a nonzero divisor. Then there is associated to s a subscheme div(s) (or (s)) of X which is an eﬀective Cartier divisor. The closed subscheme div(s) has the ideal sheaf Idiv(s) which is deﬁned on each Ui by Idiv(s) Ui = φ−1 i (s)OUi where φi : OUi → LUi are local trivializations of L. The eﬀective divisor div(s) associated to s is welldeﬁned; it is independent of choice of local trivialization. This generalizes the construction of div(s) on a nonsingular variety, given after Example 13.38.
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15. Schemes
It is possible for Idiv(s),p to have embedded primes, where p ∈ div(s), even when X is a variety. This is illustrated by Example 1.75. Suppose that I is an ideal sheaf on a projective variety X. Let x0 , . . . , xn be homogeneous coordinates on X, and let S(X) = k[x0 , . . . , xn ] be the coordinate ring of X. Let m = (x0 , . . . , xn ) be the graded maximal ideal. There exists a homogeneous ideal I in S(X) such that I is the sheaﬁﬁcation I˜ of I (Proposition 11.48). Let I = Q0 ∩ Q1 ∩ · · · ∩ Qs be a homogeneous primary decomposition of I, where Q0 is the mprimary component√of I if m is an associated prime of I and Q0 = S(X) otherwise. Let Pi = Qi be the associated (homogeneous) prime ideals to Qi . We have that I sat = Q1 ∩ · · · ∩ Qs . Let Yi = Z(Pi ). Then sat ) = Q ˜1 ∩ · · · ∩ Q˜s I = I˜ = (I
˜ i is IY = P˜i primary. is a primary decomposition of I, where the ideal sheaf Q i Let Z be the subscheme of X with ideal sheaf IZ = I. We deﬁne the homogeneous ideal of Z to be I(Z) = I sat . The homogeneous ideal I(Z) is uniquely determined by Z. Deﬁnition 15.5. Suppose that L is an invertible sheaf on a projective scheme V , F is a coherent sheaf on V , and s ∈ Γ(V, L). We will say that s is not a zero divisor on F if for all p ∈ V , φ−1 (s) is not a zero divisor on Fp , where φ : OU → LU is a trivialization of L in a neighborhood U of p. Recall (after Deﬁnition 15.4) that for s ∈ Γ(V, L) a nonzero divisor, div(s) is the subscheme of V deﬁned by Idiv(s)  U = φ−1 (s)OU , where U is an open neighorhood of p in V admitting a local isomorphism φ : OU → LU . As V is a closed subscheme of a projective space Pn , we have that OV = ) where S(V ) = S(Pn )/J for some homogeneous ideal J in S(Pn ). We S(V can realize F = F˜ where F is a graded S(V )module. Let I be the saturation of the annihilator of F as a S(V )module. Let I = Q1 ∩ · · · ∩ Qt be an irredundant primary decomposition of I, and let Pi be the prime ideals associated to Qi . Suppose that s ∈ Γ(V, L) is a nonzero divisor. Then s is a nonzero divisor on F if and only if the subvarieties Vi = Z(Pi ) of V satisfy
15.2. Blowups of ideals and associated graded rings of ideals
293
Vi ⊂ Supp(div(s)) for 1 ≤ i ≤ t. We will call the Vi the associated varieties of F . Suppose that U is an aﬃne open subset of V and φ is a local trivialization φ : OU → LU . Let S ⊂ {1, . . . , t} be the indices such that U ∩ Vi = ∅. ˜ ) which has the Then the annihilator of the module Γ(U, F ) is the ideal I(U irredundant primary decomposition ˜ i (U ) ˜ )= Q I(U i∈S

˜ i (U ) = I(U ∩ Vi ). The element s is a nonzero divisor of F on U if with Q and only if φ−1 (s) ∈ I(U ∩ Vi ) for i ∈ S. Exercise 15.6. Suppose that X is a normal aﬃne variety with regular (D) is functions R = k[X]. Suppose that D is a divisor on X such that OX invertible and f ∈ Γ(X, OX (D)) is nonzero. Suppose that (f ) + D = ai Ei with Ei prime divisors and ai ∈ Z+ . Show that, regarding f as an element of Γ(X, OX (D)), we have (a )
(a )
Idiv(f ) = IE11 ∩ · · · ∩ IErr , (a )
where IEi are the prime ideals of Ei in R and IEii = (IEaii RIEi ) ∩ R is the ai th symbolic power of IEi . Hint: Use Theorem 1.79. Exercise 15.7. Suppose that X is nonsingular in Exercise 15.6. Show that we then have Idiv(f ) = IEa11 IEa22 · · · IEarr . Exercise 15.8. Use Example 1.75 to give an example of a projective variety X with coordinate ring S such that if L = Div(F ) for some nonzero homogeneous linear form F ∈ S, then I(L) = (F ).
15.2. Blowups of ideals and associated graded rings of ideals Suppose that X is an aﬃne variety with regular functions R = k[X] and J ⊂ R is an ideal. Let π : B(J) → X be the blowup of J. From Theorem 6.4, we know that a coordinate ring of B(J) is the graded ring S(B(J)) ∼ = i . Let Z be the closed aﬃne subscheme of X with ideal sheaf I = J˜. J Z i≥0 From Theorem 6.4, we see that the coordinate ring of the ﬁber B(J)Z of Z in B(J) is J i /J i+1 . S(B(J)Z ) ∼ = S(B(J))/JS(B(J)) ∼ = i≥0
294
15. Schemes
This last ring is (by deﬁnition) the associated graded ring grJ R of J. In the case that J = m is a maximal ideal of R (which corresponds to a point p of X) we have that S(B(J)Z ) ∼ mi /mi+1 = grm (R). = i≥0
When J = m, the scheme B(J)Z (and the ring S(B(J)Z )) is often called the tangent cone of p (the tangent cone of Rm ). There is a nice way to compute the tangent cone which we now indicate. Let k[x1 , . . . , xn ] be a polynomial ring. Given 0 = f ∈ k[x1 , . . . , xn ], there is an expression f = fd + fd+1 + · · · + fr where the fi are homogeneous forms of degree i for all i and fd = 0. The initial form of f is deﬁned to be in(f ) = fd . Given an ideal I in k[x1 , . . . , xn ], the initial ideal of I is in(I) = (in(f )  0 = f ∈ I), the homogeneous ideal in k[x1 , . . . , xn ] generated by the initial forms of elements of I. Suppose that R ∼ = k[x1 , . . . , xn ]/I and m = (x1 , . . . , xn )R. Then (as explained on the bottom of page 249 through the top of page 250 of [161]), we have a graded isomorphism mi /mi+1 ∼ = k[x1 , . . . , xn ]/in(I). i≥0
We have that in(I) = I if I is a homogeneous ideal and in(I) = (in(f )) if I = (f ) is a principal ideal. However, if I is generated by f1 , . . . , fm , it is not true in general that in(I) is generated by in(f1 ), . . . , in(fm ). In fact, this can fail even if f1 , . . . , fm is a complete intersection. Suppose that p ∈ Z(J) ⊂ X, with corresponding maximal ideal n in R. Then the ﬁber B(J)p of p in B(J) has the coordinate ring J i /nJ i . S(B(J)p ) ∼ = i≥0
This ring is sometimes called the ﬁber cone. Exercise 15.9. Let X be the reduced subscheme X = Z(x1 x2 ) of P2 (a union of two lines). Compute the blowup π : X → X of the line Z(x1 ) in X. Show that X is isomorphic to P1 . Exercise 15.10. Suppose that k has characteristic > 3 and let f = x2 + y 2 + z 2 + x3 + y 3 + z 3 , an irreducible polynomial in the polynomial ring k[x, y, z]. Let S = Z(f ) ⊂ A3 . Let π : S → S be the blowup of the origin p of S. Show that the ﬁber S p over p is isomorphic to Z(F ) ⊂ P2 , where F = x2 + y 2 + z 2 ∈ k[x, y, z] = S(P2 ).
15.3. Abstract algebraic varieties
295
15.3. Abstract algebraic varieties Deﬁnition 15.11. An abstract prevariety is an irreducible scheme (Deﬁnition 15.2). A topological space X is called Noetherian if it satisﬁes the descending chain condition for closed subsets. A prevariety X is a Noetherian topological space [116, Proposition 1 and Proposition 2, pages 48–49]. The function ﬁeld k(X) of a prevariety X is the quotient ﬁeld of OX (U ) for any aﬃne open subset U of X If X is a prevariety, then the product X × X is naturally a prevariety (it is covered by the aﬃne varieties Ui × Uj where {Ui } is an aﬃne cover of X). Deﬁnition 15.12. An abstract prevariety X is an abstract variety if Δ(X) = {(p, p)  p ∈ X} is closed in X × X. Example 15.13. Let p be a point in U1 = A1 and let q be a point in U2 = A1 . Deﬁne a prevariety X by gluing U1 to U2 by identifying the open subsets U1 \ {p} and U2 \ {q}. Then X is a prevariety which is not a variety. The following theorem follows from the valuative criterion of separatedness [73, Theorem II.4.3]. Theorem 15.14. An abstract prevariety X is an abstract variety if and only if for p, q ∈ X the stalks OX,p ⊂ k(X) and OY,q ⊂ k(X) are equal if and only if p = q. A quasiprojective variety is an abstract variety since it is separated by Proposition 3.36. Deﬁnition 15.15. An abstract variety X is complete if for all abstract varieties Y the projection π : X × Y → Y is a closed map. A projective variety is complete. This follows from Corollary 5.13 or the valuative criterion of properness [73, Theorem II.4.7]. An example by Hironaka of a nonsingular, complete abstract variety which is not projective is given in [73, Example 3.4.1 of Appendix B]. Theorem 15.16. Suppose that X is an abstract prevariety over C. Then: 1) The abstract prevariety X is Hausdorﬀ in the Euclidean topology if and only if X is an abstract variety. 2) The abstract prevariety X is compact (and Hausdorﬀ ) in the Euclidean topology if and only if X is a complete abstract variety.
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15. Schemes
Proof. This is proven in [116, Section 10 of Chapter 1]. Part 1) is established for quasiprojective varieties over C in Theorem 10.41. Exercise 15.17. Show that an aﬃne variety X is complete if and only if X is a point. Exercise 15.18. Show that the prevariety constructed in Example 15.13 is not an abstract variety.
15.4. Varieties over nonclosed ﬁelds Suppose that V ⊂ An is an aﬃne variety over our algebraically closed ﬁeld k. Then k[V ] = k[x1 , . . . , xn ]/P where P is the prime ideal I(V ) of V . If k0 is a subﬁeld of k such that the prime ideal P0 = P ∩ k0 [x1 , . . . , xn ] satisﬁes P = P0 k[x1 , . . . , xn ], then we say that V is deﬁned over k0 or that k0 is a ﬁeld of deﬁnition of V , and we deﬁne k0 [V ] = k0 [x1 , . . . , xn ]/P0 . We deﬁne the k0 rational points V (k0 ) of V to be the set of points p = (α1 , . . . , αn ) ∈ k0n such that p ∈ V . These points correspond to the maximal ideals in k0 [V ] whose residue ﬁeld is k0 . The rational function ﬁeld of V , regarded as a variety over k0 , written as k0 (V ), is the quotient ﬁeld of k0 [V ]. We can also consider ﬁeld extensions k1 of k0 in k. Then we can regard V as a variety over k1 with k1 [V ] = k1 [x1 , . . . , xn ]/P k1 [x1 , . . . , xn ] ∼ = k[V ] ⊗k k1 . If V ⊂ Pn , we can extend these notions to deﬁne the concept of a ﬁeld of deﬁnition k0 of V . We then have the set of k0 rational points V (k0 ) of V , which are the set of points (α0 : . . . : αn ) ∈ Pn such that α0 , . . . , αn ∈ k0 . This philosophy, including the notions of generic points, independent generic points, and specialization of points is developed in [146]. An excellent discussion on this topic is in [116, Section 4 of Chapter 2]. An inherent diﬃculty with this approach is that if we start with a prime ideal P in a polynomial ring k0 [x1 , . . . , xn ], giving a variety Xk0 in Ank0 , we might have that P k[x1 , . . . , xn ] is no longer a prime ideal if k is an algebraic closure of k0 . In this case there is not a corresponding variety Xk in Ank . If P k[x1 , . . . , xn ] is a prime ideal, then Xk0 is said to be “absolutely irreducible”.
15.5. General schemes A very general deﬁnition of schemes is given in [65] and [69]. Some sections in books giving good introductions to this are [73, Section 2 of Chapter II], [116, Chapter II], and [53]. An aﬃne scheme is deﬁned to be the spectrum Spec(R) of a commutative ring R (Exercise 1.11). The points of Spec(R) are the prime ideals of R,
15.5. General schemes
297
and it is given the Zariski topology. The closed sets are Z(I) = {P ∈ Spec(R)  I ⊂ P } where I is an ideal of R. The closed points are then the maximal ideals of R. An aﬃne scheme X = Spec(A) has a structure sheaf OX which generalizes the notion of regular functions on an aﬃne variety. The structure sheaf has the property that the stalk OX,p = Ap for all p ∈ X. A homomorphism φ : A → B of commutative rings determines a continuous map f : Spec(B) → Spec(A) by f (p) = φ−1 (p) for p ∈ Spec(B). These maps φ are the morphisms (as locally ringed spaces) from Spec(B) to Spec(A), with f # induced by φ. If R is the ring of regular functions k[V ] of an aﬃne variety V , then the closed points of Spec(k[V ]) are the ideals I(p) of the points p ∈ V . A scheme is a topological space X with a sheaf of rings OX such that there exists an open cover {Uα } of X such that (Uα , OX Uα ) ∼ = (Spec(Rα ), OSpec(Rα ) ) for some commutative rings Rα . A morphism of schemes is a morphism of locally ringed spaces. A scheme X is separated if the image of the diagonal map Δ : X → X × X is closed. An aﬃne scheme is separated. Let S = n≥0 Sn be a graded ring where S0 = R is a commutative ring. The projective scheme Proj(S) is the topological space which consists of the graded prime ideals of S which do not contain S+ = n>0 Sn . The closed sets are Z(I) = {P ∈ Proj(S)  I ⊂ P } where I is a graded ideal of R. For F ∈ S homogeneous, Spec(S(F ) ) is an aﬃne open subset of Proj(S), and such open subsets are a basis for the topology. Our deﬁnition of a scheme follows the deﬁnitions in the later, second version of EGA I [69] and in [73]. In the earlier edition of EGA I [65] and in [116] a scheme is called a prescheme and a scheme is a separated prescheme.
Chapter 16
The Degree of a Projective Variety
In this chapter we deﬁne the degree of a projective variety X embedded in a projective space Pn . Classically, the degree is deﬁned to be the number of intersection points of X with a general linear subvariety of Pn of codimension equal to the dimension of X. In a more algebraic approach, the degree is deﬁned from the Hilbert polynomial of Y . We will indicate why these two deﬁnitions are in fact equal and derive a classical bound on the degree of a nondegenerate variety (a variety which is not contained in a linear hyperplane of Pn ). Let Pn be projective space over an algebraically closed ﬁeld k. Let x0 , . . . , xn be homogeneous coordinates on Pn , so that the coordinate ring of Pn is S = S(Pn ) = k[x0 , . . . , xn ]. The linear hyperplanes in Pn are parametrized by the projective space V = Pn by associating to A = (a0 : . . . : an ) ∈ V the hyperplane LA which is the subscheme of Pn with underlying topological space Z( ni=0 ai xi ) and ideal sheaf
ai xi . ILA = We say that a property holds for a general linear hyperplane if it holds for all LA with A in a dense open subset of V . A hyperplane LA with A in this dense open set is called a general hyperplane. 299
300
16. The Degree of a Projective Variety
The intersection Y ∩ Z of two closed subschemes will be the schemetheoretic intersection deﬁned in Section 15.1. The proof of the next theorem follows directly from [55, Corollary 3.4.14 and Theorem 3.4.10]. Theorem 16.1. Suppose that Y ⊂ Pn is a closed algebraic set (a reduced subscheme). Then for a general hyperplane H ⊂ Pn , we have that the schemetheoretic intersection Y ∩ H does not contain any irreducible component of Y and Y ∩ H is reduced (an algebraic set). If all irreducible components of Y have dimension m, then all irreducible components of Y ∩H have dimension m − 1. Further, if Y is nonsingular, then Y ∩ H is nonsingular, and if Y is normal, then Y ∩ H is normal. If Y is a variety (integral) of dimension ≥ 2, then Y ∩ H is a variety (integral). Deﬁnition 16.2. Suppose that Y is a closed algebraic set in Pn (a reduced closed subscheme) such that all irreducible components of Y have a common dimension m. A linear subvariety L of Pn of codimension m is said to be general for Y if there exist linear hyperplanes L1 , . . . , Lm of Pn such that L = L1 ∩ · · · ∩ Lm and for 1 ≤ r ≤ m, X ∩ L1 ∩ · · · ∩ Lr is a closed algebraic set in Pn such that all irreducible components have dimension m − r. If Y is a variety, we further require that for 1 ≤ r ≤ m − 1, X ∩ L1 ∩ · · · ∩ Lr is a variety. The next corollary follows from successive application of Theorem 16.1. Corollary 16.3. Suppose that Y is a closed algebraic set in Pn (a reduced closed subscheme) such that all irreducible components of Y have dimension m > 0. Then there exists a linear subvariety L of Pn of codimension m which is general for Y . Deﬁnition 16.4. A numerical polynomial is a polynomial P (z) ∈ Q[z] such that P (n) ∈ Z for all integers n 0. For n ∈ N, deﬁne nz ∈ Q[z] by z0 = 1 and z 1 = z(z − 1) · · · (z − n + 1) n! n for n ≥ 1. Lemma 16.5. Suppose that P ∈ Q[z] is a numerical polynomial of degree r. Then there are integers c0 , c1 , . . . , cr ∈ Z such that z z + c1 + · · · + cr . (16.1) P (z) = c0 r r−1 Proof. We prove the lemma by induction on r = deg P , the case r = z 0 certainly being true. Since deg r = r, we can express any numerical
16. The Degree of a Projective Variety
301
polynomial P (z) in the form (16.1), with c0 , . . . , cr ∈ Q. Deﬁne the diﬀerence z z polynomial ΔP (z) = P (z + 1) − P (z). Since Δ r = r−1 , z z + c1 + · · · + cr−1 . ΔP = c0 r−1 r−2 By induction, c0 , . . . , cr−1 ∈ Z. But then cr ∈ Z since P (n) ∈ Z for n 0. Theorem 16.6 (Hilbert, Serre). Let M = n∈Z Mn be a ﬁnitely generated graded S = k[x0 , . . . , xn ]module. Then there is a unique polynomial PM (z) ∈ Q[z] such that PM (n) = dimk Mn for n 0. The degree of the polynomial PM (z) is dim Z(Ann(M )) (viewed as an algebraic set in Pn ), where Ann(M ) = {f ∈ S  f M = 0}. Proof. [161, Theorem 41 on page 232 and Theorem 42 on page 235] or [50, Section 12.1]. The polynomial PM is called the Hilbert polynomial of M . We deﬁne the degree of M to be r! times the leading coeﬃcient of PM , where r = dim Z(Ann(M )) is the degree of PM . Suppose that I, J are homogeneous ideals in S such that I sat = J sat . Then PS/I = PS/J by Exercise 16.13. Deﬁnition 16.7. Suppose that Y ⊂ Pn is a closed subscheme of dimension r. We deﬁne the Hilbert polynomial PY of Y to be the Hilbert polynomial PS/I(Y ) . By Theorem 16.6, this polynomial has degree r. We deﬁne the degree deg(Y ) of Y to be r! times the leading coeﬃcient of PY . Proposition 16.8. The following are true. 1) The degree of a nonempty closed subsheme Y of Pn is a positive integer. 2) Suppose that Y is a closed algebraic set (a closed reduced subscheme) in Pn and Y = Y1 ∪ Y2 is the union of closed algebraic sets of the same dimension r such that dim Y1 ∩ Y2 < r. Then deg(Y ) = deg(Y1 ) + deg(Y2 ). 3) deg(Pn ) = 1. 4) If H ⊂ Pn is a hypersurface whose ideal is generated by a homogeneous polynomial of degree d, then deg(H) = d.
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16. The Degree of a Projective Variety
Proof. 1) The polynomial PY is nonzero of degree r = dim Y by Theorem 16.6. By Lemma 16.5, we have a unique expression x x + c1 + · · · + cr PY (x) = c0 r r−1 where c0 , . . . , cr ∈ Z and c0 = 0. Thus deg(Y ) = c0 is a nonzero integer. We have that c0 > 0 since PY (n) = dimk S(Y )n > 0 for n 0. 2) Let I1 = I(Y1 ), I2 = I(Y2 ), and I = I(Y ), so that I = I(Y1 ∪ Y2 ) = I1 ∩ I2 . We have an exact sequence of graded S = S(Pr )modules 0 → S/I
h→(h,−h)
→
S/I1 ⊕ S/I2
(f,g)→f +g
→
S/(I1 + I2 ) → 0.
The algebraic set Z(I1 + I2 ) = Y1 ∩ Y2 has dimension < r. Hence PS/(I1 +I2 ) has degree < r, and the leading coeﬃcient of PS/I is the sum of the leading coeﬃcients of PY1 and PY2 . 3) Let S = S(Pn ) = k[x0 , . . . , xn ]. For m > 0, dimk Sm = m+n so n x+n 1 PS (x) = n . The leading coeﬃcient of this polynomial is n! , so deg Pn = 1. 4) Let F ∈ S be a degree d homogeneous polynomial which generates the ideal of the scheme H. We have an exact sequence of graded Smodules F
0 → S(−d) → S → S/(F ) → 0. Thus dimk S(H)m = dimk Sm − dimk Sm−d , so PH (x) =
x+n x+n−d d xn−1 + lowerorder terms. − = (n − 1)! n n
Thus deg H = d.
Theorem 16.9. Suppose that Y is a closed algebraic set in Pn such that all irreducible components of Y have dimension m and L is a linear subvariety of Pn of codimension m which is general for Y . Let d be the number of irreducible components (points) of the reduced zerodimensional scheme L ∩ Y . Then d = deg(Y ). Proof. We ﬁrst show that deg(Y ) = deg(Y ∩ L). By induction on m, we need only verify that deg(Y ) = deg(Y ∩LA ) where LA is a hyperplane section of Pn such that all irreducible components of LA ∩ Y have dimension m − 1 and LA ∩ Y is reduced. With these assumptions, the radical ideal I(LA ∩ Y ) = (I(Y ) + I(LA ))sat .
16. The Degree of a Projective Variety
303
We have that I(Y ) = Qi where the Qi are the homogeneous prime ideals of the irreducible components of Y . Let T = S/I(Y ), a graded Smodule where S = S(Pn ). Let FA = ni=0 ai xi . By our assumptions on A, FA ∈ Qi for any i, so we have a short exact sequence of graded Smodules F
A 0 → T (−1) → T → T /FA T → 0.
Thus PT /FA T (n) = PT (n) − PT (n − 1) =
deg(Y ) m−1 n + lowerorder terms in n. (m − 1)!
Since sat I(Y ∩ LA )n = (I(Y ) + I(LA ))sat n = (I(Y ) + (FA ))n = (I(Y ) + (FA ))n
for n 0 and S/(I(Y ) + (FA )) ∼ = T /FA T , we have that PS/I(Y ∩LA ) = PS/(I(Y )+(FA )) = PT /FA T and thus deg(Y ∩ LA ) = deg(Y ). By induction, we have that deg(Y ∩ L) = deg(Y ). By our assumption, I(Y ∩ L) = di=1 Pi where the Pi are the homogeneous ideals of points. Thus S/Pi ∼ = k[x] is a standard graded polynomial ring in one variable. Thus PS/Pi (n) is the constant polynomial 1. By 2) of Proposition 16.8, deg(Y ∩ L) = d.
Theorem 16.10. Suppose that X is a projective subvariety of Pn and X is not contained in a linear hyperplane (X is nondegenerate). Then deg(X) ≥ codim(X) + 1. Proof. Let m = codim(X). By Corollary 16.3, we can construct a linear subspace L of Pn of dimension m such that I(L) = (F1 , . . . , Fm ) with F1 , . . . , Fm in S = S(Pn ) linear forms such that the ideals I0 = I(X) and Ii = (Ii−1 + (Fi ))sat for i < m are prime ideals and Im = (Im−1 + (Fm ))sat = (I(L) + I(X))sat = I(X ∩ L). Let Yi = Z(Ii ) and Li = Z(F1 , . . . , Fi ) ∼ = Pn−i . The variety Y0 = X, Yi is a subvariety of Li for i < m, and Ym = X ∩ L is a closed algebraic set in L = Lm (a union of d points). We will prove that Yi is nondegenerate in Li for all i by induction on i.
304
16. The Degree of a Projective Variety
Suppose that i < m. Sheaﬁfy the graded short exact sequence of S(Li ) = S/(F1 , . . . , Fi )modules Fi+1
0 → (S/Ii )(−1) → S/Ii → S/(Ii + (Fi+1 )) → 0 to obtain a short exact sequence of sheaves of OLi modules 0 → OYi (−1) → OYi → OYi+1 → 0, and obtain, after tensoring with OYi (1), a commutative diagram of sheaves of OLi modules with exact rows 0 → O Li ↓ 0 → OYi
→ OLi (1) → OLi+1 (1) → 0 ↓ ↓ → OYi (1) → OYi+1 (1) → 0.
Taking global sections, we obtain a commutative diagram of kvector spaces 0 → Γ(Li , OLi ) → Γ(Li , OLi (1)) → Γ(Li+1 , OLi+1 (1)) → 0 α↓ β ↓ γ ↓ 0 → Γ(Yi , OYi ) → Γ(Yi , OYi (1)) → Γ(Yi+1 , OYi+1 (1)) where the top and bottom rows are exact. The map α is an isomorphism since Yi is a variety (and k is algebraically closed) by Theorem 3.35. By the induction hypothesis, Yi is nondegenerate in Li ; that is, β is injective. After a diagram chase, we see that γ must also be injective, so that Yi+1 is nondegenerate in Li+1 . We have established that the algebraic set Ym , which is a union of d points in Lm ∼ = Pcodim(X) , is nondegenerate. Thus Ym must contain at least codim(X) + 1 points, so d = deg(X) ≥ codim(X) + 1.
A nondegenerate subvariety X of Pn such that deg(X) = codim(X) + 1 is called a variety of minimal degree. There is a beautiful classiﬁcation of varieties of minimal degree by Del Pezzo [48] and Bertini [19]. A modern proof is given by Eisenbud and Harris in [52]. Exercise 16.11. Let X be a proper subvariety of Pn . Give a simple direct proof that a general linear hyperplane of Pn does not contain X. Exercise 16.12. Suppose that f : Z → Z is a function such that the ﬁrst diﬀerence function Δf (n) = f (n+1)−f (n) is a numerical polynomial. Show that there exists a numerical polynomial P (z) such that f (n) = P (n) for n 0. Exercise 16.13. Suppose that S = k[x0 , . . . , xn ] and I ⊂ S is a homogeneous ideal. Show that PS/I = PS/I sat .
16. The Degree of a Projective Variety
305
Exercise 16.14. Suppose that X is a subvariety of Pn which has degree 1. Show that X is a linear subvariety. Exercise 16.15. Show that the degree of the dth Veronese embedding of Pn in PN is dn . Exercise 16.16. Show that the degree of the Segre embedding of Pr × Ps r+s N in P is r . Exercise 16.17. Let Y ⊂ Pn be an rdimensional variety of degree 2. Show that Y is contained in a linear subvariety of dimension r + 1 in Pn , and that Y is isomorphic to a quadric hypersurface in Pr+1 .
Chapter 17
Cohomology
ˇ In this chapter we study basic properties of sheaf and Cech cohomology and discuss some applications.
17.1. Complexes We begin with some preliminaries on homological algebra. A complex A∗ is a sequence of homomorphisms of Abelian groups (or modules over a ring R): di
di+1
· · · → Ai → Ai+1 → · · · for i ∈ Z with di+1 di = 0 for all i. The homomorphisms di are called diﬀerentials or coboundary maps. Associated to a complex A∗ are cohomology groups (modules) H i (A∗ ) = Kernel(di )/Image(di−1 ). Deﬁnition 17.1. Suppose that A∗ and B ∗ are complexes. A chain map (or map of complexes) f : A∗ → B ∗ is a sequence of homomorphisms f i : Ai → B i for all i such that the following diagram commutes: di
di+1
Ai+2 → · · · ↓ f i+2
ei
ei+1
B i+2
· · · → Ai → Ai+1 ↓ f i+1 ↓ fi · · · → Bi
→ B i+1
→
→
→ ··· .
A chain map f : A∗ → B ∗ is nullhomotopic if there are homomorphisms si : Ai → B i−1 such that f i = ei−1 si + si+1 di 307
308
17. Cohomology
for all i. The homomorphisms si are called a homotopy. If f and g are two chain maps from A∗ to B ∗ , then f is homotopic to g if f −g is nullhomotopic. Suppose that f is a chain map between complexes A∗ and B ∗ . Then there are induced homomorphisms f ∗ : H i (A∗ ) → H i (B ∗ ). Proposition 17.2. Suppose that f and g are homotopic chain maps between complexes A∗ and B ∗ . Then f ∗ = g ∗ : H n (A∗ ) → H n (B ∗ ) for all n ∈ Z. Proof. Let si be the homotopy. Suppose that z ∈ Kernel(di ). Then f z − gz = ei−1 si z + si+1 di z = ei−1 si z ∈ Image(ei−1 ) so f ∗ = g ∗ .
17.2. Sheaf cohomology In this section, we summarize some material from [73, Chapter III]. Associated to a sheaf of Abelian groups F on a topological space X are sheaf cohomology groups H i (X, F ) for all nonnegative integers i, which have the properties that we have a natural isomorphism H 0 (X, F ) ∼ = Γ(X, F ), and if (17.1)
0 → A → A → A → 0
is a short exact sequence of sheaves of Abelian groups on X, then there is a long exact cohomology sequence of Abelian groups (17.2) 0 → H 0 (X, A) → H 0 (X, A ) → H 0 (X, A ) → H 1 (X, A) → H 1 (X, A ) → H 1 (X, A ) → H 2 (X, A) → · · · . Further, given a commutative diagram of homomorphisms of sheaves of Abelian groups on X, (17.3)
0 → A → A → A → 0 ↓ ↓ ↓ 0 → B → B → B → 0
where the horizontal diagrams are short exact, we have an induced commutative diagram (17.4) 0 →
H 0 (X, A) → H 0 (X, A ) ↓ ↓ 0 → H 0 (X, B) → H 0 (X, B )
H 0 (X, A ) → H 1 (X, A) → · · · ↓ ↓ → H 0 (X, B ) → H 1 (X, B) → · · · .
→
The sheaf cohomology groups are constructed by an injective resolution. A sheaf of Abelian groups I on X is called injective if the functor Hom(·, I) is exact on exact sequences of sheaves of Abelian groups on X. An injective resolution of a sheaf of Abelian groups F on X is an exact sequence (17.5)
0 → F → I0 → I1 → · · ·
17.2. Sheaf cohomology
309
of sheaves of Abelian groups on X such that each I i is injective. It follows from [73, Corollary III.2.3] that every sheaf of Abelian groups F on X has an injective resolution. The sheaf cohomology groups H i (X, F ) are computed by choosing an injective resolution (17.5) and taking the cohomology of the associated complex (di+1 di = 0 for all i) (17.6)
d0
d1
Γ(X, I 0 ) → Γ(X, I 1 ) → · · · .
The sheaf cohomology groups H i (X, F ) are deﬁned by (17.7)
H i (X, F ) = Kernel(di )/Image(di−1 )
for i ≥ 0. These cohomology groups are independent of choice of injective resolution [128, Theorem 6.14]. Given a commutative diagram (17.3), we have an associated diagram (17.4) by [128, Theorem 6.26]. Lemma 17.3. Let X be a Noetherian topological space, and let F = j Fj be a direct sum of sheaves of Abelian groups Fj on X. Then H i (X, Fj ) H i (X, F ) ∼ = j
for all i. This is established in [73, Proposition III.2.9 and Remark III.2.9.1]. If X is a locally ringed space, then we can construct corresponding cohomology groups for sheaves of OX modules. In this case, the cohomology groups are constructed by taking an injective resolution (17.5) where the I i are injective OX modules, taking the complex of global sections (17.6), and then taking the cohomology (17.7) of this complex. The fact that every sheaf of OX modules has an injective resolution follows from [73, Proposition III.2.2]. The cohomology groups of an OX module F are in fact the same as those that we compute regarding F as a sheaf of groups, by [73, Proposition III.2.6]. Lemma 17.4. Let Y be a closed subscheme of a scheme X, let F be a sheaf of Abelian groups on Y , and let j : Y → X be the inclusion. Then H i (Y, F ) = H i (X, j∗ F ) for all i. This is proved in [73, Lemma III.2.10]. Suppose that Y is a closed subscheme of a scheme X and F is a sheaf of Abelian groups on X, such that the support of F is contained in the underlying topological space of Y . Then (17.8)
H i (X, F ) = H i (Y, F Y )
for all i. This follows, since letting j : Y → X be the inclusion, we have that j∗ (F Y ) = F .
310
17. Cohomology
Theorem 17.5. Suppose that X is a Noetherian topological space of dimension n. Then for all i > n and all sheaves of Abelian groups F on X, we have that H i (X, F ) = 0. This is proven in [73, Theorem III.2.7]. Corollary 17.6. Suppose that X is a scheme and F is a coherent sheaf on X such that the support of F (which is a Zariski closed subset of X by Exercise 11.36) has dimension n. Then H i (X, F ) = 0 for all i > n. Theorem 17.7. Suppose that X is an aﬃne scheme and F is a quasicoherent sheaf on X. Then H i (X, F ) = 0 for all i > 0. ˜ where M is This follows from [73, Theorem III.3.5]. In fact, if F = M an A = k[X]module and d0
d1
0 → M → I0 → I1 → · · · is an injective resolution of M as an Amodule, then H i (X, F ) = Kernel(di )/Image(di−1 ).
ˇ 17.3. Cech cohomology ˇ In practice, the most eﬀective way to compute cohomology is by Cech cohomology. Suppose that X is a topological space and F is a sheaf of Abelian groups on X. Let U = {Ui }i∈I be an open cover of X. Fix a wellordering on I. For a ﬁnite set i0 , . . . , ip ∈ I, let Ui0 ,...,ip = Ui0 ∩ · · · ∩ Uip . We deﬁne a complex of Abelian groups C ∗ = C ∗ (U , F ) on X by $ C p (U , F ) = F (Ui0 ,...,ip ). i0 <···
Deﬁne the coboundary map d = dp : C p (U , F ) → C p+1 (U , F ) by (17.9)
(dp (α))i0 ,...,ip+1 =
p+1
(−1)k αi0 ,...,ˆik ,...,ip+1 Ui0 ,...,ip+1
k=0
C p (U , F ).
for α = (αi0 ,...,ip ) ∈ The notation ˆik means omit ik . We have that 2 ∗ d = 0, so that C is a complex. We write out the ﬁrst part of the complex as $ $ $ d0 d1 F (Ui ) → F (Uj ∩ Uk ) → F (Ul ∩ Um ∩ Un ) (17.10) i
j
l<m
ˇ 17.3. Cech cohomology
311
where the indicies i, j < k and l < m < n range over I. If α = (αi ) ∈ C 0 (U , F ), then d0 (α)j,k = αk − αj , and if α = (αj,k ) ∈ C 1 (U , F ), then d1 (α)l,m,n = αl,m − αl,n + αm,n . Remark 17.8. We will sometimes ﬁnd it useful to extend the symbol αi0 ,i1 ,...,ip to be deﬁned for all (p + 1)tuples of elements of I. To do this, we deﬁne αi0 ,...,ip = 0 if any of the ij are equal, and if the ij are all distinct, deﬁne αi0 ,...,ip = (−1)sign(σ) ασ(i0 ),...,σ(ip ) where σ is the permutation such that σ(i0 ) < · · · < σ(ip ). With this convention, the formula (17.9) holds for any (p + 1)tuple i0 , . . . , ip+1 of elements of I. ˇ Deﬁnition 17.9. We deﬁne the pth Cech cohomology group of F with respect to the covering U to be ˇ p (U , F ) = H p (C ∗ (U , F )) = Kernel(dp )/Image(dp−1 ). H From the sheaf axioms, we have the following lemma. Lemma 17.10. We have that ˇ 0 (U , F ) = Γ(X, F ) = H 0 (X, F ). H Let V = {Vj }j∈J be another open cover of X. The cover V is a reﬁnement of U if there is an orderpreserving map of index sets λ : J → I such that Vj ⊂ Uλ(j) for all j ∈ J. If V is a reﬁnement of U , then there is a natural map of complexes φ : C ∗ (U , F ) → C ∗ (V , F ) deﬁned by φp (α)j0 ,...,jp = αλ(j0 ),...,λ(jp ) Vj0 ,...,jp for α ∈ C p (U , F ), where d∗ is the diﬀerential of C ∗ (U , F ) and e∗ is the diﬀerential of C ∗ (V , F ). The map φ is a map of complexes since ep φp = φp+1 dp . Thus we have natural homomorphisms of cohomology groups ˇ p (V , F ). ˇ p (U , F ) → H H This map is independent of choice of function λ, as the maps of complexes φ and ψ are homotopic (Proposition 17.2) if ψ is the induced map of complexes obtained from another orderpreserving map τ : J → I such that Vj ⊂ Uτ (j) . As the coverings of X form a partially ordered set under reﬁnement, we may make the following deﬁnition.
312
17. Cohomology
ˇ Deﬁnition 17.11. The pth Cech cohomology group of F is ˇ p (U , F ) ˇ p (X, F ) = lim H H →
where the limit is over the (ordered) open covers U of X. ˇ i (X, F ) → H i (X, F ), Theorem 17.12. There are natural homomorphisms H which are isomorphisms if i ≤ 1. Proof. [59, Corollary, page 227] or [73, Exercise III.4.4] (which gives a sketch of the proof). Theorem 17.13. Suppose that X is a scheme, U is an aﬃne cover of X, and F is a quasicoherent sheaf on X. Then ˇ p (U , F ) ∼ ˇ p (X, F ) ∼ H =H = H p (X, F ) for all p ≥ 0. Theorem 17.13 is proven in [73, Theorem III.4.5]. The key point of the proof in comparing the cohomologies is that for p > 0, the cohomology groups H p (U, F ) for p > 0 of a quasicoherent sheaf on an aﬃne scheme U vanish (by Theorem 17.7).
17.4. Applications Now we give some applications of cohomology. Theorem 17.14. Let X = Pr and hi (n) = dimk H i (X, OX (n)). Then ⎧ r+n for i = 0 and n ≥ 0, ⎪ ⎪ r ⎨ 0 for i = 0 and n < 0, i h (n) = 0 for 0 < i < r and all n ∈ Z, ⎪ ⎪ ⎩ 0 h (−n − r − 1) if i = r. Proof. Let F be the quasicoherent sheaf F = n∈Z OX (n). Cohomology commutes with direct sums (by Lemma 17.3) so H i (X, OX (n)). H i (X, F ) ∼ = n∈Z
Let Ui = Xxi = D(xi ) for 0 ≤ i ≤ r. Here U = {Ui } is an aﬃne cover of X ˇ i (U , F ) (by Theorem 17.13). We have that so H i (X, F ) = H Γ(Ui0 ...ip , OX (n)). F (Ui0 ...ip ) ∼ = n∈Z
Let S be the graded ring S = S(X) = k[x0 , . . . , xr ].
17.4. Applications
313
We have that Γ(Uip , OX (n)) = S(n)(xip ) , the elements of degree 0 in the localization S(n)xip , which is equal to the elements of degree n in Sxip . Writing Ui0 ...ip = Ui0 ∩ · · · ∩ Uip = (Uip ) xi0 ∩ · · · ∩ (Uip ) xip−1 xip
= (Uip )
x xi 0 ··· ip−1 xip xip
xip
,
we have that Γ(Ui0 ...ip , OX (n)) = [S(n)(xip ) ] xi0
xip
···
xi p−1 xip
which is the set of elements of degree n in Sxi0 ···xip . We thus have that the ˇ Cech complex C ∗ (U , F ) is isomorphic as a graded Smodule to the complex ∗ C : d0 d1 dr−1 Sxi0 → Sxi0 xi1 → · · · → Sx0 x1 ···xr i0
i0
where ds (α)i0 ...is+1 =
s+1 (−1)k αi0 ...ˆik ...is Ui0 ...is+1
k=0 of d0
C s (U , F ).
for α ∈ The kernel can be identiﬁed with the intersection S in the quotient ﬁeld of S, and this intersection is just S since S = i xi k[x0 , . . . , xr ] is a polynomial ring. Thus H 0 (X, F ) = S, which establishes the theorem for i = 0. The group H r (X, F ) is the cokernel of Sx0 ···ˆxk ···xr → Sx0 ···xr . dr−1 : k
The kvector space Sx0 ···xr has the basis xl00 · · · xlrr with li ∈ Z. The image of dr−1 is generated by the monomials xl00 · · · xlrr such that at least one li ≥ 0. Thus the (classes) of the monomials {xl00 · · · xlrr li < 0 for all i} is a kbasis of H r (X, F ). We then have that a basis of H r (X, OX (n)) is −1 m0 mr {x−1 0 · · · xr x0 · · · xr mi ≤ 0 for all i and m0 + · · · + mr = n + r + 1}
so dimk H r (X, OX (n)) = dimk S−n−r−1 = dimk H 0 (X, OX (−n − r − 1)) establishing the theorem for i = r. We now prove the theorem when 0 < i < r by induction on r. This statement is vacuously true for r = 1, so we may assume that r > 1. Localizing ˇ complex for the sheaf the complex C ∗ with respect to xr , we obtain the Cech
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17. Cohomology
F Ur on Ur with respect to the aﬃne open cover {Ui ∩ Ur i = 0, . . . , r}. By Theorem 17.13, the cohomology of the complex Cx∗r is the sheaf cohomology of F Ur on Ur , which is zero for i > 0 by Theorem 17.7. Since localization is exact, we conclude that H i (X, F )xr = 0 for i > 0. Thus every element of H i (X, F ) for i > 0 is annihilated by some power of xr . To complete the proof that H i (X, F ) = 0 for 0 < i < r, we will show that multiplication by xr induces a bijection of H i (X, F ) into itself, from which it follows that H i (X, F ) = 0. We have an exact sequence of graded Smodules x
0 → S(−1) →r S → S/xr S → 0. Sheaﬁfying gives an exact sequence of sheaves 0 → OX (−1) → OX → OH → 0 where H ∼ = Pr−1 is the hyperplane Z(xr ). Tensoring this last sequence with OX (n) and taking the direct sum over n ∈ Z gives us a short exact sequence of quasicoherent OH modules where FH = sequence
0 → F (−1) → F → FH → 0
n∈Z OH (n).
Taking cohomology, we obtain a long exact x
· · · → H i (X, F (−1)) →r H i (X, F ) → H i (X, FH ) → · · · . Now
H i (H, OH (n)), OH (n) ∼ H i (X, FH ) = H i H, =
so by induction on r, H i (X, FH ) = 0 for 0 < i < r − 1. For i = 0 and n ∈ Z, the left exact sequence 0 → H 0 (X, OX (n − 1)) → H 0 (X, OX (n)) → H 0 (H, OH (n)) → 0 is actually exact (the rightmost map is a surjection) since from our calculations earlier in this proof it is the short exact sequence x
0 → Sn−1 →r Sn → (S/xr S)n → 0. We also have at the end of the long exact sequence the right exact sequence δ
x
H r−1 (X, OH (n)) → H r (X, OX (n)) →r H r (X, OX (n)) → 0. From earlier in this proof, we know that H r (X, OX (n)) has the kbasis of (classes) of monomials {xl00 · · · xlrr  li < 0 for all i and l0 + · · · + lr = n}. The kernel Kn of xr : H r (X, OX (n − 1)) → H r (X, OX (n)) has the kbasis r−1 −1 0 {xm 0 · · · xr−1 xr  mi < 0 for all i and m0 + · · · + mr−1 − 1 = n − 1}.
m
17.4. Applications
315
Thus dimk Kn = dimk H r−1 (H, OH (n)) so δ is injective. Thus multiplication by xr : H i (X, F (−1)) → H i (X, F ) is bijective for 0 < i < r and so H i (X, F ) = 0 for 0 < i < r. Lemma 17.15. Suppose that X is a variety and k(X) is the function ﬁeld of X. Let F be the presheaf deﬁned by F (U ) = k(X) for U an open subset of X. Then F is a sheaf. We will also write the sheaf F as the constant sheaf k(X). We then have that H 0 (X, k(X)) = k(X)
H 1 (X, k(X)) = 0.
and
Proof. F is a sheaf by Proposition 11.14, since all open subsets of the variety X are irreducible and hence connected. We have that H 0 (X, k(X)) = F (X) = k(X) since the presheaf F is a sheaf. Let U = {Ui }i∈I be an open cover of X. We form an augmented complex G∗ : 0 → G−1 → G0 → G1 → · · · of the complex C ∗ (U , F ) by G∗ : 0 → k(X) → C 0 (U , F ) →0 C 1 (U , F ) →1 · · ·
d
d
where is the product of the identiﬁcations of k(X) with F (Ui , F ), for i ∈ I. Fix j ∈ I. For p ≥ 1, deﬁne λ : C p (U , F ) → C p−1 (U , F ) by (λα)i0 ,...,ip−1 = αj,i0 ,...,ip−1 , using the convention of Remark 17.8. We deﬁne λ : C 0 (U , F ) → k(X) by (λα) = αj . Suppose α ∈ C p (U , F ) with p ≥ 1. We compute dλ(α)i0 ,...,ip
p p k = (−1) λ(α)i0 ,...,ˆik ,...,ip = (−1)k αj,i0 ,...,ˆik ,...,ip k=0
k=0
and λd(α)i0 ,...,ip = d(α)j,i0 ,...,ip = αi0 ,...,ip −
p
(−1)k αj,i0 ,...,ˆik ,...,ip
k=0
so (dλ + λd)(α) = α. Suppose α ∈ C 0 (U , F ). We compute dλ(α)i = (dαj )i = (αj )i = αj and λd(α)i = d(α)j,i = αi − αj so we again have that (dλ + λd)(α) = α. Thus λ is a homotopy operator for the complex G∗ , and we have shown that the identity map is homotopic to the zero map. Thus the identity map
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17. Cohomology
and the zero map are the same maps H i (G∗ ) → H i (G∗ ) (by Proposition 17.2), so H i (G∗ ) = 0 for all i and G∗ is an exact complex. We conclude that H 0 (U , F ) = k(X) and H i (U , F ) = 0 for i > 0. Thus ˇ 1 (X, F ) = lim H 1 (U , F ) = 0 H →
and by Theorem 17.12, we have that ˇ 1 (X, F ) = 0. H 1 (X, F ) ∼ =H
Suppose that X is a variety and U = {Ui }i∈I is an open cover of X. The ∗ of units in O ˇ sheaf OX X is a group under multiplication. Thus the Cech ∗ ∗ complex C (U , OX ) is a complex of Abelian groups under multiplication. ∗ in (17.10), we have Taking F = OX d0 (α)j,k = αk αj−1 and
∗ ) for α = (αi ) ∈ C 0 (U , OX
−1 αm,n d1 (α)l,m,n = αl,m αl,n
∗ ) for α = (αj,k ) ∈ C 1 (U , OX
(recall the convention on indexing of Remark 17.8). ∗ ). Theorem 17.16. Suppose that X is a variety. Then Pic(X) ∼ = H 1 (X, OX
Proof. Let U = {Ui } be an open cover of X. Let P (U ) = {L ∈ Pic(X)  U is a trivializing open cover of X for L}. Suppose that [L] ∈ P (U ) is a class. Let φi : OX Ui → LUi be a trivialization of L. Let gij = φ−1 j φi . Then by formula (13.17), we have −1 gjk = 1 gij gik ∗ ), giving a class [{g }] in for all i, j, k so {gij } is a cocycle in C 1 (U , OX ij 1 ∗ ˇ (U , O ). Taking M ∼ H = L in Lemma 13.37, we see that for a trivialization X
ψi : OX Ui → MUi , the transition functions hij = ψj−1 ψi diﬀer from the ˇ 1 (U , O∗ ) given by gij by a coboundary. Thus the map ΨU : P (U ) → H X ΨU ([L]) = [{gij }] is welldeﬁned.
∗ ) We now establish that ΨU is onto. Suppose that {gij } ∈ C 1 (U , OX −1 gjk = 1 by (17.10). Fix k ∈ I and deﬁne L by is a cocycle, so that gij gik LUi = gik OUi . We compute
gik OUi ∩Uj = gij gjk OUi ∩Uj = gjk OUi ∩Uj ∗ ) is a unit on U ∩U , so L is a welldeﬁned invertible since gij ∈ Γ(Ui ∩Uj , OX i j sheaf and [L] ∈ P (U ). We have φi = gik : OX Ui → LUi is an isomorphism for i ∈ I, so −1 ΨU ([L]) = [{φ−1 j φi }] = [{gjk gik }] = [{gij }].
Thus ΨU is onto. The map ΨU is injective by Lemma 13.37.
17.4. Applications
317
−1 If [L], [M] ∈ P (U ) with transition functions gij and hij , then the gij are −1 transition functions of L and the gij hij are transition functions of L ⊗ M. Thus ΨU is a group isomorphism.
Suppose that [L] ∈ Pic(X) and U = {Ui }i∈I is a trivializing open cover of X for L, with isomorphisms φi : OX Ui → LUi . Let V = {Vj }j∈J be a reﬁnement of U , so there is an orderpreserving map of index sets λ : J → I such that Vj ⊂ Uλ(j) for all j ∈ J. The map ∗ ∗ ) → C ∗ (V , OX ) φ : C ∗ (U , OX
of complexes deﬁned after Lemma 17.10 takes a 1cycle {gkl } to {gλ(k)λ(l) }. Now ψi = φλ(i) Vi gives a trivialization of LVi with transition functions ψj−1 ψi = gλ(i)λ(j) . Since Pic(X) = P (U ) = lim→ P (U ), the ΨU patch by Theorem 17.12 to a group isomorphism ˇ 1 (U , O∗ ) = H 1 (X, O∗ ). Pic(X) → lim H X X →
Remark 17.17. Theorem 17.16 is true when X is an arbitrary scheme and in fact for an arbitrary locally ringed space X. The construction of an invertible sheaf L with a given cocycle {gij } as its transition functions is a little more delicate in this case because of the possibility of zero divisors in OX . Using the cocycle {gij }, we realize L as an inverse limit of sheaves (λi )∗ OUi and (λij )∗ OUi ∩Uj where λi : Ui → X and λij : Ui ∩ Uj → X are the inclusions. Theorem 17.18. Let X ⊂ Pr be a projective scheme and let OX (n) = OPr (n) ⊗OPr OX for n ∈ Z. Let F be a coherent sheaf on X. Then: 1) For each i ≥ 0, H i (X, F ) is a ﬁnitedimensional kvector space. 2) There is an integer n0 , depending only on F , such that for each i > 0 and each n ≥ n0 , H i (X, F (n)) = 0 where F (n) = F ⊗OX (n). Proof. Let i : X → Pr be the given closed embedding. Let IX be the ideal sheaf of the closed subscheme X in OPr . Then i∗ OX = OPr /IX is a coherent OPr module, so i∗ F is a coherent OPr module. For U an open subset of Pr , we have that Γ(U, i∗ F ) = Γ(U ∩ X, F ). ˇ Computing cohomology by Cech cohomology, we see that H i (X, F ) = H i (Pr , i∗ F ) for all i (alternatively, we get this isomorphism directly from Lemma 17.4). Thus we may assume that X = Pr . Now conclusions 1) and 2) of Theorem 17.18 are true when F = OX (m) for some m ∈ Z by Theorem 17.14.
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17. Cohomology
We prove the theorem for arbitrary coherent sheaves F on X by descending induction on i. For i > r we have that H i (X, F (n)) = 0 for all n ∈ Z by Corollary 17.6 or by Theorem 17.13, since X can be covered by r + 1 open aﬃne subschemes. The theorem is thus true in this case. m We can write F as a quotient sheaf of a sheaf E = i=0 OX (qi ) for suitable qi ∈ Z by Exercise 11.56. Let K be the kernel of this quotient, giving an exact sequence (17.11)
0 → K → E → F → 0.
Then K is also coherent. For instance, letting S be the coordinate ring of X, we can realize E → F as the sheaﬁﬁcation of a surjection of ﬁnitely generated graded Smodules Ψ : S(qi ) → M . Then the kernel K of Ψ is ˜ is a ﬁnitely generated graded Smodule since S is Noetherian. Thus K = K coherent. We have an exact sequence of kvector spaces · · · → H i (X, E) → H i (X, F ) → H i+1 (X, K) → · · · . The module on the left is a ﬁnitedimensional kvector space since E is a ﬁnite direct sum of sheaves OX (qi ) and cohomology commutes with direct sums. The vector space on the right is ﬁnite dimensional by induction. Thus H i (X, F ) is a ﬁnitedimensional vector space, establishing 1). To prove 2), tensor (17.11) with OX (n) and then take the long exact cohomology sequence · · · → H i (X, E(n)) → H i (X, F (n)) → H i+1 (X, K(n)) → · · · . For i > 0 and n 0, the vector space on the left is zero by Theorem 17.14 and the vector space on the right is zero by induction. Thus H i (X, F (n)) = 0 for n 0. Remark 17.19. Suppose that Y is an aﬃne variety, X ⊂ Y × Pr is a closed subscheme, and F is a coherent sheaf on X. Then the proof of Theorem 17.18 extends to show that H i (X, F ) is a ﬁnitely generated k[Y ]module for all i and H i (X, F (n)) = 0 for all i > 0 and n 0. Corollary 17.20. Suppose that F is a coherent sheaf on a projective scheme X. Then F ⊗ OX (n) is generated by global sections for all n 0. Proof. For p ∈ X, tensor the short exact sequence 0 → Ip F → F → F /Ip F → 0 with OX (n) for n 0 and take cohomology to conclude that Γ(X, F ⊗ OX (n)) surjects onto Fp /Ip Fp . The stalk Fp is thus generated by Γ(X, F ⊗ OX (n)) as an OX,p module by Nakayama’s lemma. The conclusions of the corollary then follow from the facts that X is a Noetherian
17.4. Applications
319
space and the closed set of points where F (n) is not generated by global sections contains the closed set of points where F (n + 1) is not generated by global sections. We write hi (X, F ) = dimk H i (X, F ) if F is a coherent sheaf on a projective scheme X. The Euler characteristic of a coherent sheaf F on an ndimensional projective scheme X is n (−1)i hi (X, F ). χ(F ) = i=0
Corollary 17.21. Let Z be a projective scheme. Then χ(OZ (n)) = PZ (n) for n 0, where PZ (n) is the Hilbert polynomial of the homogeneous coordinate ring S(Z) of Z. Proof. The statement of Theorem 11.47 is valid for arbitrary projective schemes (although the proof requires a little modiﬁcation). Thus for n 0, we have PZ (n) = dimk S(Z)n = dimk Γ(Z, OZ (n)) = χ(OZ (n)) by Theorems 16.6, 11.47, and 17.18.
Theorem 17.22 (Serre duality). Let X be a nonsingular projective variety of dimension n, let D be a divisor on X, and let KX be a canonical divisor on X. Then for all i, dimk H i (X, OX (D)) = dimk H n−i (X, OX (−D + KX )). This follows from [73, Corollary III.7.7]. We will present a proof for curves (by Serre in [134]) in Section 18.2. Theorem 17.23 (K¨ unneth formula). Suppose X and Y are schemes and F and G are quasicoherent sheaves on X and Y , respectively. Let π1 : X × Y → X and π2 : X × Y → Y be the projections. Then H p (X, F ) ⊗k H q (Y, G) ∼ = H n (X × Y, π1∗ F ⊗OX×Y π2∗ G). p+q=n
Proof. Let U be an aﬃne open cover of X and V be an aﬃne open cover of Y . Then W = {Ui × Vj } is an aﬃne open cover of X × Y . Deﬁne complexes An = C −n (U , F ) and Cn = C −n (V , G) for n ∈ Z (where C ∗ (U , F )
320
17. Cohomology
ˇ and C ∗ (V , G) are the Cech complexes). Deﬁne B to be the complex B = A ⊗k C, so that Bn = p+q=n Ap ⊗ Cq for n ∈ Z. Taking homology of these complexes, we have that for p ∈ Z Hp (A∗ ) ∼ = H −p (C ∗ (U , F )) ∼ = H −p (X, F ), Hp (C∗ ) ∼ = H −p (C ∗ (V , G)) ∼ = H −p (Y, G), and
Hp (B∗ ) ∼ = H −p (C ∗ (W , π1∗ F ⊗ π2∗ G)).
By [128, Corollary 11.29 on page 340], we have Hp (A∗ ) ⊗ Hq (C∗ ) ∼ = Hn (A∗ ⊗ C∗ ) p+q=n
for n ∈ Z, giving us the conclusions of the theorem.
Exercise 17.24. Suppose that X is a projective scheme and 0 → M0 → M1 → · · · → Mn → 0 is an exact sequence of coherent OX modules. Show that n
(−1)i χ(Mi ) = 0.
i=0
Exercise 17.25. Let X = Pm × Pn with projections π1 and π2 onto the ﬁrst and second factors, respectively. For a, b ∈ Z let OX (a, b) = π1∗ OPm (a) ⊗ π2∗ OPn (b). Compute hi (X, OX (a, b)) for a, b ∈ Z and i ∈ N.
17.5. Higher direct images of sheaves Deﬁnition 17.26. Suppose that φ : X → Y is a continuous map of topological spaces and F is a sheaf of groups on X. For i ≥ 0, the ith direct image sheaf of F is the sheaf Ri φ∗ F on Y associated to the presheaf U → H i (φ−1 (U ), F ) for U an open subset of Y . The sheaf R0 φ∗ F is equal to the sheaf φ∗ F . Given an open cover U of a topological space X and a sheaf F of Abelian groups on X, deﬁne a complex of sheaves of Abelian groups C ∗ = C ∗ (U , F ) on X by $ F Ui0 ,...,ip ∩ V (17.12) C p (U , F )V = i0 <···
for V an open subset of X, with coboundary map d : C p → C p+1 as deﬁned by (17.9).
17.5. Higher direct images of sheaves
321
Proposition 17.27. Suppose that φ : X → Y is a regular map of varieties and F is a quasicoherent sheaf on X. Then the sheaves Ri φ∗ F are quasicoherent on Y . In particular, for V an aﬃne open subset of Y , ˜ Ri φ∗ F  V = M where M = H i (φ−1 (V ), F ). If φ is the composition of a closed embedding X → Y × Pn for some n, followed by projection onto the ﬁrst factor, then the sheaves Ri φ∗ F are coherent on Y . Proof. Let U = {Ui } be an aﬃne open cover of X. The sheaves C p (U , F ) deﬁned in (17.12) are quasicoherent on X so the complex φ∗ C ∗ (U , F ) of OY modules is quasicoherent by Theorem 11.50, and thus the sheaves H p (φ∗ C ∗ (U , F )) are quasicoherent on Y by Exercise 11.35. Suppose V ⊂ Y is an aﬃne open set. Then the open subset Ui × V of X × Y is aﬃne, so Γφ ∩ (Ui × V ) ∼ = Ui ∩ φ−1 (V ) is an open aﬃne subset of −1 ∼ X = Γφ . Let Wi = Ui ∩ φ (V ). Then W = {Wi } is an aﬃne open cover of φ−1 (V ). We have that H p (φ∗ C ∗ (U , F ))(V ) = H p (Γ(V, φ∗ C ∗ (U , F ))) = H p (C ∗ (W , F )) = H p (φ−1 (V ), F ) by Exercise 11.35 and Theorem 17.13, and so ˜ Rp φ∗ F V = M where M = H p (φ−1 (V ), F ) by Theorem 11.32. The coherence of Rp φ∗ F in the case when X is a closed subvariety of Y × Pn follows from Remark 17.19. Lemma 17.28. Suppose that φ : X → Y is a regular map of varieties, F is a quasicoherent OX module, and E is a locally free OY module of ﬁnite rank. Then Ri φ∗ (F ⊗ φ∗ E) ∼ = Ri φ∗ F ⊗ E. Proof. Let notation be as in Proposition 17.27. Since E is locally free, φ∗ (C ∗ (U , F ⊗ φ∗ E)) ∼ = φ∗ (C ∗ (U , F ) ⊗ φ∗ E) ∼ = (φ∗ C ∗ (U , F )) ⊗ E by Exercise 11.41. Since E is locally free, H p ((φ∗ C ∗ (U , F )) ⊗ E) ∼ = H p (φ∗ C ∗ (U , F )) ⊗ E, so if V is an aﬃne open subset of Y , Γ(V, Rp φ∗ (F ⊗ φ∗ E)) = H p (φ∗ C ∗ (U , F ⊗ φ∗ E))(V ) ∼ H p (φ∗ C ∗ (U , F ))(V ) ⊗O (V ) E(V ) = ∼ = Γ(V, (Rp φ∗ F ) ⊗ E).
Y
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17. Cohomology
The Leray spectral sequence [59, II.4.7.1] 2
(17.13)
E i,j = H i (Y, Rj φ∗ F ) ⇒i H i+j (X, F )
relates the sheaf cohomology of F on X and the sheaf cohomology of the higher direct image sheaves of F on Y . The method of using a double ˇ complex to compute this spectral sequence using Cech cohomology is given a lucid explanation in [105, Section 3]. Proposition 17.29. Suppose that X and Y are projective varieties and φ : X → Y is a regular map. Let Yi = {p ∈ Y  dim φ−1 (p) ≥ i}. The Yi are closed subsets of Y by Corollary 8.14. Suppose that F is a coherent sheaf on X. Then Supp(Ri φ∗ F ) ⊂ Yi for all i.
Proof. [105, Proposition 4.3].
Proposition 17.30. Suppose that Y is a nonsingular variety and W is a nonsingular closed subvariety. Let π : X = B(W ) → Y be the blowup of W with exceptional divisor E = XW . Then E is a codimension 1 subvariety of X, X and E are nonsingular, Ri π∗ OX (mE) = 0
(17.14) and
π∗ OX (mE) =
(17.15)
for i > 0 and m ≤ 0, OY −m IW
if m ≥ 0, if m < 0.
Proof. The assertions that E is a codimension 1 subvariety of X and X and E are nonsingular in Proposition 17.30 were proven earlier in Theorem 10.19. By Proposition 17.27, it suﬃces to show that there exists an aﬃne cover {V } of Y such that for all V , H i (π −1 (V ), OX (mE)) = 0 for i > 0 and m ≤ 0
(17.16) and (17.17)
0
H (π
−1
(V ), OX (mE)) =
k[V ] IV (W ∩ V )−m
if m ≥ 0, if m < 0.
Suppose that q ∈ Y \ W . Let V be an aﬃne neighborhood of q in Y \ W . Then π −1 (V ) ∼ = V since W ∩ V = ∅, so π −1 (V ) is aﬃne, and
17.5. Higher direct images of sheaves
323
thus (17.16) holds on V by Theorem 17.7. Since W ∩ V = ∅, we have that IV (W ∩ V ) = k[V ]. Further, E ∩ π −1 (V ) = ∅, so OX (mE)π −1 (V ) = Oπ−1 (V ) ∼ = OV for all m. Thus (17.17) holds on V . Suppose that q ∈ W . Since Y and W are nonsingular, Lemma 10.3 implies that there exist f0 , . . . , fr ∈ OY,q with r = codimY X − 1, such that IW,q = (f0 , . . . , fr ) and f0 , . . . , fr is an OY,q regular sequence. There then exists an aﬃne neighborhood V of q in Y such that f0 , . . . , fr ∈ k[V ], (f0 , . . . , fr ) = IV (W ∩ V ), and f0 , . . . , fr is a k[V ]regular sequence. Let J = (f0 , . . . , fr ) ⊂ k[V ], T = W ∩ V , U = π −1 (V ), and F = E ∩ U . We have that the coordinate ring S(U ) = Jn ∼ = k[V ][tf0 , . . . , tfr ] n≥0
where t is an indeterminate with deg t = 1 by Theorem 6.4, and so for m ∈ Z and 0 ≤ i ≤ r, )(m)Utfi = tm fim OUtfi = tm OU (−mF )Utfi . OU (m)Utfi = S(U Thus
OX (−mE)U = OU (−mF ) ∼ = OU (m)
for m ∈ Z. Now S(U )/JS(U ) =
J i /J i+1 = k[T ][f 0 , . . . , f r ]
i≥0
where f i is the class of fi in J/J 2 and is a polynomial ring over k[T ] = k[V ]/J in f 0 , . . . , f r by Theorem 1.76. Thus JS(U ) is a prime ideal in S(U ), and S(F ) = S(U )/JS(U ) r ∼ and so F = W × P . Since S(F ) is integrally closed, we have by Theorem 11.47 that (17.18)
H 0 (F, OF (n)) = S(F )n = J n /J n+1
for n ≥ 0. Since
OF (m) ∼ = π2∗ OPr (m) ∼ = π1∗ OT ⊗ π2∗ OPr (m) where π1 : F ∼ = W × Pr → W and π2 : F ∼ = W × Pr → Pr are the natural projections, we have that for all i ≥ 0, H α (T, OT ) ⊗ H β (Pr , OPr (m)) H i (F, OF (m)) ∼ = α+β=i
by Theorem 17.23, so (17.19)
H i (F, OF (m)) = 0
for i > 0 and m ≥ 0
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17. Cohomology
since H β (Pr , OPr (m)) = 0 for β > 0 and m ≥ 0 and H α (T, OT ) = 0 for α > 0 since T is aﬃne, and by Theorem 17.7. Further, (17.20)
H 0 (F, OF (n)) = 0
for n < 0.
We have short exact sequences (17.21)
0 → OU (m + 1) → OU (m) → OF (m) → 0
for all m ∈ Z, giving surjections, by (17.19), H i (U, OU (m + 1)) → H i (U, OU (m)) for i > 0 and m ≥ 0. Since OU (1) is ample on U , H i (U, OU (m)) = 0 for m 0 by Remark 17.19. Thus (17.16) holds on U = π −1 (V ). By (17.16) and (17.18), we have short exact sequences 0 → H 0 (U, OU (m + 1)) → H 0 (U, OU (m)) → J m /J m+1 → 0 for m ≥ 0. By Theorem 11.47, H 0 (U, OU (m)) = J m for m 0. Thus H 0 (U, OU (m)) = J m for all m ≥ 0. By (17.20) and (17.21), we have isomorphisms H 0 (U, OU (m + 1)) → H 0 (U, OU (m)) for m < 0. Since H 0 (U, OU ) = k[V ], we have that H 0 (U, OU (m)) = k[V ] for m ≤ 0. Thus (17.17) holds for U = π −1 (V ). Since equations (17.16) and (17.17) hold on an aﬃne cover {V } of Y , the conclusions (17.14) and (17.15) of this proposition hold on Y . Suppose that Y is a nonsingular aﬃne variety and W is a nonsingular subvariety of Y . Let R = k[Y ] and p = I(W ). Let π : X → Y be the blowup of W and let E = XW . Then OX,E is a valuation ring, since E is a codimension 1 subvariety of the nonsingular variety X. We then have that (17.22)
Γ(X, OX (−nE)) = {f ∈ R  νE (f ) ≥ n} = (pn Rp ) ∩ R = p(n)
is the nth symbolic power of p. Comparing (17.15) and (17.22), we have the formula, with our assumption that p is a “regular prime” (R/p is a regular ring), (17.23)
p(n) = pn .
This formula (17.23) was previously derived in Proposition 10.4. For many prime ideals p in a regular local ring, this formula is not true and we do not have equality of ordinary and symbolic powers of prime ideals. (n) is in general not Noetherian. An p In fact, the symbolic algebra n≥0 example of this is given by Roberts [127], using an earlier example of Nagata [122]. A nonNoetherian example with p being a rational monomial curve is given by Goto, Nishida, and Watanabe in [60]. We now give some more useful formulas on cohomology.
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325
Theorem 17.31. Suppose that Y is a nonsingular variety and φ : X → Y is the blowup of a nonsingular subvariety W of Y and L is an invertible sheaf on Y . Then H i (X, φ∗ L) = H i (Y, L) for all i. Proof. We have that Ri φ∗ (φ∗ L) = 0 for i > 0 and φ∗ (φ∗ L) ∼ = L, by Proposition 17.30 and Lemma 17.28. Thus the Leray spectral sequence 2
E i,j = H i (Y, Rj φ∗ φ∗ L) ⇒i H i+j (X, φ∗ L)
degenerates at the 2 E level, so that H i (Y, L) ∼ = H i (X, φ∗ L) as desired.
Theorem 17.32. Suppose that φ : X → Y is a birational regular map of nonsingular projective varieties over a ﬁeld of characteristic 0 and L is an invertible sheaf on Y . Then H i (X, φ∗ L) = H i (Y, L) for all i. Proof. By resolution of indeterminancy ([79] or [35, Theorem 6.39]), there exists a regular birational map f : Z → X from a nonsingular projective variety Z such that g = φ ◦ f factors as a product of blowups of nonsingular subvarieties gn
gn−1
g2
g1
g : Z = Zn → Zn−1 → · · · → Z1 → Z0 = Y. By Theorem 17.31 we have that g ∗ : H i (Y, L) ∼ = H i (Z, g ∗ L) for all i. The isomorphisms g ∗ : H ∗ (Y, L) → H i (Z, g ∗ L) factor as φ∗
f∗
H i (Y, L) → H i (X, φ∗ L) → H i (Z, g ∗ L). Thus φ∗ is injective. By resolution of indeterminancy [79] there exists a projective variety W and a birational regular map γ : W → Z such that β = f ◦ γ is a product of blowups of nonsingular subvarieties, so we have isomorphisms β ∗ : H i (X, φ∗ L) ∼ = H i (W, β ∗ L) for all i. Thus f ∗ is also injective, and the theorem follows.
17.6. Local cohomology and regularity Suppose that R is a Noetherian ring and M is an Rmodule. Suppose that I is an ideal in R, with generators I = (f1 , . . . , fn ). Consider the modiﬁed ˇ Cech complex C∗ : 0 → C0 → C1 → · · · → Cd → 0 where C 0 = R and Ct =
1≤i1
Rfi1 fi2 ···fit .
326
17. Cohomology
The local cohomology of M is HIi (M ) = H i (M ⊗R C ∗ ).
(17.24) We have that
HI0 (M ) = {f ∈ M  I k f = 0 for some k ≥ 0} = ΓI (M ), the set of elements of M which have support in I. HIi (M ) does not depend on the choice of generators of I. Further, i (M ). HIi (M ) = H√ I
If 0 → A → B → C → 0 is a short exact sequence of Rmodules, then there is a long exact sequence 0 → HI0 (A) → HI0 (B) → HI0 (C) → HI1 (A) → · · · . An important use of local cohomology is to compute depth (Deﬁnition 1.67). We have the following interpretation of depth in terms of local cohomology ([27], [72], [73, Exercise III.3.4], [50, Theorem A4.3]). Proposition 17.33. Suppose that M is a ﬁnitely generated Rmodule and n ≥ 0. Then the following are equivalent: 1) depthI M ≥ n. 2) HIi (M ) = 0 for all i < n. If R is a regular local ring of dimension d and I is the maximal ideal of R, then depthI R = d since a regular system of parameters in R is a maximal Rregular sequence in R [107, Theorem 36, page 121]. Suppose that X is a Noetherian aﬃne scheme with R = Γ(X, OX ) and ˜ be the quasicoherent sheaf on X associated M is an Rmodule. Let M = M to M . Suppose that I = (f1 , . . . , fn ) is an ideal in R. ˇ Consider the Cech complex F ∗ : F 0 → F 1 → · · · → F n−1 , where Ft =
1≤i1
Rfi1 fi2 ···fit+1 .
˜ on U = X \ Z(I) is The sheaf cohomology of M ˜ ) = H i (M ⊗R F ∗ ). H i (U, M
17.6. Local cohomology and regularity
327
ˇ The modiﬁed Cech complex C ∗ , used to compute local cohomology, is ˇ ˇ complex one to the obtained from the Cech complex F ∗ by shifting the Cech 0 right and setting C = R. From this we see that there is an exact sequence ˜ ) → H 1 (M ) → 0 (17.25) 0 → H 0 (M ) → M → H 0 (U, M I
I
and isomorphisms ˜) ∼ H i (U, M = HIi+1 (M ) for i ≥ 1,
(17.26)
where U = X \ Z(I). We have the interpretation of ˜) ∼ H 0 (U, M = lim HomR (I n , M ) →
as an “ideal transform”. A particularly important case of this is when X = An+1 so that κ[X] = R = κ[x0 , . . . , xn ] is a polynomial ring over a ﬁeld κ. We give R the standard grading. Let m = (x0 , . . . , xn ). Let Q be the point Z(m) in An+1 . Suppose ˜ be the sheaﬁﬁcation of M that M is a graded module over R. Let M on the aﬃne variety X. Then the local and sheaf cohomology modules ˜ ) and H i (R) are graded, and the maps of equations (17.25) H i (An+1 \ Q, M m and (17.26) are graded. From the natural surjection of the aﬃne cone An+1 \ Q onto the projective space Pn , we obtain graded isomorphisms ˜) ∼ ˜ (j)). H i (An+1 \ Q, M H i (Pn , M = j∈Z
˜ is the sheaf associated to M on An+1 . In the ﬁrst cohomology module, M ˜ (j) is the sheaf associated to M (j) on In the second cohomology module, M n P (M (j)d = Mj+d for d ∈ Z). We thus have a degreepreserving exact sequence of graded Rmodules ˜ (j)) → H 1 (M ) → 0 (17.27) 0 → Hm0 (M ) → M → H 0 (Pn , M m j∈Z
and isomorphisms ˜ (j)) ∼ H i (Pn , M (17.28) = Hmi+1 (M ) for i ≥ 1. j∈Z
We have the interpretation ˜ (j)) ∼ H 0 (Pn , M = lim HomR (mn , M ) j∈Z
→
as an ideal transform. We continue to study the graded polynomial ring R = κ[x0 , . . . , xn ] and assume that M is a ﬁnitely generated graded Rmodule. Deﬁne
sup{j  Hmi (M )j = 0} if Hmi (M ) = 0, i a (M ) = −∞ otherwise.
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17. Cohomology
The regularity of M is deﬁned to be reg(M ) = max{ai (M ) + i}. i
Interpreting R as the coordinate ring S(Pn ) of Pn and considering the ˜ to be ˜ on Pn associated to M , we can deﬁne the regularity of M sheaf M ˜ ) = max{m  H i (Pn , M ˜ (m − i − 1)) = 0 for some i ≥ 1} reg(M i = maxi≥2 {a (M ) + i}. Thus ˜ ) ≤ reg(M ). reg(M The classical interpretations of regularity are for sections of invertible sheaves on a projective variety X. Deﬁnition 17.34. Suppose that F is a coherent sheaf on Pn . The sheaf F is said to be mregular if H i (Pn , F (m − i)) = 0 for all i > 0. Thus reg(F ) is the smallest m such that F is mregular. The following theorem by Mumford ([118, page 99]) generalizes a classical result of Castelnuovo. Theorem 17.35 (Geometric regularity theorem). Suppose that F is an mregular coherent sheaf on Pn . Then: a) H 0 (Pn , F (k)) is spanned by H 0 (Pn , F (k − 1)) ⊗ H 0 (Pn , O(1)) if k > m. b) H i (Pn , F (k)) = 0 whenever i > 0, k + i ≥ m. c) F (k) is generated by global sections if k ≥ m. d)
H 0 (Pn , F (d))
d∈Z
is generated as an R = ≤ m.
d≥0 H
0 (Pn , O
Pn (d))module
in degrees
The following proof is based on [118]. Proof. The proof is by induction on n. If n = 0, the result is immediate. For n > 0, let H be a hyperplane of Pn not containing any of the associated varieties of F (deﬁned after Deﬁnition 15.5 in Section 15.1). Tensor the exact sequence 0 → OPn (−H) ∼ = OPn (−1) → OPn → OH → 0
17.6. Local cohomology and regularity
329
with F (k). For all p ∈ Pn , if f is a local equation of H at p, then multiplication by f is injective on Fp . Thus the sequence (17.29)
0 → F (k − 1) → F (k) → FH (k) → 0,
where FH (k) = (F ⊗ OH )(k) is short exact. Taking cohomology, we obtain an exact sequence H i (Pn , F (m − i)) → H i (H, FH (m − i)) → H i+1 (Pn , F (m − i − 1)). Thus if F is mregular, then the sheaf FH on H ∼ = Pn−1 is mregular. Thus the induction hypothesis gives us the conclusions of the theorem for FH . From the short exact sequence (17.29), we obtain an exact sequence H i+1 (Pn , F (m − i − 1)) → H i+1 (Pn , F (m − i)) → H i+1 (H, FH (m − i)). If i ≥ 0, by b) for FH , the last group is (0). By mregularity, the ﬁrst group is zero. Thus the middle group is (0) and F is (m + 1)regular. By induction on k ≥ m − i, we obtain the conclusion b) for F . We now prove a). Consider the commutative diagram H 0 (Pn , F (k − 1))
H 0 (Pn , F (k − 1)) ⊗ H 0 (Pn , OPn (1))
μ
λ
H 0 (H, FH (k − 1)) ⊗ H 0 (H, OH (1))
τ
/ H 0 (Pn , F (k))
ν
/ H 0 (H, FH (k))
where the bottom row is exact. The map λ is surjective. Further, τ is surjective if k > m by conclusion a) for FH . Thus ν(Image(μ)) = H 0 (H, FH (k)); that is, H 0 (Pn , F (k)) is spanned by Image(μ) and H 0 (Pn , F (k − 1)). Let σ ∈ H 0 (Pn , OPn (1)) be such that div(σ) = H. Then the image of H 0 (Pn , F (k − 1)) in H 0 (Pn , F (k)) is equal to σ ⊗ H 0 (Pn , F (k − 1)) which is contained in Image(μ). Thus μ is surjective and a) is proven for F . By Corollary 17.20, F (k) is generated by its global sections if k is suﬃciently large. Thus by a), (17.30) H 0 (Pn , F (m)) ⊗ H 0 (Pn , OPn (k − m)) generates the sheaf F (k) if k 0. Let p ∈ Pn , and ﬁx a local isomorphism of OPn (1) and OPn in a neighborhood of p. For k ≥ m, this identiﬁes OPn (k − m) with OPn in a neighborhood of p. Then H 0 (Pn , OPn (k − m)) is identiﬁed with a vector space of elements of the local ring OPn ,p which generate OPn ,p , and (17.30) tells us that H 0 (Pn , F (m)) generates F (m)p as an OPn ,p module; that is, F (m) is generated by its global sections.
330
17. Cohomology
Continuing to assume that M is a ﬁnitely generated graded module over the polynomial ring R = κ[x0 , . . . , xn ], with maximal ideal m = (x0 , . . . , xn ), let F ∗ : 0 → · · · → Fj → · · · → F1 → F0 → M → 0 be a minimal free resolution of M as a graded Rmodule. Let bj be the maximum degree of the generators of Fj . Then reg(M ) = max{bj − j  j ≥ 0}. In fact, we have (Eisenbud and Goto [51], Bayer and Mumford [17]) that reg(M ) = max{bj − j  j ≥ 0} = max{n  ∃j such that TorR j (κ, M )n+j = 0} j = max{n  ∃j such that Hm (M )n−j = 0}. The equality of the ﬁrst and second of the righthand sides of these equations follows since TorR j (κ, M ) = Hj (F∗ ⊗ R/m), and as F∗ is minimal, we have that the maps of the complex F∗ ⊗ R/m are all zero. To obtain the equality of the ﬁrst and third conditions, we take the cohomology of the dual of F∗ , to compute ExtjR (M, R), and then apply graded local duality. The righthand side of the third equation is equal to reg(M ) by the deﬁnition of regularity. We may also deﬁne local cohomology for sheaves of Abelian groups on a topological space. Let X be a topological space, Y be a closed subset, and F be a sheaf of Abelian groups on X. Let ΓY (X, F ) be the subgroup of Γ(X, F ) consisting of all sections whose support is contained in Y . If 0→A→B→C→0 is a short exact sequence of sheaves of Abelian groups on X, then 0 → ΓY (X, A) → ΓY (X, B) → ΓY (X, C) is exact, so we can deﬁne local cohomology groups HYi (X, F ) by taking a resolution 0 → F → I0 → I1 → · · · by injective sheaves of groups I i , taking the associated complex d0
d1
ΓY (X, I 0 ) → ΓY (X, I 1 ) → · · · and deﬁning HYi (X, F ) = Kernel(di )/Image(di+1 ) for i ≥ 0. We summarize some properties of local cohomology, whose proofs can be found in [72] and are derived in exercises in [73, Chapter III]. If X is a locally ringed space and F is a sheaf of OX modules, then HYi (X, F ) can be computed by taking an injective resolution by OX modules.
17.6. Local cohomology and regularity
331
Continuing to suppose that F is a sheaf of Abelian groups on a topological space, let U be the open subset U = X \ Y of X. Then we have a long exact sequence of cohomology groups 0 → HY0 (X, F ) → H 0 (X, F ) → H 0 (U, F U ) → HY1 (X, F ) → H 1 (X, F ) → H 1 (U, F U ) → · · · . Proposition 17.36 (Excision). Let V be an open subset of X containing Y . Then for all i ≥ 0 there are natural isomorphisms H i (X, F ) ∼ = H i (V, F V ). Y
Y
Proposition 17.37 (MayerVietoris sequence). Let Y1 , Y2 be two closed subsets of X. Then there is a long exact sequence · · · → HYi 1 ∩Y2 (X, F ) → HYi 1 (X, F ) ⊕ HYi 2 (X, F ) → HYi 1 ∪Y2 (X, F ) (X, F ) → · · · . → HYi+1 1 ∩Y2 Now suppose that X is a Noetherian aﬃne scheme with R = Γ(X, OX ) ˜ where M is an Rmodule. Let I ⊂ R be an ideal, and let Y and F = M be the subscheme of X with Γ(X, OY ) = R/I. Then (by [72] and [27] or [73, Exercise III.3.3] and [50, Appendix 4]) we have that HYi (X, F ) = HIi (M ) as deﬁned in equation (17.24).
Chapter 18
Curves
In this chapter we consider the geometry of nonsingular projective curves. In Sections 18.1–18.3, we prove the RiemannRoch theorem on a nonsingular projective curve X, Theorem 18.13, which gives a formula for the dimension of the vector space Γ(X, OX (D)) of functions whose poles are bounded by a given divisor D on X in terms of the genus g of X, the degree deg D of D, and the dimension h0 (X, OX (KX −D)), where KX is a canonical divisor on X. The RiemannRoch theorem follows from the RiemannRoch inequality, Theorem 18.2 and Corollary 18.3, proven in Section 18.1 and from Serre duality, Corollary 18.10, proven in Section 18.2. The RiemannRoch inequality, Theorem 18.2 and Corollary 18.3, give a lower bound for h0 (X, OX (D)), which is only in terms of g and deg D and which is an equality if and only if h1 (X, OX (D)) = 0. Cliﬀord’s theorem, Theorem 18.20, gives an upper bound for h0 (X, OX (D)) which only depends on deg D if h1 (X, OX (D)) > 0. As a consequence of the RiemannRoch theorem, we show in Theorem 18.21 that if deg D ≥ 2g + 1, then D is very ample and the complete linear system D induces a closed embedding of X into a projective space. We deduce in Theorem 18.22 a subdivision of curves by Kodaira dimension: the curves of genus larger than 1, for which KX is ample; the elliptic curves (genus 1), for which KX ∼ 0; and P1 (genus 0), for which −KX is ample. The theory of Kodaira dimension generalizes to higherdimensional varieties [18], [91], [112], [113], [114], and [20]. In Section 18.4, we consider the RiemannRoch problem, which is the problem of computing the function h0 (X, OX (nD)) for large n, where X is a nonsingular projective variety and D is a divisor on X.
333
334
18. Curves
In Sections 18.5 and Section 18.6, we consider regular maps f : X → Y of nonsingular projective curves and ﬁnd formulas relating the genus of X, the genus of Y , and the ramiﬁcation of f . We work out the basic geometric theory of elliptic curves in Section 18.7, study the topology of complex curves in Section 18.8, and introduce the theory of Abelian varieties and Jacobians of curves in Section 18.9.
18.1. The RiemannRoch inequality Suppose that X is a nonsingular projective curve. The genus of X is g = g(X) = h0 (X, OX (KX )). We have that g(X) = h1 (X, OX )
(18.1)
as follows from Serre duality (Corollary 18.10) which will be established in Section 18.2. Recall the deﬁnition of the degree of a divisor on a curve from Section 13.5. If D1 and D2 are linearly equivalent divisors on X, then deg D1 = deg D2 by Corollary 13.19. Lemma 18.1. Let D be a divisor on X. If h0 (X, OX (D)) > 0, then deg(D) ≥ 0. If h0 (X, OX (D)) > 0 and deg D = 0, then D ∼ 0. Proof. If h0 (X, OX (D)) > 0, then there exists 0 = f ∈ Γ(X, OX (D)). Then E = (f ) + D is an eﬀective divisor, so that deg E ≥ 0. We have deg D = deg E ≥ 0 by Corollary 13.19. If deg D = 0, then D is linearly equivalent to an eﬀective divisor of degree 0. The only such divisor is 0. For a coherent sheaf F on X, we have χ(F ) = h0 (X, F ) − h1 (X, F ) by Theorem 17.5. Theorem 18.2. Let D be a divisor on a nonsingular projective curve X of genus g. Then χ(OX (D)) = h0 (X, OX (D)) − h1 (X, OX (D)) = deg D + 1 − g. Proof. We must show that (18.2)
χ(OX (D)) = deg D + 1 − g
for every divisor D on X. The formula is true for D = 0 by Theorem 3.35, the deﬁnition of genus, and equation (18.1). Let D be any divisor, and let p ∈ X be a point. We will show that the formula is true for D if and only if it is true for D + p. Since any divisor
18.2. Serre duality
335
on X can be obtained by a ﬁnite sequence of addition and subtraction of points, this will establish the formula (18.2) and prove the theorem. Let I(p) be the ideal sheaf of the point p ∈ X. Using the fact that I(p) = OX (−p) (a point is a divisor on a curve), we have a short exact sequence of sheaves of OX modules 0 → OX (−p) → OX → OX /I(p) → 0. Now tensor with OX (D + p) to get a short exact sequence (18.3)
0 → OX (D) → OX (D + p) → OX /I(p) → 0.
The sequence is short exact since OX (D + p) is a locally free (and thus ﬂat) OX module (in fact, locally, this is just like tensoring with OX ). The support of OX /I(p) is just the point p, so that (OX /I(p))⊗OX OX (D +p) ∼ = OX /I(p). Taking the long exact cohomology sequence associated to (18.3) and using Corollary 17.6, we get an exact sequence 0 → H 0 (X, OX (D)) → H 0 (X, OX (D + p)) → H 0 (X, OX /I(p)) ∼ =k → H 1 (X, OX (D)) → H 1 (X, OX (D + p)) → H 1 (X, OX /I(p)) = 0.
Thus χ(OX (D + p)) = χ(OX (D)) + 1. Since deg(D + p) = deg(D) + 1, we obtain the formula (18.2).
Corollary 18.3 (The RiemannRoch inequality). Suppose that D is a divisor on a nonsingular projective curve X of genus g. Then h0 (X, OX (D)) ≥ deg(D) + 1 − g.
18.2. Serre duality The proof in this section follows Serre [134]. We continue to assume that X divisor D = ai pi on X where pi deﬁne for a point q ∈ X
ai νq (D) = 0
is a nonsingular projective curve. For a are distinct points of X and ai ∈ Z, we if q = pi , if q =
pi for all i.
A r´epartition r is a family {rp }p∈X of elements of k(X) such that rp ∈ OX,p for all but ﬁnitely many p ∈ X. The set of all r´epartitions is an algebra R over k. Suppose that D is a divisor on X. Then deﬁne R(D) to be the ksubspace of R consisting of all r = {rp } such that νp (rp ) ≥ −νp (D) for all p ∈ X. To every f ∈ k(X), we associate the r´epartition {rp } such that rp = f for every p ∈ X, giving an injection of k(X) into R. We may then view k(X) as a subring of R.
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18. Curves
Proposition 18.4. Suppose that D is a divisor on X. Then the kvector space I(D) = H 1 (X, OX (D)) is canonically isomorphic to R/(R(D)+k(X)). Proof. For p ∈ X and r ∈ R, let [rp ] be the class of rp in k(X)/OX (D)p . Deﬁne a kvector space homomorphism Λ : R → p∈X k(X)/OX (D)p by r → {[rp ]}. Here Λ is welldeﬁned since rp ∈ OX (D)p for all but ﬁnitely many p ∈ X. We have that Λ(r) = 0 if and only if rp ∈ OX (D)p for all p ∈ X which holds if and only if νp (rp ) ≥ −νp (D) for all p ∈ X. Thus Λ induces an isomorphism k(X)/OX (D)p . (18.4) R/R(D) ∼ = p∈X
Let A be the sheaf k(X)/OX (D). By Lemma 17.15, we have natural exact sequences 0 → Γ(V, OX (D)) → Γ(V, k(X)) = k(X) → Γ(V, A) for all open subsets V of X. Suppose that U is a neighborhood of a point p ∈ X and s ∈ A(U ). There exists t ∈ k(X) such that the image of t in Ap is equal to sp . Let t be the image of t in A(U ). Then the germ of s − t in Ap is zero. Since Ap is the limit of A(V ) over open sets V containing p, we have that there exists an open neighborhood V of p in U such that the restriction of s − t in A(V ) is zero. Then replacing U with V and s with its restriction to V , we have that s is the class of t ∈ k(X), which is necessarily in OX (D)q for all but ﬁnitely many q ∈ U . Thus there exists a neighborhood U of p such that s = 0 on U \ {p}. In particular, every s ∈ H 0 (X, A) has ﬁnite support, so Ap (18.5) Φ : H 0 (X, A) → p∈X
deﬁned by s → {sp } is a welldeﬁned homomorphism. By the sheaf axioms, Ap lifts to a section of H 0 (X, A), and the kernel of every element {αp } ∈ Φ is zero. Thus Φ is an isomorphism. We have that Ap = k(X)/OX (D)p for p ∈ X so (18.4) and (18.5) give us an isomorphism (18.6)
R/R(D) ∼ = H 0 (X, A).
The sheaf OX (D) is a subsheaf of the constant sheaf k(X), so there is an exact sequence 0 → OX (D) → k(X) → k(X)/OX (D) → 0.
18.2. Serre duality
337
By Lemma 17.15, we have that H 0 (X, k(X)) = k(X) and H 1 (X, k(X)) = 0 so we have an exact sequence of cohomology modules k(X) → H 0 (X, A) → H 1 (X, OX (D)) → 0. Now using the isomorphism (18.6), we have the desired isomorphism H 1 (X, OX (D)) ∼ = R/(R(D) + k(X)).
From now on, we identify H 1 (X, OX (D)) and R/(R(D) + k(X)) which we will denote by I(D). Let J(D) be the dual of the kvector space I(D) = R/(R(D) + k(X)). An element of J(D) is thus a linear form on R which vanishes on k(X) and R(D). Suppose that D ≥ D. Then R(D ) ⊃ R(D) so that J(D) ⊃ J(D ). The union of the J(D) for D running through the divisors of X will be denoted by J. Let f ∈ k(X) and α ∈ J. The map r → α(f r) is a linear form on R vanishing on k(X), which we will denote by f α. If α ∈ J, then f α ∈ J. This follows since if α ∈ J(D) and f ∈ Γ(X, OX (Δ)), then the linear form f α vanishes on R(D − Δ) and thus belongs to J(D − Δ). The operator (f, α) → f α gives J the structure of a vector space over k(X). Proposition 18.5. The dimension of J as a k(X)vector space is ≤ 1. Proof. Suppose that α, α ∈ J are linearly independent over k(X). There exists a divisor D such that α ∈ J(D) and α ∈ J(D). Let d = deg(D). For every integer n ≥ 0, let Δn be a divisor of degree n (for example, Δn = np, where p is a ﬁxed point of X). Suppose that f, g ∈ Γ(X, OX (Δn )). Then f α, gα ∈ J(D − Δn ). Since α, α are linearly independent over k(X), any relation f α + gα = 0 implies f = g = 0. Thus the map (f, g) → f α + gα is an injective kvector space homomorphism Γ(X, OX (Δn )) ⊕ Γ(X, OX (Δn )) → J(D − Δn ), so we have the inequality (18.7)
dimk J(D − Δn ) ≥ 2 dimk Γ(X, OX (Δn ))
for all n. We will now show that (18.7) leads to a contradiction as n → ∞. The lefthand side is dimk I(D − Δn ) = h1 (X, OX (D − Δn )) = −deg(D − Δn ) + g − 1 + h0 (X, OX (D − Δn )) = n + (g − 1 − d) + h0 (D, OX (D − Δn )) by Theorem 18.2.
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When n > d, deg(D−Δn ) < 0 so that h0 (X, OX (D−Δn )) = 0 by Lemma 18.1. Thus for large n, the lefthand side of (18.7) is equal to n + A0 , A0 a constant. The righthand side of (18.7) is equal to 2h0 (X, OX (Δn )). By Theorem 18.2, h0 (X, OX (Δn )) ≥ deg(Δn ) + 1 − g = n + 1 − g. Thus the righthand side of (18.7) is ≥ 2n + A1 for some constant A1 , giving a contradiction for large n. The sheaf ΩX/k is a subsheaf of Ωk(X)/k . If p ∈ X and t is a regular parameter in OX,p , then ΩX/k,p = OX,p dt by Proposition 14.15. We further have that Ωk(X)/k = k(X)dt. We deﬁne νp (ω) = νp (f ) if ω = f dt ∈ k(X)dt. Recall (Section 14.3) that the divisor (ω) of ω ∈ Ωk(X)/k is νp (ω)p. (ω) = p∈X
Thus νp (ω) = νp (K) where (ω) = K is the divisor of ω. The quotient ˆX,p = k[[t]] (Proposition 21.41) is the ﬁeld of Laurent series k((t)). ﬁeld of O Identifying f with its image in k((t)) by the inclusion k(X) → k((t)) induced by the inclusion OX,p → k[[t]], we have an expression an tn f= n−∞
with all an ∈ k (n −∞ in the summation means that an = 0 for n 0). The coeﬃcient a−1 of t−1 in f is called the residue of ω = f dt at p, denoted by Resp (ω). The following proposition shows that the deﬁnition is welldeﬁned. Proposition 18.6 (Invariance of the residue). The preceding deﬁnition is independent of the choice of regular parameter t in OX,p . Proposition 18.6 is proven in [134, Section 11 of Chapter II]. Proposition 18.7 (Residue formula). For every ω ∈ Ωk(X)/k , Resp (ω) = 0. p∈X
Proposition 18.7 is proven in [134, Sections 12 and 13 of Chapter II]. The proof is by taking a projection to P1 and showing that it reduces to verifying the formula for P1 . Given a divisor D on X, let ΩX/k (D) be the subsheaf of Ωk(X)/k deﬁned by Γ(U, ΩX/k (D)) = {ω ∈ Ωk(X)/k  (ω) ∩ U ≥ D ∩ U } for U an open subset of X.
18.2. Serre duality
339
Let ω0 be a nonzero rational diﬀerential form, and let K = (ω0 ). Every rational diﬀerential form ω can be written as ω = f ω0 for some f ∈ k(X) and (ω) ∩ U ≥ D ∩ U if and only if (f ) ∩ U + (ω0 ) ∩ U ≥ D ∩ U , which holds if and only if f ∈ Γ(U, OX (K − D)). Thus ΩX/k (D) ∼ = ΩX (K − D) ∼ = ΩX/k ⊗ OX (−D). Let Ω(D) = Γ(X, ΩX/k (D)). We deﬁne a product ω, ! of diﬀerentials ω ∈ Ωk(X)/k and r´epartitions r ∈ R by the following formula: Resp (rp ω). ω, r! = p∈X
This formula is welldeﬁned since rp ω ∈ (ΩX/k )p for all but ﬁnitely many p ∈ X. The product has the following properties: a) ω, r! = 0 if r ∈ k(X). b) ω, r! = 0 if r ∈ R(D) and ω ∈ Ω(D). c) If f ∈ k(X), then f ω, r! = ω, f r!. Property a) follows from the residue formula (Proposition 18.7) and property b) follows since then rp ω ∈ (ΩX/k )p for all p ∈ X. For every ω ∈ Ωk(X)/k , let θ(ω) be the linear form on R deﬁned by θ(ω)(r) = ω, r!. If ω ∈ Ω(D), then θ(ω) ∈ J(D) by properties a) and b) since J(D) is the dual of R/(R(D) + k(X)). Lemma 18.8. Suppose that ω ∈ Ωk(X)/k is such that θ(ω) ∈ J(D). Then ω ∈ Ω(D). Proof. Suppose that ω ∈ Ω(D). Then there is a point p ∈ X such that νp (ω) < νp (D). Set n = νp (ω) + 1, and let r be the r´epartition deﬁned by
0 if q = p, rq = 1 where t is a regular parameter at p if q = p. tn We have νp (rp ω) = −1 so that Resp (rp ω) = 0 and ω, r! = 0. But n ≤ νp (D) so r ∈ R(D) (νq (0) = ∞). This is a contradiction since θ(ω) is assumed to vanish on R(D). Theorem 18.9 (Serre duality). For every divisor D, the map θ is a kvector space isomorphism from Ω(D) to J(D). Proof. Suppose that ω ∈ Ω(D) is such that θ(ω) = 0 in J(D). Then θ(ω) ∈ J(Δ) for all divisors Δ so ω ∈ Ω(Δ) for all divisors Δ by Lemma 18.8 so that ω = 0. Hence θ is injective.
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By property c), θ is a k(X)linear map from Ωk(X)/k to J. As Ωk(X)/k has dimension 1 and J has dimension ≤ 1 as k(X)vector spaces by Proposition 18.5, θ maps Ωk(X)/k onto J. Thus if α ∈ J(D), there exists ω ∈ Ωk(X)/k such that θ(ω) = α and Lemma 18.8 then shows that ω ∈ Ω(D). Corollary 18.10. Suppose that D is a divisor on X. Then h1 (X, OX (D)) = h0 (X, OX (KX − D)) where KX is a canonical divisor of X. Exercise 18.11. Prove the residue formula of Proposition 18.7 for X = P1 . Exercise 18.12. Strengthen the conclusions of Proposition 18.5 to show that dimk(X) J = 1.
18.3. The RiemannRoch theorem Theorem 18.13 (RiemannRoch theorem). Let D be a divisor on a nonsingular projective curve X of genus g. Then h0 (X, OX (D)) = h0 (X, OX (KX − D)) + deg D + 1 − g. Proof. The theorem follows from Theorem 18.2 and Serre duality (Corollary 18.10). Corollary 18.14. Suppose that X is a nonsingular projective curve of genus g. Then the degree of the canonical divisor is deg KX = 2g − 2. Proof. Take D = KX in the RiemannRoch theorem.
Corollary 18.15. Suppose that D is a divisor on a nonsingular projective curve X of genus g such that deg D > 2g − 2. Then h0 (X, OX (D)) = deg D + 1 − g. Proof. Since deg(KX − D) < 0, we have that h0 (X, OX (KX − D)) = 0 by Lemma 18.1. Corollary 18.16. Suppose that D is a divisor on a nonsingular projective curve X of genus g such that deg(D) > 0. Then h0 (X, OX (nD)) = n deg(D) + 1 − g for n >
2g−2 deg(D) .
18.3. The RiemannRoch theorem
341
Theorem 18.17. Suppose that X is a nonsingular projective curve. Then X∼ = P1 if and only if g(X) = 0. Proof. Theorem 17.14 implies that g(P1 ) = h1 (P1 , OP1 ) = 0. Suppose g(X) = 0 and p ∈ X is a point. Then h0 (X, OX (p)) = 2 by the RiemannRoch theorem (Theorem 18.13), Corollary 18.14, and Lemma 18.1. Now the complete linear system p consists of eﬀective divisors of degree equal to 1 = deg p by Corollary 13.19 and so X ∼ = P1 by Corollary 13.20. A nonsingular projective curve X is called an elliptic curve if g(X) = 1. Corollary 18.18. A nonsingular projective curve X is an elliptic curve if and only if KX ∼ 0. Proof. If g(X) = 1, then deg KX = 0 by Corollary 18.14. Since h0 (X, OX (KX )) = 1, we have KX ∼ 0 by Lemma 18.1. If KX ∼ 0, then g = 1 by Corollary 18.14
Theorem 18.19. Suppose that X is an elliptic curve and p0 ∈ X is a point. Then the map X → Cl0 (X) deﬁned by p → [p − p0 ] is a bijection. Proof. Suppose that D is a divisor of degree 0 on X. Then h0 (X, OX (KX − D − p0 )) = 0 since deg(KX − D − p0 ) = −1. By the RiemannRoch theorem, we then have that h0 (X, OX (D + p0 )) = 1. Thus there is a unique eﬀective divisor linearly equivalent to D + p0 which must be a single point p, since deg(D + p0 ) = 1. In particular, there exists a unique point p ∈ X such that p−p0 ∼ D from which the theorem follows. If D is a divisor on a nonsingular projective curve X and h1 (X, OX (D)) = 0, then the RiemannRoch theorem gives the dimension of h0 (X, OX (D)) but only gives a lower bound if h1 (X, OX (D)) = 0. The following theorem gives an upper bound for h0 (X, OX (D)) when D is eﬀective and h1 (X, OX (D)) = 0. The bound is sharp, and in fact the curves X and divisors D for which the upper bound is achieved are extremely special and are completely characterized (this is part of Cliﬀord’s original theorem). A proof of this characterization is given in [73, Theorem IV.5.4].
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Theorem 18.20 (Cliﬀord’s theorem). Suppose that D is a divisor on a nonsingular projective curve X such that h0 (X, OX (D)) > 0
and
h1 (X, OX (D)) > 0.
Then h0 (X, OX (D)) ≤
1 deg(D) + 1. 2
Proof. Let g = g(x). After possibly replacing D with a divisor linearly equivalent to D and KX with a divisor linearly equivalent to KX , we may assume that D ≥ 0 and D = KX − D ≥ 0. Further, we may assume that h0 (X, OX (D − p)) = h0 (X, OX (D)) for all p ∈ X since otherwise we can replace D with D − p and get a stronger inequality. We can then choose g ∈ Γ(X, OX (D)) = {f ∈ k(X)  (f ) + D ≥ 0} such that g ∈ Γ(X, OX (D − p)) for all p ∈ Supp(D ). Consider the klinear map φ : Γ(X, OX (D ))/Γ(X, OX ) → Γ(X, OX (KX ))/Γ(X, OX (D)) deﬁned by φ(f ) = f g, where bar denotes residue. The map φ is welldeﬁned, since for f ∈ Γ(X, OX (D )), (f g) ≥ −D − D = −KX and since k = Γ(X, OX ) = {f ∈ k(X)  (f ) ≥ 0} by Theorem 3.35 and Lemma 13.3, so we have that (gf ) + D ≥ 0 if f ∈ Γ(X, OX ). Suppose φ(f ) = 0 for some f ∈ Γ(X, OX (D )). Then (f ) + D ≥ 0 so if p ∈ Supp(D ), then νp (f ) ≥ 0. Suppose p ∈ Supp(D ). Then νp (g) = −νp (D) by our choice of g. Since (gf ) + D ≥ 0, we have νp (f g) ≥ −νp (D) and so νp (f ) ≥ −νp (D) − νp (g) = 0. Thus νp (f ) ≥ 0 for all p ∈ X and so f ∈ Γ(X, OX ) and we have that φ is injective. Thus (18.8)
h0 (X, OX (D )) − 1 ≤ g − h0 (X, OX (D)).
By the RiemannRoch theorem, (18.9)
h0 (X, OX (D )) = deg(D ) + 1 − g + h0 (X, OX (KX − D )) = g − 1 − deg(D) + h0 (X, OX (D))
since 2g − 2 = deg KX = deg D + deg D . Combining equations (18.8) and (18.9), we obtain the conclusions of the theorem.
18.4. The RiemannRoch problem on varieties
343
Theorem 18.21. Let D be a divisor on a nonsingular projective curve X of genus g. Then: 1) If deg D ≥ 2g, then D is base point free. 2) If deg D ≥ 2g + 1, then D is very ample, so that the regular map 0 (X,O
φD : X → Ph
X (D))
is a closed embedding. Proof. Conclusion 1) of this theorem follows from Corollary 18.15, which tells us that h0 (X, OX (D − p)) = h0 (X, OX (D)) − 1 for all p ∈ X, and 1) of Corollary 13.34. Conclusion 2) follows from Corollary 18.15, which shows that h0 (X, OX (D − p − q)) = h0 (X, OX (D)) − 2 for all p, q ∈ X, and 3) of Corollary 13.34.
Theorem 18.22. Suppose that X is a nonsingular projective curve. Then KX is ample if g(X) > 1, KX ∼ 0 if X is an elliptic curve (g(X) = 1), and −KX is ample if X ∼ = P1 (g(X) = 0). Proof. This follows from Corollary 18.14, Theorem 18.21, Theorem 18.17, and Corollary 18.18. Theorem 18.22 generalizes to the theory of Kodaira dimension for higherdimensional varieties. This is especially worked out in the classiﬁcation of surfaces [18]. Some papers on the theory in higher dimensions are [91], [112], [113], [114], and [20].
18.4. The RiemannRoch problem on varieties From Theorems 18.21, 17.18, and 17.35 we obtain the following theorem. Theorem 18.23. Suppose that D is a divisor on a nonsingular projective curve X such that deg D > 0. Then Γ(X, OX (nD)) R[D] = n≥0
is a ﬁnitely generated kalgebra. Thus Corollary 18.16 is not so surprising, since h0 (X, OX (nD)) is the Hilbert function of R[D]. However, it may be that R[D] is not generated in degree 1, so just knowing that R[D] is a ﬁnitely generated kalgebra is not enough to conclude that its Hilbert function is eventually a polynomial. We do have that the Hilbert function of a ﬁnitely generated graded kalgebra is
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eventually a quasipolynomial, which has an expression P (n) = ad (n)nd + ad−1 (n)nd−1 + · · · + a0 (n) where the coeﬃcients ai (n) are periodic functions. The RiemannRoch problem is to compute the function PD (n) = h0 (X, OX (nD)) for large n where D is a divisor on a nonsingular projective variety X. It will follow from Theorem 19.1 that χ(OX (nD)) is a polynomial in n. Thus if D is ample, we have that PD (n) is a polynomial for n 0, as PD (n) = χ(OX (nD)) for n 0 by Theorem 17.18. If D is a divisor of degree 0 on a nonsingular projective curve X, then h0 (X, OX (nD)) > 0 if and only if nD ∼ 0 by Lemma 18.1. We thus have the following complete solution to the RiemannRoch problem on a curve. Theorem 18.24. Suppose that X is a nonsingular projective curve and D is a divisor on X. Then for n 0, ⎧ ⎨ n deg D + 1 − g(X) 0 a periodic function in n h (X, OX (nD)) = ⎩ 0
if deg D > 0, if deg D = 0, if deg D < 0.
There are examples of eﬀective divisors D on a nonsingular projective surface S such that R[D] = n≥0 Γ(X, OX (nD)) is not a ﬁnitely generated kalgebra. This was shown by Zariski in [159]; we will construct Zariski’s example in Theorem 20.14. It may thus be expected that (the sometimes not ﬁnitely generated kalgebra) R = n≥0 Γ(S, OS (nD)) will not always have a good Hilbert function, that is, that h0 (S, OS (nD)) will not be polynomiallike. However, Zariski showed in [159] that this function is almost a polynomial on a surface. Theorem 18.25 (Zariski). Let D be an eﬀective divisor on a nonsingular projective surface S over an algebraically closed ﬁeld k. Then there exists a quadratic polynomial P (n) and a bounded function λ(n) such that h0 (S, OS (nD)) = P (n) + λ(n). for n ≥ 0. In this same paper, Zariski asked if λ(n) is always eventually a periodic function of n (a periodic function in n for n 0). This question is answered in [44].
18.5. The Hurwitz theorem
345
Theorem 18.26. Let D be an eﬀective divisor on a nonsingular projective surface S. Let λ(n) be the function of Theorem 18.25. Then: 1. If k has characteristic 0 or is the algebraic closure of a ﬁnite ﬁeld, then λ(n) is eventually a periodic function. 2. There are examples where λ(n) is not eventually periodic if k is of positive characteristic and is not the algebraic closure of a ﬁnite ﬁeld. Proof. Cutkosky and Srinivas [44, Theorems 2 and 3 and Example 3].
While the function h0 (X, OX (nD)) is almost a polynomial function when X is a surface, the behavior of the function h0 (X, OX (nD)) in higher dimensions can be much more complicated. Example 18.27. Over any algebraically closed ﬁeld k, there exists a nonsingular projective 3fold X and an eﬀective divisor D on X such that h0 (X, OX (nD)) n→∞ n3 is an irrational number. In particular, h0 (X, OX (nD)) is not eventually a polynomiallike function. lim
Proof. Cutkosky and Srinivas [44, Example 4].
The volume of an invertible sheaf L on a ddimensional projective variety X is deﬁned as h0 (X, Ln ) . Vol(L) = lim sup n→∞ nd /d! The volume always exists as a limit over an algebraically closed ﬁeld k (by Lazarsfeld [98] and Lazarsfeld and Mustat¸˘a [99]) and over an arbitrary ﬁeld (by Cutkosky [42]) but can be an irrational number (by Example 18.27). Exercise 18.28. Find an example of a divisor D on a nonsingular projective curve such that D is not base point free but n0 D is base point free for some positive multiple n0 . Exercise 18.29. Give an example of a ﬁnitely generated graded kalgebra such that its Hilbert function is not eventually a polynomial.
18.5. The Hurwitz theorem Suppose that φ : X → Y is a dominant regular map of nonsingular projective curves. Recall that φ is then ﬁnite (Corollary 10.26). Suppose that P ∈ X. The ramiﬁcation index eP of φ at P is deﬁned as follows. Let Q = φ(P ). Recall that the valuation νQ is a valuation of k(Y ) whose valuation ring is
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18. Curves
OY,Q and νP is a valuation of k(X) whose valuation ring is OX,P . Thus νP is an extension of νQ to k(X). Let x be a regular parameter in OX,P and y be a regular parameter in OY,Q . Then y = uxeP for some unit u ∈ OX,P and positive integer eP . The number eP is called the ramiﬁcation index of νP over νQ or the ramiﬁcation index of P over Q. Since y = 0 is a local equation for the divisor Q on Y , we have that eP P, φ∗ (Q) = P ∈φ−1 (Q)
and by Theorem 13.18, eP = deg(φ∗ (Q)) = deg(φ) = [k(X) : k(Y )] P ∈φ−1 (Q)
does not depend on Q. We will say that φ is ramiﬁed at P if eP > 1, tamely ramiﬁed at P if the characteristic p of k does not divide eP , and wildly ramiﬁed at P if p divides eP . We can then consider the set of all ramiﬁcation points of φ in X. We will say that a dominant regular map φ : X → Y of varieties is separable if the induced extension of ﬁelds k(Y ) → k(X) is ﬁnite and separable. Proposition 18.30. Suppose that φ : X → Y is a ﬁnite regular map of nonsingular curves and that φ is separable. Then there is an exact sequence of OX modules (18.10)
0 → φ∗ ΩY /k → ΩX/k → ΩX/Y → 0.
Proof. By formula (14.4), the sequence (18.10) is right exact, so we need only show that the map (18.11)
φ∗ ΩY /k → ΩX/k
is injective. Since φ∗ ΩY /k and ΩX/k are invertible sheaves of OX modules, we need only show that the map (18.11) is nonzero. Tensoring over OX with k(X), we reduce by Lemma 14.8 to showing that the natural map Ωk(Y )/k ⊗k(Y ) k(X) → Ωk(X)/k is nonzero, which will follow if the natural map (18.12)
Ωk(Y )/k → Ωk(X)/k
is nonzero. The ﬁeld k(Y ) is separably generated over the algebraically closed ﬁeld k (by Theorem 1.14). Let z ∈ k(Y ) be a transcendental element over k such that k(Y ) is separable over k(z). Then k(X) is separable over
18.5. The Hurwitz theorem
347
k(z), so z is also a separable transcendence basis of k(X) over k. By Theorem 21.75, dK(Y )/k (z) is a generator of Ωk(Y )/k and dk(X)/k (z) is a generator of Ωk(X)/k . Thus (18.12) is an injection, and Ωk(Y )/k ⊗k(Y ) k(X) → Ωk(X)/k is nonzero. Suppose that P ∈ X. Let Q = φ(P ). Let x be a regular parameter in OX,P and y be a regular parameter in OY,Q . Taking stalks at P in (18.10) gives us (by Proposition 14.15) the short exact sequence (18.13) We deﬁne
0 → OX,P dy → OX,P dx → (ΩX/Y )P → 0. dy ∈ OX,P dx
by dy dx. dx where u is a unit in OX,P . Since d is a derivation, dy =
We have that y = uxeP we have that dy = eP uxeP −1 dx + xeP du.
Now du = adx for some a ∈ OX,P , so dy = (ep uxep −1 + axep )dx. We thus obtain the following proposition. Proposition 18.31. Let φ : X → Y be a separable ﬁnite regular map of nonsingular curves. Then: 1. The support of ΩX/Y is the ﬁnite set of ramiﬁcation points of φ in X. 2. For each P ∈ X, (ΩX/Y )P is a cyclic OX,P module (generated by dy ). one element) of kdimension equal to νP ( dx 3. If φ is tamely ramiﬁed at P , then dimk (ΩX/Y )P = eP − 1. 4. If φ is wildly ramiﬁed at P , then dimk (ΩX/Y )P > eP − 1. Let DX/Y be the ideal sheaf in OX which is the annihilator of ΩX/Y . Let R be the eﬀective divisor such that DX/Y = OX (−R). We then have that DX/Y is the annihilator of ΩX/Y ⊗ Ω−1 X/k . Tensoring (18.10) with the invertible sheaf Ω−1 X/k , we obtain the short exact sequence
−1 0 → φ∗ ΩY /k ⊗ Ω−1 X/k → OX → ΩX/Y ⊗ ΩX/k → 0
so that
∼ φ∗ ΩY /k ⊗ Ω−1 X/k = OX (−R)
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and
∼ OR = OR /OX (−R) ∼ = ΩX/Y ⊗ Ω−1 X/k = ΩX/Y
since ΩX/Y has ﬁnite support. Thus dimk (ΩX/Y )P P. (18.14) R= p∈X
Taking degrees of divisors in ∼ O (φ∗ (KY ) − KX ), OX (−R) ∼ = φ∗ ΩY /k ⊗ Ω−1 X/k = X we have that deg R = deg KX − deg φ∗ (KY ) = deg KX − deg(φ) deg KY = (2g(X) − 2) − deg(φ)(2g(Y ) − 2) by Theorem 13.18 and Corollary 18.14. We thus have the following theorem. Theorem 18.32 (Hurwitz). Let φ : X → Y be a dominant separable regular map of nonsingular projective curves. Then 2g(X) − 2 = deg(φ)(2g(Y ) − 2) + deg(R), where R is the ramiﬁcation divisor (18.14). If φ has only tame ramiﬁcation, then (eP − 1). deg(R) = P ∈X
In the case that X and Y are aﬃne, with coordinate rings A = k[Y ] and B = k[X], the annihilator Γ(X, DX/Y ) = Γ(X, OX (−R)) of Γ(X, ΩX/Y ) = ΩB/A is the diﬀerent DB/A ([135, Proposition 14]). The diﬀerent is deﬁned in [135, Chapter III] and [160, Section 11 of Chapter V], using the trace of the quotient ﬁeld of B over A. Proposition 18.31 is proven in [160, Theorem 28 of Section 11, Chapter V] and [135, Proposition 13]. On [160, page 312], a derivation of “Hilbert’s formula” is given to compute deg(R) in the case of a Galois extension, even in the presence of wild ramiﬁcation.
18.6. Inseparable maps of curves Recall that a dominant regular map φ : X → Y of varieties is separable if the induced ﬁeld extension k(Y ) → k(X) is ﬁnite and separable. We will say that φ : X → Y is inseparable if k(Y ) → k(X) is not separable and that φ : X → Y is purely inseparable if k(Y ) → k(X) is purely inseparable. If K → L is a ﬁnite ﬁeld extension, then there exists a (unique) intermediate ﬁeld M (called the separable closure of K in L) such that L is purely inseparable over M and M is separable over K (Theorem 1.15). It
18.6. Inseparable maps of curves
349
follows that any dominant ﬁnite regular map of algebraic varieties factors as a purely inseparable ﬁnite map, followed by a separable ﬁnite map. Suppose that κ is a perfect ﬁeld of characteristic p > 0 and R is a κalgebra. Let F r : R → R be the Frobenius homomorphism, deﬁned by F r(x) = xp for x ∈ R. The map F r is a ring homomorphism but it is not a κalgebra homomorphism. Let Rp be the ring R with the κalgebra structure · given by a · x = ap x for a ∈ κ and x ∈ R. Then F r : R → Rp is a κalgebra homomorphism. Since Rp is equal to R as a ring, Rp is a domain if and only if R is a domain, Rp is normal if and only if R is normal, and Rp is regular if and only if R is regular. Now suppose that R is also a domain with quotient ﬁeld K. We can express R as a κalgebra by R = κ[S] for some subset S of K. Let Ω be an 1 algebraic closure of K. Deﬁne Λ : Rp → Ω by Λ(f ) = f p . For a ∈ κ and x ∈ Rp , we have that 1
Λ(a · x) = Λ(ap x) = ax p = aΛ(x), so Λ is a κalgebra homomorphism, which identiﬁes Rp with the κsubalgebra 1
Λ(Rp ) = κ[S p ] of Ω (we have that Λ(κ) = κ as κ is perfect). The composition ΛF r(x) = x for x ∈ R, so F r : R → Rp is identiﬁed with the natural inclusion of κalgebras (18.15)
1
κ[S] → κ[S p ]. 1
In particular, the quotient ﬁeld of Rp is identiﬁed with K p as a κalgebra. Now suppose that X is an aﬃne variety over an algebraically closed ﬁeld k of characteristic p > 0. Let R = k[X]. The above construction gives us a ﬁnitely generated kalgebra Rp which is a domain and a kalgebra homomorphism F r : R → Rp . Thus there is an aﬃne variety Xp and a regular map F : Xp → X such that F ∗ = F r (by Proposition 2.40). If X is a quasiprojective variety, we can apply the above construction on an aﬃne open cover of X to obtain by Proposition 3.39 a quasiprojective variety Xp with regular map F : Xp → X. (If X is embedded in Pn , then Xp is embedded in Pnp which is isomorphic to Pn as a variety over k.) Applying 1 the construction (18.15) to F r : k(X) → k(X), we see that k(Xp ) ∼ = k(X) p 1
and F ∗ : k(X) → k(Xp ) is the natural inclusion k(X) ⊂ k(X) p . The regular map F : Xp → X is called the klinear Frobenius map. If X is normal, then Xp is also normal, since Xp has an aﬃne cover by normal varieties. Theorem 18.33. Suppose that X is a variety of dimension n. Then 1
[k(X) p : k(X)] = pn .
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18. Curves
Proof. An algebraic function ﬁeld over a perfect ﬁeld k is separably generated over k (by Theorem 1.14). The theorem then follows from 2) of Theorem 21.76 since trdegk k(X) = dim X = n. Theorem 18.34. Suppose that f : X → Y is a ﬁnite purely inseparable regular map of nonsingular projective curves. Then f is a composition of klinear Frobenius maps. In particular, g(X) = g(Y ). Proof. Let the degree of f be [k(X) : k(Y )] = pr . Suppose that g ∈ k(X). i The minimal polynomial of g over k(Y ) is z p −h for some h ∈ k(Y ) and i ∈ N with i ≤ r since g is purely inseparable over k(Y ) and [k(Y )[g] : k(Y )] divides 1 r pr . Thus k(X)p ⊂ k(Y ) so k(X) ⊂ k(Y ) pr . Let F be the composition of klinear Frobenius maps F
F
Ypr → Ypr−1 → · · · → Yp → Y where Ypi = (Ypi−1 )p . Here F has degree pr by Theorem 18.33. Since 1
1
k(X) ⊂ k(Y ) pr and both k(X) and k(Y ) pr have the same degree over 1 k(Y ), we have that k(X) = k(Y ) pr . Since a nonsingular projective curve is uniquely determined by its function ﬁeld (by Corollary 10.25), we have that X∼ = Ypr , and thus f = F . Let U = {Ui } be an aﬃne open cover of Y with corresponding aﬃne open cover V = {Vi } of Ypr . Now each Γ(Ui , OY ) is isomorphic to Γ(Vi , OYpr ) as ˇ a ring (but not as a kalgebra) and the Cech complexes C ∗ (U , OY ) and C ∗ (V , OYpr ) are isomorphic complexes of rings. Thus the cohomology is isomorphic, so H 1 (Ypr , OYpr ) is H 1 (Y, OY ) with the vector space operation r a · v = ap v for a ∈ k. Since k is perfect, we have that h1 (Y, OY ) = dimk H 1 (Y, OY ) = dimk H 1 (Ypr , OYpr ) = h1 (Ypr , OYpr ) and g(Y ) = g(Ypr ).
Exercise 18.35. Let Pn be projective space over an algebraically closed ﬁeld k of characteristic p > 0. Show that (Pn )p is isomorphic to Pn (as varieties over k). Exercise 18.36. Suppose that f : X → Y is a ﬁnite regular map of nonsingular projective curves. Show that g(X) ≥ g(Y ). Exercise 18.37 (L¨ uroth’s theorem). Suppose that k is an algebraically closed ﬁeld and L is a subﬁeld of a onedimensional rational function ﬁeld k(t) over k such that L contains k and is not equal to k. Show that L is a onedimensional rational function ﬁeld over k. Exercise 18.38. Give an example of a ﬁnite purely inseparable regular map of nonsingular projective surfaces which is not a composition of Frobenius maps.
18.7. Elliptic curves
351
18.7. Elliptic curves Recall that a nonsingular projective curve X is called an elliptic curve if it has genus g(X) = 1. An elliptic curve is characterized by KX ∼ 0 by Corollary 18.18. The theory of elliptic curves is particularly remarkable and extensive. We give a brief introduction here. The group of regular isomorphisms of a variety X will be denoted by Aut(X). Every nonsingular cubic curve X in P2 is an elliptic curve. This follows since by adjunction, Theorem 14.21, OX (KX ) ∼ = OP2 (KP2 + X) ⊗ OX and since KP2 = −X by Example 14.20. The reader should peruse the deﬁnitions and statements of Section 21.7 on the Galois theory of varieties before reading the proofs of this section. Lemma 18.39. Suppose that X is an elliptic curve and P, Q are two not necessarily distinct points in X. Then there exists a regular automorphism σ : X → X such that σ 2 = id, σ(P ) = Q, and for any R ∈ X, R + σ(R) ∼ P + Q. Proof. We have that h0 (X, OX (P +Q)) = 2 by Corollary 18.15 and P +Q is base point free by Theorem 18.21. We thus have a regular map φ = φP +Q : X → P1 . A linear hyperplane section H on P1 is a point and φ∗ (H) is an eﬀective divisor linearly equivalent to P + Q by Lemma 13.28. Thus deg(φ) = [k(X) : k(P1 )] = 2 by Theorem 13.18. The ﬁeld extension k(X)/k(P1 ) is separable by Theorem 18.34, since otherwise g(x) = g(P1 ) = 0. Thus k(X) is a Galois extension of k(P1 ), so X is Galois over P1 by Theorem 21.69, with G(X/P1 ) ∼ = Z2 by Proposition 21.67. Let σ ∈ G(X/P1 ) be a generator. Since X is Galois over P1 , σ interchanges the two points of a ﬁber. There exists S ∈ P1 such that φ∗ (S) = P + Q by Lemma 13.28, so σ(P ) = Q. We have that F = φ∗ (S) = X, F ∈P +Q
S∈P1
so for any R ∈ X, R + σ(R) ∈ P + Q, and thus R + σ(R) ∼ P + Q.
Corollary 18.40. The group Aut(X) of regular automorphisms of an elliptic curve X is transitive on X. Lemma 18.41. Suppose that φ1 : X → P1 and φ2 : X → P1 are two regular maps of degree 2 from an elliptic curve X to P1 . Then there exist automorphisms σ ∈ Aut(X) and τ ∈ Aut(P1 ) such that φ2 σ = τ φ1 . Proof. Let P1 ∈ X be a ramiﬁcation point of φ1 and P2 ∈ X be a ramiﬁcation point of φ2 (which exist by Theorem 18.32). By Corollary 18.40 there is σ ∈ Aut(X) such that σ(P1 ) = P2 . Since P1 is a ramiﬁcation point of the
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18. Curves
degree 2 map φ1 and h0 (X, OX (2P1 )) = 2, we have that φ1 = φ2P1  , and since P2 is a ramiﬁcation point of the degree 2 map φ2 , φ2 = φ2P2  . Since σ takes P1 to P2 , φ1 and φ2 σ are induced by base point free linear systems which are contained in 2P1 . But 2P1  is the only such linear system. Thus φ1 and φ2 σ are induced by the same linear system, so they diﬀer only by an automorphism of P1 . Proposition 18.42. Suppose that X is an elliptic curve over an algebraically closed ﬁeld k of characteristic = 2 and let P0 ∈ X be a point. Then there is a closed embedding φ : X → P2 such that the image is the curve with the homogeneous equation (18.16)
x2 x21 − x0 (x0 − x2 )(x0 − λx2 ) = 0
for some λ ∈ k \ {0, 1} and φ(P0 ) = (0 : 1 : 0). The aﬃne equation of the image φ(X)\{(0 : 1 : 0)} of X \P0 in P2x2 ∼ = A2 is (18.17)
y 2 = x(x − 1)(x − λ),
where x = xx02 , y = xx12 . We think of P0 as being the “point at inﬁnity” on X under this embedding, since φ(P0 ) = (0 : 1 : 0) = φ(X) ∩ Z(x2 ). Proof. We have that h0 (X, OX (nP0 )) = n for n > 0 by the RiemannRoch theorem. The linear system 3P0  gives a closed embedding φ = φ3P0  of X into P2 by Theorem 18.21. Within the function ﬁeld k(X), we have inclusions k = Γ(X, OX ) = Γ(X, OX (P0 )) ⊂ Γ(X, OX (2P0 )) ⊂ · · · . Choose x ∈ k(X) so that 1, x are a basis of Γ(X, OX (2P0 )) and choose y ∈ k(X) so that 1, x, y are a basis of Γ(X, OX (3P0 )). Since h0 (X, OX (6P0 )) = 6, there is a linear relation between the seven functions 1, x, y, x2 , xy, x3 , y 2 ∈ Γ(X, OX (6P0 )). Further, x3 and y 2 must both have nonzero coeﬃcients in this relation, since otherwise the relation will have a pole of ﬁnite order at P0 , as x has a pole of order 2 at P0 and y has a pole of order 3 at P0 . (1 has no pole at P0 and xy has a pole of order 5 at P0 .) Replacing x and y by suitable scalar multiples, we may assume that we have a relation y 2 + b1 xy + b2 y = f (x) where f (x) is a degree 3 monic polynomial in x and b1 , b2 ∈ k. Completing the square in y by replacing y with 1 y = y + (b1 x + b2 ), 2
18.7. Elliptic curves
353
we obtain the relation y 2 = g(x)
(18.18)
where g(x) = x3 + a1 x2 + a2 x + a3 for some a1 , a2 , a3 ∈ k (this is where we need characteristic = 2). ( xx20
We represent the closed embedding φ : X → P2 by φ = (x : y : 1) = : xx12 : 1). The relation (18.18) becomes
(18.19)
x21 x2 = x30 + a1 x20 x2 + a2 x0 x22 + a3 x32 .
Thus φ(X) ⊂ Z(F ) where F = x21 x2 − x30 − a1 x20 x22 − a2 x0 x2 − a3 x32 . The image of φ is a closed irreducible curve φ(X), which has codimension 1 in P2 . Since F is irreducible in the coordinate ring k[x0 , x1 , x2 ] of P2 , (F ) = I(φ(X)). The regular functions on the aﬃne open subset P2 ∼ = A2 of P2 are k[P2x2 ] = k[x, y] where x = by f = y 2 − g(x).
x0 x2 , y
=
x1 x2 .
x2
The ideal of φ(X) ∩ P2x2 is generated
Since φ(X) is nonsingular, g(x) can have no multiple roots (by the Jacobian criterion of Proposition 10.14). Thus we can make a change of variables x = αx + β for some α = 0, β ∈ k, and replace y with a scalar multiple of y to obtain an expression (18.18) with g(x) = x(x − 1)(x − λ) for some λ ∈ k with λ = 0 or 1. Finally, we see that the set of points at inﬁnity on φ(X) is the algebraic set Z(x2 ) ∩ φ(X) = Z(x2 , x30 ) = {(0 : 1 : 0)}. Since x has a pole of order 2 at P0 and y has a pole of order 3 at P0 , 1 x (P0 ) : 1 : (P0 ) = (0 : 1 : 0). φ(P0 ) = y y We can regard P1 as P1 = A1 ∪{∞}, with k[A1 ] = k[z] and k(P1 ) = k(z). The group of automorphisms of P1 consists of the linear automorphisms (Exercise 13.47), so they can be represented as fractional linear transformations az + b cz + d with a, b, c, d ∈ k and ad − bc = 0. The corresponding transformation in homogeneous coordinates is (ax0 + bx1 : cx0 + dx1 ).
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18. Curves
Let G be the subgroup of Aut(P1 ) consisting of the automorphisms which permute {0, 1, ∞}. Then G ∼ = S3 with
1 1 z z−1 (18.20) G = z, , 1 − z, , , . z 1−z z−1 z We have that the group G is generated by
1 z
and 1 − z.
Suppose that X is an elliptic curve with char k = 2 and P0 ∈ X. Consider the linear system 2P0  which gives a regular map Ψ = φ2P0  : X → P1 which is Galois of degree 2 (as we saw in the proof of Lemma 18.39). By Hurwitz’s theorem (Ψ is tamely ramiﬁed since char k = 2), Ψ is ramiﬁed over four points: a, b, c, d ∈ P1 with Ψ(P0 ) = d. There exists a unique linear automorphism τ of P1 which takes d to ∞, a to 0, and b to 1. Let λ be the image of c. Then τ Ψ is ramiﬁed over 0, 1, λ, ∞. Deﬁne the j invariant of X as j(λ) = 28
(λ2 − λ + 1)3 . λ2 (λ − 1)2
We have (we need only check the generators that (18.21)
j(σ(λ)) = j(λ)
1 z
and 1 − z of G in (18.20))
for σ ∈ G.
Write P2 = A2 ∪ H where H = Z(x2 ) is “the hyperplane at inﬁnity”. The closed embedding φ = φ3P0  of Proposition 18.42 gives an expression of X as (isomorphic) to the union of the aﬃne curve C = Z(y 2 − x(x − 1)(x − λ)) ⊂ A2 and the point “at inﬁnity” P0 = (0 : 1 : 0). The degree 2 map Ψ = φ2P0  is the linear projection to P1 which takes (a, b) ∈ C to a ∈ A1 and P0 to ∞. Theorem 18.43. Suppose that k is an algebraically closed ﬁeld of characteristic not equal to 2. For λ1 , λ2 ∈ k \ {0, 1}, if X1 is an elliptic curve which gives λ1 in the above construction of λ and X2 is an elliptic curve which gives λ2 , then X1 is isomorphic to X2 if and only if j(λ1 ) = j(λ2 ). Further, every element of k is the j invariant of some elliptic curve X. Proof. We will ﬁrst show that j(λ) is uniquely determined by an elliptic curve X. Suppose P1 , P2 ∈ X and Ψ1 : X → P1 is induced by 2P1  so that the ramiﬁcation points of Ψ1 in P1 are 0, 1, λ1 , ∞ with Ψ1 (P1 ) = ∞ and Ψ2 : X → P1 is induced by 2P2  so that the ramiﬁcation points of Ψ2 in P1 are 0, 1, λ2 , ∞ with Ψ2 (P2 ) = ∞. By Lemma 18.41 and its proof, there exist automorphisms σ ∈ Aut(X) and τ ∈ Aut(P1 ) such that Ψ2 σ = τ Ψ1 with σ(P1 ) = P2 so that τ (∞) = ∞ and τ sends the other ramiﬁcation
18.7. Elliptic curves
355
points {0, 1, λ1 } to {0, 1, λ2 } in some order. Let γ(z) be the fractional linear transformation of P1 deﬁned by γ(z) =
z − τ (0) . τ (1) − τ (0)
Then γτ (0) = 0, γτ (1) = 1, and γτ (∞) = ∞, so γτ is the identity map. Thus τ (λ1 ) − τ (0) . λ1 = γτ (λ1 ) = τ (1) − τ (0) Since the sets {τ (0), τ (1), τ (λ1 )} and {0, 1, λ2 } are equal, we have that
λ1 − 1 1 1 λ1 , , λ2 ∈ λ1 , , 1 − λ1 , , λ1 1 − λ1 λ1 − 1 λ1 and so j(λ1 ) = j(λ2 ) by (18.21). Now suppose that X1 and X2 are two elliptic curves, giving λ1 and λ2 , respectively, and such that j(λ1 ) = j(λ2 ). The regular map j : A1 \ {0, 1} → A1 extends to a regular map j = (j : 1) : P1 → P1 with j −1 (∞) = {0, 1, ∞}, which induces j ∗ : k(P1 ) = k(j) → k(P1 ) = k(λ) deﬁned by j → 28
(λ2 − λ + 1)3 . λ2 (λ − 1)2
For j0 ∈ A1 , 28 (λ2 − λ + 1)3 − j0 λ2 (λ − 1)2 = 0 is an equation of degree 6 in λ, so counting multiplicities, it has six roots. Thus 6 = deg(j ∗ (j0 )) = [k(λ) : k(j)] by Theorem 13.18. By (18.21), substituting λ for z, G acts on k(λ) by kautomorphisms which leave k(j) invariant. Since [k(λ) : k(j)] = 6 and G = 6, we have that k(λ) is Galois over k(j) with Galois group G. Thus j : P1 → P1 is Galois with Galois group G (Theorem 21.69), and so j(λ1 ) = j(λ2 ) if and only if there exists τ ∈ G such that τ (λ1 ) = λ2 . By Proposition 18.42, X1 and X2 can be embedded in P2 with respective aﬃne equations (18.22)
y 2 = x(x − 1)(x − λ1 )
and (18.23)
y 2 = x(x − 1)(x − λ2 ).
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18. Curves
Since j(λ1 ) = j(λ2 ), there exists τ ∈ G such that τ (λ1 ) = λ2 , and thus τ permutes 0, 1, ∞. Let Ψ1 : X1 → P1 be the 21 cover induced by projection onto the xaxis in (18.22). Then τ Ψ1 : X1 → P1 is a degree 2 map which is ramiﬁed over 0, 1, λ2 , ∞ in P1 . Let Q = (τ Ψ1 )−1 (∞). Then x ∈ H 0 (X1 , OX1 (2Q)) where τ Ψ1 = (x : 1). Proceeding as in the proof of Proposition 18.42, we ﬁnd y ∈ H 0 (X1 , OX1 (3Q)), giving the relation y 2 = g(x) of (18.18) and such that φ3Q = (x : y : 1) is a closed embedding of X1 into P2 . The points in X1 ⊂ P2 where τ Ψ1 : X1 → P1 is ramiﬁed are Q and the points in X1 ∩ A2 where y = 0. Since τ Ψ1 is ramiﬁed over 0, 1, λ2 , and ∞ and g is monic of degree 3, we see that g(x) = x(x − 1)(x − λ2 ). Thus X1 is isomorphic to the cubic curve with aﬃne equation (18.23), so X1 is isomorphic to X2 . Now given j0 ∈ k, we can solve the equation 28 (λ2 − λ + 1)3 − j0 λ2 (λ − 1)2 = 0 to ﬁnd a solution λ0 ∈ k, which cannot be 0 or 1. The elliptic curve with aﬃne equation y 2 = x(x − 1)(x − λ0 ) deﬁnes a nonsingular cubic curve of degree 3 in P2 which is an elliptic curve that has j0 as its j invariant. Let X be an elliptic curve with a ﬁxed point P0 ∈ X. By Theorem 18.19, the map P → [P − P0 ] is a bijection from X to Cl0 (X). This induces a group structure on X with P0 as the zero element and with addition ⊕ deﬁned by P ⊕ Q = R if and only if P + Q ∼ R + P0 as divisors on X. Proposition 18.44. Suppose that X is an elliptic curve with the group structure given as above by the choice of a point P0 ∈ X. Then the addition map X × X → X and the inverse map X → X are regular maps. Proof. We will denote the addition of P and Q in X by P ⊕ Q and the inverse of P by "P . By Lemma 18.39, taking P = Q = P0 , there is an automorphism σ of X such that for any R ∈ X, R + σ(R) ∼ 2P0 . Thus "R = σ(R), and so the inverse map " is a regular map. Let P ∈ X. By Lemma 18.39, there is an automorphism τ of X such that R + τ (R) ∼ P + P0 for all R ∈ X. Thus P " R = τ (R), and since " is a regular map, we have that translation R → R ⊕ P is a regular map for ﬁxed P ∈ X. Embed X into P2 by φ3P0  . Let F = 0 be the homogeneous cubic equation of X in P2 . If L is a linear form on P2 , then L intersects X in three points with multiplicity, considering the restriction of F to L as a degree
18.7. Elliptic curves
357
3 form on L ∼ = P1 . (This is a special case of B´ezout’s theorem, which we will prove later in Theorem 19.20.) Thus we have a map λ : X × X → X obtained by letting the image of (P, Q) be the third point of intersection of the line through P and Q with X (if P = Q, the line through P is required to be tangent to X at p). We will establish that this map is in fact regular everywhere. It will follow that addition, (P, Q) → P ⊕ Q, is a regular map, since P + Q + λ(P, Q) ∼ 3P0 and P + Q ∼ (P ⊕ Q) + P0 , so (P ⊕ Q) ⊕ λ(P, Q) = P0 and thus P ⊕ Q = "λ(P, Q). Given P, Q ∈ X, there exists a linear form H of P2 such that all three intersection points of the line through P and Q with X lie in P2H . Thus we are reduced to showing that if f = 0 is the equation of C = X ∩ P2H in P2H ∼ = A2 and if P, Q are points of C such that the line through P and Q in C has three intersections with C in A2 (counting multiplicity), then the rational map λ is regular near (P, Q). We now consider P = (α, β) and Q = (γ, δ) as variable points in A2 . The line through P and Q in A2 can be parameterized as x = α + t(γ − α),
y = β + t(δ − β).
The intersection points of this line with C are obtained from the solutions in t to g(t) = f (α + t(γ − α), β + t(δ − β)) = 0. Write g(t) = at3 +bt2 +ct+d with a, b, c, d in the polynomial ring k[α, β, γ, δ]. We now constrain P = (α, β) and Q = (γ, δ) to lie on C. Thus we consider the residues a, b, c, d of a, b, c, d in R = k[α, β, γ, δ]/(f (α, β), f (γ, δ)) (which is a domain by Proposition 5.7). Let α, β, γ, δ be the residues of α, β, γ, δ in R, so that R = k[α, β, γ, δ]. Let g(t) = at3 + bt2 + ct + d be the residue of g(t) in R[t]. We have that g(0) = f (α, β) = 0 and g(1) = f (γ, δ) = 0. Thus d = 0 and a + b + c = 0, and we have a factorization g(t) = t(t − 1)(at + (a + b)). We see that if a = 0, then λ is the regular map deﬁned by λ ((u1 , v1 ) × (u2 , v2 )) a+b a+b (u1 , v1 , u2 , v2 )(u2 − u1 ), v1 − (u1 , v1 , u2 , v2 )(v2 − v1 ) . = u1 − a a In the language of [146], (α, β) and (γ, δ) in the above proof are “independent generic points”.
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18. Curves
Lemma 18.45 (Rigidity lemma). Let X be a projective variety, Y and Z be quasiprojective varieties, and f : X × Y → Z be a regular map such that for some Q ∈ Y , f (X × {Q}) = P is a single point of Z. Then there is a regular map g : Y → Z such that if π2 : X × Y → Y is the projection, we have that f = g ◦ π2 . Proof. Let R ∈ X be a point and deﬁne g : Y → Z by g(y) = f (R, y). Since two regular maps on a variety are equal if they agree on a nontrivial open set, we need only show that f and g ◦ π2 agree on some open subset of X × Y . Let U be an aﬃne open neighborhood of P in Z, F = Z \ U , and G = π2 (f −1 (F )). The set G is closed in Y since f −1 (F ) is closed in X × Y and X is projective, and hence π2 is a closed map (Corollary 5.13). We have that Q ∈ G since f (X × {Q}) = P ∈ F . Thus V = Y \ G is a nonempty open neighborhood of Q in Y . For each y ∈ V , the projective variety X × {y} is mapped by f into the aﬃne variety U and hence to a single point of U (by Corollary 5.16). Thus for any x ∈ X and y ∈ V , we have that f (x, y) = f (R, y) = g ◦ π2 (x, y),
proving the lemma.
Corollary 18.46. Let X be an elliptic curve with group structure deﬁned by a point P0 ∈ X and let Y be an elliptic curve with group structure deﬁned by Q0 ∈ Y . Suppose that Φ : X → Y is a regular map such that Φ(P0 ) = Q0 . Then Φ is a group homomorphism. Proof. Consider the regular map Ψ : X × X → Y deﬁned by Ψ(x, y) = Φ(x ⊕ y) " Φ(x) " Φ(y). Then Ψ(X × {P0 }) = Ψ({P0 } × X) = Q0 , so Ψ(x, y) = Q0 for all x, y ∈ X by Lemma 18.45.
18.8. Complex curves A nonsingular projective curve X over k = C has the structure of a Riemann surface, and g = g(X) is the topological genus of X (X is topologically a sphere with g handles). This is discussed, for instance, in [115] and [62]. Now such X has the Euclidean topology. We proved that when G is an Abelian group, then Γ(U, G) ∼ = Gr where r is the number of connected components of U (by Proposition 11.14). This is the same as the ﬁrst singular 0 (U, G). Now the Cech ˇ complex computes singular cohocohomology HSing mology of X, since X can be triangulated ([49, Section 9 of Chapter X]
18.8. Complex curves
359
or [62]) and computes sheaf cohomology, so we obtain that the sheaf cohoi (X, Z). We write Zan to indicate mology H i (X, Zan ) is isomorphic to HSing that we are in the Euclidean topology. Now we regard X as a compact twodimensional oriented real manifold, and then we have (for instance by [103]) that ⎧ Z if i = 0, ⎪ ⎪ ⎨ 2g Z if i = 1, i HSing (X, Z) = Z if i = 2, ⎪ ⎪ ⎩ 0 if i > 2. an be the sheaf of analytic functions on X and (O an )∗ be the sheaf Let OX X of nonvanishing analytic functions. Then we have a short exact sequence of sheaves an an ∗ → (OX ) → 0, 0 → Z → OX e
where e denotes the exponential map f → ef . It follows from GAGA [133] that if Y is a complex projective variety and F is a coherent sheaf on Y , then the cohomology of the extension F an of F to an analytic sheaf is the same as the cohomology of F . Thus an ) ∼ H i (X, O ) for all i. Taking sheaf cohomology, we get the H i (X, OX = X long exact sequence e
0 → Z → C → C∗ → H 1 (X, Z) → H 1 (X, OX ) c an )∗ ) → H 2 (X, Z) → H 2 (X, OX ). → H 1 (X, (OX Now X has genus g and dimension 1, so that H 2 (X, OX ) = 0. Further, e : C → C∗ is onto. Thus from our above exact sequence, we deduce that we have an exact sequence of groups an ∗ 0 → Cg /Z2g → H 1 (X, (OX ) ) → Z → 0, c
since H 1 (X, OX )/H 1 (X, Z) ∼ = Cg /Z2g . From the argument of Theorem 17.16, we have that an ∗ ∼ ) ) = Pican (X), H 1 (X, (OX
the group of invertible analytic sheaves on X, modulo isomorphism. Now again by GAGA, we know that all global analytic sheaves on X are isomorphic to algebraic sheaves, and this isomorphism takes global analytic homomorphisms to algebraic homomorphisms. Thus the natural map Pic(X) → Pican (X)
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is an isomorphism. In conclusion, we have obtained the following theorem: Theorem 18.47. Suppose that X is a nonsingular projective curve of genus g over the complex numbers. Then there is a short exact sequence of groups c
0 → G → Pic(X) → Z → 0, where G is a group Cg /Z2g . The subset H 1 (X, Z) ∼ = Z2g of H 1 (X, OX ) ∼ = Cg is in fact a lattice, if we regard Cg as a 2gdimensional real vector space. Thus in the Euclidean topology, G ∼ = (S 1 )2g where S is the circle R/Z and G is a “torus”. This group G naturally has the structure of an analytic manifold (of complex dimension g), and it is even an algebraic variety (of dimension g). The group structure on G is algebraic. The map c is just the degree map, and the exact sequence of the theorem is just the exact sequence deg
0 → Cl0 (X) → Cl(X) → Z → 0 of (13.11). Using our natural isomorphism of Pic(X) with Cl(X), the map c (for Chern) is actually the degree of a divisor which we studied on a curve earlier. We can thus identify the algebraic group G with the group Cl0 (X) of linear equivalence classes of divisors of degree 0 on X. This group G is called the Jacobian of X (in honor of Jacobi). Fixing a point P0 ∈ X, we obtain a map X → J deﬁned by mapping a point P to the class of P − P0 . This map is a regular map and is a closed embedding if g > 0.
18.9. Abelian varieties and Jacobians of curves In this section we discuss the algebraic construction of the Jacobian. We need to introduce a couple of new concepts ﬁrst. An Abelian variety A (in honor of Abel) is a projective variety with a group structure such that the multiplication m : A×A → A is a regular map and the inverse map i : A → A is a regular map. There is an extensive literature on these remarkable varieties. A few references are [146], [96], [119], and [110]. The elliptic curves are the onedimensional Abelian varieties. An Abelian variety is commutative and nonsingular (as is shown in any of these references). A gdimensional Abelian variety over the complex numbers is isomorphic by an analytic isomorphism to a complex torus Cg /Λ, where Λ is a lattice in Cg . Suppose that X is a variety and r is a positive integer. The symmetric group Sr acts on the product X r by permuting factors. There exists a variety X (r) whose function ﬁeld is k(X r )Sr which is a quotient X r /Sr [119, II, Section 7 and III Section 11]. In the case when X is a nonsingular projective
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361
curve, X (r) is nonsingular [111, Proposition 3.2]. The points of X (r) can be considered as eﬀective divisors p1 + p2 + · · · + pr of degree r on X. We have the following theorem. Theorem 18.48. Suppose that X is a nonsingular projective curve of genus g. Then there exists an Abelian variety J of dimension g and a regular map φ : X → J such that: 1. φ is a closed embedding. 2. φ induces a birational regular map X (g) → J by p1 + · · · + pg →
g
φ(pi ).
i=1 0 3. φ induces a group isomorphism Cl (X) → J by [D] → D= ni pi .
ni φ(pi ) if
The variety J of Theorem 18.48 is called the Jacobian of X. An Abelian variety A of positive dimension n > 0 over the complex numbers has lots of points of inﬁnite order (under the group law of A). This follows from the fact that there is an analytic isomorphism of A with the quotient of Cg by a lattice of Cg . However, if A is an Abelian variety over the algebraic closure of a ﬁnite ﬁeld, then every element of A has ﬁnite order. We see this as follows. Suppose that k is the algebraic closure of a ﬁnite ﬁeld and A is an Abelian variety over k. Then A is a subvariety of a projective space Pnk . Suppose that x ∈ A. Then there exists a ﬁnite ﬁeld k such that the embedding of A into Pn is deﬁned over k, x is a rational point over k , and the addition on A is deﬁned over k . There are only ﬁnitely many points of Pn which are rational over k so there are only ﬁnitely many points of A which are rational over k . All multiples of x are rational over k since the multiplication is deﬁned over k and x is rational over k . Thus x has ﬁnite order in the group A. Let A be an Abelian variety over an algebraically closed ﬁeld k. Let A(k) be the group of points of A, so that A(k) is a Zmodule. The rank of A(k) is rank(A(k)) = dimQ A(k) ⊗ Q. We have seen that if A is an Abelian variety over the algebraic closure k of a ﬁnite ﬁeld, then rank(A(k)) = 0. However, we have the following theorem ensuring us that there are lots of points of inﬁnite order on an Abelian variety of positive dimension over any other algebraically closed ﬁeld. Theorem 18.49. Suppose that A is a positivedimensional Abelian variety deﬁned over an algebraically closed ﬁeld k which is not the algebraic closure of a ﬁnite ﬁeld. Then the rank of A(k) is equal to the cardinality of k.
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Proof. [56, Theorem 10.1].
We also have the following proposition describing the points of ﬁnite order in an Abelian variety. Proposition 18.50. Let A be an Abelian variety of dimension g over an algebraically closed ﬁeld k. For n ∈ Z≥0 , let An (k) = {x ∈ A  nx = 0}. Suppose that the characteristic p of k does not divide n. Then An (k) ∼ = (Z/nZ)2g . Proof. [119, Proposition, page 64].
In the case when k = C, so that there is an analytic isomorphism A ∼ = g where Λ is a lattice in C , the proposition follows since An (C) ∼ = 1 ( n Λ)/Λ.
Cg /Λ
The history of Abelian varieties and Jacobian varieties is outlined at the end of Milne’s article [111]. Milne proves many interesting facts about the Jacobian in [111], including giving in [111, Section 7] a proof in modern language of Weil’s proof in [146] of Theorem 18.48, the original proof using the language of Foundations of Algebraic Geometry [145]. Milne refers to Section 2 of Artin [12] for a proof in modern language of Weil’s theorem that a “birational group” is isomorphic to an algebraic group [146]. Throughout these exercises C will denote a nonsingular projective curve of genus g. Exercise 18.51. Show that KC  is base point free if g ≥ 1. Exercise 18.52. Show that mKC is very ample if g ≥ 3 and m ≥ 2. Exercise 18.53. If g = 2, show that mKC is very ample for m ≥ 3 and φ2KC  : C → P2 is a degree 2 regular map onto a quadric curve in P2 (which is isomorphic to P1 ). Exercise 18.54. Suppose that C is deﬁned over an algebraically closed ﬁeld k of characteristic 0. Suppose that 0 = f ∈ k(C) and that the regular map φ = (f : 1) : C → P1 has degree n. Show that ⎛ ⎞ 1 (ep − 1)⎠ − n + 1. g= ⎝ 2 p∈C
Exercise 18.55. A curve C is called a hyperelliptic curve if there exists a degree 2 regular map φ : C → P1 . Suppose that k is an algebraically closed ﬁeld of characteristic 0 and a1 , . . . , al ∈ k are distinct. Let γ be the aﬃne
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363
curve γ = Z(y 2 − f (x)) ⊂ A2 where f (x) = li=1 (x − ai ). Let C be the resolution of singularities of the Zariski closure of γ in P2 , and let π : C → P1 be the regular map which when restricted to γ is the projection of γ onto the xaxis. Show that π is a degree 2 map. Compute the ramiﬁcation of π and show that C has genus g = l − 1. Exercise 18.56. Suppose that C is a plane curve of degree 4. a) Show that the eﬀective canonical divisors on C are the divisors i∗ (L) where i : C → P2 is inclusion and L is a line on P2 . b) Show that the genus of C is 3. c) If D is any eﬀective divisor of degree 2 on C, show that h0 (C, OC (D)) = 1. d) Conclude that C is not hyperelliptic. Exercise 18.57. Suppose that C is not a hyperelliptic curve. Show that KC is very ample. Exercise 18.58. Suppose that C is a hyperelliptic curve with degree 2 regular map φ : C → P1 and p ∈ C is a ramiﬁcation point. Show that ⎧ if m = 2i, 1 ≤ i ≤ g, ⎨ i+1 i+1 if m = 2i + 1, 1 ≤ i ≤ g, h0 (C, OC (mp)) = ⎩ m + 1 − g if m ≥ 2g. Exercise 18.59. Suppose that g ≥ 2 and φ : C → C is a dominant regular map. Show that φ is an isomorphism.
Chapter 19
An Introduction to Intersection Theory
We give a treatment of intersection theory, based on the Snapper polynomial ([139]). Let V be a ddimensional projective scheme (over an algebraically closed ﬁeld k). The results in this chapter are mostly from [89, Section 1]. More general intersection theories are presented in [57]. We conclude this chapter with some examples and applications. In Theorem 19.20, we give a proof of B´ezout’s theorem, showing that two projective plane curves intersect in m points counting multiplicity, where m is the product of the degrees of the two curves. In Theorem 19.21, Corollary 19.22, and Theorem 19.23, we give formulas relating the degrees of projective varieties W and π(W ) under a projection π. Suppose that V is a nonsingular projective variety and W is a tdimensional closed subvariety of V . Associated to divisors D1 , . . . , Dt on V we have the intersection product I(D1 · · · Dt · W ) of (19.9). This product satisﬁes the conditions that (19.1)
it is multilinear in divisors on V
by Proposition 19.6, and if D1 , . . . , Dt are divisors on V such that D1 ∼ D1 , . . . , Dt ∼ Dt , then (19.2)
I(D1 · · · Dt · W ) = I(D1 · · · Dt · W )
by the comment after Deﬁnition 19.2. 365
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By Theorem 16.1, if H1 , . . . , Ht are very ample divisors on V , then there exist eﬀective divisors H1 , . . . , Ht on V such that H1 ∼ H1 , . . . , Ht ∼ Ht and the scheme H1 ∩ · · · ∩ Ht ∩ Y is reduced and ﬁnite (a zerodimensional algebraic set). Then if d is the number of points in H1 ∩ · · · ∩ Ht ∩ Y , we have by Propositions 19.8 and 19.5 that (19.3)
I(H1 · · · Ht ) = d.
Using the three properties (19.1), (19.2), and (19.3), we can compute the intersection product I(D1 · · · Dt · W ) for any divisors D1 , . . . , Dt on V , since any divisor on V is the diﬀerence of very ample divisors by 2) of Theorem 13.32. Alternatively, we can use the three properties (19.1), (19.2), and (19.3) to deﬁne the intersection product I(D1 · · · · Dt · W ). This is the way that intersections products were ﬁrst deﬁned. Of course we must then prove that our construction is welldeﬁned. A ﬁrst step on this is given by Theorem 16.9, which shows that the number of points d in the reduced scheme H1 ∩ · · · ∩ Ht ∩ Y does not depend on the choice of general H1 , . . . , Ht which are linearly equivalent to H1 , . . . , Ht .
19.1. Deﬁnition, properties, and some examples of intersection numbers Suppose that V is a projective scheme (over an algebraically closed ﬁeld k). Theorem 19.1 (Snapper). Let F be a coherent sheaf on V and let s = dim Supp F . Let L1 , . . . , Lt be t invertible sheaves on V . Then the Euler characteristic ∞ (−1)i hi (V, F ⊗ Ln1 1 ⊗ · · · ⊗ Lnt t ) χ(F ⊗ Ln1 1 ⊗ · · · ⊗ Lnt t ) = i=0
is a numerical polynomial in n1 , . . . , nt of total degree s. A numerical polynomial in n1 , . . . , nt is a polynomial f (n1 , . . . , nt ) in n1 , . . . , nt with rational coeﬃcients such that f (n1 , . . . , nt ) is an integer whenever n1 , . . . , nt are integers. A coherent sheaf G of OV modules is a torsion sheaf if for all p ∈ V and a ∈ Gp , there exists a nonzerodivisor b ∈ OV,p such that ab = 0. Proof. We will prove the theorem by induction on s. The theorem is trivial when s = −1. Assume s ≥ 0. We may replace V with Supp(F ), given the subscheme structure deﬁned by Ann(F ). We then have reduced to the case that the theorem holds for torsion sheaves on V .
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367
Let K be the set of (F , L1 , . . . , Lt ) on V such that the theorem holds. Since χ is exact on exact sequences of coherent modules, by Grothendieck’s theory of d´evissage (unscrewing) in [67, Section 3.1], we need only prove the theorem for F = OV when V is a projective variety and if we assume that all torsion sheaves G on V satisfy the conclusions of the theorem. We proceed by induction on t, the case t = 0 being trivial. Since V is a projective variety, there exists a > 0 such that the sheaf L1 ⊗ OV (a) is generated by global sections by Theorem 11.45. Since L1 ⊗ OV (a) is generated by global sections, there exists 0 = σ ∈ Γ(V, L1 ⊗ OV (a)). We thus have a short exact sequence (19.4)
0 → OV (−a) ⊗ L−1 1 → OV → B → 0 σ
where B = OV /σOV (−a) ⊗ L−1 1 . We can assume that a is suﬃciently large that there exists 0 = τ ∈ Γ(V, OV (a)) giving a short exact sequence (19.5)
τ
0 → OV (−a) → OV → C → 0
where C = OV /τ OV (−1). Tensor (19.4) with L1n1 +1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t and tensor (19.5) with Ln1 1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t and take the long exact cohomology sequences to get χ(Ln1 1 +1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) = χ(OV (−a) ⊗ Ln1 1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) + χ(B ⊗ L1n1 +1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) and χ(Ln1 1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) = χ(OV (−a) ⊗ Ln1 1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) + χ(C ⊗ Ln1 1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ). Since dim Supp B < dim V and dim Supp C < dim V , we have that Q(n1 , . . . , nt ) = χ(L1n1 +1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) − χ(Ln1 1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) = χ(B ⊗ L1n1 +1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) − χ(C ⊗ Ln1 1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) is a numerical polynomial in n1 , . . . , nt of total degree < s. Expand Q ∈ Q[x1 , . . . , xt ] as x1 x1 + c1 + · · · + cr Q = c0 r r−1 where ci ∈ Q[x2 , . . . , xt ] and r < s is the degree of Q in x1 . Set x1 x1 + · · · + cr . P = c0 r+1 1
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The ﬁrst diﬀerence ΔP := P (n1 + 1, n2 , . . . , nt ) − P (n1 , n2 , . . . , nt ) = Q(n1 , . . . , nt ) x1 . Expand since Δ xj1 = j−1 χ(Ln1 1 ⊗ Ln2 2 ⊗ · ·· ⊗ Lnt t ) − P (n1 , . . . , nt ) n1 −1 Δ χ(Li1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) − P (i, n2 , . . . , nt ) = i=0 +χ(Ln2 2 ⊗ · · · ⊗ Lnt t ) − P (0, n2 , . . . , nt ) = χ(Ln2 2 ⊗ · · · ⊗ Lnt t ) − P (0, n2 , . . . , nt ). Since χ(Ln2 2 ⊗ · · · ⊗ Lnt t ) is a numerical polynomial of total degree s by induction on t, we have that χ(Ln1 1 ⊗Ln2 2 ⊗· · ·⊗Lnt t ) is a numerical polynomial of total degree s. Deﬁnition 19.2. Let L1 , . . . , Lt be t invertible sheaves on V , and let F be a coherent sheaf on V such that dim Supp F ≤ t. The intersection number (L1 · · · Lt ; F )V of L1 , . . . , Lt with F is the coeﬃcient of the monomial n1 · · · nt in χ(F ⊗ Ln1 1 ⊗ · · · ⊗ Lnt t ). Observe that (L1 · · · Lt ; F )V is independent of isomorphism class of L1 , . . . , Lt and F . Lemma 19.3. Suppose that F is a coherent sheaf on V and L1 , . . . , Lt are invertible sheaves on V with dim Supp F ≤ t. Then −1 −1 (L1 · · · Lt ; F )V = χ(F ) − ti=1 χ(F ⊗ L−1 i<j χ(F ⊗ Li ⊗ Lj ) i )+ −1 −1 t − · · · + (−1) χ(F ⊗ L1 ⊗ · · · ⊗ Lt ). Proof. The polynomial P (n1 , . . . , nt ) = χ(F ⊗ Ln1 1 ⊗ · · · ⊗ Lnt t ) has an expansion aj1 ,...,jt nj11 · · · njt t P (n1 , n2 , . . . , nt ) = j1 +···+jt ≤t
with aj1 ,...,jt ∈ Q. Now for i1 < i2 < · · · < is ≤ t, −1 ⊗ L−1 χ(F ⊗ L−1 i2 · · · ⊗ Lis ) i1 j 1 = j1 +···+js ≤t (−1) +···+js a0,...,0,j1 ,0,...,0,j2 ,0,...,0,js ,0,...,0 ,
from which the conclusion of the lemma follows, using the identity d d d (−1)i . 0 = (1 − 1) = i
i=0
Proposition 19.4. The intersection number (L1 · · · Lt ; F )V is an integer.
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369
Proof. This follows from Lemma 19.3, since an Euler characteristic is an integer. Proposition 19.5. The intersection number
0 if dim Supp F < t, (L1 · · · Lt ; F )V = h0 (V, F ) if dim Supp F = t = 0. Proof. If dim Supp F < t, then χ(F ⊗ Ln1 1 ⊗ · · · ⊗ Lnt t ) is polynomial of degree less than t, and if dim Supp F = t = 0, then (F )V = χ(F ) = h0 (F ).
Suppose that F is a coherent sheaf on V whose support has dimension 0. Let Supp(F ) = {Q1 , . . . , Qm }. Then the intersection multiplicity (F )V = h0 (V, F ) =
(19.6)
m
dimk FQi .
i=1
Proposition 19.6. The intersection number (L1 · · · Lt ; F )V is a symmetric tlinear form in L1 , . . . , Lt . Proof. Let M and N be invertible sheaves on V . Taking successively n = 0 and m = 0, we have that χ(F ⊗ Mm ⊗ (N −1 )n ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) = (M · L2 · · · Lt · F )V mn2 · · · nt − (N · L2 · · · Lt · F )V nn2 · · · nt + · · · . Now taking m = n = n1 , we obtain χ(F ⊗ (M ⊗ N −1 )n1 ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) = ((M · L2 · · · Lt · F )V − (N · L2 · · · Lt · F )V )n1 n2 · · · nt + · · · ,
establishing linearity. Proposition 19.7. If 0 → F → F → F → 0 is a short exact sequence of coherent OV modules, then (L1 · · · Lt ; F )V = (L1 · · · Lt ; F )V + (L1 · · · Lt ; F )V .
Proof. This follows from the additivity of the Euler characteristic on short exact sequences. Let W be a closed subscheme of V of dimension less than or equal to t. Then we deﬁne (L1 · · · Lt · W ) = (L1 · · · Lt ; OW )V . We deﬁne (L1 · · · Lt ) = (L1 · · · Lt · V ).
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In Proposition 19.8, we use notation introduced in Section 15.1. Proposition 19.8. Suppose that s ∈ Γ(V, L1 ) is a nonzero divisor and also that s is not a zero divisor on F . Then (L1 · · · Lt ; F )V = (L2 · · · Lt ; F ⊗ OΔ )V where Δ = div(s) and OΔ = OV /Idiv(s) . In particular, if dim V ≤ t, then (L1 · · · Lt ) = (L2 · · · Lt · Δ). Proof. With our assumptions on s, we have short exact sequences 0 → F ⊗L1n1 −1 ⊗Ln2 2 ⊗· · ·⊗Lnt t → F ⊗Ln1 1 ⊗· · ·⊗Lnt t → (F ⊗OΔ )⊗Ln1 1 ⊗· · ·⊗Lnt t → 0. s
Hence χ(F ⊗ OΔ ⊗ Ln1 1 ⊗ · · · ⊗ Lnt t ) = χ(F ⊗ Ln1 1 ⊗ · · · ⊗ Lnt t ) − χ(F ⊗ L1n1 −1 ⊗ · · · ⊗ Lnt t ) = (L1 · · · Lt · F )V n1 n2 · · · nt + · · · − (L1 · · · Lt · F )V (n1 − 1)n2 · · · nt + · · · = (L1 · · · Lt · F )V n2 · · · nt + · · · . Further, taking n1 = 0, we have that χ((F ⊗ OΔ ) ⊗ Ln2 2 ⊗ · · · ⊗ Lnt t ) = (L2 · · · Lt · FΔ )V n2 · · · nt + · · · .
Proposition 19.9. Suppose that W is a closed subscheme of V which contains the subscheme X = Supp(F ), where X is provided with the subscheme structure deﬁned by the annihilator of F ; that is, there is a natural surjection OW → OX = OV /Ann(F ). Then F may be considered as an OW module, and (L1 · · · Lt ; F )V = (L1 ⊗ OW · · · Lt ⊗ OW ; F )W . In particular, (L1 · · · Lt · W ) = (L1 ⊗ OW · · · Lt ⊗ OW ). Proof. If L is any invertible sheaf on V , then F ⊗L∼ = F ⊗ OW ⊗ L. Now set L = Ln1 1 ⊗ · · · ⊗ Lnt t , and take Euler characteristics to obtain the conclusions of the proposition. Corollary 19.10. Suppose that V = Supp(F ). Let V1 , . . . , Vs be the irreducible components of V (which might not be reduced), and let Fi = F ⊗ OVi for 1 ≤ i ≤ s. Then (L1 · · · Lt ; F )V = (L1 · · · Lt ; F1 )V + · · · + (L1 · · · Lt ; Fs )V .
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371
Proof. The canonical homomorphisms OV → OVj for j = 1, . . . , s give an exact sequence of coherent OV modules 0→A→F → Fj → B → 0
with Supp A, Supp B ⊂ Propositions 19.5 and 19.7.
i=j (Vi
j
∩ Vj ). The corollary now follows from
If V is a scheme and W is a subvariety, then we compute the local ring OV,W as in Deﬁnition 12.17. If U is an open aﬃne subset of V which intersects W , then OV,W = Γ(U, OV )P where P is the prime ideal of W in ˜ for some Γ(U, OV )Γ(U, OV ). If F is a coherent sheaf on V and F U = M module M , then we deﬁne FW = MP . Corollary 19.11. Suppose that V is irreducible and dim V ≤ t. Let = OV,Vred (FVred ). Then (L1 · · · Lt ; F )V = (L1 · · · Lt ; OVred ). Proof. We have that OV,Vred is an Artin local ring and FVred is a ﬁnite OV,Vred module. Let K be the set of F for which the corollary is true. The corollary is certainly true when F = OVred and when dim Supp F < t. The conclusions of the corollary now follow from d´evissage ([67, Section 3.1]). We deﬁne the degree of a regular map φ : V → V of varieties (Deﬁnition 21.25) as follows:
[k(V ) : k(V )] if dim V = dim V and φ is dominant, deg(φ) = 0 otherwise. Proposition 19.12. Let φ : V → V be a regular map of projective varieties and assume that t ≥ max{dim V, dim V }. Let L1 , . . . , Lt be invertible sheaves on V and set Li = φ∗ Li for 1 ≤ i ≤ t. Then (L1 · · · Lt ) = deg(φ) (L1 · · · Lt ). Proof. [89, Proposition 6].
If D1 , D2 , . . . , Dn are divisors on an ndimensional nonsingular projective variety V , then we deﬁne (19.7)
(D1 · D2 · · · Dn ) = (OV (D1 ) · OV (D2 ) · · · OV (Dn )).
Observe that if Di ∼ Di for 1 ≤ i ≤ n, then (19.8)
(D1 · D2 · · · Dn ) = (D1 · D2 · · · Dn ).
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Suppose that W is a tdimensional subvariety of V . Then we deﬁne (D1 · · · Dt · W ) = (OV (D1 ) · · · OV (Dt ) · OW ). Example 19.13. Suppose that L ∼ = OX (D) is an invertible sheaf on a nonsingular projective curve X. Then the intersection multiplicity (19.9)
(L) = (L · X) = deg(D). If D is eﬀective, this follows since (L) = (OX (D); OX )X = (OD )X = h0 (X, OD ) by Propositions 19.8 and 19.5. For general D, write D = D1 − D2 where D1 and D2 are eﬀective. Then (OX (D)) = (OX (D1 − D2 )) = (OX (D1 )) − (OX (D2 )) = deg(D1 ) − deg(D2 ) = deg(D). Alternatively, (L · X) is the linear term of χ(OX (nD)) = n deg(D) + 1 − g(X) by the RiemannRoch theorem. Example 19.14. Suppose that C is a nonsingular curve on a nonsingular projective surface X and D is a divisor on X. Then by Propositions 19.8 and 19.9 and Example 19.13, (D · C) = (OX (D) ⊗ OC ) = deg(OX (D) ⊗ OC ). We will denote the selfintersection number (L · · · L) of an invertible sheaf L on a ddimensional scheme V , d times with itself by (Ld ). Example 19.15. The intersection product (OPd (1)d ) = 1 for d ≥ 1. We prove this formula by induction on d. When d = 1, we have (OP1 (1)) = deg OP1 (1) = 1 by Example 19.13. If d > 1, by Propositions 19.8 and 19.9 we have (OPd (1)d ) = (OPd (1)d−1 · OPd (1)) = (OPd (1)d−1 · L) = (OPd−1 (1)d−1 ) where L ∼ = Pd−1 is a hyperplane section of Pd . Theorem 19.16. Let Z be a projective scheme of dimension d and let L be an invertible sheaf on Z. Then (Ld ) d n + Q(n) d! where Q(n) is a polynomial with rational coeﬃcients of degree ≤ d − 1. χ(Ln ) =
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Proof. There exists b ∈ Q and a polynomial Q(n) with rational coeﬃcients of degree ≤ d − 1 such that χ(Ln ) = bnd + Q(n)
(19.10) by Theorem 19.1. We have
(19.11) (−1)d (Ld )nd = (L−n · · · L−n )
d χ(L2n ) − · · · + (−1)d χ(Ldn ) = χ(OZ ) − dχ(L ) + 2 n
by multilinearity of the intersection product, Proposition 19.6, and Lemma 19.3. Substituting (19.10) into (19.11), we obtain d d d d d d d d d i d Q(in). (−1) (L )n = −d + 2 − · · · + (−1) d bn + (−1) i 2 i=0
Thus
(19.12)
b=
−1 d d d d (−1)d (Ld ). −d + 2 − · · · + (−1) d 2
In the case that Z = Pd and L = OPd (1), we have that (OPd (1)d ) = 1 by Example 19.15 and d+n−1 nd n + lowerorder terms, = χ(L ) = d! n−1 so that b =
1 d!
in this case. We then see from (19.12) that −1 1 d d d d (−1)d = , −d + 2 + · · · + (−1) d d! 2
so that for general L, b= in (19.10).
(Ld ) d!
Suppose that φ : X → Y is a regular map of normal varieties such that φ = φD where D is a divisor on X such that D is base point free. We have that D = φ∗ (H) where H is a hyperplane section of Y . Since the linear system D has no base points, we have that (C · D) ≥ 0 for all closed curves C on X. An invertible sheaf L such that (L · C) ≥ 0 for all closed curves C on X is called numerically eﬀective (nef). For a closed curve C on X, we have that φ(C) is contracted to a point on Y if and only if (C · D) = 0. A nef divisor F does not always have the property that some positive multiple mF gives a base point free linear system mF . We will give an example
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in Theorem 20.14 (and Exercise 20.17). More examples and explanations of this can be found in [159], [44], [98]. Theorem 19.17. Suppose that Z is an arbitrary closed subscheme of Pn of dimension r. Then (19.13)
deg(Z) = (OPn (1)r · Z) = (OZ (1)r ).
Proof. The quickest proof of the theorem follows from Corollary 17.21, which tells us that χ(OZ (n)) is the Hilbert polynomial PZ (n) of Z, and from Theorem 19.16. We also give an alternate, more conceptual, proof. Let I ⊂ S = S(Pn ) ˜ By prime avoidance be the saturated homogeneous ideal such that IZ = I. and since k is inﬁnite ([28, Lemma 1.5.10–Proposition 1.5.12]), there exists a linear form F ∈ S1 such that F1 is not contained in any associated prime ideals of I. Thus the sequence F
0 → (S/I)(−1) →1 S/I → S/(I + (F1 )) → 0 is short exact. Let L1 = Z(F1 ) ⊂ Pn . Comparing Hilbert functions of S/I and S/(I + (F1 )) as in the proof of Theorem 16.9, we have that PS/I (n) − PS/I (n − 1) = PS/(I+(F1 )) (n) so that deg Z = deg Z ∩ L1 . With our assumptions, we have that F1 ∈ Γ(Pn , OPn (1)) is not a zero divisor on OZ . Thus (OPn (1)r · Z) = (OZ (1)r−1 · Z∩L1 ) by Propositions 19.8 and 19.9. By induction, we can choose F1 , . . . , Fr ∈ S with r = dim Z such that if Li = Z(Fi ), we have that deg Z = deg Z ∩ L1 ∩ · · · ∩ Lr and (OPn (1)r · Z) = (OZ∩L1 ∩···∩Lr )Pr . The Hilbert polynomial of the zerodimensional scheme Z ∩ L1 ∩ · · · ∩ Lr is the constant dimension dimk (S/I + (F1 ) + · · · + (Fr ))m
for m 0,
which is h0 (Pn , OZ∩L1 ∩···∩Lr (m)) = h0 (Pn , OZ∩L1 ∩···∩Lr ) by Theorem 11.47, which is valid for arbitrary projective schemes. Since Z ∩ L1 ∩ · · · ∩ Lr is a zerodimensional scheme, we also have that (OZ∩L1 ∩···∩Lr )Pr = h0 (Pn , OZ∩L1 ∩···∩Lr ) by Proposition 19.5, from which the theorem follows.
Corollary 19.18. Suppose that X is a nonsingular projective variety of dimension d, H is an ample divisor on X, and D is a nonzero eﬀective divisor on X. Then (H d−1 · D) > 0.
19.2. Applications to degree and multiplicity
375
Proof. There exists a positive integer n0 such that n0 H is very ample on X, and so there is a closed embedding X ⊂ Pr such that OPr (1) ⊗ OX ∼ = OX (n0 H). Let A = n0 H and suppose that E is a codimension 1 subvariety of X. Then (Ad−1 · E) = (OX (A)d−1 · OX (E)) = (OX (A)d−1 · OE ) by Proposition 19.8 = (OPr (1)d−1 · OE ) by Proposition 19.9 = (OPr (1)d−1 · E) = deg(E) by Theorem 19.17. Writing D = ai Ei with Ei prime divisors on X and ai > 0, we have (H d−1 · D) = =
1 (Ad−1 · D) nd−1 0 1 ( ai deg(Ei )) nd−1 0
>0
by Proposition 19.6.
Remark 19.19. Suppose that X is a nonsingular projective variety of dimension d and D1 , · · · , Dd are divisors on X with Dd ∼ 0. Then (D1 · · · · · Dd ) = 0. The remark follows since χ(OX (D1 )n1 ⊗ · · · ⊗ OX (Dd )nd ) = χ(OX (n1 D1 + · · · + nd−1 Dd−1 )) does not depend on nd , so (OX (D1 ) · · · · · OX (Dd )) = (OX (D1 ) · · · · · OX (Dd ); OX )X = 0 by Deﬁnition 19.2 of the intersection product.
19.2. Applications to degree and multiplicity Theorem 19.20 (B´ezout’s theorem). Let Y and Z be distinct irreducible, closed curves in P2 , having degrees d and e, respectively. Let the intersection points of Y and Z be {Q1 , . . . , Qs }. Then s
dimk OY ∩Z,Qi = de
i=1
where Y ∩ Z is the schemetheoretic intersection. In particular, if Qi are nonsingular points of both Y and Z and if Y and Z have distinct tangent spaces at these points ( Y and Z intersect transversally), then s = de. Proof. We have that de = de(OP2 (1) · OP2 (1)) = (OP2 (d) · OP2 (e))
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by Example 19.15 and Proposition 19.6. We also calculate (OP2 (d) · OP2 (e)) = (OP2 (Y ) · Z) = (O Y ∩Z )P2 = h0 (P2 , OY ∩Z ) = si=1 dimk OY ∩Z,Qi by Propositions 19.8, 19.9, and 19.5.
Theorem 19.21. Suppose that W is an mdimensional closed subvariety of Pn . Let P ∈ Pn be a point not in W and let π : Pn Pn−1 be projection from the point P . Let W1 = π(W ) and let φ : W → W1 be the induced regular map. Suppose that dim W1 = dim W . Then deg(W1 ) = deg(φ)deg(W ). Proof. Since φ∗ OW1 (1) ∼ = OW (1), we have by Proposition 19.12 and Theorem 19.17 that deg(W1 ) = (OPn−1 (1)m · W1 ) = (OW1 (1)m ) = deg(φ)(OW (1)m ) = deg(φ)(OPn (1)m · W ) = deg(φ)deg(W ).
Corollary 19.22. Suppose that W is a closed subvariety of Pn of dimension m and L is a linear subspace of Pn of dimension n−m−1 such that W ∩L = ∅ and the projection W1 of W from L to Pm+1 is birational. Then deg(W1 ) = deg(W ). In particular, the degree of the homogeneous form deﬁning W1 in Pm+1 is degree d = deg W . Suppose that R is a ddimensional local ring with maximal ideal m. Then the length R (R/mt+1 ) is a polynomial in t of degree d for t 0, called the HilbertSamuel polynomial. The leading coeﬃcient times d! is an integer, called the mutiplicity e(R) of R. This theory is explained in [161, Chapter VIII]. Theorem 19.23. Suppose that W is a projective variety of dimension d which is a closed subvariety of Pn . Suppose that P ∈ W . Let π : Pn Pn−1 be the rational map which is the projection from the point P . Let W1 be the projective subvariety of Pn−1 which is the closure of π(W \ {P }). Let μ be the multiplicity of the local ring OW,P . Then: 1. μ ≤ deg(W ). 2. Suppose that μ < deg(W ). Then dim W = dim W1 and (19.14)
[k(W ) : k(W1 )]deg(W1 ) = deg(W ) − μ.
3. Suppose that μ = deg(W ). Then dim W > dim W1 and W is a cone over W1 . Proof. Let σ : Z → Pn be the blowup of P with exceptional divisor E = ZP . Let λ : Z → Pn−1 be the induced regular map (a resolution of
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377
indeterminacy of π). Let H0 be a hyperplane of Pn and H1 be a hyperplane of Pn−1 . We have a linear equivalence of divisors (19.15)
σ ∗ (H0 ) − E ∼ λ∗ (H1 ).
Let W be the strict transform of W on Z. By Proposition 19.6, we have that (19.16)
(λ∗ (H1 )d · W ) = (σ ∗ (H0 )d · W ) + ((−E)d · W ).
Since deg(σW ) = 1, by Proposition 19.12 and Theorem 19.17, we have that (σ ∗ (H0 )d · W ) = (H0d · W ) = deg(W ). Let y0 , . . . , yn be homogeneous coordinates on Pn . After a linear change of variables, we may assume that P = (0, . . . , 0) ∈ U = An = Pn \ Z(y0 ). Let Z = σ −1 (U ), W = W ∩ U , and W = W ∩ σ −1 (U ). The variables y1 , . . . , yn are homogeneous coordinates on Pn−1 and π = (y1 : · · · : yn ). Let mP be the maximal ideal of P in k[U ]. We have an embedding of Z in U × Pn−1 . The variety U × Pn−1 has the homogeneous coordinate ring k[U ][y1 , . . . , yn ]. The ﬁber ZP = E ∼ = Pn−1 has the homogeneous ideal IE = mp k[U ][y1 , . . . , yn ] and coordinate ring S(E) = k(P )[y1 , . . . , yn ].
Let mP = mP k[W ]. Then the coordinate ring of W is the coordinate ring of the blowup of mP , which is S(W ) = t≥0 mtp by Theorem 6.4 and
Proposition 6.6. The homogeneous ideal of W P = E ∩ W is mP S(W ), so the homogeneous coordinate ring of E ∩ W = E ∩ W is S(E ∩ W ) =
mtP /mt+1 P .
t≥0
The embedding of E ∩ W in E is realized by the natural surjection k(P )[y1 , . . . , yn ] →
mtP /mt+1 P
t≥0
which maps the yi to the corresponding generators of mP . Under this embedding, we have that deg(W ∩ E) is (d − 1)! times the leading coeﬃcient of the Hilbert polynomial of S(E ∩W ), since dim W ∩E = dim W P = d − 1.
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Let R = OW,P with maximal ideal mR . Since (R/mt+1 R )=
t
dimk msP /ms+1 P ,
s=0
we have that deg(W ∩ E) = e(R).
As in Example 13.38, we calculate that OE ⊗OZ (−E) ∼ = OPn−1 (1). Thus by Proposition 19.8 and (19.13), ((−E)d · W ) = −((−E)d−1 · (W ∩ E)) = −deg(W ∩ E) = −e(R) = −μ. We can thus rewrite (19.16) as (19.17)
(λ∗ (H1 )d · W ) = deg(W ) − μ.
By Proposition 19.12, we have that
[k(W ) : k(W1 )]deg(W1 ) ∗ d (λ (H1 ) · W ) = 0
if dim W = dim W1 , if dim W > dim W1 .
Substituting into (19.17), we conclude that μ ≤ deg(W ) and μ = deg(W ) if and only if W1 has dimension < d, which holds if and only if W is a cone with vertex P . If μ < deg(W ), then we obtain (19.14). Exercise 19.24. Suppose that X is an rdimensional variety of degree 2 in Pn . Show that X is birationally equivalent to Pr . Hint: We know that X is a quadric hypersurface in a linear subvariety L ∼ = Pr+1 of Pn by Exercise 16.17. Let p ∈ X be a nonsingular point, and consider the projection of L from p to Pr and its eﬀect on X. Is X necessarily isomorphic to Pr ?
Chapter 20
Surfaces
In this chapter we will derive some basic properties of nonsingular projective surfaces. For further reading on this topic, the books [18], [16], and [118] are recommended.
20.1. The RiemannRoch theorem and the Hodge index theorem on a surface Associated to divisors D1 and D2 on a nonsingular projective surface S, we have the intersection product (D1 · D2 ) deﬁned in (19.7). The product (D1 · D2 ) is symmetric and bilinear by Proposition 19.6. If D1∼ D1 and D2 , then (D1 · D2 ) = (D1 · D2 ) by (19.8). If D1 = ai Ei and D2 ∼ bj Ej , where Ei are prime divisors, then D2 = ai bj (Ei · Ej ). (D1 · D2 ) = i,j
Thus the computation of (D1 · D2 ) reduces to the case when D1 and D2 are prime divisors. In the case when D1 , D2 are distinct prime divisors, the intersection product has a nice interpretation, analogous to B´ezout’s theorem on P2 . Lemma 20.1. Suppose that D1 , D2 are distinct prime divisors on the nonsingular projective surface S. Then dimk (OD1 ∩D2 ,p ) (D1 · D2 ) = p∈D1 ∩D2
where D1 ∩ D2 is the schemetheoretic intersection. 379
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Proof. By Propositions 19.8 and 19.5, we have that (D1 · D2 ) = (OS (D1 ); OD2 )S = (OD1 ∩D2 )S = h0 (S, OD1 ∩D2 ) = dimk (OD1 ∩D2 ,p ).
p∈D1 ∩D2
Theorem 20.2 (Adjunction). Suppose that C is a nonsingular projective curve of genus g on a nonsingular projective surface X with canonical divisor KX . Then (C · (C + KX )) = 2g − 2. Proof. We have (C · (C + KX )) = deg(OX (C + KX ) ⊗ OC ) = deg OC (KC ) = 2g − 2 by Theorem 14.21, Corollary 18.14, and Example 19.14.
The following theorem is the RiemannRoch theorem on a a surface. It is often used with Serre duality (Theorem 17.22) which implies that h2 (X, OX (D)) = h0 (X, OX (KX − D)). Theorem 20.3. Suppose that X is a nonsingular projective surface and D is a divisor on X. Then χ(OX (D)) = h0 (X, OX (D)) − h1 (X, OX (D)) + h2 (X, OX (D)) = 12 (D · (D − KX )) + χ(OX ) where KX is a canonical divisor on X. Proof. We have that hi (X, OX (D)) = 0 for i > 2 by Theorem 17.5. Write D = D2 − D1 where D1 and D2 are eﬀective divisors. Let H be an ample divisor on X. By Theorem 13.32, there exists a positive integer n0 such that n0 H + D1 and n0 H + D2 are very ample. By Theorem 16.1, there exist nonsingular curves C1 and C2 such that n0 H + D1 ∼ C1 and n0 H + D2 ∼ C2 so D ∼ C2 − C1 . We have short exact sequences 0 → OX (C2 − C1 ) → OX (C2 ) → OX (C2 ) ⊗ OC1 → 0 and 0 → OX → OX (C2 ) → OX (C2 ) ⊗ OC2 → 0. Since χ is additive on short exact sequences (Exercise 17.24), we have (20.1)
χ(OX (D)) = χ(OX (C2 − C1 )) = χ(OX ) + χ(OX (C2 ) ⊗ OC2 ) − χ(OX (C2 ) ⊗ OC1 ).
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381
By the RiemannRoch theorem for curves (Theorem 18.2) and Example 19.14, we have (20.2)
χ(OX (C2 ) ⊗ OC2 ) = (C2 )2 + 1 − g(C2 )
and (20.3)
χ(OX (C2 ) ⊗ OC1 ) = (C1 · C2 ) + 1 − g(C1 ).
By Theorem 20.2, we have g(C1 ) =
(20.4)
1 (C1 · (C1 + KX )) + 1 2
and 1 (C2 · (C2 + KX )) + 1. 2 The theorem now follows from formulas (20.1)–(20.5). g(C2 ) =
(20.5)
Corollary 20.4. Suppose that D is a divisor on a nonsingular projective surface S. Then h0 (X, OX (D)) ≥
1 (D · (D − KX )) + χ(OX ) − h0 (X, OX (KX − D)). 2
This follows by combining Theorems 20.3 and 17.22. Lemma 20.5. Let H be an ample divisor on a nonsingular projective surface X. Then there is an integer n0 such that for any divisor D on X, if (D·H) > n0 , then H 2 (X, OX (D)) = 0. Proof. Let n0 = (KX · H). By Serre duality, h2 (X, OX (D)) = h0 (X, OX (KX − D)). If (D · H) > n0 and h0 (X, OX (KX − D)) > 0, then KX − D is linearly equivalent to an eﬀective divisor and thus ((KX − D) · H) ≥ 0 by Corollary 19.18 and Remark 19.19. But ((KX − D) · H) = (KX · H) − (D · H) < 0 if (D · H) > n0 = (KX · H), giving a contradiction.
Corollary 20.6. Let H be an ample divisor on a nonsingular projective surface X and let D be a divisor on X such that (D · H) > 0 and (D 2 ) > 0. Then for all n 0, nD is linearly equivalent to an eﬀective divisor. Proof. Let n0 = (KX · H) be the constant of Lemma 20.5. Then there exists n1 > 0 such that for n ≥ n1 , we have (nD · H) = n(D · H) > n0
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20. Surfaces
so h0 (X, OX (KX − nD)) = 0. By Corollary 20.4, we have 1 1 h0 (X, OX (nD)) ≥ n2 (D 2 ) − n(D · KX ) + χ(OX ) 2 2 2 0 for n ≥ n1 . Since (D ) > 0, h (X, OX (nD)) > 0 for n 0.
Deﬁnition 20.7. A divisor D on a nonsingular projective surface X is said to be numerically equivalent to zero, written D ≡ 0, if (D · E) = 0 for all divisors E on X. We say that divisors D and E are numerically equivalent, written as D ≡ E if D − E ≡ 0. If D and E are linearly equivalent divisors, then OX (D − E) ∼ = OX , so that D ≡ E by Remark 19.19. Theorem 20.8 (Hodge index theorem). Let H be an ample divisor on a nonsingular projective surface X and suppose that D is a divisor on X such that D ≡ 0 and (D · H) = 0. Then (D 2 ) < 0. Proof. Suppose that (D 2 ) ≥ 0. We will derive a contradiction. First suppose that (D 2 ) > 0. Let H = D + nH. For n 0, H is ample by Theorem 13.32. Since (D · H ) = (D 2 ) > 0, we have that mD is linearly equivalent to a nonzero eﬀective divisor for m 0 by Corollary 20.6 and Remark 19.19. Then (mD · H) = m(D · H) > 0 by Corollary 19.18. Hence (D · H) > 0, giving a contradiction. Now suppose that (D 2 ) = 0. Since D ≡ 0, there exists a divisor E such that (D · E) = 0. Let E1 = (H 2 )E − (E · H)H. Then (D · E1 ) = 0 ((H 2 ) > 0 by Corollary 19.18) and (E1 · H) = 0. Let D1 = nD + E1 for n ∈ Z. Then (D1 · H) = 0 and (D12 ) = 2n(D · E1 ) + (E12 ). Since (D · E1 ) = 0, there exists n ∈ Z such that (D12 ) > 0. Now the ﬁrst case of the proof applied to D1 gives a contradiction. Let X be a nonsingular projective surface. Deﬁne Num(X) = Pic(X)/ ≡ . The group Num(X) is a ﬁnitely generated group [97] without torsion, so Num(X) ∼ = Zρ for some ρ. Let N (X) = Num(X) ⊗ R, which is a ﬁnitedimensional vector space. The intersection pairing is a nondegenerate bilinear form on N (X). By Sylvester’s theorem [95, Theorem 4.1, page 577],
20.2. Contractions and linear systems
383
the form can be diagonalized with ±1’s on the diagonal, and the diﬀerence of the number of +1’s minus the number of −1’s is an invariant called the signature of the form. The Hodge index theorem tells us that the diagonalized intersection form on a surface has exactly one +1. In particular, the signature of the form is 2 − ρ.
20.2. Contractions and linear systems The selfintersection number (C 2 ) = (C · C) of a curve C on a nonsingular projective surface can by negative, as is shown by the following example. Example 20.9. Let S be a nonsingular projective surface, and let p ∈ S be a point. Let π : S1 → S be the blowup of p with exceptional divisor E. Then (E 2 ) = (E · E) = −1. Proof. We have that (E · E) = deg(OS1 (E) ⊗ OE ) by Example 19.14. Now E∼ = OP1 (−q) where q is a point on P1 by Example = P1 and OS1 (E) ⊗ OE ∼ 13.38, so (E · E) = −1. A remarkable fact is that there is a converse to this example. Suppose that X is a normal projective surface and C = {C1 , . . . , Cn } is a ﬁnite set of closed curves on X. A contraction of C is a regular birational map φ : X → Y such that Y is normal, there exists a point q ∈ Y such that φ(Ci ) = q for all i, and φ : X \ C → Y \ {q} is an isomorphism. The contraction φ : X → Y of C, if it exists, must be unique. To see this, suppose that φ1 : X → Y1 and φ2 : X → Y2 are two contractions of C. Let Ψ : Y1 Y2 be the induced birational map and let ΓΨ ⊂ Y1 × Y2 be the graph. Let π1 : ΓΨ → Y1 and π2 : ΓΨ → Y2 be the projections. Let q1 = φ1 (C) and q2 = φ2 (C). Then Ψ : Y1 \{q1 } → Y2 \{q2 } is an isomorphism, so π1−1 (q1 ) = π2−1 (q2 ) = {(q1 , q2 )}. By Zariski’s main theorem, Theorem 9.3, both projections π1 and π2 are isomorphisms, so Y1 ∼ = Y2 . By Zariski’s connectedness theorem, Theorem 9.7, a set of curves C must have a connected union to be contractible. Theorem 20.10 (Castelnuovo’s contraction theorem). Suppose that S is a nonsingular projective surface and E ∼ = P1 is a curve on S such that (E 2 ) = −1. Then there exists a birational regular map φ : S → T where T is a nonsingular projective surface such that φ(E) = p is a point on T and φ : S → T is isomorphic to the blowup π : B(p) → T of p. Proof. We will ﬁrst prove the existence of a contraction T of E. Let H be a very ample divisor on S. After possibly replacing H with a positive multiple of H, we may assume that H 1 (S, OS (H)) = 0 (by Theorem 17.18).
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Let m = (H · E) > 0 by Corollary 19.18. Then (E · (H + mE)) = 0. We have an exact sequence 0 → OS (−mE) → OS → OmE → 0, where mE is the subscheme of S with OmE = OS /IEm . Tensoring with OS (H + mE) and taking global sections, we have an exact sequence (20.6)
H 0 (S, OS (H + mE)) → H 0 (S, OS (H + mE) ⊗ OmE ) → H 1 (S, OS (H)) = 0.
Let L = OS (H + mE). For all n ≥ 1, we have short exact sequences 0 → OE ⊗ OS (−nE) → O(n+1)E → OnE → 0 since
OS (−nE)/OS (−(n + 1)E) ∼ = OE ⊗ OS (−nE).
Tensoring with L and taking global sections, we have exact sequences (20.7)
H 0 (S, L ⊗ O(n+1)E ) → H 0 (S, L ⊗ OnE ) → H 1 (E, OE ⊗ OS (−nE) ⊗ L).
Now (E · (−nE)) = n and (OS (E) · L) = 0 so OE ⊗ OS (−nE) ⊗ L ∼ = OE (n) = OP1 (n). Thus H 1 (E, OS (−nE) ⊗ L ⊗ OE ) = 0 for n ≥ 1 by Theorem 17.14. Combining (20.6) and (20.7), we have a surjection H 0 (S, OS (H + mE)) → H 0 (E, OS (H + mE) ⊗ OE ) ∼ = H 0 (E, OE ). Thus Base(H + mE) ∩ E = ∅, and so Base(H + mE) = ∅ since H is very ample. Let φ : S → Pr be the regular map induced by H + mE. Suppose C is a curve on S. Then φ(C) is a point if and only if ((H + mE) · C) = 0, so the only curve contracted by φ is E. Let T be the normalization of the image φ(S) in the function ﬁeld of S (Theorem 7.17). Then we have a factorization Ψ : S → T by Exercise 9.9 since S is normal. Let p = Ψ(E). Then S \E → T \{p} is an isomorphism by Zariski’s main theorem (Theorem 9.3). We have that Ψ∗ OS ∼ = OT be Proposition 11.52. By the theorem on formal functions ([156] or [73, Theorem III.11.1]) ˆT,p ∼ O = lim H 0 (En , OE ) ←
n
where En is the closed subscheme of S with the ideal sheaf mnp OS (the completion Aˆ of a local ring A is deﬁned in Section 21.5). Since Ψ−1 (p) = E, the sequence of ideal sheaves mnp OS is conﬁnal with respect to IEn , so ˆT,p ∼ O = lim H 0 (S, OS /IEn ). ←
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385
We will show that H 0 (S, OnE ) ∼ = An = k[x, y]/(x, y)n so ˆT,p ∼ O = lim An ∼ = k[[x, y]]. ←
We then have that OT,p is a regular local ring (by Theorem 21.36). For n = 1, H 0 (E, OE ) ∼ = k. For n > 1, we have short exact sequences 0 → IEn /IEn+1 → OEn+1 → OEn → 0, with
IEn /IEn+1 ∼ = OS (−nE) ⊗ OE ∼ = OP1 (n). 0 1 For n = 1, H (P , OP1 (1)) is a twodimensional vector space. Let x, y be a basis. Then H 0 (S, OE2 ) ∼ = A2 . Inductively, if H 0 (S, OEn ) is isomorphic to 0 An , lift x, y to H (S, OEn+1 ). Since H 0 (P1 , OP1 (n)) is the vector space with basis xn , xn−1 y, . . . , y n , we have that H 0 (S, OEn+1 ) ∼ = An+1 . The contraction S → T is then the blowup of p by Theorem 10.32.
The above theorem may lead one to hope that any curve with negative selfintersection number on a nonsingular projective surface can be contracted by a regular map, but this is not true, as we show in the following example. Example 20.11. Suppose that the algebraically closed ﬁeld k is not an algebraic closure of a ﬁnite ﬁeld. Then there exists a nonsingular projective surface X over k and a nonsingular closed curve C on X with (C 2 ) = −1 such that C is not contractible. Proof. Let γ be a nonsingular cubic curve in S = P2C and L be a line in S such that γ∩L is reduced. Since (γ·L) = 3, γ∩L is a divisor γ∩L = q1 +q2 +q3 for distinct points q1 , q2 , q3 on γ. Since the curve γ is nonsingular of degree 3, its canonical divisor is Kγ = 0 by Corollary 14.22. Thus γ has genus 1 by Corollary 18.14. The elliptic curve γ is isomorphic as a group to Cl0 (γ), by Theorem 18.19, and as explained before Proposition 18.44. We will show that we can ﬁnd a point p0 ∈ γ such that q1 + q2 + q3 − 3p0 has inﬁnite order in Cl0 (γ). Let p be a point in γ. If q1 + q2 + q3 − 3p has inﬁnite order, then take p0 = p. Otherwise, there exists a positive integer m such that m(q1 + q2 + q3 − 3p) ∼ 0. By Theorem 18.49, there exists p0 ∈ γ such that p0 − p has inﬁnite order. We will show that q1 + q2 + q3 − 3p0 has inﬁnite order. Suppose it does not. Then there eixsts a positive integer n such that n(q1 + q2 + q3 − 3p0 ) ∼ 0, so 0 ∼ mn(q1 +q2 +q3 −3p0 ) = nm(q1 +q2 +q3 −3p)+mn3(p−p0 ) ∼ mn3(p−p0 ), a contradiction since p0 − p is assumed to have inﬁnite order. By Theorem 18.49, there exist distinct points p1 , . . . , p10 ∈ γ such that the classes of
386
20. Surfaces
q1 +q2 +q3 −3p0 , p1 −p0 , . . . , p10 −p0 are linearly independent in the rational vector space Cl0 (γ) ⊗ Q. After possibly replacing L with a diﬀerent line linearly equivalent to L, we may assume that q1 , q2 , q3 , p1 , . . . , p10 are distinct points of γ. Suppose that there exist n, m1 , . . . , m10 ∈ Z such that n(q1 + q2 + q3 ) ∼ m1 p1 +· · ·+m10 p10 on γ. Then from m1 +· · ·+m10 = deg(n(q1 +q2 +q3 )) = 3n we have that m1 (p1 − p0 ) + · · · + m10 (p10 − p0 ) ∼ n(q1 + q2 + q3 − 3p0 ) so that m1 = · · · = m10 = n = 0. Let π : X → S be the blowup of the ten points p1 , . . . , p10 . Let H = π ∗ (L) and Ei be the rational curves Ei = π −1 (pi ) for 1 ≤ i ≤ 10. We have that (20.8)
Cl(X) = [H]Z ⊕ [E1 ]Z ⊕ · · · ⊕ [E10 ]Z
by Exercise 13.48 and Example 13.10. Let C be the strict transform of γ on X. Since γ is a nonsingular curve, π : C → γ is an isomorphism (for instance since γ is nonsingular and π : C → γ is birational). This allows us to identify C with γ. Since p1 , . . . , p10 ∈ L, we can identify H ∩ C = L ∩ γ = q1 + q2 + q3 . We also have Ei ∩ C = pi for 1 ≤ i ≤ 10. We have that C = π ∗ (γ) − E1 − · · · − E10 ∼ π ∗ (3L) − E1 − · · · − E10 ∼ 3H − E1 − · · · − E10 . We compute (C 2 ) = (C · (3H − E1 − · · · − E10 )) = −1. Suppose that there exists a contraction φ : X → Y of C. Let q = φ(C). Let A be a very ample eﬀective divisor on Y which does not contain q. Let A = φ∗ (A). Then A ∩ C = ∅ so OX (A) ⊗ OC ∼ = OC . There exist m1 , . . . , m10 , n ∈ Z such that A ∼ nH + m1 E1 + · · · + m10 E10 by (20.8). Thus OC ∼ = OX (A) ⊗ OC ∼ = OX (nH + m1 E1 + · · · + m10 E10 ) ⊗ OC ∼ = Oγ (n(q1 + q2 + q3 ) + m1 p1 + · · · + m10 p10 ) so n(q1 + q2 + q3 ) + m1 p1 + · · · + m10 p10 ∼ 0 on γ. Thus m1 = · · · = m10 = n = 0, so that A = 0. Since φ∗ : Cl(Y ) → Cl(X) is injective (by Proposition 13.39), we have that A ∼ 0 on Y . But this is impossible since A is a hyperplane section of Y . We have the following necessary condition for the contractibility of a union of curves. Theorem 20.12. Suppose that S is a nonsingular projective surface and C = {C1 , . . . , Cn } is a ﬁnite set of curves on S such that C is contractible. Then the intersection matrix A = ((Ci · Cj )) is negative deﬁnite.
20.2. Contractions and linear systems
387
Proof. Let φ : S → S be the contraction of C and let H be a very ample divisor on S , which we may assume does not contain the point which is the image of the Ci (Lemma 13.12). Let D = φ∗ (H). Then (D 2 ) = (H 2 ) > 0 (Proposition 19.12 and Corollary 19.18) and (D · Ci ) = 0 for all i. With the notation following the Hodge index theorem, Theorem 20.8, let L = {v ∈ Num(S)  (v · D) = 0}. The restriction of the intersection form to L is negative deﬁnite as commented after the proof of Theorem 20.8. If C = {C1 , . . . , Cn } is a ﬁnite set of curves on a nonsingular projective surface S over C such that the union of the curves in C is connected and the intersection matrix (Ci · Cj ) is negative deﬁnite, then although there may not be a contraction of these curves by a regular map, there does exist an analytic map φ : S → T where T is a complex analytic (but not necessarily algebraic) normal surface such that φ is an analytic contraction of the C. This is proven by Grauert in [61]. We also have [11] that C is contractible on a nonsingular projective surface S over a ﬁeld k which is an algebraic closure of a ﬁnite ﬁeld if and only if the intersection matrix (Ci · Cj ) is negative deﬁnite (and the union of the curves in C is connected), showing that the assumption that k is not an algebraic closure of a ﬁnite ﬁeld is necessary in Example 20.11. Theorem 20.10 is the ﬁrst result in a general philosophy that curves with negative intersection number with the canonical divisor play a major role in geometry; a nonsingular closed rational curve E on a nonsingular projective surface S with (E 2 ) = −1 satisﬁes (E · KS ) = −1 by adjunction (Theorem 20.2). This philosophy has been realized to a remarkable degree. The theory for projective surfaces is classical. In higher dimensions, much of the theory has been developed, although many questions still remain (especially in positive characteristic). A few papers and articles on this subject, which contain detailed references, are Mori’s papers [112], [113], and [114], the book [91] by Koll´ ar and Mori, and the article [20] by Birkar, Cascini, Hacon, and McKernan. Let D be an eﬀective divisor on a normal projective variety X. The ﬁxed component Bn of the complete linear system nD is the largest eﬀective divisor Bn ≤ nD such that Γ(X, OX (nD − Bn )) = Γ(X, OX (nD)). Lemma 20.13. Suppose that D is an eﬀective divisor on a normal projective variety X. Let R[D] = n≥0 Γ(X, OX (nD)). Then R[D] is not a ﬁnitely generated kalgebra if 1) Bn = 0 for all n > 0, 2) Bn is bounded from above.
388
20. Surfaces
Proof. Assume that 1) and 2) hold and that R[D] is a ﬁnitely generated kalgebra. Then there exists a positive integer N , λ(i) ∈ Z+ and ui ∈ Γ(X, OX (λ(i)D)) for 1 ≤ i ≤ N such that Γ(X, OX (nD)) is spanned by the products #N % $ ν ui i  νi ≥ 0 for all i and λ(i)νi = n . i=1
Thus Bn ≥ glb
#N
ν i Bi 
% λ(i)νi = n
i=1
(glb means “greatest lower bound”). We have mBn ≥ Bmn for all m, n ∈ N. Thus Ni ! Bi ≥ BN ! for i = 1, 2, . . . , N . The divisor BN ! is nonzero by 1). Thus B1 , B2 , . . . , BN have at least one prime divisor as a common component, so condition 2) cannot be satisﬁed, as νi → ∞ as n → ∞. The following example is given by Zariski in [159]. Theorem 20.14 (Zariski). Suppose that k is a ﬁeld which is not an algebraic closure of a ﬁnite ﬁeld. Then there is an eﬀective divisor D on a nonsingular projective surface X over k such that R[D] = n≥0 Γ(X, OX (nD)) is not a ﬁnitely generated kalgebra. Proof. Let X = P2k , let E be a nonsingular cubic curve on X , and let H be a line on X . Let α be a divisor on E such that OE (α) ∼ = OX (H +E )⊗OE . 0 The elliptic curve E is isomorphic to Cl (E ) as a group by Theorem 18.19 and is explained before Proposition 18.44. By Theorem 18.49, there exists a divisor β on E of degree 0 such that the class of β has inﬁnite order in Cl0 (E ). The genus g of E is 1, as commented before Lemma 18.39. We compute deg α = ((H + E ) · E ) = 12 (by Example 19.14), so deg(α − β) = 12 > 2g + 1 = 5, so α − β is a very ample divisor on E by Theorem 18.21. Thus there exist distinct points p1 , . . . , p12 ∈ E such that α − β ∼ p1 + · · · + p12 by Theorem 16.1. Now n(p1 + · · · + p12 ) − nα ∼ −nβ for all n ∈ Z, so (20.9)
n(p1 + · · · + p12 ) ∼ nα
for all 0 = n ∈ Z. Let π : X → X be the blowup of the points p1 , . . . , p12 , with exceptional divisors F1 , . . . , F12 . Let Γ ∈ H  be an irreducible curve which does not pass through any of the points p1 , . . . , p12 (Theorem 16.1 and since H  is base point free). Let Γ = π ∗ (Γ ) and let E be the strict transform of E on X, so that π ∗ (E ) = E + F1 + · · · + F12 . Set D = Γ + E. We will show that the ﬁxed component Bn of nD is precisely E (for n ∈ Z+ ). The theorem will then follow from Lemma 20.13.
20.2. Contractions and linear systems
389
The restriction of π to E is an isomorphism onto E . We compute (E · E) = ((3Γ − F1 − · · · − F12 ) · E) = 3(H · E ) − 12 = −3, so for m, n ∈ Z, ((nΓ + mE) · E) = 3n − 3m. The canonical divisor KE is zero, since E is an elliptic curve (Corollary 18.18). By Serre duality, Corollary 18.10, h1 (E, OX (nΓ + mE) ⊗ OE ) = 0
if n ≥ 1 and 0 ≤ m ≤ n − 1
and (20.10) OX (nΓ+mE)⊗OE is generated by global sections if n ≥ 1 and 0 ≤ m ≤ n − 1 by Theorem 18.21. However, when 0 = m = n, we have OX (n(Γ + E)) ⊗ OE ∼ = OX (n(Γ + π ∗ (E ) − F1 − · · · − F12 )) ⊗ OE ∼ = OX (n(H + E )) ⊗ OE (−n(p1 + · · · + p12 )) ∼ OE (n(α − p − · · · − p )). = 1
12
Thus h0 (E, OX (nΓ + nE) ⊗ OE ) = 0 by (20.9) and h1 (E, OX (nΓ + nE) ⊗ OE ) = 0 by Theorem 18.2. We have (20.11)
H 1 (X, OX (nΓ)) = H 1 (X , OX (nΓ )) = 0
and (20.12)
H 0 (X, OX (nΓ)) = H 0 (X , OX (nΓ ))
for all n ≥ 0 by Theorem 17.32 and Theorem 17.14. We have a short exact sequence of OX modules 0 → OX (−E) → OX → OE → 0. Tensor this short exact sequence with OX (nΓ + mE) and take cohomology to obtain long exact sequences 0 → H 0 (X, OX (nΓ + (m − 1)E)) → H 0 (X, OX (nΓ + mE)) → H 0 (E, OX (nΓ + mE) ⊗ OE ) → H 1 (X, OX (nΓ + (m − 1)E)) → H 1 (X, OX (nΓ + mE)) → H 1 (E, OX (nΓ + mE) ⊗ OE ) = 0 for n ≥ 1 and 0 ≤ m ≤ n. By induction on m and (20.11), we have that H 1 (X, OX (nΓ + mE)) = 0
for n ≥ 1 and 0 ≤ m ≤ n.
The OX module OX (nΓ) is generated by global sections by (20.12), so OX (nΓ + E) is generated by global sections except possibly at points of E. But H 0 (X, OX (nΓ + E)) surjects onto H 0 (E, OX (nΓ + E) ⊗ OE ) and
390
20. Surfaces
OX (nΓ + E) ⊗ OE is generated by global sections by (20.10) if n ≥ 2, so OX (nΓ + E) is generated by global sections (if n ≥ 2). By induction and (20.10), OX (nΓ + (n − 1)E) is generated by global sections for all n ≥ 1. Now H 0 (X, OX (nΓ + nE) ⊗ OE ) = 0, so the ﬁxed component Bn of nD = nΓ + nE is E. This type of example cannot occur if k is an algebraic closure of a ﬁnite ﬁeld. If D is an eﬀective divisor on a nonsingular projective surface X over a ﬁeld k which is the algebraic closure of a ﬁnite ﬁeld, then Γ(X, OX (nD)) R[D] = n≥0
is a ﬁnitely generated kalgebra by [44, Theorem 3]. Exercise 20.15. Suppose that C is a nonsingular curve of degree d in P2 . Show that 1 g(C) = (d − 1)(d − 2). 2 Exercise 20.16. Let C be a nonsingular curve and S = C × P1 . Let π : S → C be the projection onto C. a) Suppose that p, q ∈ C. Show that π ∗ (p) ≡ π ∗ (q). b) Let C be an elliptic curve over an algebraically closed ﬁeld k which is not the algebraic closure of a ﬁnite ﬁeld, so there exist p, q ∈ C such that the class of p−q in Cl0 (C) has inﬁnite order (by Theorem 18.49). Show that nπ ∗ (p) ∼ mπ ∗ (q) for all m, n ∈ Z. Exercise 20.17. Let D be the divisor constructed in Theorem 20.14. Show that D is numerically eﬀective; that is, (D · C) ≥ 0 for all curves C on X. Exercise 20.18. Compute the functions hi (X, OX (nD)) for i = 0, 1, 2 and n ∈ N for the divisor D constructed in Theorem 20.14. Exercise 20.19. Modify the proof of Theorem 20.14 by starting with a divisor β on E which has degree 0 and ﬁnite order r in Cl0 (E ). (We can always ﬁnd such a β if r is relatively prime to the characteristic of k by Proposition 18.50). Let X and D be the surface and divisor which we construct. Is R[D] a ﬁnitely generated kalgebra? Is it generated in degree 1? Compute the functions hi (X, OX (nD)) for i = 0, 1, 2 and n ∈ N.
Chapter 21
Ramiﬁcation ´ and Etale Maps In this chapter, we consider algebraic analogues of the topological covering spaces and branched (ramiﬁed) covers in the Euclidean topology. Finite maps correlate with branched covers and the algebraic counterpart of the topological covering spaces are the ´etale morphisms (Deﬁnition 21.79). A regular map φ : X → Y of nonsingular complex varieties is a topological covering space in the Euclidean topology if and only if φ is ´etale, by the analytic implicit function theorem (Exercise 21.87). The concept of a covering space in the Zariski topology is much too restrictive a notion; there are many ﬁnite maps of nonsingular complex varieties which are ´etale but are not covering spaces in the Zariski topology (Exercise 21.88). A related concept to ´etale is the more classical notion of an unramiﬁed map (Section 21.4). If φ : X → Y is a ﬁnite regular map of varieties and Y is normal, then φ is ´etale if and only if φ is unramiﬁed (Proposition 21.84). We develop various characterizations of the concepts of ´etale and unramiﬁed maps. We deﬁne Galois maps of varieties (Section 21.7). We develop the concept of completion (Section 21.5) and prove the local form of Zariski’s main theorem (Proposition 21.54) which we earlier used in the proof of global forms of Zariski’s main theorem (Chapter 9) and prove Zariski’s subspace theorem (Proposition 21.61). We prove the purity of the branch locus, Theorem 21.92, which tells us that if φ : X → Y is a ﬁnite separable regular map, Y is nonsingular, and X is normal, then the ramiﬁcation locus of φ has pure codimension 1 (all irreducible components of the ramiﬁcation locus have codimension 1). We also prove the AbhyankarJung theorem (Theorem 21.93) showing that tamely ramiﬁed covers have a very simple form.
391
´ 21. Ramiﬁcation and Etale Maps
392
Most of the results in this chapter are due to Zariski and Abhyankar.
21.1. Norms and Traces We summarize some properties of norms and traces which will be useful in this chapter. Suppose that L is a ﬁnite extension of a ﬁeld K and K is an algebraic closure of K. Let σ1 , . . . , σr be the distinct embeddings of L into K by Khomomorphisms. For α ∈ L, the norm of α over K is deﬁned to be r [L:K]i $ σi (α) . (21.1) NL/K (α) = i=1
The trace of α over K is deﬁned to be (21.2)
TrL/K (α) = [L : K]i
r
σi (α) .
i=1
The inseparability index [L : K] is deﬁned in formula (1.2). Theorem 21.1. Let L/K be a ﬁnite extension. Then the norm NL/K is a multiplicative homomorphism of L× into K × (where L× and K × are the respective multiplicative groups of nonzero elements of L and K) and the trace is an additive homomorphism of L into K. If K ⊂ F ⊂ L is a tower of ﬁelds, then the two maps are transitive; in other words, NL/K = NF/K NL/F
and
TrL/K = TrF/K TrL/F .
If L = K(α) and f (t) = tn + an−1 tn−1 + · · · + a0 ∈ K[t] is the minimal polynomial of α over K, then NL/K (α) = (−1)n a0
and
Proof. [95, Theorem 5.1, page 285].
TrL/K (α) = −an−1 .
We now give an alternate construction of the trace and norm. Suppose that L is a ﬁnite ﬁeld extension of K and ω1 , . . . , ωn is a basis of L over K. For α ∈ L, we have expressions n aij ωj (21.3) αωi = j=1
with aij ∈ K. Let A = Aα = (aij ). From (21.3), we have Det(αIn − A) = 0. Letting X be an indeterminate, we deﬁne the ﬁeld polynomial PL/K,α (X) ∈ K[X] as PL/K,α (X) = Det(XIn − A) ∈ K[X]. By its construction, we see that the ﬁeld polynomial is independent of choice of basis of L. It will only be the minimal polynomial of α over K if L = K(α).
21.2. Integral extensions
393
Proposition 21.2. For α ∈ L, NL/K (α) = Det(Aα )
and
TrL/K (α) = trace(Aα ) =
n
aii .
i=1
Proof. This follows from [160, formulas (6) and (7) on page 87 and (19) and (20) on page 91].
21.2. Integral extensions Suppose that K is a ﬁeld and K ∗ is a ﬁnite extension ﬁeld. Suppose that A is a subring of K whose quotient ﬁeld is K and B is a subring of K ∗ whose quotient ﬁeld is K ∗ and that B is integral over A. If A is Noetherian and normal, B is the integral closure of A in K ∗ , and K ∗ is ﬁnite separable over K, then B is a ﬁnite Aalgebra [160, Corollary 1, page 265]. However, there exist examples of Noetherian domains A such that the normalization of A (in K) is not Noetherian and examples of normal Noetherian domains A whose integral closure in a ﬁnite (necessarily inseparable) ﬁeld extension is not Noetherian [121, Example 5, page 207]. This type of pathology cannot occur when A is a localization of a ﬁnite type algebra over a ﬁeld (Theorem 1.54) and, more generally, when A is excellent [107, Chapter 13]. Suppose that P and Q are respective prime ideals of A and B. We say that Q lies over P or P lies below Q if Q ∩ A = P . We have the following useful lemmas. Lemma 21.3 (Going up theorem). Let P and P ∗ be prime ideals in A such that P ∗ ⊂ P and let Q∗ be a prime ideal in B lying over P ∗ . Then there exists a prime ideal Q in B lying over P such that Q∗ ⊂ Q. Proof. [160, Corollary, page 259].
Corollary 21.4. Let notation be as above. Then: 1) Two distinct prime ideals P and Q of B such that P ⊂ Q cannot lie over the same prime ideal of A. 2) Let P be a prime ideal of B lying over P . For P to be a maximal ideal of B it is necessary and suﬃcient that P be a maximal ideal of A. Proof. [160, Complements 1) and 2) on page 259].
Lemma 21.5 (Going down theorem). Suppose that A is normal. Let P ⊂ Q be prime ideals in A and let Q∗ be a prime ideal in B lying over Q. Then there exists a prime ideal P ∗ in B lying over P with P ∗ ⊂ Q∗ .
394
Proof. [160, Theorem 6, page 262].
´ 21. Ramiﬁcation and Etale Maps
The example on page 32 [107] and the remark on page 37 [107] show that the assumption that A is normal is necessary in Lemma 21.5. Deﬁnition 21.6. An extension L of a ﬁeld K is a normal extension of K (L is normal over K) if L is an algebraic extension of K and if every irreducible polynomial f (x) ∈ K[x] which has a root in L factors completely in L[x] into linear factors. Lemma 21.7. A ﬁnite extension L of a ﬁeld K is normal over K if and only if it satisﬁes the following condition: If M is any extension of L, then any Khomomorphism of L into M is necessarily a Kautomorphism of L. Proof. [160, Theorem 15, page 77].
Lemma 21.8. Suppose that R is an integrally closed local domain with quotient ﬁeld K and T is the integral closure of R in a ﬁnite normal extension L of K. Let {mi } be the prime ideals of T lying over mR . Then the mi are the maximal ideals of T and Aut(L/K) acts transitively on the set {mi }, so that {mi } is a ﬁnite set. Proof. The fact that the mi are the maximal ideals of T follows from 2) of Corollary 21.4. Let G = Aut(L/K). We have that σ(T ) = T for σ ∈ G since T is the integral closure of R. Thus σ permutes the mi . Suppose there exists an mj such that mj is not a conjugate σ(m1 ) of m1 . We will derive a contradiction. By Theorem 1.5, there exists x ∈ mj such that x ∈ σ(m1 ) for all σ ∈ G. Thus none of the conjugates σ(x) of x are in m1 , and thus no power of a product of conjugates of x is in the prime ideal m1 . By (21.1), y = NL/K (x) ∈ K is of this form (a power of a product of conjugates of x). Thus y ∈ K ∩ T = R since R is integrally closed. Finally, y ∈ mj ∩ R = mR . Since mR ⊂ m1 and m1 is a prime ideal, this is a contradiction. With the notation introduced in the ﬁrst paragraph of this section, suppose that P is a prime ideal of A. Then there exists a prime ideal Q of B lying over P by Proposition 1.56, and if A is normal, then there are only a ﬁnite number of prime ideals in B lying over P (by Lemma 21.8, applied to the integral closure of AP in a ﬁnite normal extension of K containing K ∗ ). Let R be a normal (not necessarily Noetherian) local domain with maximal ideal mR and quotient ﬁeld K, and let R∗ be the integral closure of R in a ﬁnite extension ﬁeld K ∗ of K. Let m∗1 , . . . , m∗r be the prime ideals of R∗ which lie over mR . The m∗i are the maximal ideals of R∗ by Corollary ∗ 21.4. Let Si = Rm ∗ for 1 ≤ i ≤ r. We shall say that the Si are the local i ∗ rings of K lying above (or over) R and that R is the local ring of K lying below Si .
21.2. Integral extensions
395
Lemma 21.9. With the notation introduced in the above paragraph, we have that Si ∩ K = R for all i. ˜ be the ˜ be a ﬁnite normal extension of K containing K ∗ . Let R Proof. Let K ˜ lying ˜ and let m ˜ t be the maximal ideals of R integral closure of R in K ˜ 1, . . . , m ∗ ∗ ˜ ˜ ˜ 1 ∩ R = mi . Let Sj = Rm over mR . We may assume that m ˜ j for 1 ≤ j ≤ t. ˜ Let G = Aut(K/K). We have that R ⊂ Si ∩ K ⊂ S˜1 ∩ K so it suﬃces to show that S˜1 ∩ K = R. Let u ∈ S˜1 ∩ K. Given S˜j , there exists g ∈ G such that g(S˜1 ) = S˜j , by Lemma 21.8, and hence g(u) ∈ S˜j . But u ∈ K implies ˜ by Lemma g(u) = u. Hence u ∈ S˜j for i = 1, . . . , t so that u ∈ tj=1 S˜j = R ˜ ∩ K = R, so that S˜1 ∩ K = R, and hence Si ∩ K = R. 1.77. Thus u ∈ R Now, with the notation introduced in the ﬁrst paragraph of this section, suppose that A is normal (not necessarily Noetherian) and B is the integral closure of A in K ∗ . Then B ∩ K = A. Further, if K is an intermediate ﬁeld between K and K ∗ , then B ∩ K is the integral closure of A in K . We saw above that if P ∗ is a prime ideal of B and P = P ∗ ∩ A, then BP ∗ ∩ K = AP . Suppose that C is a normal subring of K ∗ whose quotient ﬁeld is K ∗ . Then C ∩ K is normal, but it can happen that the quotient ﬁeld of C ∩ K is not equal to K. Here is a simple example which was constructed by Bill Heinzer. Let x and y be algebraically independent over a ﬁeld κ, and let S ∗ = κ[x3 , x2 y](x3 ,x2 y) . Consider the automorphism of the quotient ﬁeld K ∗ = κ(x3 , x2 y, xy 2 , y 3 ) of the regular local ring S ∗ over κ which interchanges x and y. The image of S ∗ by this automorphism is the twodimensional regular local ring S = κ[y 3 , y 2 x](y3 ,y2 x) . Regarding S ∗ and S as subrings of the formal power series ring κ[[x, y]], we see that the intersection of S ∗ and S is κ. Hence if K is the ﬁxed ﬁeld of this automorphism of K ∗ , we have S ∗ ∩ K = S ∩ K = κ. However, every valuation ring V of K ∗ has the property that K ∩ V is a valuation ring of K, and hence its quotient ﬁeld is K. This follows since the restriction ν of a valuation ν ∗ of K ∗ to K is a valuation of K, and V ∩ K is the set of all element in K whose value is nonnegative. A valuation ν of a ﬁeld K is called a discrete valuation if its value group is Z. A local domain R is called a discrete valuation ring (dvr) if R is the valuation ring of the quotient ﬁeld of R. The divisorial valuations, deﬁned in Section 12.3, are examples of discrete valuation rings. Theorem 21.10. A domain R is a discrete valuation ring if and only if R is a onedimensional regular local ring. Proof. Let K be the quotient ﬁeld of R. First suppose that R is a onedimensional regular local ring. If 0 = f ∈ R, then there exists n ∈ N such
´ 21. Ramiﬁcation and Etale Maps
396
n that 0 = f ∈ mnR \ mn+1 R . Since mR = (t) for some t ∈ R, f = t v with v ∈ R \ mR . Thus v is a unit in R. Since K is the quotient ﬁeld of R, every nonzero element f ∈ K has a unique expression
f = tn v
(21.4)
with n ∈ Z and v ∈ R a unit. Deﬁne ν : K × → Z by ν(f ) = n if f has the expression (21.4). From (21.4), it follows that ν is a valuation of K and that R is the valuation ring of ν. Now suppose that R = Vν where ν is a discrete valuation of K. Then R = {f ∈ K  ν(f ) ≥ 0} (with the convention that ν(0) = ∞). We have that R is a Noetherian local domain by Theorems 12.15 and 12.16. There exists t ∈ R such that ν(t) = 1. Suppose that f ∈ R is nonzero. Let ν(f ) = n ≥ 0 and let u = tfn . Then ν(u) = 0 so u ∈ R. Further, ν( u1 ) = −ν(u) = 0 so u1 ∈ R and thus u is a unit in R. It follows that mR = (t) and so dimR/mR mR /m2R = 1. Now R = K since 1t ∈ R. Thus 1 ≤ dim R ≤ dimR/mR mR /m2R = 1 by Theorem 1.81, so R is a onedimensional regular local ring.
From the above proof, we have the following remark. Remark 21.11. The maximal ideal mR in a discrete valuation ring R is a principal ideal, mR = (t), and the nonzero ideals in R are the principal ideals (tn ) for n ∈ N. Proposition 21.12. Let K be a ﬁeld, ν a valuation of K, K ∗ an algebraic extension of K, and R1∗ , R2∗ , . . . the local rings in K ∗ lying over Vν . Then Ri∗ is the valuation ring Vνi∗ of a valuation νi∗ of K ∗ where ν1∗ , ν2∗ , . . . are exactly the extensions of ν to K ∗ .
Proof. [6, Proposition 2.38]
Let R be a onedimensional regular local ring with quotient ﬁeld K and let L be a ﬁnite extension of K. Let S1 , . . . , Sg be the local rings of L which lie over R. The Si are onedimensional regular local rings by [160, Theorem 19 on page 281], [160, Theorem 13, page 275], and Theorem 1.87. Since the maximal ideals mR of R and mSi of the Si are principal, there exist positive integers ni such that mR Si = mnSii for 1 ≤ i ≤ g. We deﬁne (21.5)
e(Si /R) = ni
and
f (Si /R) = [Si /mSi : R/mR ].
The index f (Si /R) is ﬁnite by [160, Lemma 1 on page 284]. The index e(Si /R) is called the reduced ramiﬁcation index of Si over R and f (Si /R) is called the relative degree of Si over R. The reduced ramiﬁcation index has already been encountered in Section 18.5.
21.2. Integral extensions
397
We deﬁne (21.6)
e(ν ∗ /ν) = e(Si /R)
and
f (ν ∗ /ν) = f (Si /R)
if R = Vν and Si = Vν ∗ where ν is a discrete valuation of K and ν ∗ is an extension of ν to K ∗ (which is necessarily a discrete valuation). Lemma 21.13 (Kronecker). Let A be a normal domain with quotient ﬁeld K. Let f (t), g(t) be monic polynomials in K[t] and let h(t) = f (t)g(t). Then h(t) ∈ A[t] implies f (t), g(t) ∈ A[t]. Let K ∗ be an overﬁeld of K. If u ∈ K ∗ is such that u is integral over A, then the minimal polynomial of u over K is in A[t]. Proof. The second statement of the lemma follows from the ﬁrst. We will prove the ﬁrst statement. Let L be a splitting ﬁeld of h(t) over K. Let f (t) =
n i=0
i
fn−i t =
n $
(t − ui )
i=1
where fi ∈ K, f0 = 1, and ui ∈ L. Let pi (Y1 , . . . , Yn ) be (−1)i times the ith elementary symmetric function in Y1 , . . . , Yn , so that fi = pi (u1 , . . . , un ). Now f (ui ) = 0 implies h(ui ) = 0 which implies that the ui are integral over A. Since the integral closure of A in L is a domain, the fact that u1 , . . . , un are integral over A implies fi = pi (u1 , . . . , un ) are integral over A. Since A is normal and fi ∈ K, we must have fi ∈ A for i = 1, . . . , n, so f (t) ∈ A[t]. Similarily, g(t) ∈ A[t]. Theorem 21.14. Let A be an integral domain, let K be its quotient ﬁeld, and let x be an element of an extension of K. Suppose that x is integral over A. Then x is algebraic over K, and the coeﬃcients of the minimal polynomial f (t) of x over K, in particular the norm and trace of x over K, are elements of K which are integral over A. If A is integrally closed, these coeﬃcients are in A. Proof. Let L be the algebraic closure of K. Since x is integral over A, it is necessarily algebraic over K. Let n be the degree of the minimal polynomial f (x) of x over K. Then f (t) = (t − xi ) ∈ L[t] where the xi are conjugates of x in L over K. Since an equation of integral dependence of x over A is satisﬁed by all the conjugates xi of x over K, the coeﬃcients of f (t) are integral over A by Corollary 1.50. In particular, the norm NK(x)/K (x) and trace TK(x)/K (x) are in K and are integral over A by Theorem 21.1. It follows from (21.1) and (21.2) that if L is any ﬁnite extension of K containing x, then NL/K (x) and TL/K (x) are in K and are integral over A. Exercise 21.15. Let f (t) ∈ tk[[t]] be transcendental over k(t). Deﬁne an injective kalgebra homomorphism φ : k[x, y] → k((t)), where k((t)) is the
´ 21. Ramiﬁcation and Etale Maps
398
quotient ﬁeld of k[[t]], by φ(g(x, y)) = g(t, f (t)) for g ∈ k[x, y], with induced inclusion φ : k(x, y) → k((t)). Deﬁne a valuation ν of k(x, y) by ν(g) = ordt (g(t, f (t))) for g ∈ k(x, y). Show that νk × = 0 so that ν is a kvaluation and ν is a dvr, but ν is not a divisorial valuation.
21.3. Discriminants and ramiﬁcation We ﬁrst deﬁne the discriminant ideal of a normal domain. The construction in this section follows [92]. It generalizes the classical construction of discriminants of ﬁeld extensions ([131], [160, Section 11 of Chapter 2]). Our treatment is based on the section “The discriminant of an ideal” in [6]. Let K be a ﬁeld and L be a ﬁnite ﬁeld extension of K. Let {w1 , . . . , wn } be a basis of L/K. For a ∈ L, let Ta : L → L be the Klinear transformation deﬁned by Ta (b) = ab for b ∈ L. For a ∈ L, the trace of a relative to L/K (Section 1.2) is n kii TrL/K (a) = i=1
where Ta (wi ) =
n
kij wj
with kij ∈ K.
j=1
For a1 , . . . , an ∈ L, the discriminant of (a1 , . . . , an ) relative to L/K is deﬁned to be DL/K (a1 , . . . , an ) = Det(TrL/K (ai aj )). Basic properties of the discriminant are derived in [160, Section 11 of Chapter 2]. The condition of the discriminant DL/K (a1 , . . . , an ) being zero or not zero is independent of choice of basis {a1 , . . . , an } of L over K, so it makes sense to say that the discriminant DL/K (a1 , . . . , an ) of a basis is zero or not zero ([160, Corollary, page 93]). The discriminant DL/K (a1 , . . . , an ) of a basis {a1 , . . . , an } of L over K is nonzero if and only if L is separable over K ([160, Theorem 22, page 95] or equation (21.2)). If L/K is separable, then L has a primitive element b over K (Theorem 1.16), and then if n = [L : K], $ (21.7) DL/K (1, b, b2 , . . . , bn−1 ) = (bi − bj )2 i<j
where the bi are the n distinct roots of the minimal polynomial f (x) of b over K in an algebraic closure of K containing L by [160, formula (7) on page 95]. Let f (x) = xn + a1 xn−1 + · · · + an be the minimal polynomial of b over K. We can then compute the discriminant DL/K (1, b, b2 , . . . , bn−1 ) as (21.8)
DL/K (1, b, b2 , . . . , bn−1 ) = (−1)
n(n−1) 2
Res(f, f )
21.3. Discriminants and ramiﬁcation
399
by [95, Proposition 8.5, page 204], where the resultant Res(f, f ) is the determinant of the (2n − 1) × (2n − 1) matrix ⎛ 1 ⎜0 ⎜ ⎜.. ⎜. ⎜ ⎜0 ⎜ ⎜n ⎜ ⎜0 ⎜ ⎜. ⎝.. 0
⎞
a1 1
a2 a1
... a2
an−1 ...
an an−1
0... an
... 0...
0 0
... (n − 1)a1 n
0 (n − 2)a2 (n − 1)a1
1 ... (n − 2)a2
a1 an−1 ...
a2 0 an−1
... ... 0
an−1 ... ...
an 0 0
0
...
0
n
(n − 1)a1
(n − 2)a2
...
an−1
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
We have that (21.9)
Res(f, f ) =
n $
f (bi )
i=1
by [95, formula (4), page 204]. The formula (21.8) is also called the discriminant of f . Let A be a normal domain with quotient ﬁeld K, and let L be a ﬁnite extension of K with [L : K] = n. Let B be a domain with quotient ﬁeld L which contains A and is integral over A. If a1 , . . . , an ∈ B, then TrL/K (ai aj ) ∈ A for all i, j by Theorem 21.14, so that DL/K (a1 , . . . , an ) ∈ A. We deﬁne the discriminant ideal D(B/A) of B over A to be the ideal in A generated by the discriminants DL/K (w1 , . . . , wn ) of all the bases {w1 , . . . , wn } of L/K which are in B. If B is the integral closure of A in L, we will sometimes write D(L/A) to denote D(B/A). The following formula is useful in computing discriminants. Suppose that {a1 , . . . , an } and {b1 , . . . , bn } are bases of L/K. Deﬁne kij ∈ K by bi = j kij aj . Then ([160, equation (2), page 93] or [6, equation (5), page 26]) (21.10)
DL/K (b1 , . . . , bn ) = Det(kij )2 DL/K (a1 , . . . , an ).
We thus have that D(B/A) localizes; that is, if S is a multiplicative set in A, then (21.11)
D(S −1 B/S −1 A) = S −1 D(B/A).
Deﬁnition 21.16. Suppose R is a local domain and S is a local domain which dominates R (R ⊂ S and mS ∩ R = mR ). We say that the extension R → S is unramiﬁed if mR S = mS and S/mS is ﬁnite and separable over R/mR . Otherwise, we say that the extension is ramiﬁed.
´ 21. Ramiﬁcation and Etale Maps
400
This deﬁnition generalizes our deﬁnition of ramiﬁcation for a regular map of nonsingular curves in Section 18.5. A dominant regular map of nonsingular projective curves φ : X → Y is ramiﬁed at P ∈ X if and only if the extension OY,φ(P ) → OX,P is ramiﬁed. Let A be a normal local domain with quotient ﬁeld K. Let L be a ﬁnite extension of K, and let B be a domain with quotient ﬁeld L which contains A and is integral over A. Let Q1 , . . . , Qt be the prime ideals in B lying over the maximal ideal mA . If the BQi are unramiﬁed over A for 1 ≤ i ≤ t, then we say that B is unramiﬁed over A. Otherwise, we will say that B is ramiﬁed over A. We will say that A is unramiﬁed in L if the integral closure B of A in L is unramiﬁed over A. Suppose that L is a ﬁnite extension ﬁeld of a ﬁeld K. Let M be the maximal separable extension ﬁeld of K in L. The separable degree of L over K is [L : K]s = [M : K] (Section 1.2). Let A be a ring containing a ﬁeld κ such that A is a ﬁnitedimensional vector space over κ. We can extend the deﬁnitions of the trace and discriminant to this situation to deﬁne the discriminant DA/κ (a1 , . . . , an ) of a basis a1 , . . . , an of A over κ. The formula (21.10) holds in this situation, so the question of whether a discriminant of A over κ is zero or nonzero is independent of the choice of a basis of A over κ. Since A is an Artinian ring, A∼ = A1 ⊕ · · · ⊕ Am where the Ai are Artin local rings by [13, Theorem 8.7]. Proposition 21.17. With the above assumptions on A and κ, a discriminant of A over κ is nonzero if and only if each Ai is a ﬁeld which is a separable ﬁeld extension of κ. Proof. Let ei = (0, . . . , 0, 1Ai , 0, . . . , 0) ∈ A (where 1Ai is in the ith position) for 1 ≤ i ≤ m. Let {wij  1 ≤ j ≤ ni } be a κbasis of Ai . Then {ei wij  1 ≤ i ≤ m, 1 ≤ j ≤ ni } is a κbasis of A. Let a ∈ A and expand a = ae1 + · · · + aem Then aei wij =
l
al el
ei wij = awij ei =
with ai ∈ Ai .
ni
(i) kju wiu
ei
(i)
for some kju ∈ κ.
u=1
Thus TrA/κ (a) = TrA1 /κ (a1 ) + · · · + TrAm /κ (am )
21.3. Discriminants and ramiﬁcation
401
and i = u implies ei wij eu wuv = 0, so that TrA/κ (ei wij eu wuv ) = 0. Thus DA/k (e1 w11 , e1 w12 , . . . , e1 w1n1 , e2 w21 , . . . , em wmnm ) = Det(TrA/κ (ei wij eu wuv )) = =
m $ i=1 m $
Det(TrA/κ (ei wij ei wiv )) DAi /κ (wi1 , . . . , wini ),
i=1
so that the discriminant of A over κ is nonzero if and only if the discriminant of Ai over κi is nonzero for i = 1, . . . , m. Now suppose that the radical N = (0) of A is nonzero, and let s > 1 be the integer which satisﬁes N s−1 = 0 and N s = 0. For 0 ≤ i ≤ s − 1, N i is a ﬁnitedimensional vector space over κ, so there exist pi1 , . . . piqi in N i whose residue classes mod N i+1 form a κbasis of N i /N i+1 . Then {pij  0 ≤ i ≤ s − 1, 1 ≤ j ≤ qi } is a κbasis of A. Now a ∈ N implies apij = kijvu pvu (21.12)
v>i
for some kijvu ∈ κ, which implies TrA/κ (a) = 0. Thus since there exists a basis {w1 , . . . , wn } of A over κ with at least the last element wn ∈ N if N = 0, (21.13)
N = 0 implies the discriminant of A/κ is zero.
The proposition now follows from (21.12), (21.13), and the fact that the discriminant of a ﬁnite ﬁeld extension L over κ is nonzero if and only if L is separable over κ [160, Theorem 22, page 95]. We will say that a ∈ A is a primitive element of A over κ if A = κ[a]. Lemma 21.18. Suppose that κ is an inﬁnite ﬁeld and A ∼ = A1 ⊕ · · · ⊕ Am where the Ai are ﬁnite separable ﬁeld extensions of κ. Then A has a primitive element over κ. Proof. Let ei = (0, . . . , 0, 1Ai , 0, . . . , 0) ∈ A (where 1Ai is in the ith position) for 1 ≤ i ≤ m. Let aj be primitive elements of Aj over κ (Theorem 1.16) and let gj (x) ∈ κ[x] be the respective minimal polynomials of the aj over κ. Since κ is assumed to be inﬁnite and two polynomials in κ[x] are coprime if and only if they have no common roots in an algebraic closure of κ, after possibly multiplying the aj by suitable nonzero elements of κ, we may assume that the gj (x) are pairwise coprime in κ[x]. Let a = e1 a1 +· · ·+em am . Suppose f (x) = b0 xq + b1 xq−1 + · · · + bq ∈ κ[x] satisﬁes f (a) = 0. Then
402
´ 21. Ramiﬁcation and Etale Maps
0 = ej f (a) = ej f (aj ) so that f (aj ) = 0 for 1 ≤ j ≤ m and thus gj (x) divides f (x) in κ[x] for 1≤ j ≤ m, so that the minimal m polynomial g(x) of a over g (x). Since dim A = κ is divisible by m j κ j=1 i=1 deg(gi (x)), we have that a is a primitive element of A over κ. Lemma 21.19. Suppose that the ring A is a ﬁnitedimensional vector space over an inﬁnite ﬁeld κ, so that A = A1 ⊕ · · · ⊕ Am where the Ai are Artin local rings. Suppose that some Ai is not a separable ﬁeld extension of κ. Let Ni be the maximal ideal of Ai . Then there exists a ∈ A such that the minimal polynomial of a in κ[x] has degree > m i=1 [Ai /Ni : κ]s . Proof. We will ﬁrst ﬁnd ai ∈ Ai such that the minimal polynomial of ai over κ has degree ≥ [Ai /Ni : κ]s , with a strict inequality if Ai /Ni is not a separable ﬁeld extension of κ. Let bi ∈ Ai be such that the minimal polynomial of the residue bi of bi in Ai /Ni has degree [Ai /Ni : κ] if Ai /Ni is separable over κ and has degree ≥ [Ai /Ni : κ]s p, where p is the characteristic of κ if Ai /Ni is not separable over κ (by the primitive element theorem in [142, page 139]). If Ni = 0, then we take ai = bi . Suppose Ni = 0 and Ai /Ni is separable over κ. Let hi (x) be the minimal polynomial of bi over κ. If hi (bi ) = 0, then we can take ai = bi , so suppose hi (bi ) = 0. There exists 0 = ci ∈ Ni . Let ai = bi + ci . Then hi (bi )r = 0 for all r, where hi (x) is the derivative of hi (x), so that hi (bi ) ∈ Ni . We have hi (ai ) = ci [hi (bi )+ci f ] with f ∈ Ai . Thus hi (bi ) + ci f ∈ Ni which implies that hi (bi ) + ci f is a nonzero divisor in Ai , so that hi (ai ) = 0. Since hi (ai ) = hi (bi ) = 0, hi (ai )r = 0 and hi (ai )r−1 = 0 for some r > 1 and so hri (x) is the minimal polynomial of ai over κ, since hi (x) is irreducible in κ[x]. Now by the argument of the proof of Lemma 21.18, we can ﬁnd a ∈ A satisfying the conclusions of the lemma. Theorem 21.20. Let R be a normal local domain with quotient ﬁeld K and K ∗ be a ﬁnite algebraic extension of K. Let R∗ be a domain with quotient ﬁeld K ∗ such that R∗ contains and is integral over R. Then D(R∗ /R) = R implies R∗ is unramiﬁed over R. Proof. Let N ∗ = mR R∗ , κ = R/mR and A = R∗ /N ∗ . We will denote the class of the residue of an element x ∈ R∗ in A by x. Since D(R∗ /R) = R, there exists a basis w1 , . . . , wn of K ∗ over K in R∗ such that D(w1 , . . . , wn ) is a unit in R. Given w ∈ K ∗ , we have that w = a1 w1 + · · · + an wn with aj ∈ K. If w ∈ w1 R + · · · + wn R, then some aj ∈ R, say a1 ∈ R. It follows that a21 ∈ R, since if a21 ∈ R, then a1 is integral over R, which is impossible since R is normal. Then D(w, w2 , . . . , wn ) = a21 D(w1 , w2 , . . . , wn ) ∈ R
21.3. Discriminants and ramiﬁcation
403
implies w is not integral over R, since if w is integral over R, then D(w, w2 , . . . , wn ) ∈ R ∩ K = R, where R is the integral closure of R in K ∗ (by Theorem 21.14). Since w1 R + · · · + wn R ⊂ R∗ , we have that R∗ = w1 R + · · · + wn R is the integral closure of R in K ∗ . Thus w1 , . . . , wn is a free Rbasis of R∗ , and so N ∗ = mR R∗ = w1 mR + · · · + wn mR and A is a free κmodule of rank n with basis w1 , . . . , wn . Let (wi wj )wp =
n
aijpq wq
q=1
with aijpq ∈ R. Then (wi wj )wp =
n
aijpq wq .
q=1
Thus D(w1 , . . . , wn ) is the discriminant of D(w1 , . . . , wn ) modulo mR . Hence D(w1 , . . . , wn ) = 0, since D(w1 , . . . , wn ) is a unit in R. Thus R∗ is unramiﬁed over R by Proposition 21.17. Theorem 21.21. Let R be a Noetherian normal local domain and let K be the quotient ﬁeld of R. Let K ∗ be a ﬁnite extension of K, and let R∗ be the integral closure of R in K ∗ . Suppose that R∗ is a ﬁnitely generated Rmodule. Then R∗ is unramiﬁed over R if and only if the discriminant ideal D(R∗ /R) = R. The separability index [L : K]s of a ﬁnite extension of ﬁelds L/K is deﬁned in (1.2). Proof. The “if” direction follows from Theorem 21.20. We will prove the “only if” direction. With our assumptions, R∗ is a ﬁnite Rmodule. Let κ R/mR and A = R∗ /mR R∗ . The ring A is a ﬁnite κvector space, so A = = r ∗ i=1 Ai where the Ai are Artin local rings. Since R is unramiﬁed over R, we have that each Ai is a separable ﬁeld extension of κ. Let u1 , . . . , un ∈ R∗ be such that their residues u1 , . . . , un in A are a κbasis of A. By Nakayama’s lemma (Lemma 1.18), (21.14)
R∗ = u1 R + · · · + un R.
´ 21. Ramiﬁcation and Etale Maps
404
Let L = u1 K + · · · + un K, a subring of K ∗ by (21.14). Since L is a ﬁnitedimensional Kvector space, L is a subﬁeld of K ∗ , so that L = K ∗ . Thus ∗
[K : K] ≤ n = [A : κ] =
t
[Ai : k ]s .
i=1
The proof for general κ now follows from [6, Theorem 1.42] (referring to [92] in the case when κ is ﬁnite). We will present here a proof with the assumption that κ is an inﬁnite ﬁeld. Since the Ai are separable over the inﬁnite ﬁeld κ, there exists a primitive element v ∈ A of A over κ by Lemma 21.18. Let v be a lift of v to R∗ . Let f (x) ∈ K[x] be the minimal polynomial of v over K. Then f (x) ∈ R[x] by Theorem 21.14 since R is normal. Let h(x) ∈ κ[x] be the minimal polynomial of v over κ. Then h divides the reduction f of f in κ[x], so (21.15)
[K ∗ : K] ≥ deg(f ) ≥ deg(h) = [A : κ] = n.
Thus [K ∗ : K] = [A : κ] = n. Now Proposition 21.17 implies that the discriminant D(u1 , . . . , un ) of A over κ is nonzero. Thus D(u1 , . . . , un ) ∈ mR , which we have shown is in the discriminant ideal of R∗ over R, so D(R∗ /R) = R. Theorem 21.22. Let A be a normal domain with quotient ﬁeld K. Let L be a ﬁnite extension of K, and let B be the integral closure of A in L. Let P be a prime ideal in A. Let Q1 , . . . , Qt be the prime ideals in B lying over P . Let κ = AP /P AP and κi = BQi /Qi BQi for 1 ≤ i ≤ t. Then (21.16)
t [κi : κ]s ≤ [L : K], i=1
and equality holds if and only if the discriminant ideal D(B/A) ⊂ P . Proof. We will prove the theorem with the assumptions that A is Noetherian, κ is inﬁnite, and B is ﬁnite over A, referring to [6, Theorem 1.45] for the general case. Write P BP = I1 ∩ · · · ∩ It where the Ii are Qi primary ideals by Lemma 21.8. There exists r > 0 such that Qri ⊂ Ii for 1 ≤ i ≤ t. We have that t BP /P BP ∼ = i=1 BP /(Qri BP + P BP ) by Theorem 1.5 t ∼ = i=1 BQi /(Qri BQi + P BQi ) by Lemma 1.28 t ∼ = i=1 BQi /Ii BQi . We have a natural surjection BP /P BP → ti=1 κi . By the theorem of the primitive element (Theorem 1.16) there exist ui ∈ κi such that the minimal polynomial gi (x) ∈ κ[x] of ui has degree equal to [κi : κ]s . Since κ is assumed to be inﬁnite, we can replace the ui with suitable products of the ui with elements of κ to get that the minimal polynomials gi (x) are pairwise coprime
21.3. Discriminants and ramiﬁcation
405
in κ[x]. Let v = u1 + · · · + ut . Then the minimal polynomial g(x) ∈ κ[x] of v is divisible by ti=1 gi (x). Let v be a lift of v to BP . Let f (x) ∈ K[x] be the minimal polynomial of v over K. Then f (x) ∈ AP [x] by Theorem 21.14. Let f (x) be the reduction of f (x) in κ[x]. Then f (v) = 0, so that g(x) divides f (x) and thus [L : K] ≥ [K(v) : K] ≥ deg g(x) ≥
t
[κi : κ]s .
i=1
Let R = AP and S be the integral closure of R in L. We have that D(B/A)P = D(S/R) by (21.11) since S = T −1 B where T = A \ P . Thus D(S/R) = R if and only if D(B/A) ⊂ P . By Theorem 21.21, D(S/R) = R implies R → S is unramiﬁed, and it is shown in the proof of Theorem 21.21 that R → S unramiﬁed implies equality in (21.16). Assume equality holds in t (21.16). Then BP /P BP ∼ = i=1 κi and each κi is a separable extension of κ by Lemma 21.19. Thus BP /P BP has a primitive element u over κ by Lemma 21.18, and the discriminant D(1, u, . . . , un−1 ) = 0 by Proposition 21.17. Let u be a lift of u to BP . Then the residue of D(1, u, . . . , un−1 ) ∈ AP in κ is D(1, u, . . . , un−1 ), so that D(1, u, . . . , un−1 ) is a unit in AP . Now u is a primitive element of L over K since 1, u, . . . , un−1 are linearly independent over κ, since D(1, u, . . . , un−1 ) = 0 and by (21.10), so that L = K(u) since [L : K] = n. Thus D(S/R) = R. A particularly strong form of Theorem 21.22 holds if A is a Dedekind domain, as we have seen in the case of curves, Theorem 13.18, and more generally as shown in [160, Section 9 of Chapter V]. Proposition 21.23. Suppose that R is a onedimensional regular local ring with quotient ﬁeld K and L is a ﬁnite separable extension of K. Let S1 , . . . , Sg be the normal local rings of L which lie over R. Then the Si are onedimensional regular local rings, and the indices e(Si /R) and f (Si /R) deﬁned in equation (21.5) satisfy g
e(Si /R)f (Si /R) = [L : K],
i=1
and if L is a Galois extension of K, then the e(Si /R) all have a common value e and the f (Si /R) all have a common value f , so that ef g = [L : K]. Proof. [160, Corollary, page 287] and [160, Theorem 22, page 289].
Suppose that R is a onedimensional regular local ring with quotient ﬁeld K and L is a ﬁnite extension of K. Suppose that S is a normal local ring of L which lies over R. Then R → S is unramiﬁed if and only if e(S/R) = 1 and S/mS is a separable ﬁeld extension of R/mR .
406
´ 21. Ramiﬁcation and Etale Maps
21.4. Ramiﬁcation of regular maps of varieties Proposition 21.24. Let A be a normal domain with quotient ﬁeld K and let L be a ﬁnite extension ﬁeld of K. Suppose that the integral closure of A in L is a ﬁnite Amodule. Let U = {P ∈ Spec(A)  AP is unramiﬁed in L}. Then U = Spec(A) \ Z(D(B/A)) is an open subset of Spec(A). Proof. By Theorem 21.21, for P ∈ Spec(A), AP is unramiﬁed in L if and only if D(BP /AP ) = AP . By (21.11), D(BP /AP ) = D(B/A)P . Thus AP is unramiﬁed in L if and only if D(B/A) ⊂ P . If R is a normal local ring and L is a ﬁnite extension ﬁeld of the quotient ﬁeld of R such that the integral closure of R in L is a ﬁnite Rmodule and R is unramiﬁed in L, then RP is unramiﬁed in L for all P ∈ Spec(R). We may thus deﬁne a normal domain A to be unramiﬁed in a ﬁnite extension L of the quotient ﬁeld of A if AP is unramiﬁed in L for all P ∈ Spec(A). Deﬁnition 21.25. Let X and Y be varieties, and let φ : X → Y be a regular map. The degree of φ is
[k(X) : k(Y )] if dim X = dim Y and φ is dominant, deg(φ) = 0 otherwise. This deﬁnition was anticipated in Chapter 19. Theorem 21.26. Suppose that φ : X → Y is a dominant ﬁnite map of varieties and Y is normal. Then the number of points above any point y ∈ Y is less than or equal to deg(φ). Proof. Let U be an aﬃne neighborhood of y in Y and let V = φ−1 (U ). Then V is aﬃne and φ : V → U is ﬁnite (by Theorem 7.5). Let W be the normalization of V (Theorem 7.17). The variety W is also aﬃne and the induced map ψ : W → U is ﬁnite and factors through V . Further, deg(ψ) = deg(φ) since k(W ) = k(V ) = k(X). It thus suﬃces to prove the theorem with Y replaced by U and X replaced by W . The theorem now follows from Theorem 21.22, taking P to be the prime ideal of y in A = k[U ]. Deﬁnition 21.27. Suppose that φ : X → Y is a dominant ﬁnite map of normal varieties and y ∈ Y . We will say that φ is ramiﬁed at y if OY,y → B is ramiﬁed, where B is the integral closure of OY,y in k(X). The map φ is said to be unramiﬁed at y if this extension is not ramiﬁed. We will say that the map φ is unramiﬁed if φ is unramiﬁed at y for all y ∈ Y . We will say that φ is unramiﬁed at x ∈ X if OY,φ(x) → OX,x is unramiﬁed.
21.4. Ramiﬁcation of regular maps of varieties
407
Theorem 21.28. Suppose that φ : X → Y is a dominant ﬁnite map of normal varieties and y ∈ Y . Then φ is unramiﬁed at y if and only if the number of points in the preimage φ−1 (y) is equal to the degree deg(φ). Proof. As in the proof of Theorem 21.26, we may assume that Y and X are aﬃne. The theorem then follows from Theorems 21.22 and 21.21, with the observation that in the language of these theorems, k ∼ = A/P ∼ = B/Qi for all Qi , since P is the ideal of y in A = k[Y ] and the Qi are the ideals in B = k[X] of the points in the preimage of y by φ. Theorem 21.29. Suppose that φ : X → Y is a dominant ﬁnite map of normal varieties. Then the set of points in Y at which φ is unramiﬁed is open and is nonempty if and only if k(X) is a separable extension of k(Y ). Proof. It suﬃces to prove this in the case when Y and X are aﬃne. Let A = k[Y ] and B = k[X]. Then φ is unramiﬁed at y ∈ Y if and only if the ideal P = I(y) of y in A does not contain the discriminant ideal D(B/A) by Theorem 21.21, that is, if and only if y ∈ Z(D(B/A). Thus the set of unramiﬁed points is open. By Theorem 21.21 we have that the set of points in Y at which φ is unramiﬁed is nonempty if and only if Z(D(B/A)) = Y , which holds if and only if D(B/A) = (0). By Theorem 21.22, taking P to be the prime ideal (0) in A, we have that D(B/A) = (0) if and only if [k(X) : k(Y )]s = [k(X) : k(Y )].
Deﬁnition 21.30. Suppose that φ : X → Y is a dominant ﬁnite map of normal varieties. The closed set of points p ∈ Y at which φ is ramiﬁed is called the ramiﬁcation locus of φ in Y . Exercise 21.31. Suppose that R = κ[x1 , . . . , xm ] is a polynomial ring over a ﬁeld κ and f = z n + a1 z n−1 + · · · + an with ai ∈ R is an irreducible polynomial in R[z]. Let S = R[z]/(f ) = R[z] where z is the residue of z in S. Then S is a domain which is ﬁnite over R. Let K be the quotient ﬁeld of R and L be the quotient ﬁeld of S. Using the deﬁnition of D(S/R), show that D(S/R) is generated by DL/K (1, z, . . . , z n−1 ). Exercise 21.32. Let R = k[x] be a polynomial ring over an algebraically closed ﬁeld k of characteristic = 2 or 3 and let S = R[y]/(f ). Let X be the aﬃne variety with regular functions S and induced regular map Φ : X → A1 . In each of the following problems, show that S is normal and ﬁnite over R. Compute the ideal D(S/R) in terms of Res(f, f ) using formula (21.8). Compute the ramiﬁcation locus of the map φ in A1 . a) f = y 2 + 3xy + (x2 + 3). b) f = y 3 + x3 + 1.
´ 21. Ramiﬁcation and Etale Maps
408
21.5. Completion In this section, we will give a brief survey of the topic of completion of a ring, referring to [161, Chapter VIII], [107, Chapter 9], and [50, Chapter 7] for more details. Let A be a ring with a given topology. The ring A is called a topological ring if the ring operations are continuous. Let A be a topological ring. An Amodule E, with a given topology, is said to be a topological Amodule if the module operations on E are continuous. Let E be a topological Amodule and let Σ(E) be a system of open sets in E which contain the zero 0 of E and such that any open set in E containing 0 contains a set of the system Σ(E). Then the system of sets of the form x + U where x ∈ E and U ∈ Σ(E) is a basis for the topology of E. Such a set Σ(E) is called a basis of neighborhoods of 0 for the topological module E. It follows that if Σ(E) is a basis of neighborhoods of the zero of a topological Amodule E, then E is a Hausdorﬀ (separated) space if and only if U = {0}. U ∈Σ(E)
Let I be an ideal of a ring R. The system {I n  n ∈ N} is a basis of neighborhoods of 0 of a topology of R called the Iadic topology. If E is an Rmodule, then the Iadic topology of E is deﬁned by taking {I n E  n ∈ N} as a basis of neighborhoods of 0. A submodule F of E has the Iadic topology and the induced topology which has {(I n E) ∩ F  n ∈ N} as a basis of neighborhoods of 0. Since I n F ⊂ (I n E) ∩ F for all n ≥ 0, the inclusion map F → E is continuous, with the respective Iadic topologies. Theorem 21.33. Suppose that A is a Noetherian ring, I is an ideal of A, and E is a ﬁnite Amodule. Then for every submodule F of E, the Iadic topology of F is induced by the Iadic topology of E. Proof. Since E is a ﬁnite Amodule, the ArtinRees lemma, [161, Theorem 4, page 255], tells us that there exists an integer k such that I n E ∩ F = I n−k (I n E ∩ F ) ⊂ I n−k F for every n ≥ k.
Suppose that E is an Amodule with the Iadic topology. A sequence (xn ) in E is called a Cauchy sequence in E if xn − xn+i ∈ I N (n) E for all i ≥ 0, where N (n) → ∞ as n → ∞. A limit of a Cauchy sequence (xn ) in E is an element y of E such that given m ∈ N, there exists n0 ∈ N such that xn − y ∈ I m E whenever n ≥ n0 . If E is separated, a limit, if it exists, is
21.5. Completion
409
unique. The module E is said to be complete if every Cauchy sequence in E converges to a limit in E. Theorem 21.34. Let A be a ring with an ideal I and M be an Amodule ˆ which is complete with the Iadic topology. Then there exists an Amodule M and separated for the Iadic topology, with a continuous homomorphism φ : ˆ , which satisﬁes the following universal property: for every AM → M module M which is complete and separated for the Iadic topology, and for any continuous homorphism f : M → M , there exists a unique continuous ˆ → M satisfying fˆ ◦ φ = f . homomorphism fˆ : M ˆ is unique up to isomorIt follows from the universal property that M ˆ phism. The module M is called the completion of M . It is a topological ˆ Amodule. Several proofs of the theorem are given in [107, Section (23.H), n ˆ is page 165]. The proof shows that the kernel of φ : M → M n≥0 I M , so φ is injective if and only if M is separated. One of the constructions shows that ˆ ∼ M = lim M/I n M. ←
ˆ is deﬁned as follows. By (11.2), M ˆ is naturally a subThe topology on M n ˆ has the module of n M/I M which has the product topology, and M subspace topology. We have the following useful properties. Theorem 21.35. Suppose that R is a Noetherian ring and I is an ideal in ˆ be the Iadic completion of R. Then R ˆ is Noetherian. R. Let R Proof. [13, Theorem 10.26].
Theorem 21.36. Suppose that R is a Noetherian local ring and I is a proper ˆ be the Iadic completion of R. Then dim R ˆ = dim R. ideal of R. Let R Proof. [106, Theorem 15.7].
Lemma 21.37. Suppose that A is a Noetherian local ring and Aˆ is the mA adic completion of A. Let I be an ideal of A. Then I Aˆ ∩ A = I. Proof. The map A → Aˆ is faithfully ﬂat by [107, (24.B), page 173, and Theorem 56 (5), page 172]. The conclusions of the lemma now follow from [107, (4.C)(ii), page 28]. ˆ will denote the mR adic completion of R, If R is a local ring, then R unless we explicitly say otherwise. Proposition 21.38. A Noetherian local ring R is separated in its mR adic ˆ is an inclusion. topology. Thus the natural map R → R Proof. [161, Theorem 9, page 262].
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´ 21. Ramiﬁcation and Etale Maps
A local ring R is said to be equicharacteristic if R and its residue ﬁeld have the same characteristic. The ring R is equicharacteristic if and only if R contains a ﬁeld. A coeﬃcient ﬁeld of a local ring R is a subﬁeld L of R which is mapped onto the residue ﬁeld R/mR . A coeﬃcient ﬁeld of R is thus isomorphic to R/mR . A theorem of fundamental importance is the following theorem of Cohen. Proofs can be found in [30] and [161, Theorem 27]. Theorem 21.39. An equicharacteristic complete Noetherian local ring R has a coeﬃcient ﬁeld. If R has equicharacteristic 0, then the existence of a coeﬃcient ﬁeld follows from Hensel’s lemma [161, Corollary 2, page 280]. We have the following corollary of Theorem 21.39 ([30] or [161, Corollary on page 307]). Corollary 21.40. An equicharacteristic complete isomorphic to a formal power series ring over a ﬁcient ﬁeld and x1 , . . . , xd is a regular system of R = K[[x1 , . . . , xd ]] is a ddimensional power series
regular local ring R is ﬁeld. If K is a coefparameters in R, then ring over K.
If R is a regular local ring of mixed characteristic (the characteristic ˆ is a of R is 0 but the characteristic of its residue ﬁeld is p > 0), then R power series ring over a complete discrete valuation ring if R is unramiﬁed (p ∈ m2R ). However, if R is ramiﬁed (p ∈ m2R ), this may not be true. Examples of this are given on [30, pages 93–94]. The following two propositions are very helpful in computing completions. The following proposition follows from Corollary 21.40, as we always have that our algebraically closed ﬁeld k is a coeﬃcient ﬁeld of the local ring of a point on a variety. Proposition 21.41. Suppose that X is a variety and p ∈ X is a nonsinˆX,p = gular point. Let x1 , . . . , xn be regular parameters in OX,p . Then O k[[x1 , . . . , xn ]] is a power series ring over k in x1 , . . . , xn . The following proposition follows from [161, Theorem 6, page 257]. Proposition 21.42. Suppose that R is a Noetherian ring and I and J / ∼ ˆ R ˆ where the completion is the Iadic are ideals in R. Then R/J = R/J completion. Lemma 21.43. Suppose that R and S are local rings and φ : R → S is a homomorphism such that φ(mR ) ⊂ mS . Then there exists a unique ˆ ˆ ˆ ) ⊂ m ˆ and φ(x) ˆ → Sˆ such that φ(m = φ(x) for homomorphism φˆ : R R S x ∈ R.
21.5. Completion
411
ˆ Let (yn ) be a Cauchy Proof. We ﬁrst prove existence. Suppose that y ∈ R. sequence in R which has y as its limit. Then (φ(yn )) is a Cauchy sequence in S since φ(mR ) ⊂ mS . Thus there exists z ∈ Sˆ which is the limit of (φ(yn )). We have that z does not depend on the choice of Cauchy sequence (yn ) ˆ ˆ → Sˆ which has y as its limit. We may thus deﬁne φ(y) = z. The map φˆ : R ˆ ˆ ˆ ) ⊂ m ˆ and φ(x) = φ(x) for x ∈ R. is a homomorphism such that φ(m R S ˆ ˆ We now prove uniqueness. Let ψ : R → S be a homomorphism such ˆ and suppose that ψ(mRˆ ) ⊂ mSˆ and ψ(x) = φ(x) for x ∈ R. Let y ∈ R, that (yn ) is a Cauchy sequence in R which has y as its limit. Then 0 is the limit of the Cauchy sequence (y − yn ). Since ψ(mRˆ ) ⊂ mSˆ , we have that the Cauchy sequence (ψ(y − yn )) has 0 as its limit. Now ψ(y − yn ) = ψ(y) − ψ(yn ) = ψ(y) − φ(yn ) so that the Cauchy sequence (φ(yn )) has ψ(y) as its limit, so that ψ(y) = ˆ ˆ φ(y) by our construction of φ. Lemma 21.44. Suppose that R and S are equicharacteristic regular local rings and φ : R → S is a homomorphism such that φ(mR ) ⊂ mS . Suppose that x1 , . . . , xm are regular parameters in R and y1 , . . . yn are regular parameters in S and S/mS is ﬁnite and separable over R/mR . Then there ˆ and κ2 of Sˆ such that R ˆ = κ[[x1 , . . . , xn ]] and exist coeﬃcient ﬁelds κ1 of R ˆ 1 ) ⊂ κ2 . Sˆ = κ2 [[y1 , . . . , yn ]] are power series rings and φ(κ Proof. This follows from Hensel’s lemma ([161, Theorem 17, page 279]) and Corollary 21.40. The conclusions of Lemma 21.44 may be false if S/mS is not separable ˆ and κ2 of over R/mR ; that is, there may not exist coeﬃcient ﬁelds κ1 of R ˆ ˆ S, respectively, such that φ(κ1 ) ⊂ κ2 . An example is given in [35, page 23]. A semilocal ring is a ring with a ﬁnite number of maximal ideals. Theorem 21.45 (Chevalley). Let A be a complete Noetherian semilocal ring, let m be the intersection of its maximal ∞ ideals, and let (an ) be a descending sequence of ideals in A such that n=0 an = (0). Then there exists an integralvalued function s(n) which tends to inﬁnity with n, such that an ⊂ ms(n) for all n ≥ 0. Proof. [161, Theorem 13, page 270].
Suppose that R and S are Noetherian local rings such that R is a subring of S. We say that R is a subspace of S if R, with its mR adic topology, is a subspace of S with its mS adic topology. This is so if and only if S dominates R and there exists a sequence of nonnegative integers a(n) such a(n) that a(n) → ∞ as n → ∞ and R ∩ mnS ⊂ mR for all n 0. It follows
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412
from Theorem 21.45 that if R is a complete local ring and S is local ring dominating R, then R is a subspace of S. Lemma 21.46. Suppose that R and S are Noetherian local rings such that ˆ and Sˆ be the respective S dominates R, with inclusion f : R → S. Let R ˆ → Sˆ be the induced homomorphism. completions of R and S and let fˆ : R Then R is a subspace of S if and only if fˆ is an injection. ˆ and Proof. First assume that R is a subspace of S. Suppose that y ∈ R ˆ f (y) = 0. There exists a Cauchy sequence (yn ) in R such that y is the limit of (yn ). Then (f (yn )) is a Cauchy sequence in S whose limit is fˆ(y) = 0. Since R is a subspace of S, given m > 0, there exists t0 such that R ∩ mtS ⊂ mm R if t ≥ t0 , and there exists m0 such that n > m0 implies f (yn ) ∈ mtS0 so that n > m0 implies yn ∈ mm R . Thus y = 0 is the limit of the Cauchy sequence (yn ) and so fˆ is an injection. ˆ Now suppose that fˆ is injective. Then in R, (0) = Kernel fˆ =
∞
an
n=0
ˆ ∩ mn . By Theorem 21.45, there exists an integervalued where an = R Sˆ s(n)
function s(n) which tends to ∞ with n, such that an ⊂ m ˆ R Thus s(n) R
R ∩ mnS = R ∩ (S ∩ mSˆn ) = R ∩ an ⊂ R ∩ m ˆ
for all n 0. s(n)
= mR
by Lemma 21.37, and so R is a subspace of S.
Exercise 21.47. In this exercise, consider two power series rings κ[[x1 , . . . , xm ]] and
κ[[y1 , . . . , yn ]]
over a ﬁeld κ. a) Suppose that h1 , . . . , hm are in the maximal ideal of κ[[y1 , . . . , yn ]]. Show that there is a unique local κalgebra homomorphism φ : κ[[x1 , . . . , xm ]] → κ[[y1 , . . . , yn ]] such that φ(xi ) = hi . Explain why all local κalgebra homomorphisms φ : κ[[x1 , . . . , xm ]] → κ[[y1 , . . . , yn ]] have this form. Hint: Let R = κ[x1 , . . . , xm ]. Use the universal property of polynomial rings to deﬁne a (unique) κalgebra homomorphism ψ : R → κ[[y1 , . . . , yn ]] such that ψ(xi ) = hi . Let I be the maximal ideal I = (x1 , . . . , xn ) of R. Explain why ψ extends to a homomorphism of local rings ψ : RI → κ[[y1 , . . . , yn ]], which has the property that ψ(IRI ) ⊂ (y1 , . . . , yn ). By Lemma 21.43, ψ extends to a local 0I = κ[[x1 , . . . , xm ]] → κ[[y1 , . . . , yn ]]. homomorphism ψˆ : R
21.6. Zariski’s main theorem and Zariski’s subspace theorem
413
b) Can there be local homomorphisms φ : κ[[x1 , . . . , xm ]] → κ[[y1 , . . . , yn ]] which are not κalgebra homomorphisms? Can there be ring homomorphisms φ : κ[[x1 , . . . , xm ]] → κ[[y1 , . . . , yn ]] such that φ is not a local homomorphism (φ does not map the maximal ideal of κ[[x1 , . . . , xm ]] into the maximal ideal of κ[[y1 , . . . , yn ]])? c) Suppose that φ : κ[[x1 , . . . , xn ]] → κ[[x1 , . . . , xn ]] is a local κalgebra homomorphism. Show that φ is an isomorphism if and only if Det(aij ) = 0 where a(i)i1 ,...,in y1i1 · · · ynin φ(xi ) = ai1 y1 + · · · + ain yn + i1 +···+in >1
with aij , a(i)i1 ,...,in ∈ κ is the series expansion of φ(xi ) for q ≤ i ≤ n.
21.6. Zariski’s main theorem and Zariski’s subspace theorem In this section we give some generalizations by Abhyankar in [4, Section 10] of some theorems of Zariski in [155]. Proposition 21.48. Suppose that R and S are complete Noetherian local rings such that S dominates R, S/mS is ﬁnite algebraic over R/mR , and mR S is mS primary. Then S is a ﬁnite Rmodule. If S/mS = R/mR and mR S = mS , then R = S. Proof. The quotient S/mS is a ﬁnite length Smodule since mR S is mS primary. Since S/mS is a ﬁnite R/mR vector space, we have that S/mR S is a ﬁnitely generated Rmodule. By Proposition 21.38 and since S is a Noetherian local ring with mR S ⊂ mS , we have that ∞ mnR S = (0). n=1
Let u1 , . . . , us ∈ S generate S/mR S as an Rmodule. Let N =
s
i=1 Rui .
Suppose a ∈ S. We will show that there exists a sequence of elements a1 , . . . , an , . . . in N such that an = with mni ∈
mn−1 R
and a −
n
j=1 aj
s
mni ui
i=1
∈ mnR S.
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We will prove this by induction on n. The case n = 1 is immediate from our assumptions. Assume that a1 , . . . , an are deﬁned. Then a−
n
aj =
mi bi
j=1
with mi ∈ mnR , bi ∈ S. Let ci be elements of N such that bi − ci ∈ mR S and set mi ci = mn+1,i ui . an+1 = Then an+1 is the required element and the sequence in welldeﬁned. Set m∗i
=
∗ mi ui ∈ N . Then a and a∗ = thus S = N .
∞
mni ∈ R
n=1 − a∗ ∈
mnR S for all n, so that a = a∗ and
ˆ is a domain. A local ring R is said to be analytically irreducible if R Proposition 21.49. Let R and S be analytically irreducible Noetherian local domains such that S dominates R. Assume that there exists a subring T of S with R ⊂ T such that T is a ﬁnite Rmodule and S = TT ∩mS . Also assume ˆ and Sˆ be the completions of R and S. Let that R is a subspace of S. Let R ∗ ∗ ˆ and S, ˆ respectively, K, L, K , and L be the quotient ﬁelds of R, S, R, ∗ ∗ ˆ ˆ where K is identiﬁed with a subﬁeld of L . Then S = R[T ], Sˆ is a ﬁnite ˆ Rmodule, and L∗ = K ∗ (L). ˆ ] is a ﬁnite Rmodule ˆ Proof. We have that R[T and hence is a complete  Noeˆ ] dominates R ˆ and m ˆ = m ˆ R[T ˆ ] therian local domain such that R[T R[T ]
R
by [161, Theorem 15, page 276] and [161, Corollary 2 on page 283]. Thus ˆ ]. In particular, Sˆ dominates R[T T ∩ mR[T ˆ ] = T ∩ mSˆ = T ∩ mS ˆ ] dominates S. Consequently Sˆ and R[T ˆ ] have isomorphic and hence R[T ˆ ˆ residue ﬁelds and mR[T ˆ ] S = mSˆ . By Proposition 21.48 we have that S = ˆ ]. Thus Sˆ is a ﬁnite Rmodule ˆ R[T and L∗ = K ∗ (L). Proposition 21.50. Suppose that R is a normal local ring which is essenˆ is a normal local tially of ﬁnite type over a ﬁeld. Then its completion R ring. Proof. [121, Theorem 37.5] or [107, Theorem 79, page 258].
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Proposition 21.51. Let R be an analytically irreducible Noetherian local ˆ be the completion of R, let K and K ˆ be the respective quotient domain, let R ˆ ﬁelds of R and R, let V be a local (not necessarily Noetherian) domain with quotient ﬁeld K such that V dominates R, and let H be the smallest subring ˆ such that H contains V and R. ˆ Then mV H = H and there exists a of K ∗ ∗ ˆ ˆ valuation ring V of K such that V dominates V and R. Proof. Suppose that mV H = H. We will derive a contradiction. Since H ˆ we have an is the set of all ﬁnite sums ai bi with ai ∈ V and bi ∈ R, ˆ expression 1 = x1 y1 + · · · + xn yn where x1 , . . . , xn ∈ mV and y1 , . . . , yn ∈ R. zi Since R and V have the same quotient ﬁeld, we have expressions xi = z , where z, z1 , . . . , zn ∈ R with z = 0. Then
ˆ = (z1 , . . . , zn )R z = z1 y1 + · · · + zn yn ∈ R ∩ (z1 , . . . , zn )R by Lemma 21.37. Hence z = z1 r1 + · · · + zn rn with r1 , . . . , rn ∈ R. Then 1 = x1 r1 + · · · + xn rn ∈ mV , a contradiction. Thus mV H = H, and so by the existence theorem of ˆ valuations [161, Theorem 4, page 11] there exists a valuation ring V ∗ of K ∗ ∗ such that H ⊂ V and mV H ⊂ mV ∗ . The valuation ring V thus dominates ˆ V and R. Theorem 21.52. Let A be a subring of a ﬁeld K and let P be a prime ideal of A. Then V = the integral closure of AP in K, V ∈N
where N is the set of all valuation rings V of K such that V dominates AP . Proof. [161, Theorem 8, page 17].
Proposition 21.53. Let R and S be Noetherian local domains such that R is analytically irreducible, S dominates R, dim R = dim S, S/mS is ﬁnite over ˆ and Sˆ be the respective completions R/mR , and mR S is mS primary. Let R of R and S. Let f : R → S be the inclusion, with induced homomorphism ˆ → S. ˆ Then fˆ is an injection (and hence R is a subspace of S by fˆ : R Lemma 21.46). If R is normal and R and S have the same quotient ﬁelds, then R = S. ˆ is a complete local ring since it is isomorphic to a Proof. The ring fˆ(R) ∼ ˆ By Proposition 21.48, Sˆ is a ﬁnite fˆ(R)module ˆ ˆ quotient of R. as S/m Sˆ = ∼ ˆ ˆ ˆ S/mS is ﬁnite over fˆ(R)/m ˆ S = mR S is mSˆ primary. ˆ = R/mR and mf (R) fˆ(R) ˆ by Theorem 1.62. Thus dim fˆ(R) ˆ = dim R, ˆ as Hence dim Sˆ = dim fˆ(R) ˆ = dim R and dim Sˆ = dim S by Theorem 21.36. Since R ˆ is a domain, dim R ˆ f is then a monomorphism.
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Now assume that R is normal and the quotient ﬁelds of R and S coincide. ˆ be the respective quotient ﬁelds of R and R. ˆ Since R is normal, Let K and K we have by Theorem 21.52 that R is the intersection of all valuation rings of K which dominate R. Thus it suﬃces to show that if V is any valuation ring of K which dominates R and z is an element of S, then z ∈ V . Since K is the quotient ﬁeld of R, we have an expression z = xy where x, y ∈ R ˆ (by Proposition and y = 0. Since z ∈ S ⊂ Sˆ and Sˆ is integral over R n n−1 ˆ + · · · + zn = 0. 21.48), there exist z1 , . . . , zn ∈ R such that z + z1 z ˆ such that V ∗ By Proposition 21.51, there exists a valuation ring V ∗ of K ˆ Since V ∗ is normal, R ˆ ⊂ V ∗ , and z is integral over R, ˆ dominates V and R. ∗ ∗ we have that z ∈ V . Now V = V ∩ K so z ∈ V . Proposition 21.54 (Zariski’s main theorem). Let κ be a ﬁeld, let R be a normal Noetherian local domain which is a localization of a ﬁnite type κalgebra, and let S be a Noetherian local domain such that S dominates R, dim R = dim S, S/mS is ﬁnite over R/mR , mR S is mS primary, and R and S have a common quotient ﬁeld. Then R = S. Proof. The ring R is analytically irreducible by Proposition 21.50. The proposition now follows from Proposition 21.53. Proposition 21.55. Let R be a Noetherian normal local ring which is essentially of ﬁnite type over a ﬁeld κ, let T be the integral closure of R in a ﬁnite algebraic extension L of the quotient ﬁeld K of R, let P be a prime ideal in T with R ∩ P = mR , and let S = TP . Then S is normal and a localization of a ﬁnite type κalgebra, S dominates R, dim R = dim S, S/mS ˆ is ﬁnite algebraic over R/mR , mR S is primary for mS , the completions R and Sˆ are normal domains, R is a subspace of S, and upon identifying the ˆ of R ˆ with a subﬁeld of the quotient ﬁeld L ˆ of S, ˆ we have quotient ﬁeld K ˆ ˆ ˆ ˆ ˆ ˆ that S = R[T ], S is a ﬁnite Rmodule, and L = K(T ). Proof. The ring T is a ﬁnitely generated Rmodule and S is a localization of a ﬁnite type κalgebra by Theorem 1.54. Thus S/mS is ﬁnite algebraic over R/mR and mR S is primary for mS . We have that dim S = dim R by ˆ and Sˆ are normal domains by Proposition Theorem 1.62. The completions R 21.50. Proposition 21.53 implies that R is a subspace of S. The facts that ˆ ], Sˆ is a ﬁnite Rmodule, ˆ ˆ = K(L) ˆ Sˆ = R[T and L now follow from Proposition 21.49. Proposition 21.56. Let R be a Noetherian local domain with quotient ﬁeld K and let V be a valuation ring of K such that V = K, V dominates R, and trdegR/mR V /mV ≥ dim R − 1. Then trdegR/mR V /mV = dim R − 1 and V is a onedimensional regular local ring.
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Proof. We have that the value group of V is isomorphic to Z and trdegR/mR V /mV = dim R − 1 by [161, Proposition 2, page 331] and [161, Proposition 3, page 335] (which are generalizations of “Abhyankar’s inequality” in [3]). Thus V is a onedimensional regular local ring by Theorem 21.10. The following proposition generalizes Theorem 10.19. Proposition 21.57. Let R be an ndimensional Noetherian local domain with n > 1. Let x1 , . . . , xn ∈ R be a system of parameters in R and let Q = (x1 , . . . , xn ). Let A = R[ xx21 , . . . , xxn1 ] and let π : A → A/mR A be the natural √ quotient map. Then mR A is a prime ideal, dim AmR = 1, mR A = QA, R ∩ mR A = mR , and π( xx12 ), . . . , π( xxn1 ) are algebraically independent over π(R) ∼ = R/mR . Proof. We have that QA = x1 A. There exists a positive integer e such that meR ⊂ Q since Q is mR primary, so that (mR A)e ⊂ x1 A. Let X1 , . . . , Xn be indeterminates. Suppose that R ∩ mR A = mR . We will derive a contradiction. Then mR A = A and thus x1 A = A, so there exists a nonzero element y ∈ A such that x1 y = 1. There thus exists a nonzero polynomial f (X2 , . . . , Xn ) of some degree d in X2 , . . . , Xn with coeﬃcients in R such that y = f ( xx12 , . . . , xxn1 ). Thus xd1 = xd+1 1 y = x1 g(x1 , . . . , xn ) where g(X1 , . . . , Xn ) is a nonzero homogeneous polynomial of degree d in X1 , . . . , Xn with coeﬃcients in R. In particular, xd1 ∈ mR Qd , which is a contradiction by [161, Theorem 21, page 292], since x1 , . . . , xn is a system of parameters in R. Thus R ∩ mR A = mR and π(R) ∼ = R/mR . Suppose that π( xx12 ), . . . , π( xxn1 ) are algebraically dependent over R/mR . We will derive a contradiction. By our assumption that they are algebraically dependent, there exists a nonzero polynomial F (X2 , . . . , Xn ) of some degree u in X2 , . . . , Xn with coeﬃcients in R at least one of which is not in mR such that F ( xx21 , . . . , xxn1 ) ∈ mR A. Thus there exists a polynomial G(X2 , . . . , Xn ) in X2 , . . . , Xn with coeﬃcients in mR such that xn xn x2 x2 ,..., ,..., =G . F x1 x1 x1 x1 After multiplying both sides of this equation by xv1 for a suitable integer v ≥ u, we obtain that U (x1 , . . . , xn ) = V (x1 , . . . , xn ) where U (X1 , . . . , Xn ) is a nonzero homogeneous polynomial of degree v in X1 , . . . , Xn with coeﬃcients in R, at least one of which is not in mR ,
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and V (X1 , . . . , Xn ) is either the zero polynomial or a nonzero homogeneous polynomial of degree v in X1 , . . . , Xn with coeﬃcients in mR . Thus U (x1 , . . . , xn ) ∈ mR Qv , which is a contradiction by [161, Theorem 21, page 292], since x1 , . . . , xn is a system of parameters. Thus π( xx12 ), . . . , π( xxn1 ) are algebraically independent over π(R). Since x2 xn π(A) = π(R) π ,...,π , x1 x1 we have that π(A) is a domain and thus mR A is a prime ideal in A. Now (mR A)e ⊂ x1 A = QA, which implies that x1 AmR A is mR AmR A primary, which implies that dim AmR A = 1 by Krull’s principal ideal theorem, Theorem 1.65. Proposition 21.58. Let R be an ndimensional Noetherian local domain with n > 0, and let K be the quotient ﬁeld of R. Then there exists a onedimensional regular local ring V with quotient ﬁeld K such that V dominates R and trdegR/mR V /mV = n − 1. Proof. By Theorem 8.12, there exists a system of parameters x1 , . . . , xn in R, so that (x1 , . . . , xn ) is mR primary. Let A = R[ xx12 , . . . , xxn1 ]. By Proposition 21.57, mR A is a prime ideal in A, and upon letting S = AmR A we have that S is a onedimensional Noetherian local domain with trdegR/mR S/mS = n−1. Let T be the integral closure of S in K. By the KrullAkizuki theorem, [121, Theorem 33.2] (or by Theorem 1.54 if R is a localization of a ﬁnite type algebra over a ﬁeld), we have that T is Noetherian and dim T = 1. Let P be a maximal ideal of T , and let V = TP . Then V dominates S and V /mV is algebraic over S/mS . We have that V is a regular local ring by Theorem 1.87, since V is normal by Exercise 1.58. Proposition 21.59. Suppose that A and B are local domains which are localizations of ﬁnitely generated algebras over a ﬁeld, A and B have respective quotient ﬁelds K and L, and B dominates A. Then dim A + trdegK L = dim B + trdegA/mA B/mB . Proof. [107, Corollary, page 86, and Corollary 3, page 92].
Proposition 21.60. Let R and S be Noetherian local domains, with respective quotient ﬁelds K and L, such that R is analytically irreducible, S dominates R, trdegK L < ∞, and dim R + trdegK L = dim S + trdegR/mR S/mS . Then R is a subspace of S. ˆ be the quotient ﬁeld of the completion R ˆ of R. If dim R = 0, Proof. Let K then the conclusions of the proposition follow trivially. Assume dim R > 0.
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419
Then dim S > 0, and by Proposition 21.58, there exists a onedimensional regular local ring W with quotient ﬁeld L such that W dominates S and trdegS/mS W/mW = dim S − 1. Let V = K ∩ W . Since W is a valuation ring of L, we have that V is a valuation ring of K, W dominates V , and V dominates R. In particular, R ∩ mV = mR = (0), and hence mV = (0). Now W is the valuation ring of a discrete valuation ω of L, so we have that V is the valuation ring of the discrete valuation ν = ωK and V is a onedimensional regular local ring by Theorem 21.10. Thus the principal ideals i mV W = muW where u is a positive integer, and K ∩ mui W = mV for every nonnegative integer i (by Remark 21.11). We will now establish that trdegV /mV W/mW ≤ trdegK L. Suppose that t1 , . . . , tr ∈ W/mW are algebraically independent over V /mV . Let t1 , . . . , tr ∈ W be lifts of t1 , . . . , tr . Suppose that t1 , . . . , tr are algebraically dependent over K. We will derive a contradiction. With this assumption, there exists a relation ai1 ,...,ir ti11 · · · tirr = 0 (21.17) i1 ,...,ir
with the ﬁnitely many coeﬃcients ai1 ,...,ir ∈ K not all zero. Let j1 , . . . , jr be such that ν(aj1 ,...,jr ) = min{ν(ai1 ,...,ir )}. Dividing the relation (21.17) by aj1 ,...,jr , we may assume that aj1 ,...,jr = 1 and ν(ai1 ,...,ir ) ≥ 0 for all i1 , . . . , ir . Let ai1 ,...,ir be the residue of ai1 ,...,ir in V /mV . Then i i ai1 ,...,ir t11 · · · trr = 0 i1 ,...,ir
is a nontrivial relation, contradicting our assumption that t1 , . . . , tr are algebraically independent over V /mV . We have that trdegS/mS W/mW = dim S − 1, trdegV /mV W/mW +trdegR/mR V /mV = trdegS/mS W/mW +trdegR/mR S/mS , and, by assumption, dim R + trdegK L = dim S + trdegR/mR S/mS .
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Thus trdegR/mR V /mV ≥ dim R−1. By Proposition 21.51, there exists a valˆ such that V ∗ dominates V and R. ˆ Since dim R ˆ = dim R uation ring V ∗ of K ˆ V ∗ /mV ∗ ≥ (by Theorem 21.36) and R/mRˆ = R/mR , we have that trdegR/m ˆ ˆ R ˆ − 1. Thus V ∗ is a onedimensional regular local ring by Proposition dim R i 21.56. In particular, ∞ i=0 mV ∗ = (0), and hence ∞
ˆ ∩ mi ∗ ) = (0). (R V
i=0
Thus by Theorem 21.45, there exists a sequence of nonnegative integers ˆ ∩ (mV ∗ )i ⊂ ma(i) for every a(i) which tend to inﬁnity with i such that R ˆ R nonnegative integer i. We thus have a(i) a(i) ui i i i ˆ R∩mui S ⊂ R∩mW = R∩mV ⊂ R∩mV ∗ ⊂ R∩(R∩mV ∗ ) ⊂ R∩m ˆ = mR R
by Lemma 21.37. Thus there exists a sequence of nonnegative integers b(i) b(i) which tend to inﬁnity with i such that R ∩ miS ⊂ mR for every nonnegative integer i. Thus R is a subspace of S. Proposition 21.61 (Zariski’s subspace theorem). Let R and S be local domains which are localizations of ﬁnite type algebras over a ﬁeld such that R is analytically irreducible and S dominates R. Then R is a subspace of S ˆ → Sˆ is an inclusion. so that the natural map R Proof. This follows from Propositions 21.59 and 21.60.
Proposition 21.62. Suppose that φ : X → Y is a dominant regular map of varieties, p ∈ X, and q = φ(p). Assume that OY,q is analytically irreducible (which holds if OY,q is normal by Proposition 21.50) and that mq OX,p = mp ˆY,q → O ˆX,p is an isomorphism. is the maximal ideal of OX,p . Then φˆ∗ : O Proof. Let R = OY,q and S = OX,p . By Proposition 21.61, we have that ˆ → Sˆ is an injection. Since R/mR = S/mS = k and mR S = S, we φˆ : R have that φˆ∗ is an isomorphism by Proposition 21.48. Corollary 21.63. Suppose that φ : X → Y is a ﬁnite map of normal ˆY,φ(p) → O ˆX,p varieties. Suppose that φ is unramiﬁed at p ∈ X. Then φˆ∗ : O is an isomorphism. The subspace theorem is not true in complex analytic geometry. Example 21.64 (Gabri`elov, [58]). There exists an injective local Calgebra homomorphism R → S of rings of germs of convergent power series, such ˆ → Sˆ of formal power series rings is not an injection. that the induced map R
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Exercise 21.65. In this exercise, we show that the assumption that OY,q is analytically irreducible is necessary in Proposition 21.62. a) Let Y be the nodal curve Y = Z(x2 − y 2 − y 3 ) ⊂ A2k where k is an algebraically closed ﬁeld of characteristic = 2 or 3. Show that Y is a variety with an isolated singularity at the origin q. b) Let φ : X → Y be the blowup of the ideal I(q) of Y . Show that X is an aﬃne variety which is the normalization of Y and the ring of regular functions on X is k[X] = k[x1 , y1 ]/(1 − y12 − x1 y13 ), with the inclusion k[Y ] → k[X] deﬁned by the substitutions x = x1 , y = x1 y1 . c) Let p ∈ φ−1 (q). Show that OY,q → OX,p is unramiﬁed but the ˆY,q → O ˆX,p is not an isomorinduced map on completions φˆ∗ : O phism. Exercise 21.66. Let f1 = xy, f2 = x, f3 = y in the polynomial ring k[x, y]. Let R = k[x, y](x,y) and Q = (f1 , f2 , f3 )R = mR . Let A = R[ ff21 , ff31 ]. Show that mR A = A.
21.7. Galois theory of varieties Suppose that φ : Y → X is a dominant ﬁnite regular map of normal varieties. Let G(Y /X) be the group of all regular isomorphisms of Y /X, that is, the group of all regular isomorphisms τ : Y → Y such that there is a commutative diagram τ / Y Y A AA AA AA φ A
φ
X. The group of k(X)algebra isomorphisms of k(Y ) is denoted by Aut(k(Y )/k(X)) (Section 1.2). Proposition 21.67. The map Φ : G(Y /X) → Aut(k(Y )/k(X))op (where Aut(k(Y )/k(X))op is the opposite group) deﬁned by τ → τ ∗ is a group isomorphism. Proof. Suppose that τ ∈ G(Y /X). Then τ ∗ gives an isomorphism of k(Y ) which ﬁxes k(X), so τ ∗ ∈ Aut(k(Y )/k(X)). Now suppose that σ ∈ Aut(k(Y )/k(X)). Then there exists a unique birational map τ : Y Y such that τ ∗ = σ. Suppose that U ⊂ X is an
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aﬃne open subset. Let V = φ−1 (U ) which is an aﬃne open subset of Y (by Theorem 7.5). Let A = k[U ] and B = k[V ]. Here B is the integral closure of A in k(Y ) (by φ∗ : k(X) → k(Y )). Suppose f ∈ B. Then f is integral over A and σ(f ) ∈ k(Y ) must satisfy the same equation of integrality over A as f since A ⊂ k(X) is ﬁxed by σ. Thus σ(f ) ∈ B and σ(B) ⊂ B. Since σ is an isomorphism, we have σ(B) = B. Hence σ is an Aalgebra isomorphism of B so τ V ∈ G(V /U ). Since this is true for all members of an aﬃne cover {Ui } of X, we have that τ ∈ G(Y /X) by Proposition 3.39. Finally, we observe that the group structure is preserved since (τ1 τ2 )∗ = for τ1 , τ2 ∈ G(Y /X).
τ2∗ τ1∗
Suppose that H is a subgroup of G(Y /X). Then we can deﬁne (by Theorem 7.17) a normal variety Y H by taking Y H to be the normalization of X in the ﬁxed ﬁeld k(Y )H = k(Y )Φ(H) (σ ∈ H acts as σ ∗ ). We call Y H the quotient of Y by H. Deﬁnition 21.68. Suppose that φ : Y → X is a dominant ﬁnite regular map of normal varieties and k(Y ) is a separable extension of k(X). The map φ is said to be Galois and Y is said to be Galois over X if for every p ∈ X and q1 , q2 ∈ φ−1 (p) there exists τ ∈ G(Y /X) such that τ (q1 ) = q2 . Theorem 21.69. Suppose that φ : Y → X is a dominant ﬁnite regular map of normal varieties. Then Y is Galois over X if and only if k(Y ) is Galois over k(X). Proof. Suppose that Y /X is Galois. Let Z = Y G(Y /X) and let α
β
Y →Z→X be the regular maps factoring φ. By Theorems 21.29 and 21.28, there exists p ∈ X such that #{φ−1 (p)} = [k(Y ) : k(X)]. Thus β is unramiﬁed above p and α is unramiﬁed above all points of β −1 (p) by Theorem 21.26 and Theorem 21.28. Suppose that Z = X so that k(Z) = k(X). Then there exist a1 , a2 ∈ β −1 (p) which are not equal. Let q1 ∈ α−1 (a1 ) and q2 ∈ α−1 (a2 ). Since Y is Galois over X, there exists τ ∈ G(Y /X) such that τ (q1 ) = q2 . But G(Y /Z) = G(Y /X) (since τ ∈ G(Y /X) implies τ ∗ : k(Z) → k(Z) is the identity) so a1 = α(τ (q1 )) = α(q2 ) = a2 , a contradiction. Thus Z = X and so k(Y )G(Y /X) = k(X), so that k(Y ) is Galois over k(X). Now suppose that k(Y ) is Galois over k(X). Suppose that q ∈ X and φ−1 (q) = {p1 , . . . , pr }. Let T be the integral closure of OX,q in k(Y ), and let m1 , . . . , mr be the maximal ideals of T , with Tmi = OY,pi . By Lemma 21.8, if i = j, then there exists σ ∈ G(k(Y )/k(X)) such that σ(mi ) = mj . If τ ∈ G(Y /X) corresponds to σ, then we have τ (pj ) = pi . Thus Y is Galois over X.
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We now summarize some results on quotients that we found in this section. Suppose that φ : Y → X is a dominant ﬁnite map of normal varieties and U ⊂ X is aﬃne. Then V = φ−1 (U ) is aﬃne (since φ is ﬁnite) and k[V ] is the integral closure of k[U ] in k(Y ). Further, if Y is Galois over X, then G = G(Y /X) acts naturally on k[V ] and the ring of invariants is k[V ]G = k[U ]. Exercise 21.70. Suppose that X and Y are normal varieties over an algebraically closed ﬁeld of characteristic = 2 and φ : Y → X is a ﬁnite map with deg(φ) = 2. Show that φ is Galois. Exercise 21.71. Suppose that φ : Y → X is a Galois map of nonsingular curves. Suppose that p ∈ X and φ−1 (p) = {q1 , . . . , qt }. Show that the divisor φ∗ (p) = eq1 + · · · + eqt where et = deg(φ). Hint: Use Theorem 13.18. Exercise 21.72. Suppose that k is an algebraically closed ﬁeld of characteristic = 2. Let φ : A1k → A1k be deﬁned by φ(z) = (z 2 + 1)2 . Show that φ is not Galois. Hint: Use Exercise 21.71. Exercise 21.73. Suppose that φ : Y → X is a ﬁnite regular Galois map of varieties and H is a subgroup of G(Y /X). Let Z = Y H with natural regular maps α
β
Y →Z→X factoring φ. Suppose that U is an aﬃne open subset of X. Let V = φ−1 (U ) and W = β −1 (U ). Here V and W are aﬃne open subsets of Y and Z, respectively, by Theorem 7.5. a) Show that G(V /U ) = {(σV )  σ ∈ G(Y /X)}. b) Let k[V ]H = {f ∈ k[V ]  σ ∗ (f ) = f for all σ ∈ H}. Show that k[W ] = k[V ]H . c) Show that Y is Galois over Z. Exercise 21.74. Let k be an algebraically closed ﬁeld of characteristic = 3 and let φ : Y = A1k → X = A1k be the ﬁnite map deﬁned by φ(t) = t3 − 3t. Compute G(Y /X) and show that φ is not Galois. Hint: Use the fact that every automorphism of A1 extends to an automorphism of P1 .
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21.8. Derivations and K¨ ahler diﬀerentials redux We require some more results on derivations and diﬀerentials. Theorem 21.75. Suppose that κ is a ﬁeld and K is a ﬁnitely generated extension ﬁeld, of transcendence degree n over k. Then ΩK/κ is a vector space of dimension ≥ n over K. Suppose that x1 , . . . , xn ∈ K. Then dx1 , . . . , dxn is a Kbasis of ΩK/κ if and only if x1 , . . . , xn is a separating transcendence basis of K over κ. Theorem 21.75 follows from [50, Theorem 16.4 and Corollary 16.17] or the material in [160, Section 17, Chapter II] on derivations, along with the isomorphism of Kvector spaces Derκ (K, K) ∼ = HomK (ΩK/κ , K) of Lemma 14.3. Suppose that K is a ﬁnitely generated extension ﬁeld of an algebraically closed ﬁeld k and x1 , . . . , xn is a separating transcendence basis of K over k (which exists by Theorem1.14). Let L = k(x1 , . . . , xn ). By Example 14.5 n and Lemma 14.8, ΩL/k ∼ = HomL (ΩL/k , L) = i=1 Ldxi . Since Derk (L, L) ∼ by Lemma 14.3, ∂x∂ 1 , . . . , ∂x∂n is an Lbasis of Derk (L, L), and by Theorem 21.75 and since Derk (K, K) ∼ = HomK (ΩK/k , K) by Lemma 14.3, ∂x∂ 1 , . . . , ∂x∂n extend uniquely to a Kbasis of Derk (K, K). A direct proof of this is given in Theorem 39 and its corollaries in [160, Section 17, Chapter II]. Suppose that 0 = α ∈ K and δ ∈ Derk (L, L). Let g(t) ∈ L[t] be the minimal polynomial of α over L. Write g(t) = td + ad−1 td−1 + · · · + a0 with ai ∈ L. Let g δ (t) = δ(ad−1 )td−1 + δ(ad−2 )td−2 + · · · + δ(a0 ) ∈ L[t]. By the properties of derivations, we have that 0 = δ(g(α)) = g δ (α) + g (α)δ(α) where
dg = dtd−1 + (d − 1)ad−1 td−1 + · · · + a1 dt is the formal derivative of g(t). Since α is separable over L, we have that g (α) = 0. Thus g δ (α) δ(α) = − . g (α) g (t) =
If δ(α) = 0, then g δ (α) = 0. Since g δ (t) has smaller degree in t than the minimal polynomial g(t) of α, we have that g δ (t) = 0. Suppose that k has characteristic 0. If δ(α) = 0 for all δ ∈ Derk (L, L), then g(t) ∈ k[t], so that α ∈ k (as k is algebraically closed). Thus if k has characteristic 0, (21.18)
k = {f ∈ K  δ(f ) = 0 for all δ ∈ Derk (K, K)}.
21.8. Derivations and K¨ahler diﬀerentials redux
425
Suppose that K is a ﬁeld of characteristic p > 0. A ﬁnite set of elements x1 , . . . , xn in K are said to be pindependent if the np monomials xi11 xi22 · · · xinn with 0 ≤ iq < p for 1 ≤ q ≤ n are linearly independent over K p . If we also have that the set S = {x1 , . . . , xn } satisﬁes K = K p (S), then we say that S is a pbasis of K. Theorem 21.76. Suppose that K is a ﬁnitely generated extension ﬁeld of an algebraically closed ﬁeld k of characteristic p > 0 and x1 , . . . , xn is a separating transcendence basis of K over k. Then: 1) The k(x1 , . . . , xn )basis ∂ ∂ ,..., ∂x1 ∂xn of Derk (k(x1 , . . . , xn ), k(x1 , . . . , xn )) extends uniquely to a Kbasis of Derk (K, K) = DerK p (K, K). 2) [K : K p ] = pn and x1 , . . . , xn is a pbasis of K. 3) K p = {f ∈ K  δ(f ) = 0 for all δ ∈ Derk (K, K)}. Proof. Statement 1) follows from Theorem 21.75 since Derk (K, K) ∼ = Homk (ΩK/k , K) by Lemma 14.3. Suppose that f ∈ K p (x1 , . . . , xn ). Then f has an expression aI xi11 xi22 · · · xinn (21.19) f= I
where the sum is over I = (i1 , . . . , in ) ∈ Nn with 0 ≤ iq < p for 1 ≤ q ≤ n and all aI ∈ K p . Let I be such that i1 + · · · + in is maximal for aI = 0. Suppose that i1 + · · · + in > 0. Without loss of generality, i1 > 0. Then ∂xinn
∂f = i1 !i2 ! · · · in !aI = 0 · · · ∂xi11
∂f so ∂x
= 0. Thus f = 0, and so x1 , . . . , xn are pindependent, and we have 1 that [K p (x1 , . . . , xn ) : K p ] = pn .
We have that [K : K p ] = pn by [160, Theorem 41, Section 17, Chapter II] , so x1 , . . . , xn is a pbasis of K. Now 3) follows from 2) and the above calculation showing that if an element f with an expansion (21.19) has the property that all derivations vanish on f , then f ∈ K p .
´ 21. Ramiﬁcation and Etale Maps
426
Theorem 21.77. Let K be a ﬁeld and F be a ﬁnitely generated extension ﬁeld of K. Then dimK DerK (F, F ) ≥ trdegK F and F is separably generated over K if and only if dimK DerK (F, F ) = trdegK F.
Proof. [160, Theorem 41, page 127].
Exercise 21.78. Let κ = Fp (t), and let R = κ[x, y]/(xp + y p − t) be the ring considered in Exercise 10.21. Let K be the quotient ﬁeld of R. It was shown in Exercise 10.21 that R is a regular ring of dimension 1. a) Compute ΩR/κ . b) Show that K is not separably generated over κ.
´ 21.9. Etale maps and uniformizing parameters Deﬁnition 21.79. A regular map of varieties φ : X → Y is said to be ´etale if for all p ∈ X there are open neighborhoods U ⊂ X of p and V ⊂ Y of φ(p) such that φ(U ) ⊂ V and there exists a commutative diagram open embedding
U φ
V
↓
→
open embedding
→
Z ↓ W
where Z and W are aﬃne varieties, and k[Z] = R[x1 , . . . , xn ]/(f1 , . . . , fn ) ∂fi with R = k[W ], and the rank of the n × n matrix ( ∂x (p)) over k is n. j
This deﬁnition of ´etale is equivalent to the deﬁnition of ´etale in [109, page 20] and [73, Exercise III.10.3], as will be explained after Deﬁnition 22.8. The deﬁnition is valid with φ : X → Y a map of schemes (and Z, W aﬃne schemes). A reﬁnement of Deﬁnition 21.79 is given in Exercise 21.86. The following is a version of Hensel’s lemma. Lemma 21.80. Let R be a complete Noetherian local ring and let f1 , . . . , fn be elements of the polynomial ring R[x1 , . . . , xn ]. Assume a1 , . . . , an ∈ R are such that f1 (a1 , . . . , an ), . . . , fn (a1 , . . . , an ) ∈ mR and ∂fi (a1 , . . . , an ) ∈ mR . Det ∂xj Then there exist α1 , . . . , αn ∈ R such that αi − ai ∈ mR and f1 (α1 , . . . , αn ) = · · · = fn (α1 , . . . , αn ) = 0.
´ 21.9. Etale maps and uniformizing parameters
427 (r)
(r)
Proof. We inductively deﬁne approximate roots a1 , . . . , an (1) ai = ai , such that (r)
(21.20)
ai
(r−1)
≡ ai
r−1 mod mR
∈ R, with
for 1 ≤ i ≤ n and r ≥ 2
and (21.21)
(r)
(r)
(r) r f1 (a1 , . . . , a(r) n ) ≡ · · · ≡ fn (a1 , . . . , an ) ≡ 0 mod mR . (r)
(r)
Suppose a1 , . . . , an ∈ R satisfy (21.20) and (21.21). Let 1 , . . . , 2 ∈ mrR . Then n ∂fi (r) (r) (r) + ) ≡ f (a , . . . , a )+ (a1 , . . . , an )j mod mr+1 fi (a1 +1 , . . . , a(r) n i 1 n n R . ∂xj j=1
Let
−1 ∂fi B = (bij ) = , (a1 , . . . , an ) ∂xj a matrix with coeﬃcients in R. Set n (r) bij fj (a1 , . . . , a(r) i = − n ) for 1 ≤ i ≤ n j=1 (r+1)
and ai
(r)
= ai + i . Then (r)
r+1 fi (a1 + 1 , . . . , a(r) n + n ) ≡ 0 mod mR (r)
for all i. Setting αi to be the limit of the Cauchy sequence (ai ) for 1 ≤ i ≤ n, we have that ∞ miR = (0) fi (α1 , . . . , αn ) ∈ i=1
by Proposition 21.38.
Theorem 21.81. Suppose that φ : X → Y is a regular map of varieties and p ∈ X. Then φ is ´etale in some neighborhood of p if and only if the induced map on complete local rings ˆY,φ(p) → O ˆX,p φˆ∗ : O is an isomorphism. Proof. First suppose that φ : X → Y is ´etale in some neighborhood of p ∈ X. Let notation be as in Deﬁnition 21.79. Let y1 , . . . , ym be generators of the maximal ideal n of q = φ(p) in R. Let m be the maximal ideal of p in R[x1 , . . . , xn ]. Observe that every h ∈ R[x1 , . . . , xn ] has a unique expression h = λ + f with λ ∈ k and f ∈ m. We have that f1 , . . . , fn ∈ m.
´ 21. Ramiﬁcation and Etale Maps
428
Replacing the xi with xi − xi (p) for 1 ≤ i ≤ n, we may assume that m = (y1 , . . . , ym , x1 , . . . , xn ). We have that ∂fi fi − (p)xj ∈ (y1 , . . . , ym ) + m2 ∂xj j
for 1 ≤ i ≤ n. Deﬁne an n × n matrix A = (aij ) by A = for 1 ≤ i ≤ n, n m aij fj + cij yj + hi (21.22) xi = j=1
∂fi ∂xj (p)
−1
. Then
j=1
for some cij ∈ k and with hi ∈ Now substitute the n expressions (21.22) into hi in (21.22) to obtain an expression aij fj + cij yj + dijk fj fk + gijk fj yk + hijk yj yk + Ωi xi = m2 .
j
j
j,k
j,k
j,k
with dijk , gijk , hijk ∈ k and Ωi ∈ m3 . Iterating, we obtain Cauchy sequences in R[x1 , . . . , xn ] which converge to series jm ai1 ...in j1 ...jm f1i1 · · · fnin y1j1 · · · ym xi = ˆ 1 , . . . , xn ]] with ai ...in j ...jm ∈ k. Thus we have expansions for 1 ≤ in R[[x 1 1 i ≤ n, (21.23)
xi = ψi (f1 , . . . , fn ),
ˆ 1 , . . . , zn ]] (a power series ring over R). ˆ By Lemma with ψi (z1 , . . . , zn ) ∈ R[[z 21.80, there exist α1 , . . . , αn ∈ mRˆ such that f1 (α1 , . . . , αn ) = · · · = fn (α1 , . . . , αn ) = 0. ˆ be the homomorphism deﬁned by Λ(g) = ˆ 1 , . . . , xn ]] → R Let Λ : R[[x ˆ 1 , . . . , xn ]] (Lemma 21.43), which has the kernel g(α1 , . . . , αn ) for g ∈ R[[x (x1 − α1 , . . . , xn − αn ), so that (f1 , . . . , fn ) ⊂ (x1 − α1 , . . . , xn − αn ). Evaluating (21.23) at (α1 , . . . , αn ), we have that ψi (f1 , . . . , fn )(α1 , . . . , αn ) = αi , so that (f1 , . . . , fn ) = (x1 − α1 , . . . , xn − αn ). Thus ˆ∼ ˆY,q . ˆ 1 , . . . , xn ]]/(x1 − α1 , . . . , xn − αn ) ∼ ˆX,p ∼ O =R =O = R[[x ˆY,q → O ˆX,p is an isomorphism. We may assume Now suppose that φˆ∗ : O that X and Y are aﬃne. Let A = k[Y ] and B = k[X] be the respective rings ˆX,p )∩OX,p = I of regular functions. For any ideal I ⊂ OX,p , we have that (I O by Lemma 21.37. We have that ˆX,p ) ∩ OX,p = (mp O ˆX,p ) ∩ OX,p = mp . (21.24) mq OX,p = (mq O
´ 21.9. Etale maps and uniformizing parameters
429
Let n be the ideal of q in A. We have a representation B = A[x1 , . . . , xn ]/I, where I is an ideal in a polynomial ring A[x1 , . . . , xn ] over A and m = nA[x1 , . . . , xn ] + (x1 , . . . , xn ) is the ideal of p in A[x1 , . . . , xn ]. We have that ΩA[x1 ,...,xn ]m /A ∼ = ΩA[x1 ,...,xn ]/A ⊗A[x1 ,...,xn ] A[x1 , . . . , xn ]m n ∼ A[x1 , . . . , xn ]m dxi =
by Lemma 14.8
i=1
by Example 14.5. Let N = {df  f ∈ Im }, so that n A[x1 , . . . , xn ]m dxi /N ΩB /A ∼ = m
i=1
by Example 14.7. For 1 ≤ i ≤ n, there exists ai ∈ A, bi ∈ I, and ci ∈ m2 such that xi = ai + bi + ci by (21.24). Thus dxi ∈ N = mΩA[x1 ,...,xn ]m /A and so (ΩB/A ) ⊗B Bm ∼ = ΩBm /A = 0 by Nakayama’s lemma. Thus there exist ∂fi (p) = 0. f1 , . . . , fn ∈ I such that Det ∂x j Deﬁne a ring C by C = A[x1 , . . . , xn ]/(f1 , . . . , fn ). By the ﬁrst part of ˆ this proof, the completion Cˆ of C at the maximal ideal mC is equal to A. Thus the natural maps A → C → B induce isomorphisms ˆ∼ ˆX,p . ˆY,q ∼ O = Aˆ ∼ = Cˆ ∼ =B =O ˆ (by Proposition 21.38), Since C is a subring of Cˆ and B is a subring of B we have that CmC is a subring of BmB . Since B is a quotient of C, we have that BmB = CmC . Thus Im = (f1 , . . . , fm )m in A[x1 , . . . , xn ]m , from which it follows that φ is ´etale near p. Theorem 21.82. Suppose that X is an ndimensional variety and U is an open subset of X. Suppose that f1 , . . . , fn ∈ Γ(U, OX ). Let φ = (f1 , . . . , fn ) : U → An be the induced regular map. Then the following conditions are equivalent: 1) φ is ´etale. 2) For all p ∈ U , t1 = f1 − f1 (p), . . . , tn = fn − fn (p) generate mp /m2p . 3) For all p ∈ U , the kalgebra homomorphism ˆX,p k[[T1 , . . . , Tn ]] → O deﬁned by Ti → ti is an isomorphism. 4) ΩX/k U = ni=1 OU dfi (f1 , . . . , fn are uniformizing parameters on U ). 5) ΩU/An = 0.
´ 21. Ramiﬁcation and Etale Maps
430
Proof. We observe that 3) is equivalent to the statement that ˆAn ,φ(p) → O ˆU,p φˆ∗ : O is an isomorphism for all p ∈ U . The equivalence of 1) and 3) follows from Theorem 21.81. The equivalence of 4) and 5) follows from the exact sequence (14.4) and Example 14.5. If we assume 5) and N is the kernel of the surjection φ∗ ΩAn /k → ΩU/k , then we have that N ⊗ k(X) = 0 since both φ∗ ΩAn /k and ΩU/k have rank n, so that N = 0 since it is a torsion submodule of the locally free sheaf φ∗ ΩAn /k . The equivalence of 2) and 3) is by Proposition 21.62. It remains to establish that 2) is equivalent to 4). Condition 2) implies U is nonsingular by Deﬁnition 10.12, and condition 4) implies U is nonsingular by Theorem 14.14, so we may assume that U is nonsingular. Condition 2) is the statement that for all p ∈ U , t1 , . . . , tn generate mp /m2p , which is equivalent to the statement that df1 , . . . , dfn generate (ΩX/k,p ) ⊗ k(p) by Proposition 14.13. This is equivalent to the statement that df1 , . . . , dfn generate ΩX/k in some neighborhood of p (by Nakayama’s lemma). So condition 2) is equivalent to the statement that ΩU/k is a quo tient of ni=1 OU dfi . Since ΩU/k is locally free of rank n (as U is nonsingular) this is equivalent to statement 4). Ramiﬁcation can also be expressed in terms of K¨ahler diﬀerentials. Theorem 21.83. Suppose that φ : X → Y is a dominant regular map of varieties, p ∈ X, and q = φ(p). Then OY,q → OX,p is unramiﬁed if and only if (ΩX/Y )p = ΩOX,p /OY,q = 0. Proof. We may suppose that X, Y are aﬃne. We express k[X] as a quotient of a polynomial ring k[x1 , . . . , xn , . . . , xm ] so that the subring k[Y ] is a quotient of the polynomial ring k[x1 , . . . , xn ]. There exist g1 , . . . , gN ∈ k[x1 , . . . , xm ] such that k[X] = k[x1 , . . . , xm ]/(g1 , . . . , gN ). Let xi be the residues of xi in k[X]. The point p ∈ X has a maximal ideal α = I(p) in k[x1 , . . . , xm ]. Under the corresponding embedding X ⊂ Am , suppose that p is the point (ξ1 , . . . , ξm ). We will ﬁrst establish that the following condition (21.25) holds if and only if OY,q → OX,p is unramiﬁed: (21.25)
the Jacobian matrix
∂(g1 ,...,gN ) ∂(xn+1 ,...,xm ) (p)
has rank m − n.
First suppose that OY,q → OX,p is unramiﬁed. Suppose that D ∈ Derk (OX,p , k(p)) and DOY,q = 0. Suppose that f ∈ OX,p . There exists
´ 21.9. Etale maps and uniformizing parameters
431
g ∈ OY,q such that f −g ∈ m2p since mp = mq OX,p and OX,p /mp ∼ = OY,q /mq . Thus D(f ) = D(g) = 0, so D = 0. Suppose that u1 , . . . , um ∈ k. A necessary and suﬃcient condition that there exists D ∈ Derk (OX,p , k(p)) such that D(xi ) = ui is that m ∂gi (p)uj = 0 ∂xj
for 1 ≤ i ≤ N.
j=1
This follows since Derk (OX,p , k(p)) ∼ = HomOX,p (ΩOX,p /k , k(p)) by Proposition 14.13 and ΩOX,p /k
⎛
⎞ m ∂gi = OX,p dx1 ⊕ · · · ⊕ OX,p dxm / ⎝ dxj  1 ≤ i ≤ N ⎠ . ∂xj j=1
We have earlier shown that if u1 = u2 = · · · = un = 0, then necessarily un+1 = · · · = um = 0. Thus (21.25) holds. Now suppose that (21.25) holds. We may assume after reindexing the gi that ∂(g1 , . . . , gm−n ) (p) has rank m − n. ∂(xn+1 , . . . , xm ) Thus g1 (ξ1 , . . . , ξn , xn+1 , . . . , xn ), . . . , gm−n (ξ1 , . . . , ξn , xn+1 , . . . , xn ) are uniformizing parameters at the point (ξn+1 , . . . , ξm ) in the aﬃne space Am−n with coordinate ring k(p)[xn+1 , . . . , xm ] by Theorem 21.82. Thus x1 − ξ1 , . . . , xn − ξn , g1 , . . . , gm−n are uniformizing parameters in Am at p (they are in fact a kbasis of α/α2 ). Thus mq OX,p = (x1 − ξ1 , . . . , xn − ξn )OX,p = αOX,p = mp and so OY,q → OX,p is unramiﬁed. It remains to show that equation (21.25) holds if and only if ΩOX,p /OY,q = 0. From the surjection (OY,q [xn+1 , . . . , xm ])α → (OY,q [xn+1 , . . . , xm ]/(g1 , . . . , gN ))α = OX,p , we have that
⎛
⎞ m ∂g i dxj  1 ≤ i ≤ N ⎠ . ΩOX,p /OY,q = OX,p dxn+1 ⊕ · · · ⊕ OX,p dxm / ⎝ ∂xj j=n+1
Thus ΩOX,p /OY,q ⊗k(p) = 0 if and only if (21.25) holds, and this condition is equivalent to ΩOX,p /OY,q = 0 by Nakayama’s lemma, Lemma 1.18. Proposition 21.84. Suppose that φ : X → Y is a dominant regular map of varieties such that φ is ´etale. Then φ is unramiﬁed.
´ 21. Ramiﬁcation and Etale Maps
432
Proof. The proof of Theorem 21.81 shows that ΩX/Y = 0, which implies φ is unramiﬁed by Theorem 21.83. The converse of Proposition 21.84 is false. An example is given in Exercise 21.89. However, the converse is true if Y is normal, as shown by the following proposition, whose proof is Exercise 21.91. Proposition 21.85. Suppose that φ : X → Y is a dominant regular map of varieties and Y is normal. Then φ is unramiﬁed if and only if φ is ´etale. Exercise 21.86. Show that a regular map of varieties φ : X → Y is ´etale if and only if for every p ∈ X, there exist open aﬃne neighborhoods A of p and B of q = φ(p) such that k[A] is a quotient of a polynomial ring over ∂fi ) is a unit in k[A]. k[B], k[A] = k[B][x1 , . . . , xn ]/(f1 , . . . , fn ) where Det( ∂x j Hint: Use Exercise 1.7. Exercise 21.87. Suppose that X is a variety over the complex numbers and U is an open subset of X with uniformizing parameters f1 , . . . , fn on U . Let φ = (f1 , . . . , fn ) : U → An . Suppose that p ∈ U . Use the implicit function theorem (Theorem 10.42) to show that there exists an open neighborhood V of p in the Euclidean topology, which is contained in U such that φ : V → φ(V ) is an analytic isomorphism. Exercise 21.88. Show that the implicit function theorem (Theorem 10.42) is false in the Zariski topology by considering the following example. Suppose that k has characteristic = 2 Let X = Z(x21 − x2 ) ⊂ A2 and let φ : X → A1 be deﬁned by φ(a1 , a2 ) = a2 . At p = (1, 1), we have that ∂ 2 ∂x1 (x1 − x2 ) = 0. Show that φ is not 11 in any Zariski open subset U of X. Exercise 21.89. Let φ : X → Y be the regular map of Exercise 21.65. Show that ΩX/Y = 0 (so that φ is unramiﬁed) but that φ is not ´etale. Exercise 21.90. Suppose that φ : X → Y is a ﬁnite regular map of varieties. The ideal sheaf Ann(ΩX/Y ) is deﬁned (in Exercise 11.43) by Ann(ΩX/Y )(U ) = {f ∈ OX (U )  f ΩX/Y (U ) = 0} for U an open subset of X. Show that φ(Supp(OX /Ann(ΩX/Y ))) is the locus in Y above which φ is ramiﬁed. Conclude that {p ∈ X  φ is unramiﬁed at p} is an open subset of X which is nonempty if and only if k(X) is separable over k(Y ). Exercise 21.91. Prove Proposition 21.85.
21.10. Purity of the branch locus and the AbhyankarJung theorem
433
21.10. Purity of the branch locus and the AbhyankarJung theorem Suppose that φ : X → Y is a dominant ﬁnite regular map of normal varieties. Let (21.26)
Δ = {p ∈ X  φ∗ : OY,φ(p) → OX,p is ramiﬁed}
be the locus of points in X where φ is ramiﬁed. We will call Δ the ramiﬁcation locus of φ in X. By Theorem 21.83, we have that Δ = Supp(ΩX/Y ), so Δ is a closed subset of X. If we also assume that φ∗ : k(Y ) → k(X) is separable, then Δ is a proper subset of X (for instance by Theorem 21.29). We have that φ(Δ) is the ramiﬁcation locus of φ in Y (or the branch locus of φ). Since φ is ﬁnite, if Δ has pure codimension 1 in X (all irreducible components have codimension 1), then the ramiﬁcation locus φ(Δ) has pure codimension 1 in Y . The proof of the following theorem is based on the proof by Zariski in [157, Proposition 2]. Stronger forms of Theorem 21.92 are by Nagata [121, Theorem 41.1], Auslander [14], Grothendieck [64], and Bhatt, CarvajalRojas, Grant, Schwede, and Tucker [21], although Y cannot be too far from being a nonsingular variety for purity of the branch locus to hold. Theorem 21.92 (Purity of the branch locus). Suppose that X is a normal variety, Y is a nonsingular variety, and φ : X → Y is a dominant ﬁnite regular map such that k(Y ) → k(X) is separable. Then the closed set of points of X at which φ is ramiﬁed has pure codimension 1 in X (all irreducible components of the ramiﬁcation locus have codimension 1). Proof. Suppose that a ∈ X and a is not contained in an irreducible component of Δ which has codimension 1 in X. Let q = φ(a) and let x1 , . . . , xn be regular parameters in OY,q . There exists, by Proposition 14.15, an aﬃne open neighborhood U of q in Y such that x1 , . . . , xn are uniformizing parameters on U . Let α : U → An be the corresponding ´etale map (Theorem 21.82). Let V be an aﬃne neighborhood of a such that φ(V ) ⊂ U and Δ ∩ V has codimension ≥ 2 in V . The elements x1 , . . . , xn are a separating transcendence basis of k(X) over k since they are a separating transcendence basis of k(Y ) over k by Theorem 21.75 and k(X) is ﬁnite and separable over k(Y ). Thus the deriva∂ on the rational function ﬁeld k(x1 , . . . , xn ) extend uniquely to a tives ∂x i k(Y )basis of Derk (k(Y ), k(Y )) and to a k(X)basis of Derk (k(X), k(X)), as explained after Theorem 21.75. Suppose that E is a prime divisor on V . Then there exists p ∈ E \ Δ. By Theorem 21.83, we have that ΩOX,p /OY,q = 0
´ 21. Ramiﬁcation and Etale Maps
434 where q = φ(p ), so
ΩOY,q /k ⊗OY,q OX,p ∼ = ΩOX,p /k by (14.4), since ΩOY,q /k is a free module of rank equal to the dimension of X by Proposition 14.15. Now Derk (OY,q , OY,q ) is a free OY,q module with basis ∂x∂ 1 , . . . , ∂x∂n , so Derk (OX,p , OX,p ) ∼ = HomOX,p (ΩOX,p /k , OX,p ) ∼ = HomOY,q (ΩOY,q /k , OY,q ) ⊗OY,q OX,p ∼ = Derk (OY,q , OY,q ) ⊗OY,q OX,p by Lemma 14.3 and since ΩOY,q /k is a free OY,q module. Thus ∂x∂ 1 , . . . , ∂x∂n is a free basis of Derk (OX,p , OX,p ) as an OX,p module. ∂ : k(X) → k(X) map OX,p into OX,p for In particular, the derivations ∂x i ∂ all i, and so ∂xi : OX,E → OX,E for all i, since OX,E is a localization of OX,p . Now, by Theorem 1.79, OX,a = a∈E OX,E , where the intersection is over all prime divisors E of V which contain a since OX,a is integrally closed. Thus the derivations ∂ : OX,a → OX,a for all i. (21.27) ∂xi Suppose that k has characteristic 0. Then we have a natural kalgebra homomorphism ψ : OX,a → k[[x1 , . . . , xn ]] deﬁned by ψ(f ) = ci1 ,...,in xi11 · · · xinn where ci1 ,...,in
1 = i1 ! · · · in !
∂ i1 +···+in f ∂xi11 · · · ∂xinn
(a).
Now ﬁbers of the composed map α ◦ φ : V → An are ﬁnite sets, so the ideal (x1 , . . . , xn )OX,a contains a power mra of the maximal ideal ma of OX,a . Now ψ(mra ) ⊂ (x1 , . . . , xn ) implies ψ(ma ) ⊂ (x1 , . . . , xn ) since (x1 , . . . , xn ) is a prime ideal in k[[x1 , . . . , xn ]]. Thus ψ extends uniquely to a kalgebra ˆX,a → k[[x1 , . . . , xn ]] such that ψ(m ˆ a ) ⊂ (x1 , . . . , xn ) homomorphism ψ : O and ψ(f ) = ψ(f ) for f ∈ OX,a by Lemma 21.43. The natural inclusions of normal domains k[x1 , . . . , xn ](x1 ,...,xn ) = OAn ,α(q) → OY,q → OX,a induce kalgebra homomorphisms of integral domains, by Proposition 21.50, of the same dimension n by Theorem 21.36, ψ ˆAn ,α(q) → O ˆY,q → O ˆX,a → k[[x1 , . . . , xn ]] k[[x1 , . . . , xn ]] = O
21.10. Purity of the branch locus and the AbhyankarJung theorem
435
whose composite is the identity map. Thus each of these maps must be ˆX,a , so that φ is unramiﬁed at a. ˆY,q = O an equality, and in particular, O Thus a ∈ Δ, completing the proof of Theorem 21.92 in the case that k has characteristic 0. Now suppose that k has characteristic p > 0. Then x1 , . . . , xn is a pbasis of k(X) by Theorem 21.76. Let f ∈ OX,a . Since x1 , . . . , xn is a pbasis, we can write f uniquely in the form p f= Ai1 ,...,in xi11 · · · xinn with all Ai1 ,...,xn ∈ k(X) and 0 ≤ i1 + · · · + in < p for all i1 , . . . , in in the sum. Since all partials ∂ i1 +···+in f ∈ OX,a ∂xi11 · · · ∂xinn by (21.27), we have that all Api1 ,...,in ∈ OX,a , and thus all Ai1 ,...,in ∈ OX,a since OX,a is integrally closed. If f is in the maximal ideal maof OX,a , then we have that A0,...,0 ∈ ma , so Ap0,...,0 ∈ mpa ⊂ m2a . Thus ma ⊂ ni=1 xi OX,a + m2a , so ma = (x1 , . . . , xn )OX,a + m2a . Thus ma = (x1 , . . . , xn )OX,a by Nakayama’s lemma, Lemma 1.18. Thus ma = mq OX,a , where mq is the maximal ideal of OY,q and so a ∈ Δ, completing the proof of Theorem 21.92 in the case that k has positive characteristic p. The following theorem is the AbhyankarJung theorem, which generalizes a topological proof in the complex analytic case by Jung [85]. A proof based on Abhyankar’s original proof in [1] will be given in Section 21.12. The above theorem on the purity of the branch locus is an important ingredient in the proof. Generalizations of the AbhyankarJung theorem can be found in [70, Section XII], [71], and other references. Let K → K ∗ be a ﬁnite separable ﬁeld extension of algebraic function ﬁelds, R be a normal algebraic local ring of K, and S be a normal local ∗ ring of K which lies over R. Let J(S/R) = Ann(ΩS/R ), which is an ideal in S deﬁning the locus in S where R → S is ramiﬁed (Theorem 21.83 and Exercise 21.90). That is, Z(J(S/R)) = {Q ∈ Spec(S)  RQ∩R → SQ is ramiﬁed}. If S is ﬁnite over R, deﬁne I(S/R) =
J(S/R) ∩ R
´ 21. Ramiﬁcation and Etale Maps
436
which deﬁnes by Exercise 21.90 the locus in R where R → S is ramiﬁed. That is, if S is ﬁnite over R, Z(I(S/R)) = {P ∈ Spec(R)  there exists Q ∈ Spec(S) such that RP → SQ is ramiﬁed}. Theorem 21.93 (AbhyankarJung theorem). Suppose that Y is a nonsingular variety and X is a normal variety, φ : X → Y is a dominant ﬁnite regular map such that k(Y ) → k(X) is separable, and if the characteristic of the ground ﬁeld k is p > 0, then p does not divide the index [K : k(Y )] where K is a Galois closure of k(X)/k(Y ). Suppose that p ∈ X and q = φ(p) ∈ Y are such that there exist regular parameters x1 , . . . , xn ∈ OY,q such that t i=1 xi ∈ J(OX,p /OY,q ) for some t ≤ n. Let d = [K : k(Y )]. Then there 1 1 exists a subgroup Γ of Zt such that OX,p ∼ = k[[x d , . . . , x d , xt+1 , . . . , xn ]]Γ , d
t
1
1
where the basis element ei ∈ Ztd acts on xid by multiplication by a dth root 1
of unity (in the ground ﬁeld k) and is the identity on xjd if j = i. Corollary 21.94. Let k be an algebraically closed ﬁeld of characteristic 0, let R = k[[x1 , . . . , xn ]] be a power series ring over k, and let f ∈ R[z] be an irreducible monic polynomial. Suppose that the discriminant (21.8) of f is a unit in R times a monomial in x1 , . . . , xn . Then f (z) has a root 1
1
α ∈ k[[x1d , . . . , xnd ]] (a fractional power series) for some d ∈ Z>0 . We in 1 1 deg(f ) fact have a factorization f (z) = i=1 (z − αi ) where αi ∈ k[[x1d , . . . , xnd ]] for all i. Proof. Let L be the quotient ﬁeld of S = R[z]/(f ) = R[z]. By Theorem 21.20, SP is unramiﬁed over RP if P is a prime ideal in R which does not contain the monomial x1 · · · xn . Let T be the integral closure of S in L. We have that SQ → TQ is an isomorphism if Q is a prime ideal in S such that SQ is a regular local ring. Thus TQ is unramiﬁed over SQ if SQ is regular, and so TP is unramiﬁed over RP if P is a prime ideal in R which does not contain x1 · · · xn . Now the proof of Theorem 21.93 generalizes to prove the AbhyankarJung theorem in the situation of this corollary, if we use Nagata’s theorem [121, Theorem 41.1], which proves the purity of the branch locus over an arbitrary regular local ring, giving us inclusions 1
1
R → S → T → k[[x1d , . . . , xnd ]], 1
1
showing that the root α = z of f is in k[[x1d , . . . , xnd ]]. The last assertion 1
1
of the corollary follows since the quotient ﬁeld E of k[[x1d , . . . , xnd ]] is Galois over the quotient ﬁeld of R, and the irreducible polynomial f has a root in 1
1
E, so all roots of f must be contained in the integral closure k[[x1d , . . . , xnd ]] of R in E.
21.10. Purity of the branch locus and the AbhyankarJung theorem
437
Corollary 21.95 (Newton). Suppose that k is an algebraically closed ﬁeld of characteristic 0 and R = k[[x]] is a power series ring over k. Suppose 1 that f ∈ R[z] is irreducible and monic. Then f (z) has a root α ∈ k[[x d ]] (a fractional power series) for some d ∈ Z>0 . It follows that when k is algebraically closed of characteristic 0, the 1 d Laurent ﬁeld of fractional power series expansions (all series ∞ i=m ai x for some d ∈ Z>0 , m ∈ Z, and ai ∈ k) is an algebraic closure of the Laurent ﬁeld k((x)) of formal power series in x over k. Newton’s proof is constructive. His algorithm is explained in [35, Section 2.1] and in the book [26]. Some letters of Newton presenting this algorithm are translated in [26]. This result is no longer true when k has characteristic p > 0. An example showing that this fails is given in Exercise 21.98. The example computed in the exercise is essentially the worst thing that can happen. A construction of an algebraic closure of k((x)) when k is algebraically closed of positive characteristic is given in [88] The following example shows that Corollary 21.95 does not extend to power series rings of dimension greater than 1. Example 21.96. Let k be an algebraically closed ﬁeld of characteristic 0 or p > 5 and let n ≥ 2 be a positive integer which is prime to p. Let R = k[[x, y]] and f = z n + x2 − y 3 . Then there does not exist a fractional 1 1 1 1 power series solution z = α(x d , y d ) ∈ k[[x d , y d ] to f (x, y, z) = 0 for any d ∈ Z>0 . Further, this statement is true for any regular system of parameters y1 , y2 in R (so that R = k[[y1 , y2 ]]). Proof. If there were such a fractional power series solution α, then we would 1 1 have an inclusion S = R[z]/(f ) → T = k[[x d , y d ]]. The ﬁnite extension R → T is then unramiﬁed above primes that do not contain xy and thus R → S is unramiﬁed above primes that do not contain xy. This is a contradiction since R → S is ramiﬁed above the prime ideal (y 3 − x2 ). Exercise 21.97. This exercise shows that the conclusions of Theorem 21.92 may fail if Y is normal but not nonsingular. Consider the ﬁnite map φ : X → Y of aﬃne varieties where φ∗ : k[Y ] = k[x2 , xy, y 2 ] → k[X] = k[x, y] is the natural inclusion and the characteristic of k is = 2. Show that X and Y are normal and compute the locus of points in X at which φ is ramiﬁed. Conclude that the ramiﬁcation locus of φ in X does not have pure codimension 1 in X, so the conclusions of Theorem 21.92 do not hold.
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Exercise 21.98. (This is [1, Example 1].) Consider the ﬁnite map φ : X → Y of aﬃne varieties over a ﬁeld k of characteristic p > 2 where φ∗ : k[Y ] = k[x, y] → k[X] = k[x, y, z]/(z p − xp−1 z − y p−1 ) is the natural inclusion. Show that X is normal and Y is nonsingular and that k(X) is a degree p Galois extension of k(Y ) (an ArtinSchreier extension). Compute the locus of points in X at which φ is ramiﬁed. Show that all of the hypotheses of Theorem 21.8 hold, except the assumption that p does not divide the index [k(Y ) : k(X)]. Show that the conclusions of Theorem 21.8 do not hold. Exercise 21.99. Let k be a ﬁeld of positive characteristic p > 0. Show that σ=
∞
x
1−
1 pi
i=1
is algebraic over the rational function ﬁeld k(x) with minimal polynomial f (z) = z p − xp−1 z − xp−1 . Show that we have a factorization f (z) =
p $ (z − (σ + ix)). i=1
Since this factorization holds in the ﬁeld k(xQ ) of all series with exponents being wellordered subsets of Q and coeﬃcients elements of k, we see that f (z) cannot have a root which is a fractional power series in x (with bounded denominator). The ﬁeld k(x)(σ) is an example of an ArtinSchreier extension of k(x). This type of extension is ultimately responsible for all of the problems which arise in ramiﬁcation in positive characteristic.
21.11. Galois theory of local rings We introduce in this section some material from the section “Galois theory of local rings” in [6], which we will need in the proof of the AbhyankarJung theorem in Section 21.12. Let K → K ∗ be a ﬁnite Galois ﬁeld extension with Galois group G = G(K ∗ /K), let R be a normal local ring with quotient ﬁeld K, and let S be a normal local ring with quotient ﬁeld K ∗ which lies over R. The splitting group Gs (S/R) is deﬁned to be Gs (S/R) = {σ ∈ G  σ(S) = S}. The splitting ﬁeld K s = K s (S/R) is deﬁned to be K s = (K ∗ )G
s (S/R)
.
Lemma 21.100. The ﬁeld K s is the smallest ﬁeld K lying between K and K ∗ for which S is the only normal local ring in K ∗ lying above S ∩ K .
21.11. Galois theory of local rings
439
Proof. Let T be the integral closure of R in K ∗ . Let n1 , . . . , nu be the maximal ideals of T , indexed so that S = Tn1 . Let T s = T ∩K s , msi = ni ∩Ts for 1 ≤ i ≤ u. By the Chinese remainder theorem, Theorem 1.5, there exists a ∈ T such that a ≡ 0 mod n1 and a ≡ 1 mod ni if i > 1. Then the norm $ NK ∗ /K s (a) = σ(a) ∈ n1 ∩ K s = ms1 σ∈Gs (S/R)
and NK ∗ /K s (a) ≡ 1mod ni
for i > 1.
Thus for i > 1, ms1 ⊂ ni so that ni does not lie above ms1 . Hence it is enough to show that if K satisﬁes the assumption of the lemma, then K s ⊂ K . Suppose that σ ∈ G(K ∗ /K ). If σ(nj ) = n1 for some j > 1, then nj ∩ K = σ(nj ∩ K ) = n1 ∩ K , which is a contradiction to our assumptions. Thus Gs (K ∗ /K ) ⊂ Gs , and so K s ⊂ K . Lemma 21.101. Let Rs = K s ∩ S. Then mR Rs = mRs and Rs /mRs ∼ = R/mR so that R → Rs is unramiﬁed. Proof. Let T be the integral closure of R in K ∗ . Let n1 , . . . , nu be the maximal ideals of T , indexed so that S = Tn1 . Let T s = T ∩K s , msi = ni ∩T s for 1 ≤ i ≤ u. Suppose that a ∈ T s . By Lemma 21.100 and the Chinese remainder theorem (Theorem 1.5), there exists b ∈ T s such that b ≡ a mod ms1 and b ≡ 1 mod msi for i > 1, so that b ≡ a mod n1 and b ≡ 1 mod ni for all i > 1. Let σ1 = id, σ2 , . . . , σq be a complete set of representatives of the cosets of Gs (S/R) in G. Then σ1 (b), . . . , σq (b) are the K s /K conjugates of b and hence q $ σt (b) ∈ R. c = NK s /K (b) = t=1
We have that σ1 (b) = b ≡ a mod n1 , and if t > 1, then there exists an i > 1 such that σt (ni ) = n1 , so that σt (b) ≡ 1 mod n1 . Thus σ1 (b) ≡ a mod n1 and for t > 1, σt (b) ≡ 1 mod n1 . Hence c ≡ a mod n1 and c − a ∈ n1 ∩ K s = ms1 so that c ≡ a mod ms1 . Thus Rs /mRs ∼ = T s /ms1 ∼ = R/mR . Let X1 = ms1 and let X2 , . . . , Xv be the other distinct maximal ideals in so that given i > 1, msi = Xt for some t > 1 and given t > 1, Xt = msi for some i > 1. We have a primary decomposition
T s,
mR T s = Y 1 ∩ · · · ∩ Y v = Y 1 · · · Y v where the Yt are primary for Xt by Theorem 1.5. Let Z = X1 ∩ · · · ∩ Xv = X1 · · · Xv .
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Suppose that a ∈ X1 is such that a ∈ Xt for t > 1. We will show that a ∈ Y1 . To see this, let a∗ =
q $
σi (a) ∈
i=2
1 K ⊂ K s. a
Since a∗ is integral over R, a∗ ∈ T s . Since a∗ ∈ n1 , we have that a∗ ∈ X1 . Now aa∗ ∈ mR ⊂ Y1 . Thus a ∈ Y1 , since Y1 is X1 primary. By the Chinese remainder theorem (Theorem 1.5), there exists e ∈ T s such that e ≡ 0 mod X1 and e ≡ 1 mod Xt for t > 1. By the above observation, e ∈ Y1 . Suppose that f ∈ Z. Then f + e ∈ X1 and f + e ∈ Xt if t > 1. Thus f + e ∈ Y1 (again by the observation) and so f ∈ Y1 . Thus Z ⊂ Y1 and hence X1 ⊂ Y1 so that X1 = Y1 , so that mR Rs = X1 Rs = mRs . The inertia group Gi (S/R) is deﬁned to be Gi (S/R) = {σ ∈ Gs (S/R)  σ(u) ≡ u mod mS for all u ∈ S}. The inertia ﬁeld K i = K i (S/R) is deﬁned to be K i = (K ∗ )G (S/R) . i
Lemma 21.102. Let Ri = S ∩ K i . Then mR Ri = mRs Ri = mRi . Further, Ri /mRi is isomorphic to the separable closure of R/mR in S/mS , so that R → Ri and Rs → Ri are unramiﬁed. The group Gi (S/R) is a normal subgroup of Gs (S/R) and Ri /mRi is a Galois extension of R/mR whose Galois group is isomorphic to Gs (S/R)/Gi (S/R). Proof. Let κ = R/mR = Rs /mRs , κi = Ri /mRi , and κ∗ = S/mS . For a ∈ S, let a denote the residue of a in κ∗ . For g ∈ Gs (S/R), deﬁne a κalgebra automorphism Φ(g) : κ∗ → κ∗ by Φ(g)(a) = g(a). We have that Φ : Gs (S/R) → Aut(κ∗ /κ) is a group homomorphism. Let fa (t) be the minimal polynomial of a over K s and let fa (t) be the minimal polynomial of a over κ. We have that fa (t) ∈ Rs [t] by Theorem 21.14. Let f a (t) ∈ κ[t] be obtained from fa (t) by reducing its coeﬃcients modulo mR . Since K ∗ is Galois over K s , we have a factorization fa (t) =
deg $fa
(t − aj ) with aj ∈ K ∗ .
j=1
By Lemma 21.13, all aj ∈ S. Since fa (t) divides f a (t) in κ[t], we have that fa (t) =
deg $fa
(t − atu )
with atu ∈ κ∗ .
u=1
κ∗
is a normal ﬁeld extension of κ. Let κ be the separable closure of Thus ∗ κ in κ . The ﬁeld κ is ﬁnite algebraic over κ by Theorem 21.22. Thus κ is ﬁnite Galois over κ.
21.12. A proof of the AbhyankarJung theorem
441
Let a ∈ S be such that a is a primitive element of κ over κ. Then there exists σu ∈ Gs (K ∗ /K) such that σu (a) = atu . Thus Φ(σu )(a) = atu , and so Φ is surjective onto G(κ /κ). We have that Kernel(Φ) = Gi (S/R). In particular, Gi (S/R) is a normal subgroup of Gs (S/R). Let G = Gs (S/R)/Gi (S/R). We have that Aut(κ∗ /κ) = G(κ /κ) ∼ = G. Further, [K i : K s ] = [Gs (K ∗ /K) : Gi (K ∗ /K)] = [κ : κ] and K i is a Galois extension of K s , with Galois group G. Since S is the only local ring of K ∗ lying over Ri , Gs (S/Ri ) = G(K ∗ /K i ) = Gi (S/R) = Gi (S/Ri ). Thus by the ﬁrst part of the proof, Aut(κ∗ /κi ) ∼ = Gs (S/Ri )/Gi (S/Ri ) = (1). Thus κ∗ is purely inseparable over κi since κ∗ is a normal extension of κ. Hence κ ⊂ κi . Thus [κi : κ]s = [κ : κ] = [K i : K s ]. Now Ri is the unique local ring of K i lying over Rs , so Ri is the integral closure of Rs in K i . By Theorem 21.22, D(Ri /Rs ) = Rs and so by Theorem 21.20, Rs → Ri is unramiﬁed. Thus κi = κ and mRs Ri = mRi , and so R → Ri is unramiﬁed.
21.12. A proof of the AbhyankarJung theorem In this section we give a proof of Theorem 21.93, based on Abhyankar’s original proof in [1]. If K is an algebraic function ﬁeld over a ﬁeld κ, we will say that a local ring R with quotient ﬁeld K is an algebraic local ring of K if R is a localization of a ﬁnite type κalgebra. Suppose that R → S is an extension of dvrs (discrete valuation rings are discussed at the end of Section 21.2). We have that e(S/R) = n if a generator t of mR has an expansion t = un v where u is a generator of mS and v is a unit in S (Section 21.2). Thus the extension R → S is unramiﬁed if and only if e(S/R) = 1 and S/mS is a ﬁnite separable extension of R/mR . If ν is a valuation of a ﬁeld K, we will denote the valuation ring of ν by Vν . Thus Vν = {f ∈ K  ν(f ) ≥ 0}. We denote the maximal ideal of Vν by mν . Suppose that K → K ∗ is a ﬁnite ﬁeld extension. If S = Vν is the valuation ring of a valuation ν of K ∗ and
´ 21. Ramiﬁcation and Etale Maps
442
R = Vν ∩ K = Vμ is the valuation ring of the restriction μ of ν to K, then we will write e(ν/μ) = e(Vν /Vμ ). Proposition 21.103. Let R be a normal local domain which is essentially of ﬁnite type over a ﬁeld κ. Let K be the quotient ﬁeld of R. Let K ∗ be a ﬁnite separable extension of K and let n = [K ∗ : K]. Let x ∈ R be a primitive element of K ∗ over K, with minimal polynomial f (t) ∈ R[t] (such an x exists by Theorem 21.14). Let Si for 1 ≤ i ≤ h be the local rings in K ∗ ˆ and Sˆi lying over R and let S be the integral closure of R in K ∗ . Then R ˆ ˆ are normal local domains, the natural homomorphisms R → Si are injective for all i, and we have a natural isomorphism ˆ∼ S ⊗R R =
h
Sˆi .
i=1
ˆ and Ei be the quotient ﬁeld of Sˆi for 1 ≤ Let E be the quotient ﬁeld of R i ≤ h. Let ei = [Ei : E]. Then n = e1 + · · · + eh . Further, x is a primitive element of Ei over E for all i with minimal polynomial fi (t) ∈ E[t] and there is a factorization f (t) = f1 (t) · · · fh (t) in E[t]. ˆ is a normal local domain by Proposition 21.50. By Proof. We have that R [161, Theorem 16, page 277] and [161, Corollary 2, page 283], we have a natural isomorphism ˆ∼ Sˆi . S ⊗R R = We have that the Sˆi are normal local domains by Proposition 21.50. We have that K ∗ ⊗K E ∼ = E[x] ∼ = E[t]/(f (t)). Let A be a ring. The total quotient ring of A is QR(A) = S −1 A where S is the multiplicative set of all nonzero divisors of A. Let g ∈ K ∗ ⊗K E = E[x]. ˆ ˆ \ {0}. Since R ˆ is a domain, b and b ∈ R Then g = ab where a ∈ R[x] ˆ is not a zero divisor in S ⊗R R by [161, Theorem 16, page 277]. Thus ˆ Since the reduced ring S ⊗R R ˆ is naturally a K ∗ ⊗K E ⊂ QR(S ⊗R R). ˆ ˆ ⊂ K ∗ ⊗K E. subring of QR(S ⊗R R), we have a natural inclusion S ⊗R R ∗ Now f (t) is reduced in E[t] since K is separable over K. We have that ∗ ∼ E[t]/(fi (t)) is a direct sum of ﬁelds, where the fi (t) are K ⊗K E = the irreducible factors of f (t) in E[t] by the Chinese remainder theorem, ˆ = K ∗ ⊗K E. Now S ⊗R R ˆ is reduced, so Theorem 1.5. Thus QR(S ⊗R R) ∼ ˆ QR(S⊗R R) = i Ei . Thus, after reindexing, we have that Ei ∼ = E[t]/(fi (t)) and we have that deg(fi ) = deg(f ) = [K ∗ : K]. [Ei : E] =
21.12. A proof of the AbhyankarJung theorem
443
Suppose that K is an algebraic function ﬁeld over a ﬁeld κ and K is a ﬁnite separable extension of K. Suppose that R is a normal, algebraic local ring of K and R is a normal local ring of K which lies over R. Let E be ˆ . Deﬁne ˆ and let E be the quotient ﬁeld of R the quotient ﬁeld of R d(R : R) = [E : E],
g(R : R) = [R /mR : R/mR ]s ,
and r(R : R) = d(R : R)/g(R : R). We have that d(R : R), g(R : R), r(R : R) are all multiplicative in towers of ﬁelds. Now suppose that K ∗ is a ﬁnite Galois extension of K and that R∗ is a normal local ring of K ∗ which lies over R. Let Rs = R∗ ∩ K s where K s = K s (R∗ /R) is the splitting ﬁeld of R∗ over R. Let Ri = R∗ ∩ K i where K i is the inertia ﬁeld K i = K i (R∗ /R) of R∗ over R. Let E, E s , E i , E ∗ be ˆ i , and R ˆ∗. ˆ R ˆs, R the respective quotient ﬁelds of R, Since R∗ is the unique local ring of K ∗ which dominates Rs by Lemma 21.100, we have by Proposition 21.103 that (21.28)
[K ∗ : K s ] = d(R∗ : Rs )
and [K ∗ : K i ] = d(R∗ : Ri ).
By Proposition 21.103 and Lemma 21.102 we have that (21.29)
[K i : K s ] = d(Ri : Rs ) = g(R∗ : R).
ˆ by Lemma 21.101, Proposition 21.49, and Proposition ˆs ∼ We have that R =R s ∼ 21.48, and so E = E. Thus (21.30)
d(Rs : R) = 1.
Lemma 21.104. Let notation be as above. Then r(R : R) is a positive integer. Proof. Let K ∗ be a ﬁnite Galois extension of K which contains K , and let R∗ be a normal local ring of K ∗ such that R∗ lies over R . We have that Gi (R∗ /R ) = Gi (R∗ /R) ∩ G(K ∗ /K ) and Gs (R∗ /R ) = Gs (R∗ /R) ∩ G(K ∗ /K ),
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444
so (K )s = K K s and (K )i = K K i . Thus we have a commutative diagram of ﬁelds K∗ # $ i → (K )i K ↑ ↑ → (K )s Ks ↑ ↑ K → K where K s = K s (R∗ /R), K i = K i (R∗ /R), (K )s = (K )s (R∗ /R ), and (K )i = (K )i (R∗ /R ). Considering the induced commutative diagram of ﬁelds E∗ # $ i → (E )i E ↑ ↑ Es → (E )s ↑ ↑ E → E where E, E , E ∗ , E s , E i , (E )s , (E )i are the respective quotient ﬁelds of ˆ R,
1 , R
1∗ , R
1s , R
1i , R
)s , (R
)i , (R
we see from (21.29) and (21.30) that d(R : R)g(R∗ : R ) = g(R∗ : R)d((R )i : Ri ). Thus d(R : R) = g(R : R)d((R )i : Ri ).
We also read oﬀ the following formulas from the commutative diagrams of the proof of Lemma 21.104 and from (21.28), (21.29), and (21.30): [K ∗ : K i ][R∗ /mR∗ : R/mR ]s = d(R : R)[K ∗ : (K )i ][R∗ /mR∗ : R /mR ]s , so that (21.31)
[K ∗ : K i ] = r(R : R)[K ∗ : (K )i ]
and (21.32)
[K ∗ : K s ] = d(R : R)[K ∗ : (K )s ].
Lemma 21.105. Let notation be as above. Then: 1) Gi (R∗ /R) ⊂ G(K ∗ /K ) if and only if r(R : R) = 1. 2) Gs (R∗ /R) ⊂ G(K ∗ /K ) if and only if d(R : R) = 1.
21.12. A proof of the AbhyankarJung theorem
445
Proof. The lemma follows from equations (21.31) and (21.32) and the observations that Gi (R∗ /R) ⊂ G(K ∗ /K ) if and only if (K )i = K i and Gs (R∗ /R) ⊂ G(K ∗ /K ) if and only if (K )s = K s . Proposition 21.106. Suppose that R is an algebraic normal local ring with quotient ﬁeld K, K is a ﬁnite ﬁeld extension of K, and R is a normal local ring of K which lies over R. Then R → R is unramiﬁed if and only if r(R /R) = 1. ˆ →R ˆ is unProof. The extension R → R is unramiﬁed if and only if R ˆ /R) ˆ =R ˆ by Theorem 21.21. This ramiﬁed which holds if and only if D(R holds if and only if [R /mR : R/mR ]s = [E : E], ˆ and E is the quotient ﬁeld of R ˆ by where E is the quotient ﬁeld of R Theorem 21.22. But this is equivalent to the condition that r(R /R) = 1. Proposition 21.107. Let R be a normal algebraic local ring with quotient ﬁeld K. Let K be a ﬁnite separable extension of K and let K ∗ be a least Galois extension of K containing K . Let I be an ideal deﬁning the ramiﬁcation locus in R of R → K (for a prime P ∈ Spec(R), RP → K is ramiﬁed if and only if I ⊂ P ). Then I is an ideal deﬁning the ramiﬁcation locus in R of R → K ∗ . Proof. We must show that for P ∈ Spec(R), RP → K is ramiﬁed if and only if RP → K ∗ is ramiﬁed. It follows from the deﬁnition of ramiﬁcation that RP → K ∗ unramiﬁed implies RP → K is unramiﬁed. Suppose that RP → K is unramiﬁed. Let Rj∗ with 1 ≤ j ≤ t be the local rings of K ∗ lying over RP , G = G(K ∗ /K), G = (K ∗ /K ), and Gj = Gi (Rj∗ /R). Then Gj ⊂ G for 1 ≤ j ≤ t by Lemma 21.105 and Proposition 21.106. Let G be the smallest subgroup of G which contains G1 , . . . , Gt . Since the Gj are conjugate subgroups, G is a normal subgroup of G and G ⊂ G . Thus the ﬁxed ﬁeld K of G is a Galois extension of K containing K , and hence by minimality of K ∗ , we have that K = K ∗ ; that is, G = (1). Thus G1 = G2 = · · · = Gt = (1). Deﬁning Kji to be the inertial ﬁeld Kji = (K ∗ )Gi , this implies that K ∗ = Kji . Thus R is unramiﬁed in K ∗ by Lemma 21.102. Lemma 21.108. Let R be a normal algebraic local ring with quotient ﬁeld K, which is an algebraic function ﬁeld over an algebraically closed ﬁeld, and suppose that K ∗ is a ﬁnite Galois extension of K. Let p be the characteristic of K and let y1 , . . . , yh be a ﬁnite number of elements in R such that for all j, Nj = yj R is a height 1 prime ideal in R and hence that RNj is the valuation ring of a discrete valuation ωj of K. Let ωj∗ be an extension of ωj to K ∗
´ 21. Ramiﬁcation and Etale Maps
446
and let nj = e(ωj∗ /ωj ), which is deﬁned in (21.6) (nj , the residue degree of ωj∗ over ωj , and the inseparable degree of the residue ﬁeld of ωj∗ over ωj only depend on j since K → K ∗ is Galois, as shown in [160, Section 10, Chapter V]). Assume that if p = 0, then p does not divide nj for j = 1, . . . , h. Let 1 nj
yj
be an nj th root of yj . Let 1 n
1 n
K = K(y1 1 , . . . , yh h )
1 n
∗
1 n
K = K ∗ (y1 1 , . . . , yh h ).
and
Let N0 be a height 1 prime deal in R diﬀerent from N1 , . . . , Nh and let ω0 be the discrete valuation of K with valuation ring RN0 . Let ω ∗j be an extension ∗ of ωj∗ to K and let ω j be the restriction of ω ∗j to K. Let Vωj , Vwj , Vωj∗ , and Vωj be the respective valuation rings. Then: ∗
∗
∗
a) K /K, K/K, K /K ∗ , and K /K are Galois extensions. b) For 1 ≤ j ≤ h, we have that e(ω ∗j /ωj∗ ) = 1 and Vω∗j /mω∗j is separable over Vωj∗ /mωj∗ . Further, e(ω j /ωj ) = nj and Vωj /mωj is separable over Vωj /mωj . c) e(ω ∗0 /ω0∗ ) = 1 and Vω∗0 /mω∗0 is separable over Vω0∗ /mω0∗ . Further, e(ω ∗0 /ω 0 ) = e(ω0∗ /ω0 ) and the inseparability indices [Vω∗0 /mω∗0 : Vω0 /mω0 ]i = [Vω0∗ /mω0∗ : Vω0 /mω0 ]i . Proof. We assume that h = 1. The general case then follows by induction on h, using the fact that e and the inseparable degree of residue ﬁeld extensions are multiplicative for extensions of dvrs in towers of ﬁelds. Statement a) follows since the composition of two Galois extensions is again Galois (Theorem 1.14, page 267 of [95]). ∗
1 n
∗
Let n = [K : K ∗ ]. Then n  n1 . Let x = y1 1 which is a generator of K over K ∗ and let z = xn . Then f (X) = X n − z is the minimal polynomial of n(n−1)
x over K ∗ . Now z n−1 (times the unit (−1) 2 nn ) is the discriminant of f (X) (as follows from (21.7) and (21.9)). We have that z n−1 is a unit in Vω0∗ , ∗ so Vω0∗ = D(K /Vω0∗ ) and thus Vω0∗ → Vω∗0 is unramiﬁed, so e(ω ∗0 /ω0∗ ) = 1 and Vω0∗ /mω0∗ → Vω∗0 /mω∗0 is a ﬁnite separable extension. Fix u ∈ Vω1∗ which is a generator of mω1∗ . Now ux is a primitive element ∗ of K over K ∗ and g(X) = X n − zu−n is the minimal polynomial of ux over K ∗ . Now ω1∗ (z) = nω1∗ (x) and ω1∗ (y1 ) = e(ω1∗ /ω1 ) = n1 , so ω1∗ (x) = 1 and thus ω1∗ ( uzn ) = n − n = 0. We thus have that uzn is a unit in Vω1∗ and since ∗ ( uzn )n−1 is the discriminant of g(X), we have that D(K /Vω1∗ ) = Vω1∗ and thus Vω1∗ → Vω∗1 is unramiﬁed and so e(ω ∗1 /ω1∗ ) = 1 and Vω1∗ /mω1∗ → Vω∗1 /mω∗1 is a ﬁnite separable extension.
21.12. A proof of the AbhyankarJung theorem
447
We have that n1 ω 1 (x) = ω 1 (y1 ), so e(ω 1 /ω1 ) ≥ n1 . But [Vω1 /mω1 : Vω1 /mω1 ]e(ω 1 /ω1 ) ≤ [K : K] = n1 by [160, Theorem 22, page 289]. Thus e(ω 1 /ω1 ) = n1 and Vω1 /mω1 = Vω1 /mω1 . By a similar calculation to the analysis of Vω0∗ → Vω∗0 , we obtain that e(ω 0 /ω0 ) = 1 and Vω0 /mω0 → Vω0 /mω0 is ﬁnite and separable. From the identities e(ω ∗0 /ω 0 )e(ω 0 /ω0 ) = e(ω ∗0 /ω0∗ )e(ω0∗ /ω0 ) and [Vω∗0 /mω∗0 : Vω0∗ /mω0∗ ]i [Vω0∗ /mω0∗ : Vω0 /mω0 ]i = [Vω∗0 /mω∗0 : Vω0 /mω0 ]i [Vω0 /mω0 : Vω0 /mω0 ]i ,
we obtain that e(ω ∗0 /ω 0 ) = e(ω0∗ /ω0 ) and
[Vω∗0 /mω∗0 : Vω0 /mω0 ]i = [Vω0∗ /mω0∗ : Vω0 /mω0 ]i .
Lemma 21.109. Let K be an algebraic function ﬁeld and let K ∗ be a ﬁnite Galois extension of K. Let R be a normal algebraic local ring of K and let R∗ be a normal algebraic local ring of K ∗ such that R∗ lies over R. Let E ˆ and let E ∗ be the quotient ﬁeld of R ˆ ∗ . Then E ∗ is be the quotient ﬁeld of R s ∗ Galois over E with Galois group isomorphic to G (R /R) by restriction in the commutative diagram E → E∗ ↑ ↑ K → K ∗. Proof. Let K s be the splitting ﬁeld of R∗ over R. The ﬁeld K ∗ is Galois 1s = R ˆ over K s with Galois group G = Gs (R∗ /R). Let Rs = R∗ ∩K s . Then R 1s is isomorphic to E. by Lemma 21.101, and so the quotient ﬁeld E s of R ∗ ∗ s Since R is the unique local ring of K lying over R , by Lemma 21.100, we have that (21.33)
[E ∗ : E] = [K s : K]
by Proposition 21.103. Suppose that σ ∈ Gs (R∗ /R). Then there is a commutative diagram R → R∗ ↓σ R∗ . Taking completions and quotient ﬁelds, we obtain an Eautomorphism of E ∗ which extends σ, and thus we have an inclusion of Gs (R∗ /R) into Aut(E ∗ /E). Thus by (21.33), we have that E ∗ is Galois over E with Galois group Gs (R∗ /R).
´ 21. Ramiﬁcation and Etale Maps
448
We now give the proof of Theorem 21.93. Let K = k(Y ), K ∗ = k(X), R = OY,q , and R∗ = OX,p , so that R∗ lies over R. We are given the assumption that ti=1 xi ∈ J(R∗ /R) for some regular system of parameters x1 , . . . , xt , . . . , xn in R. Let K be a least Galois extension of K containing K ∗ . By assumption, the characteristic p of k(X) does not divide [K : K]. Let R be an algebraic local ring of K which lies over R∗ . Let s K s = (K )G (R /R) and let K be the composite of K ∗ and K s . Then s ∗ K = (K )G (R /R ) since Gs (R /R∗ ) = Gs (R /R) ∩ G(K /K ∗ ). Let Rs be the normal local ring of K s such that R lies over Rs and let R be the normal local ring of K such that R lies over R .
Now R → Rs and R∗ → R are unramiﬁed by Lemma 21.101, so that
J(R /Rs ) = J(R /R) ⊃ J(R∗ /R) by Theorem 21.83 and Theorem 14.9 and thus ti=1 xi ∈ I(R /Rs ), which deﬁnes the ramiﬁcation locus of Rs → K since Rs → R is ﬁnite as R is the ˜ be a least unique local ring of K lying over Rs by Lemma 21.100. Let K s ˜ Galois extension over K containing K in K . Let R be the normal local ˜ By Proposition 21.107 and since R ˜ is ˜ such that R lies over R. ring of K t s s ˜ the unique local ring of K lying over R , i=1 xi ∈ I(R /R ) = I(R /Rs ) ˜ deﬁnes the ramiﬁcation locus of Rs in K. ˆ s , let ˆ let E s be the quotient ﬁeld of R Let E be the quotient ﬁeld of R, 1 ˆ ∗ , and let E ˜ E be the quotient ﬁeld of R , let E ∗ be the quotient ﬁeld of R 1 0˜ Then R ˆ∗ ∼ ˆ∼ ˆ s and R be the quotient ﬁeld of R. = R , by Lemma 21.101, =R ˜ is Galois over K , we have that E ˜ is so E ∼ = E . Since K = E s and E ∗ ∼
Galois over E by Lemma 21.109. Thus it suﬃces to prove the theorem with R replaced with Rs and R∗ replaced with R , K replaced with K s and K ∗ replaced with K . In summary, we have reduced to the situation where we have a tower of ﬁelds ˜ K → K∗ → K ˜ is Galois over K, p does not divide [K ˜ : K], and (after possibly perwhere K ˜ = ( t xi ) where x1 , . . . , xn muting x1 , . . . , xt and decreasing t) D(K/R) i=1
˜ is the unique local is a suitable regular system of parameters in R, and R ˜ ring of K which lies over R. Let wi be the valuation of K with valuation ring Vωi = R(xi ) for 1 ≤ i ≤ ˜ for 1 ≤ i ≤ t. Let ni = e(˜ ωi /ωi ) t, and let ω ˜ i be an extension of ωi to K ˜ i of ωi , as explained for 1 ≤ i ≤ t (ni does not depend on the extension ω ˜ : K] by Proposition 21.23, so p in Proposition 21.23). Now ni divides [K
21.12. A proof of the AbhyankarJung theorem
449
1 n
1
does not divide ni for 1 ≤ i ≤ t. Let K1 = K(x1 1 , . . . , xtnt ) and K2 = 1
1
˜ and ˜ n1 , . . . , x nt ). Let R2 be a normal local ring of K2 which lies over R K(x t 1 let R1 = R2 ∩ K1 . All extensions in ˜ K → K ↓ ↓ K1 → K2 are Galois by a) of Lemma 21.108. ˆ 1 is the integral closure of R ˆ in ˆ∼ We have that R = k[[x1 , . . . , xn ]] and R 1 n
1
k((x1 1 , . . . , xtnt , xt+1 , . . . , xn )) 1
1
n n ˆ 1 is ˆ1 ∼ by Proposition 21.55. Thus R = k[[x1 1 , . . . , xt t , xt+1 , . . . , xn ]]. Since R regular, we have that R1 is a regular local ring.
Suppose that ν is a valuation of K whose valuation ring is RP where P is a height 1 prime ideal in R. Let ν2 be an extension of ν to K2 , let ν˜ ˜ and let ν1 be the restriction of ν2 to K1 . We be the restriction of ν2 to K, ˜ : K] have that p does not divide [Vν˜ /mν˜ : Vν /mν ] since p does not divide [K and by Proposition 21.23. Hence Vν˜ /mν˜ is separable over Vν /mν . If P is a height 1 prime ideal of R such that P = (xi ) for some i with 1 ≤ i ≤ t, ˜ is unramiﬁed. Thus if ν ˜ then we have that D(K/K)
⊂ P , so RP → K ν /ν) = 1. is a valuation of K whose valuation ring is RP , we have that e(˜ Thus by Lemma 21.108, R1 → R2 is unramiﬁed in codimension 1. By the purity of the branch locus, Theorem 21.92, R1 → R2 is unramiﬁed since R1 is regular. Thus by Proposition 21.62, 1
1
n n ˆ2 ∼ ˆ1 ∼ R =R = k[[x1 1 , . . . , xt t , xt+1 , . . . , xn ]]. ˆ let E ∗ be the quotient ﬁeld of R ˆ ∗ , let E1 Let E be the quotient ﬁeld of R, ˆ 1 , and let E2 be the quotient ﬁeld of R ˆ2. be the quotient ﬁeld of R
Now E2 is Galois over E ∗ with Galois group G(E2 /E ∗ ) = Gs (R2 /R∗ ) and E2 is Galois over E with Galois group G(E2 /E) = Gs (R2 /R) by Lemma 21.109. We further have that E2 ∼ = E1 is Galois over E with Galois group t n i , which acts by multiplication of roots of unity on the Z G(E2 /E) ∼ = i=1 1
n xi i . Now G(E2 /E ∗ ) ∼ = Gs (R2 /R∗ ) is a subgroup of G(E2 /E) and thus s (R /R∗ ) G 2 ˆ ˆ∗ ∼ has the desired form. R =R 2
Chapter 22
Bertini’s Theorems and General Fibers of Maps
We ﬁrst consider the question of when a variety X over a not necessarily algebraically closed ﬁeld k0 satisﬁes the property that the scheme X obtained from X by extending the ﬁeld k0 to a larger ﬁeld k is always reduced, irreducible, or integral (reduced and irreducible). If this property holds for all extension ﬁelds k of k0 , then X is said to be geometrically reduced, geometrically irreducible, or geometrically integral. In Corollary 22.3 we characterize these conditions in terms of properties of the extension ﬁeld k0 (X) of k0 , and we say that the extension k0 → k0 (X) is geometrically reduced, geometrically irreducible, or geometrically integral if the variety X has this property over k0 . In Proposition 22.14 and more generally in Theorem 22.18, it is shown that the general ﬁber of a dominant regular map φ : X → Y of varieties has one of the properties geometrically reduced, geometrically irreducible, or geometrically integral if and only if the extension of function ﬁelds k(Y ) → k(X) has this property. We establish Theorem 22.4, showing that the general ﬁber of a map from a nonsingular characteristic 0 variety is nonsingular. This property does not hold in positive characteristic, as shown by Exercise 22.20. In Deﬁnition 22.8 and Theorem 22.9 we introduce the general notion of a smooth morphism of schemes. We derive the two theorems of Bertini over algebraically closed ﬁelds k. The ﬁrst theorem of Bertini, Theorem 22.12, states that if X is a normal variety over an algebraically closed ﬁeld k of characteristic p ≥ 0 and L is a linear system on X without ﬁxed component which is not composite with a pencil (L does not induce a rational map to a curve), then there is 451
452
22. Bertini’s Theorems and General Fibers of Maps
a power pe such that a general member of L has the form pe D where D is a prime divisor. Classical algebraic proofs of the ﬁrst theorem are given by Zariski in [151] (when k has characteristic 0), Matsusaka [104], and Zariski in [158, Section I.6] (in any characteristic). Our proof of the ﬁrst theorem is based on the proof in [158, Section 1.6]. The second theorem of Bertini, Theorem 22.11, states that if X is a nonsingular variety over an algebraically closed ﬁeld k of characteristic 0 and L is a linear system without ﬁxed component on X, then a general member of L is nonsingular outside the base locus of L. The second theorem is only true in characteristic 0, as shown by Exercise 22.19.
22.1. Geometric integrality A ring A containing a ﬁeld κ is called geometrically irreducible over κ if the nilradical of A ⊗κ k is a prime ideal for all extension ﬁelds k of κ. The ring A is called geometrically reduced over κ if A ⊗κ k is reduced for all extension ﬁelds k of κ. The ring A is called geometrically integral over κ if A ⊗κ k is a domain for all extension ﬁelds k of κ. A scheme X over a ﬁeld κ (there is a natural inclusion κ → OX (U ) for all open subsets U of X) is called geometrically irreducible over κ if the nilradical of OX (U ) ⊗κ k is a prime ideal for all open aﬃne subsets U of X and extension ﬁelds k of κ. The scheme X is called geometrically reduced over κ if OX (U ) ⊗κ k is reduced for all open aﬃne subsets U of X and extension ﬁelds k of κ. The scheme X is called geometrically integral over κ if OX (U ) ⊗κ k is a domain for all open aﬃne subsets U of X and extension ﬁelds k of κ. In the language of Foundations of Algebraic Geometry [145], a geometrically integral variety is called “absolutely irreducible”. Deﬁnition 22.1. If K is an extension ﬁeld of a ﬁeld κ of characteristic p, −1 then K is said to be a separable extension of κ if K and κp are linearly disjoint over κ. If K is an algebraic extension of κ, then this deﬁnition of separability is equivalent to the deﬁnition of separability for algebraic extensions, by [160, Theorem 34, page 109]. If K is a ﬁnitely generated extension of κ, then K is separably generated over κ if and only if K is a separable extension of κ, by [160, Theorem 35, page 111]. A dominant regular map ϕ : X → Y of varieties is said to be separable if the induced extension of ﬁelds k(Y ) → k(X) is separable.
22.1. Geometric integrality
453
Proposition 22.2. Suppose that F is a ﬁnitely generated extension ﬁeld of a ﬁeld K. Then: 1) F is geometrically reduced over K if and only if F is separable over K. 2) F is geometrically irreducible over K if and only if K is separably closed in F (if α ∈ F is separably algebraic over K, then α ∈ K). 3) F is geometrically integral over K if and only if F is separable over K and K is separably closed in F . Proof. The “if” statement of 1) follows from [160, Theorem 39, Chapter 3, page 195]. For the “only if” statement of 1), suppose that F is not −1 separable over K. Then F and K p (where p is the characteristic of K) are not linearly disjoint over K (Deﬁnition 22.1), so by Theorem 5.3, the kernel −1 p of the canonical homomorphism Λ of F ⊗K K p onto the subring S of an −1 algebraic closure of F generated by F and K p is nontrivial. Now f ∈ p implies f p ∈ (F ⊗ 1) ∩ kernel(Λ) = (0) so all elements of p are nilpotent. We now prove the “if” statement of 2). In the case that K has characteristic 0, this follows from [160, Corollary 2 to Theorem 40, page 198]. Suppose that K has characteristic p > 0. Since K is separably closed in F , we have that F and K are quasilinearly disjoint over K by [160, Theorem 40, page 197], and thus all free joins of F/K and K /K are equivalent by [160, Theorem 38 page 192], and so the zero ideal of F ⊗K K is primary by [160, Corollary 2, page 195]. For the “only if” statement of 2), suppose that K is not separably closed in F . There exists α ∈ F \ K which is separably algebraic over K. Let f (x) ∈ K[x] be the minimal polynomial of α. Let L = K(α). We have that ∼ K(α)[x]/(f (x)). L ⊗K L = Since we have a factorization f (x) = (x − α)g(x) in K(α)[x] where x − α and g(x) are reduced and relatively prime, the zero ideal of L ⊗K L is not primary. Since we have a natural inclusion L ⊗K L ⊂ F ⊗K L, the zero ideal of F ⊗K L is not primary. Conclusion 3) follows from 1) and 2). Corollary 22.3. Suppose that X is a variety over a (not necessarily algebraically closed) ﬁeld k0 (as deﬁned in Section 15.4). Then: 1) X is geometrically reduced over k0 if and only if k0 (X) is separable over k0 . 2) X is geometrically irreducible over k0 if and only if k0 is separably closed in k0 (X). 3) X is geometrically integral over k0 if and only if k0 (X) is separable over k0 and k0 is separably closed in k0 (X).
454
22. Bertini’s Theorems and General Fibers of Maps
Proof. By considering an aﬃne cover of X, we reduce to the case when X is an aﬃne variety. Suppose that X is aﬃne. Let S be the multiplicative set S = k0 [X] \ {0}, and let k be an extension ﬁeld of k0 . By consideration of the natural inclusion k0 [X] ⊗k0 k → k0 (X) ⊗k0 k = S −1 (k0 [X] ⊗k0 k ), we see that k0 [X] is geometrically irreducible, geometrically reduced, or geometrically integral over k0 if and only if k0 (X) has this property over k0 .
22.2. Nonsingularity of the general ﬁber In this section, we prove the following theorem. Theorem 22.4. Let φ : X → Y be a dominant regular map of varieties over an (algebraically closed) ﬁeld k of characteristic 0, and suppose that X is nonsingular. Then there exists a nonempty open subset U of Y such that the ﬁber Xb is nonsingular for all b ∈ U ; that is, OXb ,a = OX,a /mb OX,a , where mb is the maximal ideal of OY,b , is a regular local ring for all a ∈ φ−1 (b). The conclusions of Theorem 22.4 are false in positive characteristic. Exercise 22.20 gives a positive characteristic example where k(X) is geometrically integral over k(Y ) and all ﬁbers of φ are singular. Throughout this section we will assume that the assumptions of Theorem 22.4 hold. By Theorems 8.13 and 10.16, after replacing Y with an open subset, we may assume that for all b ∈ Y , every irreducible component of φ−1 (b) has dimension n − m where m = dim Y and n = dim X and Y is nonsingular. Lemma 22.5. With the above assumptions, suppose that b ∈ Y and Ta (X) → Tb (Y ) is surjective for all a ∈ φ−1 (b). Then Xb is nonsingular. Proof. Suppose that a ∈ φ−1 (b). Let A = OXb ,a = OX,a /mb OX,a , and let n be the maximal ideal of A. We have that A/n ∼ = k(a) ∼ = k. The natural surjection of local rings OX,a → A induces a surjection of kvector spaces (22.1)
ma /m2a → n/n2 → 0
where ma is the maximal ideal of OX,a . Taking the dual of (22.1), we have an injection of kvector spaces, 0 → T (A) = Homk (n/n2 , k) → Ta (X). The argument above (10.5) (after Deﬁnition 10.8) is applicable to our slightly more general situation and shows that T (A) ⊂ Kernel(dφa ).
22.2. Nonsingularity of the general ﬁber
455
Thus dimA/n (n/n2 ) = dimk (T (A)) ≤ dim Ta (X) − dim Tb (Y ) = dim X − dim Y since a is a nonsingular point of X and b is a nonsingular point of Y . We have that dimA/n n/n2 ≥ dim A = dim φ−1 (b) = dim X − dim Y. Thus dim A = dimA/n n/n2 , and A is a regular local ring.
Lemma 22.6. Let assumptions be as above. Then there exists a nonempty open subset V of X such that dφa is a surjection for all a ∈ V . Proof. Let b ∈ Y and let u1 , . . . , um be regular parameters in OY,b . Then du1 , . . . , dum is a free OY,b basis of (ΩY /k )b by Proposition 14.15, and since Ωk(Y )/k is a localization of (ΩY /k )b , it is also a k(Y )basis of Ωk(Y )/k . Thus u1 , . . . , um is a transcendence basis of k(Y )/k by Theorem 21.75. We can extend u1 , . . . , um to a transcendence basis u1 , . . . , un of k(X)/k. Since k has characteristic 0, u1 , . . . , un is a separating transcendence basis of k(X) over k, so by Theorem 21.75, du1 , . . . , dun is a k(X)basis of Ωk(X)/k , and thus there exists a nonempty open subset U of X such that du1 , . . . , dun is a free basis of ΩU/k . By Proposition 14.17, u1 − u1 (a), . . . , un − un (a) are regular parameters in OX,a for all a ∈ U . (Here we are making the usual abuse of notation, identifying φ∗ (ui ) with ui for 1 ≤ i ≤ m; in particular, by ui (a) for 1 ≤ i ≤ m we mean φ∗ (ui )(a) = ui (φ(a))). By Propositions 14.15 and 14.17, there exists an open neighborhood W of b in Y such that u1 − u1 (c), . . . , um − um (c) are regular parameters in OY,c for all c ∈ W . Suppose that a ∈ U ∩ φ−1 (W ). Then u1 − u1 (a), . . . , um − um (a) are regular parameters in OY,φ(a) and u1 − u1 (a), . . . , un − un (a) are regular parameters in OX,a . The classes of u1 − u1 (a), . . . , um − um (a) form a kbasis of mφ(a) /m2φ(a) and the classes of u1 − u1 (a), . . . , un − un (a) form a kbasis of ma /m2a . Thus mφ(a) /m2φ(a) → ma /m2a is an inclusion of kvector spaces, so dφa : Ta (X) → Tφ(a) (Y )
is surjective. Lemma 22.7. Let assumptions be as above. Suppose that r ∈ N, and let Xr = {a ∈ X  rank dφa ≤ r}. Then dim φ(Xr ) ≤ r.
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Proof. Let Y be an irreducible component of the Zariski closure φ(Xr ) of φ(Xr ) in Y , and let X be an irreducible component of the Zariski closure X r of Xr in X which dominates Y . Let φ : X → Y be the induced dominant regular map. By Lemma 22.6, there exists a nonempty open subset V of X such that dφa is surjective for a ∈ V . Let a ∈ V ∩ Xr . We then have, by (10.5) (after Deﬁnition 10.8), a commutative diagram of kvector spaces, → Ta (X) Ta (X ) dφa ↓ ↓ dφa Tφ (a) (Y ) → Tφ(a) (Y ). The horizontal arrows are injections since X is a subvariety of X and Y is a subvariety of Y . Since rank dφa ≤ r, we have that dim Y ≤ dimk Tφ (a) (Y ) ≤ r.
Now we give the proof of Theorem 22.4. Let Xm−1 = {a ∈ X  rank dφa ≤ m − 1}. Then dim φ(Xm−1 ) ≤ m − 1 by Lemma 22.7. Recall that we have made the reduction that Y is nonsingular. Let U = Y \ φ(Xm−1 ), a nonempty open subset of Y . Suppose that b ∈ U and a ∈ φ−1 (b). Then a ∈ Xm−1 , so rank dφa ≥ m, and since b is a nonsingular point of Y , dφa is surjective. By Lemma 22.5, Xb is nonsingular. We end this section by giving the deﬁnition of a smooth morphism. A morphism of schemes is deﬁned in Section 15.5. The kmorphisms of varieties are the regular maps. Deﬁnition 22.8. A morphism of schemes φ : X → Y is said to be smooth of relative dimension m if for all p ∈ X there are open neighborhoods U ⊂ X of p and V ⊂ Y of φ(p) such that φ(U ) ⊂ V , and there exists a commutative diagram open embedding
U φ
↓
→
Z ↓
open embedding
→ W V where Z and W are aﬃne schemes, and if R = OW (W ), then OZ (Z) = R[x1 , . . . , xn+m ]/(f1 , . . . , fn ) ∂fi (p)) over κ(p) = OX,p /mp is n. and the rank of the (n + m) × n matrix ( ∂x j
This is the deﬁnition given in [116, Deﬁnition 3 on pages 436–437]. An ´etale morphism (Deﬁnition 21.79) is a smooth morphism of relative dimension 0. A reﬁnement of Deﬁnition 22.8 is given in Exercise 22.10.
22.3. Bertini’s second theorem
457
Using the general notion of a scheme of Section 15.5 (which has a larger topological space including nonclosed points) we have the following. Theorem 22.9. A morphism of schemes φ : X → Y is smooth of relative dimension m if and only if the following three conditions hold. 1. f is ﬂat, 2. if X ⊂ X and Y ⊂ Y are irreducible components such that f (X ) ⊂ Y , then dim X = dim Y + m, 3. for each point x ∈ X (closed or not) dimκ(x) (ΩX/Y ⊗ κ(x)) = m, where κ(x) is the residue ﬁeld of OX,x . The proof of Theorem 22.9 follows from [116, Theorem 3’, page 437] and [73, Theorem III.10.2]. The criterion of Theorem 22.9 is the deﬁnition of a smooth morphism given in Section 10 of Chapter III of [73]. Exercise 22.10. Show that a regular map of varieties φ : X → Y is smooth of relative dimension m if and only if for every p ∈ X, there exist open aﬃne neighborhoods A of p and B of q = φ(p) such that k[A] is a quotient of a polynomial ring over k[B] of the form k[A] = k[B][x1 , . . . , xn+m ]/(f1 , . . . , fn ) ∂fi ∂fi ) generated by the n × n minors of the matrix ( ∂x ) where the ideal In ( ∂x j j is equal to k[A]. Hint: Use Exercise 1.7 and Deﬁnition 22.8.
22.3. Bertini’s second theorem Suppose that X is a normal variety and L is a linear system on X without ﬁxed component (Base(L) has codimension ≥ 2 in X). Then there exists an eﬀective D on X and linearly independent sections s0 , . . . , sn ∈ Γ(X, OX (D)) such that L = {La = div(a0 s0 + a1 s1 + · · · + an sn ) + D  a = (a0 : . . . : an ) ∈ VL = Pn }. The projective space VL parametrizes the linear system L. The linear system L induces a rational map φL = (s0 : . . . : sn ) : X Pn . Let X be the projective variety which is the Zariski closure of φL (X) in Pn . The function ﬁeld of X is s1 sn ,..., ⊂ k(X). k(X ) = k s0 s0
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We now assume that X is nonsingular, and we associate to the linear system L a closed subvariety ZL of X × VL . Let x0 , . . . , xn be homogeneous coordinates on VL . Suppose that U ⊂ X is an aﬃne open subset such that D ∩ U has a local equation gU = 0 on U . Then OX (D)  U = g1U OU . h
Let sj  U = gU,j with hU,j ∈ Γ(U, OX ) for 0 ≤ j ≤ n. Deﬁne the closed U ideal of ZU in the graded subscheme ZU of U × VL so that the homogeneous n ring k[U ][x0 , . . . , xn ] is generated by j=0 hU,j xj . Then the coordinate ring of ZU is ⎞ ⎛ n S(ZU ) = k[U ] ⊗k k[x0 , . . . , xn ]/ ⎝ hU,j xj ⎠ . j=0
The ZU patch to determine a closed subscheme Z of X × VL . This is since gU gV is a unit on U ∩ V if U, V are aﬃne open subsets of X such that gU = 0 is a local equation of D on U and gV is a local equation of D on V . Let π : Z → VL be the natural projection. The schemetheoretic ﬁber Za by π of a ∈ VL is isomorphic to the divisor Da for all a ∈ VL . Theorem 22.11 (the second theorem of Bertini). Suppose that X is a nonsingular variety over an algebraically closed ﬁeld k of characteristic 0, and L is a linear system on X without ﬁxed component. Let W be the base locus of L. Then there exists a dense open subset U of VL such that La \ W is nonsingular for a ∈ U . This theorem is not true in positive characteristic. A counterexample is given in Exercise 22.19. Proof. Let Y = X \ W . Then LY is a base point free linear system. We construct the family π : Z → VL for LY as above. The ﬁbers Za are equal to La \ W for a ∈ VL . Here Z is locally a hypersurface in homogeneous coordinates of VL . Since LY is base point free, at every point of Z one of the coeﬃcients of the hypersurface is a unit. Thus Z is nonsingular by the Jacobian criterion. The theorem now follows from Theorem 22.4.
22.4. Bertini’s ﬁrst theorem Suppose that X is a normal variety and L is a linear system on X without ﬁxed component. Let notation be as in Section 22.3, with VL ∼ = Pm . The linear system L is said to be composite with a pencil if dim X = 1; that is, s1 sm ,..., = 1. trdegk k s0 s0 In this section, we prove the following theorem. Our proof is based on the proof by Zariski in [158, Section 1.6].
22.4. Bertini’s ﬁrst theorem
459
Theorem 22.12 (the ﬁrst theorem of Bertini). Suppose that X is a normal variety over an algebraically closed ﬁeld k of characteristic p ≥ 0 and L is a linear system on X without ﬁxed component. Suppose that L is not composite with a pencil. Deﬁne pe by pe = 1 if K has characteristic 0, and so that pe is the largest e exponent such that k(X ) ⊂ k(X)p if k has characteristic p > 0. Then there is a linear system L on X such that L = pe L and there exists a Zariski open subset C of VL such that La = pe L 1 where L 1 is a prime divisor for a ∈ C.
a pe
a pe
Suppose that κ is a ﬁeld and f (x) = f (x1 , . . . , xn ) is in the polynomial ring κ[x1 , . . . , xn ]. We say that f is geometrically irreducible over κ if f (x) is irreducible in F [x1 , . . . , xn ] for all extension ﬁelds F of K; that is, f = f1 f2 with f1 , f2 ∈ F [x1 , . . . , xn ] implies f1 or f2 has degree 0. Remark 22.13. By Propositions 1.31 and 22.2, the following are equivalent: 1) f is geometrically irreducible over κ, 2) F [x1 , . . . , xn ]/(f ) is a domain for all extension ﬁelds F of κ, 3) κ[x1 , . . . , xn ]/(f ) is geometrically integral over κ, 4) κ[x1 , . . . , xn ]/(f ) is a domain, and letting L be the quotient ﬁeld of κ[x1 , . . . , xn ]/(f ), we have that L is separable over κ and κ is separably closed in L. Proposition 22.14. Let A = k[X] be the ring of regular functions on an aﬃne variety X, and suppose that a nonzero element F in the polynomial ring A[T1 , . . . , Tn ] over A has degree d. If R is an Aalgebra, let F(R) be the image of F in R[T1 , . . . , Tn ] ∼ = A[T1 , . . . , Tn ] ⊗A R. Let K = k(X) be the quotient ﬁeld of A. Assume that F(K) is geometrically irreducible. Then there exists a nonzero element f ∈ A such that for all r ∈ Xf , F(k(r)) is irreducible. Proof. Write F = cα T α with cα ∈ A and where the indexing is over multiindices α = (α1 , . . . , αn ) with T α = T1α1 · · · Tnαn . Deﬁne α = α1 + · · · + αn . Let p, q be positive integers such that p + q = d. Let Tβ and Tγ be indeterminates indexed by multiindices β = (i1 , . . . , in ) and γ = (j1 , . . . , jn ) such that β ≤ p and γ ≤ q. Let B(p,q) be the polynomial ring B(p,q) = A[{Tβ  β ≤ p}, {Tγ  γ ≤ q}],
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which we will denote by A[Tβ , Tγ ]. Let I(p,q) be the ideal in B(p,q) generated by {Pα  α ≤ d} where Tβ Tγ − cα . Pα = β+γ=α
Let Ω be an algebraically closed ﬁeld containing K. Suppose that we have a factorization (22.2)
F(Ω) = F1 F2
for some F1 , F2∈ Ω[T1 , . . . , Tn ] with deg F2 = q. Then degγ F1 = p and tβ T β and F2 = tγ T with tβ , tγ ∈ Ω, we have that writing F1 = Pα (tβ , tγ ) = 0 for all α with α ≤ d. In fact, we have that there exists a factorization (22.2) if and only if the equations Pα (Tβ , Tγ ) = 0 have a common solution in Ω for all α with α ≤ d, so F(Ω) does not have a factorization (22.2) if and only if Z(I(p,q) ) = ∅ in AnΩ . By the nullstellensatz, this is equivalent to the statement that I(p,q) Ω[Tβ , Tγ ] = Ω[Tβ , Tγ ], which is equivalent to the statement that I(p,q) K[Tβ , Tγ ] = K[Tβ , Tγ ]. Thus if F(Ω) does not have a factorization (22.2), then there exists 0 = f(p,q) ∈ A such that I(p,q) Af [Tβ , Tγ ] = Af(p,q) [Tβ , Tγ ]. Suppose that r ∈ X has the associated maximal ideal m = I(r) ⊂ A. We have (since k(r) ∼ = k is algebraically closed) that F(k(r)) has a factorization F(k(r)) = F 1 F 2 for some F 1 , F 2 ∈ k(r)[T1 , . . . , Tn ] with deg F1 = p and deg F2 = q if and only if I(p,q) k(r)[Tβ , Tγ ] = k(r)[Tβ , Tγ ], which holds if and only if the closed set in Ank(r) Z(I(p,q) k(r)[Tβ , Tγ ]) = ∅.
Thus, letting f = p+q=d f(p,q) (with the restriction that p and q are posi tive), we have that F(k(r)) is irreducible for all r ∈ Xf . Lemma 22.15. Suppose that P is a ﬁeld which is algebraically closed in a ﬁeld Σ and Σ∗ = Σ(x1 , . . . , xm ) is a pure transcendental extension of Σ. Then P (x1 , . . . , xm ) is algebraically closed in Σ∗ . Proof. This is [160, Lemma, page 196] or [151, Lemma 2].
The following proposition and its proof are based on [158, Proposition 1.6.1].
22.4. Bertini’s ﬁrst theorem
461
Proposition 22.16. Let K/F be a ﬁeld of algebraic functions, of transcendence degree r ≥ 2. Let z1 , . . . , zm be elements of K such that the ﬁeld F (z) = F (z1 , . . . , zm ) has transcendence degree s ≥ 2 over F , let u1 , . . . , um be algebraically independent elements over K, and let zu denote the linear form u1 z1 + · · · + um zm . If F is separably closed in K, then the ﬁeld F (zu , u) = F (zu , u1 , . . . , um ) is separably closed in K(u1 , . . . , um ). Proof. Step a). We will ﬁrst reduce to the case s = r = 2 and F algebraically closed in K. Fix a transcendence basis {xr−s+1 , xr−s+2 , . . . , xr } of F (z) over F and extend it to a transcendence basis {x1 , x2 , . . . , xr } of K over F . Let F be the algebraic closure of F (x1 , . . . , xr−2 ) in K. Then trdegF K = trdegF F (z) = 2. Suppose the proposition is true if s = r = 2. Then F (zu , u) is separably closed in K(u). We will now show that u1 , . . . , um and zu are algebraically independent over F . Assume that this is not the case. We will derive a contradiction. At least one of the zi is transcendental over F . Without loss of generality, we may assume that zm is transcendental over F . Since u1 , . . . , um are transcendental over K, and hence over F , we then have a relation a0 zun + a1 zun−1 + · · · + an = 0 with ai ∈ F [u1 , . . . , um ] for all i and the ai not all zero. After possibly dividing out a common power of u1 from all of the ai , we may assume that u1 does not divide ai for some i. Then setting u1 = 0, we have a relation a0 z nu + · · · + an = 0, with z n = u2 z2 + · · · + um zm and ai = ai (0, u2 , . . . , un ) with the ai not all zero. Now, after dividing this new relation by the largest power of u2 which divides all of the ai , we may repeat this argument, eventually getting a relation (22.3)
l l−1 b0 z m + b1 zm + · · · + b0 = 0
with all bi ∈ F [um ] and the bi not all zero. Now zm ∈ K and um is transcendental over K, so expanding (22.3) as a polynomial in K[um ], the coeﬃcients must all be zero. Now these coeﬃcients are of the form hi (zm ) with hi (y) in the polynomial ring F [y] not all zero. Since hi (zm ) = 0 for all i, we have that zm is algebraic over F , a contradiction. Thus u1 , . . . , um , zu are algebraically independent over F . Since F is assumed to be separably closed in K (and therefore also in F ) it follows from Lemma 22.15 (taking Σ = F , Σ∗ = F (zu , u), and P to be the algebraic closure of F in F ) that P (zu , u) is algebraically closed in F (zu , u) and thus is separably closed in K(u). Now P is purely inseparable over F , so F (zu , u) is separably closed in P (zu , u). Thus F (zu , u) is separably closed in K(u).
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Step b). We now assume that s = r = 2 and that F is algebraically closed in K. We will next ﬁnd a reduction to the case m = 2. Let vi , wi for 1 ≤ i ≤ m and t1 , t2 be m + 2 elements which are algebraically independent over K. Set (22.4)
ui = t1 vi + t2 wi
for i = 1, 2, . . . , m,
(22.5)
zv = v1 z1 + v2 z2 + · · · + vm zm ,
(22.6)
zw = w1 z1 + w2 z2 + · · · + wm zm ,
(22.7)
zu = t1 zv + t2 zw = u1 z1 + u2 z2 + · · · + um zm .
By (22.4), we have that (22.8)
K(t1 , t2 , v, w) = K(t1 , t2 , v, u),
and thus the 2m + 2 elements t1 , t2 , vi , ui are algebraically independent over K. In particular, the ui are algebraically independent over K. Let F = F (v, w) = F ({vi }, {wj }) and K = K(v, w) = K({vi }, {wj }). We have that trdegF K = 2, and F is algebraically closed in K by Lemma 22.15. By an argument as in the ﬁrst step of the proof, regarding vi and wj as indeterminates, we see that zv and zw are algebraically independent over F . Since t1 and t2 are algebraically independent over K and since zu = t1 zv + t2 zw , we conclude, since we are assuming the proposition is true if s = m = r = 2, that F (zu , t1 , t2 ) is separably closed in K (t1 , t2 ). By (22.4), we have that the ﬁeld F (zu , t1 , t2 ) is generated over F (zu , u) by the m +2 element t1 , t2 , vi . By (22.8), these elements are algebraically independent over K(u), and hence over F (zu , u). Thus F (zu , u) is algebraically closed in F (zu , t1 , t2 ) and so F (zu , u) is separably closed in K (t1 , t2 ), and thus also in K(u), since K(u) is contained in K (t1 , t2 ). Step c). Now assume that s = r = m = 2. We will next reduce to the case that K is a Galois extension of F (z1 , z2 ). We have that K is an algebraic extension of F (z1 , z2 ). Let K0 be the separable closure of F (z1 , z2 ) in K. Then F (zu , u) is separably closed in K(u) if it is separably closed in K0 (u). Hence we may replace the ﬁeld K with K0 in the proof, and so we may assume that K is a separable extension of F (z1 , z2 ). Let K be the smallest Galois extension of F (z1 , z2 ) containing K. Then the ui are also algebraically independent over the ﬁnite extension K of K. Let F be the algebraic closure of F in K . Then, assuming that the proposition is true in the case that K is a Galois extension of F (z1 , z2 ), we have that F (zu , u) is separably closed in K (u). We will now show that F (zu , u) ∩ K(u) = F (Zu , u), which will show that F (zu , u) is separably closed in K(u).
22.4. Bertini’s ﬁrst theorem
463
Let t ∈ F (zu , u) ∩ K(u). The elements u1 , u2 and uz = u1 z1 + u2 z2 are algebraically independent over F , hence also over the ﬁnite extension F of F . Thus the expression of t as a quotient t=
f (u1 , u2 , zu ) g(u1 , u2 , zu )
with f, g ∈ F [u1 , u2 , zu ] is uniquely determined if we normalize one of these polynomials by imposing the condition that a preassigned nonzero coeﬃcient of the polynomial is 1. Suppose σ ∈ G(K /K). Then σ extends naturally to an automorphism of K (u) over K such that σ(u) = u. Hence σ(t) = t since t ∈ K(u), and thus the coeﬃcients of the polynomials must be invariant under σ. Thus these coeﬃcients must be in K, and since they are algebraic over F , they must be in F . Hence t ∈ F (zu , u), proving our assertion. Step d). We now further assume that K is a Galois extension of F (z1 , z2 ). Let zv = v1 z1 + v2 z2 be a second linear form with indeterminate coeﬃcients v1 , v2 , which are assumed to be algebraically independent over K(u). The automorphisms in G(K/F (z1 , z2 )) extend uniquely to automorphisms in G(K(u, v)/F (z1 , z2 , u, v)) and K(u, v) is Galois over F (z1 , z2 , u, v) with Galois group G(K/F (z1 , z2 )). Let Hu be the algebraic closure of F (zu , u) in K(u) and let Hv be the algebraic closure of F (zv , v) in K(v). We shall prove that (22.9)
Hu (zv , v) = Hv (zu , u).
Since both ﬁelds in (22.9) contain the ﬁeld F (u, v, z1 , z2 ), it will suﬃce to show that we have equality of Galois groups G1 = G2 , where G1 = G(K(u, v)/Hu (zv , v)) and G2 = G(K(u, v)/Hv (zu , u)). The ﬁeld K(u, v) has an automorphism σ which interchanges ui and vi for i = 1, 2 and which is the identity on K. Then σ(Hu (zv , v)) = Hv (zu , u), and G2 = σ −1 G1 σ. Now we have that σ commutes with each element of the Galois group G(K(u, v)/F (u, v, z1 , z2 )) since these automorphisms are extensions of elements of G(K/F (z1 , z2 )). Thus G1 = G2 . Since F is algebraically closed in K, we have that F is algebraically closed in K(v), and thus also in Hv . We have that zu , u1 , and u2 are algebraically independent over F (zv , v) since zu and zv are algebraically independent over F (u, v). Hence zu , u1 , u2 are also algebraically independent over Hv . Thus F (zu , u) is algebraically closed in Hv (zu , u), and thus by (22.9), we have that F (zu , u) = Hu , completing the proof of the proposition. We now give the proof of Theorem 22.12.
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Deﬁne an eﬀective divisor D0 by (s0 ) + D = D0 . Then 1, ss10 , . . . , ssm0 ∈ Γ(X, OX (D0 )) and
s1 sm + · · · + am + D0  a = (a0 : . . . : am ) ∈ VL . L = div a0 + a1 s0 s0 Thus after replacing D with D0 and the si with ss0i , we may assume that s0 = 1. If pe > 1, we construct another linear system L asfollows. Let e si ∈ k(X) be such that (si )p = si for 0 ≤ i ≤ m. Write D = ni Ei where the Ei are the integral components of D. The linear system L has no ﬁxed component, so for each i, there exists a0 , . . . , am ∈ k such that Ei is not a component of div(a0 s0 + · · · + am sm ) + D. So νEi (a0 s0 + · · · + am sm ) = −ni . Now 1 e
1 e
p sm ) νEi (a0 s0 + · · · + am sm ) = pe νEi (a0p s0 + · · · + am
so pe ni for all i. Let D =
1 pe D.
Then
L = {div(b0 s0 + · · · + bm sm ) + D } is a linear system such that L = pe L . We have thus reduced to the case that pe = 1, which we will assume for the rest of the proof. Let S(VL ) = k[u0 , . . . , um ] be the homogeneous coordinate ring of VL (a graded polynomial ring over k with deg ui = 1 for all i). Let U be a nonsingular aﬃne open subset of X such that U ∩Supp D = ∅. Then si ∈ k[U ] for all i by Lemma 13.3. Let F1 , . . . , Fs be the irreducible components of X \ U which have codimension 1 in X. Since L has no ﬁxed component, for each i there exists pi ∈ Fi which is not in the base locus of L. Let Wi = {a ∈ VL  pi ∈ La }. of VL since for each i there exists Gi ∈ L Wi is a proper linear subspace
∈ Gi . Hence VL \ si=0 Wi is a nonempty open subset of VL . such that pi For a ∈ VL \ si=0 Wi , La is a prime divisor if and only if La ∩ U is a prime divisor, since La cannot contain a codim 1 component of X \ U . The coordinate ring of U × VL is S(U × VL ) = R ⊗k k[u0 , . . . , um ] = R[u0 , . . . , um ] where R = k[U ]. The ideal ( m i=0 si ui ) ⊂ R[u0 , . . . , um ] is a homogeneous prime ideal (s0 = 1). Let Z = Z( m i=0 si ui ) ⊂ U ×VL . Here Z is a subvariety of U × VL with coordinate ring m si ui ∼ S(Z) = R[u0 , . . . , um ]/ = R[u1 , . . . , um ]. i=0
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465
Let k ∗ = k(u0 , . . . , um ) and let K ∗ be the quotient ﬁeld of S(Z). We have natural inclusions k ∗ ⊂ K ∗ and k(X) ⊂ K ∗ such that u1 , . . . , um are algebraically independent over k(X). Let F = k, zi = si for 1 ≤ i ≤ m, zu = −u0 , and K = k(X). Then K ∗ = K(u1 , . . . , um ) and k ∗ = F (zu , u1 , . . . , um ). Since F (z) = F (z1 , . . . , zm ) = k(X ) has transcendence degree ≥ 2 over F = k by assumption, we conclude from Proposition 22.16 that k ∗ is separably closed in K ∗ . Observing that k(VL ) = k( uu01 , . . . , uum0 ) and that k(Z) is the quotient ﬁeld of m ui um u1 , ,..., si / 1+ R u0 u0 u0 i=1
we see that we have a commutative diagram of inclusions of ﬁelds k(VL ) → k(Z) ↓ ↓ ∗ → K ∗. k Since k ∗ is separably closed in K ∗ , the separable closure of k(VL ) in k(Z) is contained in k ∗ . Now k ∗ = k(VL )(u0 ) is a transcendental extension of k(VL ), so k(VL ) is separably closed in k(Z). Since k is algebraically closed, there exist x1 , . . . , xr ∈ K which are a separating transcendence basis of K/k (by Theorem 1.14). After possibly replacing U with a smaller aﬃne open subset of X, we may assume that x1 , . . . , xr are uniformizing parameters in R = Γ(U, OX ) (Theorem 21.75 and Deﬁnition 14.16), so ΩR/k = Rdx1 ⊕ · · · ⊕ Rdxr and dR/k : R → ΩR/k is the map f → D1 (f )dx1 + · · · + Dr (f )dxr where {D1 , . . . , Dr } is the basis in Derk (k(X), k(X)) which is deﬁned by Di (xj ) = δij for 1 ≤ i, j ≤ r. Let S = R ⊗k k(Vl ). We have that ΩS/k(VL ) ∼ = Sdx1 ⊕ · · · ⊕ Sdxr = ΩR/k ⊗k k(VL ) ∼ by Exercise 14.11 and D1 , . . . , Dr extend naturally to a basis of Derk(VL ) (S, S). Let m u i si . A = k[Z ∩ (U × (VL )u0 )] ⊗k[ u1 ,..., um ] k(VL ) ∼ = S/ 1 + u0 u0 u0 i=1
By Theorem 14.6, we have a right exact sequence of Amodules δ
A → Adx1 ⊕ · · · ⊕ Adxr → ΩA/k(VL ) → 0
466
22. Bertini’s Theorems and General Fibers of Maps
where δ(1) =
r m uj i=1 j=1
u0
Di (sj )dxi .
Localizing at the quotient ﬁeld k(Z) of A, we have, by Lemma 14.8, a right exact sequence of k(Z)vector spaces δ
k(Z) → k(Z)dx1 ⊕ · · · ⊕ k(Z)dxr → Ωk(Z)/k(VL ) → 0. Since at least one of the sj is not contained in k(X)p (not contained in k if k has characteristic 0), we have for this sj that Di (sj ) = 0 for some i by Theorem 21.76 or (21.18). Further, since Derk(VL ) (k(Z), k(Z)) ∼ = Homk(Z) (Ωk(Z)/k(VL ) , k(Z)) by Lemma 14.3, we have that (22.10)
dimk(Z) Derk(VL ) (k(Z), k(Z)) = r − 1 = trdegk(VL ) k(Z)
since dim Z = dim VL + dim X − 1 and trdegk k(Z) = trdegk(VL ) k(Z) + trdegk k(VL ) by (1.1). We conclude from Theorem 21.77 and the equality (22.10) that k(Z) is separably generated over k(VL ). Since k(Z) is an algebraic function ﬁeld over k(VL ), we have that k(Z) is a separable extension of k(VL ) as observed before Proposition 22.2. Thus k(Z) is geometrically integral over k(VL ) by Proposition 22.2, since we earlier showed that k(VL ) is separably closed in k(Z). Let Y = Z ∩ (U × (VL )u0 ) with projection π : Y → (VL )u0 . We must ﬁnd an open subset A of (VL )u0 \ ( si=0 Wi ) such that for a ∈ A, the scheme Ya has only one irreducible component γ and OYa ,γ ∼ = Oγ . For any a ∈ (VL )u0 we have that the irreducible components of Ya all have dimension r − 1 (where r = dim X). The ﬁeld k(Z) is separably generated over k(VL ), so k(Z) has a separating transcendence basis y1 , . . . , yr−1 over k(VL ). Let L = k(VL )(y1 , . . . , yr−1 ). By the theorem of the primitive element, there exists a primitive element t of k(Z) over L. Let g1 (yr ) ∈ L[yr ] be the minimal polynomial of t over L. Then k(Z) ∼ = L[yr ]/(g1 ). By Proposition 1.31, there exists g ∈ k[(VL )u0 ][y1 , . . . , yr ] which is irreducible and gL[yr ] = (g1 ). Let B be the aﬃne variety with k[B] = k[(VL )u0 ][y1 , . . . , yr ]/(g). Then B is birationally equivalent to Z, so there exist aﬃne open subsets C of Y and E of B such that C ∼ = Ea for all a ∈ (VL )u0 . = E and Ca ∼
22.4. Bertini’s ﬁrst theorem
467
Proposition 22.14 implies that there exists an aﬃne open subset F of (VL )u0 such that Ba is a variety for all a ∈ F . Let F1 , . . . , Fe be the irreducible components of Y \ C. By Theorem 8.13, there exists an aﬃne open subset G of (VL )u0 such that for a ∈ G, (Fi )a = ∅ if π(Fi ) is not dense in (VL )u0and dim(Fi )a < r − 1 if π(Fi ) is dense in (VL )u0 . Then for a ∈ (F ∩ G) \ ( si=0 Wi ), Ya is a variety, and Ya is dense in Za , and so La is a prime divisor (since we have reduced to L = L ), and the conclusions of the theorem follow. Remark 22.17. If the assumptions of Theorem 22.12 hold, except that L is composite with a pencil, then we can still conclude that L = pe L 1 where L
a pe
1
a pe
is a reduced divisor for a in a suitable open subset C of VL
1 e ap
. If L is
composite with a pencil, then (after replacing L with L ) Proposition 22.16 is not applicable and it may be that k(VL ) is not separably closed in k(Z). However, the argument showing that k(Z) is a separable extension of k(VL ) is still valid, allowing us to conclude that k(Z) is geometrically reduced over k(VL ). Now a modiﬁcation of Proposition 22.14 and the arguments at the end of the proof of Theorem 22.12 (or Theorem 22.18 below) allow us to conclude that L 1 is reduced for a in a suitable open subset C of VL . a pe
We mention a general result, which can be deduced from the methods of this chapter. The proof follows directly from [68, Theorem IV.9.7.7]. Theorem 22.18. Suppose that φ : X → Y is a dominant regular map of varieties over an algebraically closed ﬁeld k. Then there exists a nonempty open subset U ⊂ Y such that for all p ∈ U , 1) the ﬁber Xp is irreducible if and only if k(X)/k(Y ) is geometrically irreducible, 2) the ﬁber Xp is reduced if and only if k(X)/k(Y ) is geometrically reduced, 3) the ﬁber Xp is integral if and only if k(X)/k(Y ) is geometrically integral. Proof. In this proof, we regard X and Y as general schemes (Section 15.5). Let {Vi } be an aﬃne cover of X such that there exist aﬃne open subsets Ui of Y such that f (Vi ) ⊂ Ui for all i. Let η be the generic point of Y , so that OY,η = k(Y ). Let k be an extension ﬁeld of k(Y ). The ring k[Vi ] ⊗k[Ui ] k(Y ) is the localization of k[Vi ] with respect to the multiplicative set T = k[Ui ] \ {0}, so it is a subring of k(X). Thus (k[Vi ]⊗k[Ui ] k(Y ))⊗k(Y ) k is a subring of k(X)⊗k(Y ) k and k[Vi ]⊗k[Ui ] k is irreducible (respectively, reduced, integral) if and only if k(X) ⊗k(Y ) k is irreducible (respectively, reduced, integral) and so Xη is geometrically irreducible (respectively, reduced, integral) over
468
22. Bertini’s Theorems and General Fibers of Maps
k(X) if and only if k(Y ) is geometrically irreducible (respectively, reduced, integral) over k(X). The theorem now follows from [68, Theorem IV.9.7.7]. Exercise 22.19. This exercise (from an example of Serre in [73, Exercise III.10.7]) shows that the second theorem of Bertini, Theorem 22.11, is not true in characteristic p > 0, even when the linear system is complete, base point free, irreducible, and not composite with a pencil. Let k be an algebraically closed ﬁeld of characteristic 2. Let p1 , . . . , p7 ∈ X = P2k be the seven points with coeﬃcients in Z2 . Let L be the linear system of all cubic curves passing through p1 , . . . , p7 . Prove the following statements. a) L is a linear system with the base points p1 , . . . , p7 , and φL is an inseparable regular map of degree 2 from X \ {p1 , . . . , p7 } → P2 . b) Every curve C ∈ L is singular. More precisely, either C consists of three lines all passing through one of the pi or C is an irreducible cubic curve with its only singular point some p = pi . Furthermore, the correspondence C → the singular point of C is a 11 correspondence between L and P2 . c) Let H be a cubic curve on P2k , and let φ : Y → X be the blowup of the seven points p1 , . . . , p7 with exceptional divisors E1 , . . . , E7 . Show that the complete linear system φ∗ (H) − E1 − · · · − E7  has no base points, is not composite with a pencil, all but ﬁnitely many members of the linear system are irreducible (integral), but every member of the linear system is singular. Exercise 22.20. This exercise shows that Theorem 22.4 is not true in characteristic p > 0, even when k(X) is geometrically integral over k(Y ). Let D = φ∗ (H) − E1 − · · · − E12 be the divisor on Y constructed in Exercise 22.19. Let π : Z → VD be the regular map constructed before Theorem 22.11 for the linear system L = D. a) Show that the ﬁber Zp is a singular plane cubic for all p ∈ VD . b) Show that the ﬁeld k(Z) is geometrically integral over k(VD ). Exercise 22.21. Give an example of a linear system L on Pn which does not have a ﬁxed component, is composite with a pencil, and such that there is a nontrivial open subset U of VL such that La is not integral for all a ∈ U .
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Index
(L1 · · · Lt ), 369 (L1 · · · Lt ; F )V , 368 (L1 · · · Lt · W ), 369 (φ), 286 (φ)X , 286 (f ), 241 (f )0 , 241 (f )∞ , 242 Atorsion, 208 B(I), 113, 121 B(I), 221–223 C ∗ (U , F ), 310 D(B/A), 399 D(F ), 71, 103 D(f ), 48 D ∩ U , 242 D ≡ 0, 382 D1 ≥ D2 , 240 D1 ∼ D2 , 242 DL/K (a1 , . . . , an ), 398 F : Xp → X, 349 F a , 71 F r(x), 349 G(L, K), 7 G(Y /X), 421 Gi (S/R), 440 Gs (S/R), 438 H i (A∗ ), 307 H i (X, F ), 308, 309 HYi (X, F ), 330 HIi (M ), 326 0 HSing (U, G), 358 I(S/R), 435
I(Y ), 29, 33, 69, 70, 103 I(Z), 292 I sat , 66 IX (Y ), 33, 70 J(S/R), 435 K s , 438 KX , 286 M (n), 63 M ∨ , 243 M(F ) , 64 M(p) , 64 N (X), 382 NL/K , 392 P (A), 243 P GX , 189 PM (z), 301 PY , 301 Ri φ∗ F , 320 R(d) , 64 S(I; R), 56 S(W ), 103 S(X), 70 S(X × Pn ), 105 S(Y ), 105 S(Pm × Pn ), 102 S(Pn ), 68 T (A), 24 TA (M ), 208 Tp (X), 157 Vν , 228 WF , 103 X × Y , 101 X (r) , 360
477
478
XF , 71 XZ , 290 Xf , 48 Xp , 290 Y H , 422 Z(A), 102 Z(I), 68 Z(J), 70 Z(T ), 28, 33 Z(U ), 68 Z(f ), 28 Z1 ∩ Z2 , 290 ZX (T ), 33 ZX (U ), 70 Zred , 289 [L : K]i , 7 [L : K]s , 7 Am × An , 101 An , 27 C{x1 , . . . , xn }, 176 Δ, 176 ΔX , 110 ΔPn , 106 ΓY (X, F ), 330 Γν , 228 Γφ , 106, 107 N, 1 Ωn X/k , 286 Ωn k(X)/k , 286 ΩB/A , 280 ΩX/k , 283 Pm × Pn , 102 Pn k , 67 Z+ , 1 Zan , 359 ˇ p (X, F ), 312 H ˇ p (U , F ), 311 H χ(F ), 319 deg x, 63 deg, 250 deg(D), 254 deg(Y ), 301 deg(φ), 254, 371, 406 dim R, 19 dim X, 42, 139 R (M ), 9 ˆ , 409 M ˆ → S, ˆ 411 φˆ : R lim← Ai , 184 lim→ Ai , 182 F (n), 210
Index
F ⊗ G, 204, 206 F ⊗OX G, 204, 206 F an , 359 G ∗ , 245 IOX , 205 IY , 155, 198, 200 IZ , 289 IY,p , 155 Idiv(s) , 291 L−1 , 269 OD , 290 OU (V ), 53 OX module, 191 OX module homomorphism, 191 OX submodule, 192 OX torsion, 209 OX (D), 242 OX (U ), 49, 75, 187 OX (n), 210 ∗ , 271 OX OU,p , 53 OW,(p,q) , 103 OW (U ), 103 OX,E , 240 OX,Y , 229 OX,p , 50, 75 T (F ), 209 (a ) pi i , 245 AnnX (σ), 208 Aut(X), 351 Base(L), 261 Cl(X), 242 Cl0 (X), 258 Cokernel(α), 194 DerA (B, M ), 279 Div(F ), 249 Div(X), 240 Grass(a, b), 73 Image(α), 194 Kernel(α), 194 Map(U, A1 ), 49, 75 Pic(X), 270 Pican (X), 359 Proj(S), 67, 297 Spec(R), 5, 296 Supp D, 240 bideg(x), 102 div(φ), 286 div(φ)X , 286 div(f ), 242 div(f )X , 242
Index
div(s), 291 gcd(D0 , . . . , Dn ), 259 ht(P), 19 reg(M ), 328 ˜ ), 328 reg(M trdegK L, 6 μ(I), 46 νE , 241 φ : X Y , 58, 93 φ∗ , 35, 76 φ∗ : Cl(Y ) → Cl(X), 252 φ∗ M, 205 φ−1 (Z), 290 φL , 260 φV , 260 Vol(L), 345 √ I, 3 √ I, 289 TrL/K , 392 ˜ 120, 198, 200 I, ˜ 192 J, ˜ , 196 M ˜ , 199 N D, 260 ∧n M , 286 ai (M ), 327 d(R : R), 443 dφp , 157 e(R), 376 e(Si /R), 396 e(ν ∗ /ν), 397 eP , 346 f (Si /R), 396 f (ν ∗ /ν), 397 f ∗ G, 207 f h , 71 f # , 191 f∗ F , 191 g(R : R), 443 g(X), 334 hi (X, F ), 319 k, an algebraically closed ﬁeld, 1 klinear Frobenius map, 349 k(U ), 53, 82 k(W ), 103 k(X), 49, 74 k(p), 50, 75, 283 k[X], 33 k[Am × An ], 101 k[An ], 27 mregular, 328
479
mR , the maximal ideal of a local ring, 1 mp , 50, 75, 283 mX,p , 50, 75 nfold, 45 pbasis, 425 pindependent, 425 r(R : R), 443 Aut(L/K), 7 Aut(k(Y )/k(X)), 421 HomOX (F , G), 201 HomOX (F , G), 204, 206 Num(X), 382 QF(R), 1 QR(A), 291 codimX (Y ), 45 depthI M , 326 res, 34 gr mR (R), 23 Abel, 360 Abelian variety, 360 Abhyankar, xii, 170, 227, 228, 231, 232, 234, 235, 392, 413, 435 AbhyankarJung theorem, 436 Abramovich, 234 Abstract prevariety, 295 Abstract variety, 295 Adjunction, 287, 380 Aﬃne map, 127 Aﬃne scheme, 296 Aﬃne subscheme, 289 Aﬃne subvariety, 34 Aﬃne variety, 55 Algebraic function ﬁeld, 7 Algebraic local ring, 441 Algebraic set, biprojective, 103 Algebraic set, in An , 28 Algebraic set, projective, 68 Algebraic set, quasiaﬃne, 34 Algebraic set, quasibiprojective, 103 Ample divisor, 262 Ample invertible sheaf, 270 Analytic implicit function theorem, 176 Artin, 362 Auslander, 25, 433 Base locus, of a linear system, 261 Base point free linear system, 261 Bayer, 330 Bertini, 304, 451 B´ezout, 375 B´ezout’s theorem, 375
480
Bhatt, 433 Bihomogeneous coordinate ring, 102 Birational equivalence, of aﬃne varieties, 60 Birational equivalence, of quasiprojective varieties, 94 Birational map, of aﬃne varieties, 60 Birational map, of quasiprojective varieties, 94 Birkar, 387 Blowup, of a subvariety of a projective variety, 123 Blowup, of a subvariety of an aﬃne variety, 113 Blowup, of an ideal of a projective variety, 121 Blowup, of an ideal of an aﬃne variety, 113 Blowup, of an ideal sheaf on a projective variety, 222 Blowup, of an ideal sheaf