Algebraic Geometry. Lei Fu. Tsinghua University Press. Nankai Institute of Mathematics Tianjin, P. R. China


 Donald Baker
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1 Algebraic Geometry Lei Fu Nankai Institute of Mathematics Tianjin, P. R. China Tsinghua University Press
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3 Preface In this book we study the cohomology of coherent sheaves on schemes. An excellent textbook on this topic is [Hartshorne]. But in Hartshorn s book, many important theorems which hold for proper morphisms are proved only for projective morphisms. Moreover, Hartshorne doesn t say much about the technique of spectral sequences. The main purpose of this book is to fulfil this need. Of course, everything in this book is contained in Grothendieck s [EGA]. I hope this book can make the wonderful ideas of Grothendieck which are hidden in the voluminous [EGA] more clear. To make the book selfcontained, I include many materials which are treated nicely in [Hartshorne]. So there are some overlaps with [Hartshorne] in this book, especially in sections 1.2 and 1.4. This book roughly covers the main materials in [EGA] I, II and III. A few years ago, I was invited to give a series of talks on ladic cohomolgy theory at the Morningside Center of Mathematics (MCM) at the Chinese Academy of Science. To prepare the talk, I wrote a book (unpublished) on étale cohomology theory which covered the main materials in [SGA] 1, 4, 4 1 2, 5, and 7. The current book together with [Matsumura] covers all the prerequisites for reading my manuscripts. So I think it should provide adequate preparation for learning étale cohomology theory. I only assume the reader is familiar with Chapter 18 of [AtiyahMacdonald]. All other results on algebra used in this book are either proved in this book, or can be proved by the reader without much difficulty. The reason why I include materials on algebra is my belief that the best way to learn algebra is to learn it simultaneously with geometry so that one can get geometric intuition of abstract algebraic concepts. This book is by no means a complete treatise on algebraic geometry. Nothing is said on how to apply the results obtained by cohomological method in this book to study the geometry of algebraic varieties. Serre duality is also omitted. The reader should consult [Hartshorne] and references there for these topics. I thank heartily Prof. Keqin Feng. Ever since he knew my existence, he has never stopped encouraging me. He invited me to MCM to lecture on ladic cohomology theory and other topics on algebraic geometry. Part of this book is based on some of my lecture notes. Prof. Feng also makes it possible to publish this book. Without his help, this book will never come into existence. iii
4 iv PREFACE During the preparation of this book, I was supported by the Qiu Shi Science & Technologies Foundation, by Project 973, by IHES, and by MCM. Lei Fu Nankai Institute of Mathematics
5 Contents Preface iii 1 Schemes and Coherent Sheaves Presheaves and Sheaves Schemes and Morphisms Properties of Schemes and Morphisms Coherent Sheaves Formal Completions of Schemes and Sheaves Cohomology Derived Functors Spectral Sequences Čech Cohomology Cohomology of Affine and Projective Schemes Cohomological Study of Proper Morphisms Local Freeness of Higher Direct Images Grothendieck s Existence Theorem Bibliography 217 v
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7 Chapter 1 Schemes and Coherent Sheaves 1.1 Presheaves and Sheaves Let X be a topological space. A presheaf F of sets on X consists of the following data: (a) For every nonempty open subset U of X, we have a set F(U) whose elements are called sections of F over U. (b) For every inclusion V U of nonempty open subsets of X, we have a map ρ UV : F(U) F(V ), called the restriction. These data satisfy (i) ρ UU = id F(U), (ii) if W V U are open subsets of X, then ρ UW = ρ V W ρ UV. For any section s F(U) and V U, we often denote ρ UV (s) by s V. Elements in F(U) are often denoted by (s, U) in order to make the open set U explicit in the notation. We make the convention that F( ) = for any presheaf of sets F. Define the category of open subsets of X so that its objects are nonempty open subsets of X, and for any two objects U and V, define { Hom(V, U) = if V U, {V U} if V U. Then a presheaf F of sets on X is just a contravariant functor from the category of open subsets of X to the category of sets. Similarly we define a presheaf F of abelian groups (resp. rings) on X to be a contravariant functor F from the category of open subsets of X to the category of abelian groups (resp. rings). For any sheaf of abelian groups or rings, we make the convention that F( ) = {0}. Here are some examples of presheaves: 1
8 2 CHAPTER 1. SCHEMES AND COHERENT SHEAVES 1. Let X be a topological space and A an abelian group. For every nonempty open subset U of X, define F(U) = A, and for every inclusion V U of nonempty open subsets, define ρ UV = id A. Then F is a presheaf of abelian groups, called the constant presheaf associated to A. 2. Let X be a topological space. For every open subset U of X, define C(U) to be the ring of complex valued continuous functions on U, and for every inclusion V U of nonempty open subsets, define ρ UV : C(U) C(V ) to be the restriction of functions. Then C is a presheaf of rings. 3. Let π : X X be a continuous map of topological spaces. For every nonempty open subset U of X, define S(U) to be the set of continuous sections of π over U: S(U) = {s : U π 1 (U) πs = id, and s is continuous}, and for every inclusion V U of nonempty open subsets, define ρ UV : S(U) S(V ) to be the restriction of sections. Then S is a presheaf of sets. We say a presheaf F of sets (resp. abelian groups, resp. rings) is a sheaf if it satisfies the following conditions: (i) Let s, t F(U) be two sections. If there exists an open covering {U i } i I of U such that s Ui = t Ui for any i, then s = t. (ii) Suppose {U i } i I is an open covering of U and s i F(U i ) are some sections satisfying s i Ui U j = s j Ui U j for any i, j I. Then there exists a section s F(U) such that s Ui = s i for any i I. (By (i), such s is unique.) Note that a presheaf F of abelian groups is a sheaf if and only if for any open covering {U i } i I of any open subset U, the sequence 0 F(U) F(U i ) F(U i U j ) i I i,j I is exact, where the second arrow is F(U) i I F(U i ), s (s Ui ) and the third arrow is F(U i ) F(U i U j ), (s i ) (s j Ui U j s i Ui U j ). i I i,j I The last two examples of presheaves given above are sheaves. A direct set is a partially ordered set (I, ) such that for any i, j I, there exists a k I such that i, j k. A direct system (A i, φ ij ) i I of sets consists of a family of sets A i (i I) and maps φ ij : A i A j for pairs i j such that φ ii = id Ai and φ jk φ ij = φ ik whenever i j k. For any x i A i and x j A j, we say x i is equivalent to x j if there exists a k i, j such that φ ik (x i ) = φ jk (x j ). This defines an equivalence relation on the disjoint union i A i of A i (i I).
9 1.1. PRESHEAVES AND SHEAVES 3 The direct limit inv. lim i A i of (A i, φ ij ) i I is defined to be the set of equivalence classes. Let X be a topological space and P a point in X. For any two neighborhoods U and V of P, we say V U if U V. Then the family of neighborhoods of P becomes a direct set with respect to this order. For any presheaf F on X, define the stalk F P of F at P by F P = dir. lim P U F(U), where the direct limit is taken over the family of neighborhoods of P. So elements of F P can be represented by sections of F over some neighborhoods of P. Two sections s F(U) and t F(V ) define the same element in F P if and only if there exists a neighborhood W of P such that W U V and s W = t W. For any neighborhood U of P, we have a canonical map F(U) F P. The image of a section s F(U) in F P is called the germ of s at P and is denoted by s P. Let F and G be presheaves of abelian groups on X. A morphism of presheaves φ : F G consists of a homomorphism of abelian groups φ(u) : F(U) G(U) for every open subset U such that for every inclusion V U of open subsets, the following diagram commutes: F(U) ρ UV F(V ) φ(u) G(U) ρ UV φ(v ) G(V ). For any point P X, φ induces a homomorphism on stalks φ P : F P G P. If we regard presheaves of abelian groups as contravariant functors from the category of open subsets on X to the category of abelian groups, then a morphism of presheaves is just a natural transformation. Similarly we can define morphisms between presheaves of sets or rings. For any presheaf F, we have the identity morphism id F. Given two morphisms of presheaves φ : F G and ψ : G H, we can define their composition ψφ : F H in the obvious way. We thus get the category of presheaves. A morphism of presheaves φ : F G is called an isomorphism if it has a twosided inverse, that is, there exists a morphism of presheaves ψ : G F such that ψφ = id F and φψ = id G. This is equivalent to saying that φ(u) : F(U) G(U) is an isomorphism for every open subset U. We define morphisms of sheaves as morphisms of presheaves. We thus get the category of sheaves which is a full subcategory of the category of presheaves. Proposition 1.1 Let φ : F G be a morphism of sheaves on a topological space X. Then φ is an isomorphism if and only if the induced map on stalks φ P : F P G P is an isomorphism for every P X. Proof. The only if part is obvious. Let s prove the if part. Suppose φ P : F P G P is bijective for every P X. We need to show φ(u) : F(U) G(U) is bijective for every open subset U of X.
10 4 CHAPTER 1. SCHEMES AND COHERENT SHEAVES Let s, s F(U) be two sections such that φ(s) = φ(s ). Then φ P (s P ) = φ P (s P ) for any P U. Since φ P is injective, we have s P = s P. So there exists a neighborhood U P of P contained in U such that s UP = s UP. Note that {U P } P U is an open covering of U. Since F is a sheaf, we must have s = s. So φ(u) is injective. Let (t, U) be a section in G(U). For any P U, since φ P : F P G P is surjective, we may find s P F P such that φ P (s P ) = t P. We may assume s P is the germ of a section (s, U P ) F(U P ) for some neighborhood U P of P. Note that φ(s, U P ) and (t, U) have the same germ at P. Choosing U P sufficiently small, we may assume U P U and φ(s, U P ) = (t, U) UP. Then for any two points P, Q U, we have φ(s, U P ) UP U Q = (t, U) UP U Q = φ(s, U Q ) UP U Q. By the injectivity of φ(u P U Q ) that we have proved above, we must have (s, U P ) UP U Q = (s, U Q ) UP U Q. Note that {U P } P U form an open covering of U. Since F is a sheaf, we may find a section (s, U) F(U) such that (s, U) Up = (s, U P ) for any P U. We have (φ(s, U)) UP = (t, U) UP. Since G is a sheaf, we must have φ(s, U) = (t, U). So φ(u) is surjective. Before going on, we introduce some concepts from the theory of categories. Let C be a category. A morphism f : A B in C is called a monomorphism or injective if for any two morphisms α, β : C A satisfying fα = fβ, we have α = β. An epimorphism is defined similarly by reversing the directions of arrows. More precisely, f : A B is called an epimorphism or surjective if for any two morphisms α, β : B C satisfying αf = βf, we have α = β. If a morphism is both injective and surjective, we say it is bijective. An isomorphism is a morphism with a twosided inverse. Any isomorphism is bijective. But a bijective morphism may not be an isomorphism. Let A i (i I) be a family of objects in C. The direct product of A i (i I) is an object i I A i together with a family of morphisms p i : i I A i A i (i I) called projections with the following universal property: For any object C and any family of morphisms f i : C A i (i I), there exists one and only one morphism f : C i I A i such that p i f = f i for any i. If the direct product of A i (i I) exists, it is unique up to unique isomorphism, that is, any two direct product of A i (i I) are isomorphic and the isomorphism between them is unique. The direct sum of A i (i I) is defined similarly as above by reversing the directions of arrows. More precisely, the direct sum of A i (i I) is an object i I A i together with a family of morphisms k i : A i i I A i (i I) with the following universal property: For any object C and any family of morphisms f i : A i C (i I), there exists one and only one morphism f : i I A i C such that fk i = f i for any i. If the direct product of A i (i I) exists, it is unique up to unique isomorphism. Let (I, ) be a directed set. A direct system (A i, φ ij ) i I consists of a family of objects A i (i I) and morphisms φ ij : A i A j for pairs i j such that
11 1.1. PRESHEAVES AND SHEAVES 5 φ ii = id Ai for any i and φ jk φ ij = φ ik whenever i j k. The direct limit of a direct system (A i, φ ij ) is an object dir. lim i A i together with morphisms φ i : A i dir. lim i A i (i I) satisfying φ j φ ij = φ i whenever i j and having the following universal property: For any object C and any morphisms ψ i : A i C (i I) satisfying ψ j φ ij = ψ i (i j), there exists a unique morphism ψ : dir. lim i A i C such that ψφ i = ψ i for any i. If the direct limit exists, it is unique up to unique isomorphism. Let (A i, φ ij) i I be another direct system. A morphism from (A i, φ ij ) to (A i, φ ij) is a family of morphisms u i : A i A i (i I) such that for any i j, the following diagram commutes: A i φ ij A j u i A i φ ij u j A j. It induces a morphism dir. lim i u i : dir. lim i A i dir. lim i A i. An inverse system (A i, φ ji ) i I consists of a family of objects A i (i I) and morphisms φ ji : A j A i for pairs i j such that φ ii = id Ai for any i and φ ji φ kj = φ ki whenever i j k. The inverse limit of an inverse system (A i, φ ij ) is an object inv. lim i A i together with morphisms φ i : inv. lim i A i A i (i I) satisfying φ ji φ j = φ i whenever i j and having the following universal property: For any object C and any morphisms ψ i : C A i (i I) satisfying ψ ji ψ j = ψ i (i j), there exists a unique morphism ψ : C inv. lim i A i such that φ i ψ = ψ i for any i. A morphism from an inverse system (A i, φ ji ) i I to an inverse system (A i, φ ji) i I is a family of morphisms u i : A i A i (i I) such that for any i j, the following diagram commutes: A j φ ji A i u i A j φ ji u j A i. It induces a morphism inv. lim i u i : inv. lim i A i inv. lim i A i. If (A i, φ ji ) i I is an inverse system of sets, then inv. lim i A i is the subset of i A i consisting of those elements (x i ) i A i satisfying φ ji (x j ) = x i for any i j. A category C is called an additive category if for any objects A, B and C in C, the direct product of A and B exists, Hom(A, B) is an abelian group, and the map Hom(A, B) Hom(B, C) Hom(A, C), (f, g) gf is a homomorphism. We call 0 Hom(A, B) the zero morphism. Proposition 1.2. Let C be an additive category and let A and B be two objects in C. (i) Let p 1 : A B A and p 2 : A B B be the projections. Define k 1 : A A B to be the unique morphism satisfying p 1 k 1 = id A and p 2 k 1 = 0, and define k 2 : B A B to be the unique morphism satisfying p 1 k 2 = 0 and p 2 k 2 = id B. Then we have k 1 p 1 + k 2 p 2 = id A B.
12 6 CHAPTER 1. SCHEMES AND COHERENT SHEAVES (ii) Suppose we have an object P and morphisms p 1 : P A, p 2 : P B, k 1 : A P, k 2 : B P such that p 1 k 1 = id A, p 2 k 2 = id B, k 1 p 1 + k 2 p 2 = id P. Then P together with the morphisms p 1 : P A and p 2 : P B is the direct product of A and B, P together with the morphisms k 1 : A P and k 2 : B P is the direct sum of A and B. Proof. (i) It is easy to verify that p 1 (k 1 p 1 + k 2 p 2 ) = p 1 id A B, p 2 (k 1 p 1 + k 2 p 2 ) = p 2 id A B. By the universal property of the direct product, we have k 1 p 1 + k 2 p 2 = id A B. (ii) We have p 1 k 2 = p 1 id P k 2 = p 1 (k 1 p 1 + k 2 p 2 )k 2 = (p 1 k 1 )(p 1 k 2 ) + (p 1 k 2 )(p 2 k 2 ) = p 1 k 2 + p 1 k 2 = 2p 1 k 2. So p 1 k 2 = 0. Similarly p 2 k 1 = 0. Let s prove (P, k 1, k 2 ) is the direct sum of A and B and leave to the reader to prove (P, p 1, p 2 ) is the direct product of A and B. Given any object C and any morphisms f 1 : A C and f 2 : B C, define f = f 1 p 1 + f 2 p 2. It is easy to verify that fk 1 = f 1 and fk 2 = f 2. If f : P C is a morphism such that f k 1 = f 1 and f k 2 = f 2, then we have f = f id P = f (k 1 p 1 + k 2 p 2 ) = (f k 1 )p 1 + (f k 2 )p 2 = f 1 p 1 + f 2 p 2. This proves (P, k 1, k 2 ) has the required universal property. Let C be an additive category and f : A B a morphism in C. We say a monomorphism K A is the kernel of f if the composition K A B is 0, and for any morphism K A such that the composition K A B is 0, there exists a unique morphism K K such that the diagram K K A commutes. We often denote K by kerf and call it the kernel of f. Similarly we define the cokernel of f to be an epimorphism B C such that the composition A B C is 0, and for any morphism B C such that the composition
13 1.1. PRESHEAVES AND SHEAVES 7 A B C is 0, there exists a unique morphism C C such that the diagram B C C commutes. We often denote C by cokerf and call it the cokernel of f. We define the image of f to be the kernel of the cokernel of f, and define the coimage of f to be the cokernel of the kernel of f. There exists a canonical morphism coimf imf from the coimage to the image such that the diagram A B coimf imf commutes. For example, when f : A B is a morphism in the category of abelian groups, then kerf = {a A f(a) = 0}, imf = {b B b = f(a) for some a A}, cokerf = B/imf, coimf = A/kerf, and the canonical morphism from the coimage to the image is the canonical homomorphism A/kerf imf (which is an isomorphism). Let C be an additive category. A zero object 0 in C is an object such that Hom(0, 0) = {0}. This is equivalent to saying that the identity morphism of 0 is equal to the zero morphism. For any object X in C, we have Hom(X, 0) = {0} and Hom(0, X) = {0}. Zero objects in C are isomorphic to each other. An abelian category C is an additive category with zero objects such that for any morphism f in C, the kernel and cokernel of f exist (and hence the image and coimage of f exist), and the canonical morphism coimf imf is an isomorphism. In an abelian category, a bijective morphism is an isomorphism. Indeed, if f : A B is injective, then the kernel of f is 0 A and the coimage of f is id A : A A. If f : A B is surjective, then the cokernel of f is B 0 and the image of f is id B : B B. If f : A B is bijective, then the canonical morphism coimf imf is just f : A B. Since coimf imf is an isomorphism, f : A B is an isomorphism. Suppose u : A B is a monomorphism in an abelian category. We often say A is a subobject of B. Let B C be the cokernel of u. We call C the quotient of B by A and denote it by B/A. In an abelian category, a sequence of morphisms A u B v C is called exact if vu = 0 and the canonical morphism coimu kerv is an isomorphism. An exact sequence of the form 0 A B C 0
14 8 CHAPTER 1. SCHEMES AND COHERENT SHEAVES is called a short exact sequence. This short exact sequence is called split it is isomorphic to 0 A A C C 0, where A A C and A C = A C C are the canonical morphisms. Proposition 1.3. Let i 0 A 1 p 2 1 A A2 0 be a short exact sequence in an abelian category. The following conditions are equivalent. (i) The above short exact sequence is split. (ii) There exists a morphism p 1 : A A 1 such that p 1 i 1 = id A1. (iii) There exists a morphism i 2 : A 2 A such that p 2 i 2 = id A2. Proof. (i) (ii) and (i) (iii) are obvious. (ii) (i) Consider the morphism id A i 1 p 1 : A A. We have (id A i 1 p 1 )i 1 = i 1 i 1 (p 1 i 1 ) = 0. Since A 2 is the cokernel of i 1 : A 1 A, there exists a morphism i 2 : A 2 A so that id A i 1 p 1 = i 2 p 2. Our assertion then follows from Proposition 1.2 (ii). Similarly one can prove (iii) (i). Let F : C D be a covariant functor between abelian categories. We say F is additive if for any objects A and B in C, the map is a homomorphism. We then have Hom(A, B) Hom(F (A), F (B)) F (A B) = F (A) F (B). Indeed, keeping the notations in Proposition 1.2 (i) and applying F to the equalities there, we get F (p 1 )F (k 1 ) = id F (A), F (p 2 )F (k 2 ) = id F (B), F (k 1 )F (p 1 ) + F (k 2 )F (p 2 ) = id F (A B). So by Proposition 1.2 (ii), (F (A B), F (k 1 ), F (k 2 )) is the direct sum of F (A) and F (B). Hence if 0 A B C 0 is a split short exact sequence, then is also a split short exact sequence. 0 F (A) F (B) F (C) 0
15 1.1. PRESHEAVES AND SHEAVES 9 Note that the category of abelian groups is an abelian category. We leave to the reader to prove the following proposition: Proposition 1.4. Let X be a topological space. Then the category of presheaves of abelian groups on X is an abelian category. Let φ : F G be a morphism of presheaves of abelian groups. Then the kernel, cokernel and image of φ are the presheaves defined by (kerφ)(u) = ker(φ(u) : F(U) G(U)), (cokerφ)(u) = coker(φ(u) : F(U) G(U)), (imφ)(u) = im(φ(u) : F(U) G(U)) for every open subset U of X. The stalks of these presheaves at a point P X are given by (kerφ) P = ker(φ P : F P G P ), (cokerφ) P = coker(φ P : F P G P ), (imφ) P = im(φ P : F P G P ). Proposition 1.5. Let F be a presheaf on a topological space X. There exists a pair (F +, θ) consisting of a sheaf F + and a morphism θ : F F + such that for any sheaf G and any morphism φ : F G, there exists a unique morphism ψ : F + G such that φ = ψθ. The pair (F +, θ) is unique up to unique isomorphism. For any point P X, θ P : F P (F + ) P is an isomorphism. We call F + the sheaf associated to the presheaf F. Proof. For any open subset U of X, define F + (U) to be the set of functions s : U P X F P satisfying the following two conditions: (a) For any P U, we have s(p ) F P. (b) For any P U, there exists a neighborhood U P of P contained in U and a section t F(U P ) such that for any Q U P, s(q) is the germ t Q of t at Q. Then F + is a sheaf and we have a canonical morphism θ : F F +. We leave to the reader to verify the pair (F +, θ) has the required property. Note that for any section s F + (U), we may find an open covering {U i } i I of U and sections s i F(U i ) such that θ(s i ) = s Ui, that is, sections in F + locally come from sections of F. Two sections s, t F(U) have the same image in F + (U) if and only if there exists a covering {U i } i I of U such that s Ui = t Ui for any i, that is, two sections of F which are locally equal are identified in F +. When F is a sheaf, θ : F F + is an isomorphism. Proposition 1.6. Let X be a topological space. Then the category of sheaves of abelian groups on X is an abelian category. Let φ : F G be a morphism of sheaves of abelian groups. Then kerφ is the sheaf defined by (kerφ)(u) = ker(φ(u)),
16 10 CHAPTER 1. SCHEMES AND COHERENT SHEAVES cokerφ is the sheaf associated to the presheaf U coker(φ(u)), and imφ is the sheaf associated to the presheaf U im(φ(u)). Moreover, for any P X, we have (kerφ) P = ker(φ P ), (cokerφ) P = coker(φ P ), (imφ) P = im(φ P ). Proof. It is easy to show that the presheaf defined by U ker(φ(u)) is a sheaf and is the kernel of φ. Using Proposition 1.4 and 1.5, one can show the cokernel of φ is the sheaf associated to the presheaf U coker(φ(u)). Denote by C the presheaf defined by U coker(φ(u)), and by P the presheaf defined by U im(φ(u)). Let s prove P + is the image of φ, that is, the kernel of the cokernel of φ. We have a canonical morphism of presheaves P G. It induces a morphism P + G. Since the composition of P G C is 0, the composition P G C + is also 0 and hence the composition P + G C + is 0. So P + G induces a morphism P + ker(g C + ). We claim it is an isomorphism. By Proposition 1.1, it suffices to show that for any P X, the homomorphism (P + ) P (ker(g C + )) P is an isomorphism. By Proposition 1.4 and 1.5, we have (P + ) P = P P = im(φ P ), (ker(g C + )) P = ker(g P (C + ) P ) = ker(g P C P ) = ker(g P coker(φ P )). Our claim follows immediately. We leave to the reader to prove the other assertions in the proposition. Lemma 1.7. Let f : A B be a morphism in an abelian category. Then (i) f is injective kerf = 0. (ii) f is surjective cokerf = 0 imf = B. Proof. We only prove the part in (i) and leave to the reader to prove the rest. Consider the commutative diagram A B coimf imf. If kerf = 0, then the left vertical arrow is an isomorphism. The bottom horizontal arrow is an isomorphism by the axiom of abelian category, and the right
17 1.1. PRESHEAVES AND SHEAVES 11 vertical arrow is a monomorphism. So the upper horizontal arrow is a monomorphism. Corollary 1.8. Let X be a topological space. (i) In the category of presheaves of abelian groups on X, a morphism φ : F G is injective (resp. surjective, resp. bijective) if and only if φ(u) : F(U) G(U) is injective (resp. surjective, resp. bijective) for any open subset U of X. (ii) In the category of sheaves of abelian groups on X, a morphism φ : F G is injective if and only if φ(u) : F(U) G(U) is injective for any open subset U of X. (iii) In the category of sheaves of abelian groups on X, a morphism φ : F G is injective (resp. surjective, resp. bijective) if and only if φ P : F P G P is injective (resp. surjective, resp. bijective) for any P X. (iv) In the category of sheaves of abelian groups on X, a morphism φ : F G is surjective if and only if for any open subset U and any section t G(U), there exists an open covering {U i } i I of U and sections s i F(U i ) such that φ(s i ) = t Ui for any i. Proof. (i) follows from Proposition 1.4 and Lemma 1.7. (ii) follows from Proposition 1.6 and Lemma 1.7. (iv) follows from (iii). Let s prove the statement about surjectivity in (iii) and leave to the reader to prove the rest. By Lemma 1.7, φ is surjective if and only if imφ = G. By Proposition 1.1, this is equivalent to saying (imφ) P = GP for any P X. By Proposition 1.6, this is equivalent to saying im(φ P ) = G P, that is, φ P is surjective for any P X. We leave to the reader to prove the following corollary: Corollary 1.9. Let X be topological space. (i) In the category of presheaves of abelian groups on X, a sequence is exact if and only if F φ G ψ H F(U) φ(u) G(U) ψ(u) H(U) is exact for any open subset U of X. (ii) In the category of sheaves of abelian groups on X, a sequence is exact if and only if F φ G ψ H is exact for any P X. F P φ P GP ψ P HP Let f : X Y be a continuous map of topological spaces. For any sheaf F on X, the direct image f F of F is the sheaf on Y defined by (f F)(V ) = F(f 1 (V ))
18 12 CHAPTER 1. SCHEMES AND COHERENT SHEAVES for any open subset V of Y. Note that f is an additive functor from the category of sheaves of abelian groups on X to the category of sheaves of abelian groups on Y. For any sheaf G on Y, the inverse image f 1 G of G is the sheaf on X associated to the presheaf defined by U dir. lim G(V ) f(u) V for every open subset U of X, where the limit is taken over the family of open subsets V of Y containing f(u). Note that f 1 is an additive functor from the category of sheaves of abelian groups on Y to the category of sheaves of abelian groups on X. When f is an imbedding, we often denote f 1 G by G X and call it the restriction of G to X. For any sheaf F on X, define a canonical morphism f 1 f F F as follows: Since f 1 f F is the sheaf associated to the presheaf U dir. lim (f F)(V ) = dir. lim F(f 1 (V )), f(u) V U f 1 (V ) by Proposition 1.5, it suffices to define a canonical morphism from the presheaf U dir. lim U f 1 (V ) F(f 1 (V )) to the sheaf F. We define dir. lim F(f 1 (V )) F(U) U f 1 (V ) to be the map induced by the restrictions F(f 1 (V )) F(U). For any sheaf G on Y, define a canonical morphism G f f 1 G as follows: For any open subset W of Y, since f(f 1 (W )) W, we have a map G(W ) dir. lim f(f 1 (W )) V G(V ). Composing with the canonical homomorphism dir. lim f(f 1 (W )) V G(V ) f 1 G(f 1 (W )), we get a homomorphism G(W ) f 1 G(f 1 (W )) = (f f 1 G)(W ), and hence a morphism of sheaves G f f 1 G. For any sheaf F on X and G on Y, define α F,G : Hom(G, f F) Hom(f 1 G, F) as follows: For any morphism φ : G f F, define α F,G (φ) to be the composition Define f 1 G f 1 φ f 1 f F F. β F,G : Hom(f 1 G, F) Hom(G, f F) as follows: For any morphism ψ : f 1 G F, define β F,G (ψ) to be the composition G f f 1 G f ψ f F.
19 1.1. PRESHEAVES AND SHEAVES 13 One can verify that α F,G and β F,G are inverse to each other, and they are functorial with respect to F and G, that is, for any morphism F 1 F 2 of sheaves on X and any morphism G 1 G 2 of sheaves on Y, the following diagram commutes: α F1,G Hom(G 2, f F 1 ) 2 Hom(f 1 G 2, F 1 ) Hom(G 1, f F 2 ) α F2,G 1 Hom(f 1 G 1, F 2 ), where the vertical arrows are induced by F 1 F 2 and G 1 G 2. Moreover a similar diagram for β F,G commutes. In general, let C and D be categories and let u : C D and v : D C be functors. We say u is left adjoint to v or v is right adjoint to u if for any object C in C and D in D, we have a bijection α C,D : Hom(C, v(d)) = Hom(u(C), D) which is functorial in C and D. We have seen that G f 1 G is left adjoint to F f F. Proposition 1.5 shows that the functor F F + from the category of presheaves to the category of sheaves is left adjoint to the inclusion functor from the category of sheaves to the category of presheaves. If a functor left adjoint to v exists, then it is unique up to isomorphism. Indeed, let u and u be two functors left adjoint to v. Then for any object C in C, we have a functorial bijection Hom(u(C), D) = Hom(u (C), D) for any object D in D. Taking D = u(c), we see that there exists a morphism φ C Hom(u (C), u(c)) corresponding to id u(c) Hom(u(C), u(c)). Taking D = u (C), we see that there exists a morphism ψ C Hom(u(C), u (C)) corresponding to id u (C) Hom(u (C), u (C)). One can verify that φ and ψ are natural transformations between u and u and they are inverse to each other. Similarly, if a functor right adjoint to u exists, it is unique up to isomorphism. Let f : X Y and g : Y Z be two continuous maps. Obviously we have (gf) = g f. So for any sheaf F on X and H on Z, we have that is, Hom((gf) 1 H, F) = Hom(H, (gf) F) = Hom(H, g f F) = Hom(g 1 H, f F) = Hom(f 1 g 1 H, F), Hom((gf) 1 H, F) = Hom(f 1 g 1 H, F). Hence (gf) 1 = f 1 g 1. Let P be a point in X and let i : {P } X be the inclusion. Then the inverse image i 1 F of a sheaf F on X can be identified with the stalk F P. So for any continuous map f : X Y and any sheaf G on Y, we have (f 1 G) P = i 1 f 1 G = (fi) 1 G = G f(p ),
20 14 CHAPTER 1. SCHEMES AND COHERENT SHEAVES that is, (f 1 G) P = G f(p ). We end this section with two lemmas which hold in any abelian categories. We only need the special case where the abelian category is the category of sheaves. In this case, the lemmas can be easily proved by diagram chasing. Lemma (The Snake Lemma) Let 0 A B C 0 u v w 0 A B C 0 be a commutative diagram in an abelian category such that the two rows are exact. Then there exists a morphism δ : kerw cokeru such that the following sequence is exact: 0 keru kerv kerw δ cokeru cokerv cokerw 0. Lemma (The Five Lemma) Let A 1 A 2 A 3 A 4 A 5 f 1 f 2 f 3 f 4 f 5 B 1 B 2 B 3 B 4 B 5 be a commutative diagram in an abelian category such that the two rows are exact. If f 1, f 2, f 4 and f 5 are isomorphisms, then f 3 is also an isomorphism. 1.2 Schemes and Morphisms Throughout this book, all rings are assumed to be commutative and have identity element 1, and all homomorphisms of rings are assumed to map 1 to 1. For every ring A, let SpecA be the set of prime ideals of A. For every ideal a of A, define V (a) = {p SpecA a p}. We leave to the reader to prove the following proposition: Proposition 2.1. (i) V (0) = SpecA and V (A) =. (ii) If a b, then V (a) V (b). (iii) i I V (a i ) = V ( i I a i) for any family of ideals a i (i I) of A. (iv) V (a) V (b) = V (ab) = V (a b). This proposition shows that the family of subsets of SpecA of the form V (a) is closed under the operations of intersection and finite union, and includes the
21 1.2. SCHEMES AND MORPHISMS 15 empty set and the total space SpecA. So SpecA is a topological space whose closed sets are of the form V (a) for ideals a of A. This topology is called the Zariski topology on SpecA. Proposition 2.2. (i) We have SpecA = if and only if 0 = 1 in A. (ii) For any ideal a of A, define the nilpotent radical of a to be the ideal a = {a A a n a for some natural number n}. Then we have V (a) = V ( a). (iii) We have a = p. p V (a) (iv) For any ideals a and b of A, we have V (a) V (b) if and only if a b. Proof. (i) If 0 = 1, then A = 0 and SpecA =. Suppose 0 1. Let S = {a a is an ideal of A and 1 a}. Then (0) S and hence S is nonempty. One can easily show that any totally ordered subset of S with respect to the order defined by inclusion has an upper bound in S. So by Zorn s Lemma, S has a maximal element, that is, A has a maximal ideal. Any maximal ideal is prime. So SpecA is nonempty. (ii) Follows directly from the definition of V (a). (iii) Obviously we have a p. Suppose f a. Then in (A/a) f, we p V (a) have 0 1. By (i), we may find a prime ideal q of (A/a) f. Let p be the inverse image of q under the canonical homomorphism A (A/a) f. Then p V (a) but f p. Hence f p. p V (a) (iv) Follows from (ii),(iii), and Proposition 2.1 (ii). Proposition 2.3. For any f A, the subset D(f) = SpecA V ((f)) = {p SpecA f p} is open. Open subsets of this type form a basis for the Zariski topology of SpecA, that is, any open subset of SpecA is a union of such open subsets. Moreover, D(f) is quasicompact. Proof. Since D(f) is the complement of V ((f)), it is open. Let U be an open subset of SpecA and p a point in U. Then U = SpecA V (a) for some ideal a of A. We have a p. So there exists an f a p. Then D(f) is a neighborhood of p contained in U. So open subsets of the type D(f) (f A) form a basis for SpecA. Let s prove D(f) is quasicompact. Suppose D(f) i I D(f i ) for some family of elements f i (i I) in A. Then we have V ((f)) i I V ((f i )) = V ( i I (f i)). By Proposition 2.2 (iv), we have (f) i I (f i). So we may find some natural number n such that f n = a i1 f i1 + + a ik f ik for some i 1,..., i k I
22 16 CHAPTER 1. SCHEMES AND COHERENT SHEAVES and a i1,..., a ik A. This implies that D(f) D(f i1 ) D(f ik ). Hence D(f) is quasicompact. Define a sheaf of rings O SpecA on SpecA as follows: For any open subset U of SpecA, O SpecA (U) consists of functions s : U p SpecA A p satisfying the following two conditions: (a) For any p U, we have s(p) Ap. (b) For any p U, there exist a neighborhood Up of p contained in U and a, f A such that for any q Up, we have f q and s(q) = a f in A q. For any inclusion of open subsets V U, we define O SpecA (U) O SpecA (V ) to be the restriction of functions. We call the pair (SpecA, O SpecA ) the spectrum of A. We often denote O SpecA by O if this doesn t cause any confusion. Proposition 2.4. (i) For any p SpecA, we have a canonical isomorphism Op = Ap. (ii) For any f A, we have a canonical isomorphism O(D(f)) = A f. In particular, taking f = 1, we get O(SpecA) = A. Proof. (i) For any neighborhood U of p, we have a homomorphism O(U) Ap defined by s s(p). These homomorphisms induce a homomorphism Op = dir. lim p U O(U) Ap. One can easily show it is surjective. To show it is injective, assume (s, U) O(U) satisfies s(p) = 0. Let be a neighborhood of p contained in U and let a, f A such that for any q Up, we have f q and s(q) = a f in A q. Since s(p) = 0, we have a f = 0 in A p. So there exists a t p such that ta = 0. Then for any q D(f) D(t), we have a f = ta tf = 0 in A q. Hence we have s U p D(t) = 0. Note that Up D(t) is a neighborhood of p. This shows that Op Ap is injective. (ii) Let s prove the homomorphism A f O(D(f)) which sends a A f k f to the section D(f) p SpecA A p, p a f Ap is k bijective. Injectivity: Suppose a lies in the kernel of the above homomorphism. Then f k for any p D(f), we have a = 0 in Ap. So for any p D(f), there exists a f k t p such that ta = 0. Hence we have p D(f) Ann(a) p, where Ann(a) = {r A ra = 0} is the annihilator of a. So p V ((f)) p V (Ann(a)). Therefore V (Ann(a)) V ((f)).
23 1.2. SCHEMES AND MORPHISMS 17 By Proposition 2.2 (iv), we have f Ann(a), that is, f n a = 0 for some natural number n. So a = 0 in A f k f. This proves the injectivity. Surjectivity: Let s : D(f) p SpecA A p be a section in O(D(f)), let {U i } i I be an open covering of D(f), and let a i, f i A such that for any q U i, we have f i q and s(q) = ai f i in Aq. By Proposition 2.3, we may assume U i = D(g i ) for some g i A, and we may assume the covering is a finite covering. Then for any q D(g i ), we have f i q. Hence D(g i ) D(f i ). By Proposition 2.2 (iv), we have (g i ) (f i ). So g ki i = h i f i for some natural number k i and h i A. Note that for any q D(g i ), we have h i f i q and s(q) = a i f i = h ia i h i f i = h ia i g ki i in Aq. Moreover, we have D(g i ) = D(g ki i ). Replacing g i by g ki i and a i by h i a i, we may assume that we have a finite covering {D(g i )} i I of D(f) and elements a i A such that for any q D(g i ), we have s(q) = ai g i in Aq. For any q D(g i ) D(g j ) = D(g i g j ), we have ai g i = aj g j in Aq. So for any q D(g i g j ), there exists a t q such that t(a i g j a j g i ) = 0. Hence for any q D(g i g j ), we have Ann(a i g j a j g i ) q, that is, if q V ((g i g j )), then q V (Ann(a i g j a j g i )). Therefore V (Ann(a i g j a j g i )) V ((g i g j )). By Proposition 2.2 (iv), we have g i g j Ann(a i g j a j g i ). Choose a large integer k, we may assume (g i g j ) k (a i g j a j g i ) = 0 for every pair i, j. (Our covering is a finite covering.) We have ai g i = aigk i. Replacing g g k+1 i by g k+1 i and a i by a i gi k, we i may assume {D(g i )} i I is a finite covering of D(f), and for any q D(g i ), we have s(q) = ai g i in Aq, and a i g j = a j g i for every pair i, j. From D(f) = i D(g i ), we get V ((f)) = i V ((g i )) = V (( i g i)). By Proposition 2.2 (iv), we have f k = i b ig i for some natural number k and b i A. For any j, we have a j f k = a j b i g i = i i b i a j g i = i b i a i g j = ( i b i a i )g j. So we have a j i = b ia i g j f k. Hence under the homomorphism A f O(D(f)) defined at the beginning, bia i f k is mapped to the section s O(D(f)). This proves the surjectivity. A ringed space is a pair (X, O X ) consisting of a topological space X and a sheaf of rings O X on X. We say (X, O X ) is a locally ringed space if for any P X, the stalk O X,P of O X at P is a local ring. (Recall that a ring A is called local if it has only one maximal ideal. This is equivalent to saying that elements in A which are not unit form an ideal of A.) By Proposition 2.4 (i), for any ring A, the spectrum (SpecA, O SpecA ) is a locally ringed space. If X is a topological
24 18 CHAPTER 1. SCHEMES AND COHERENT SHEAVES space and C X is the sheaf of continuous function on X, then (X, C X ) is also a locally ringed space. Let (X, O X ) and (Y, O Y ) be two ringed spaces. A morphism from (X, O X ) to (Y, O Y ) is a pair (f, f ) consisting of a continuous map f : X Y and a morphism of sheaves f : O Y f O X. For any point P X, we have a homomorphism (f O X ) f(p ) O X,P defined by (f O X ) f(p ) = dir. lim (f O X )(V ) = dir. f(p ) V dir. lim P U O X (U) = O X,P. lim P f 1 (V ) O X (f 1 (V )) Composing this homomorphism with the homomorphism O Y,f(P ) (f O X ) f(p ) induced by f, we get a homomorphism f P : O Y,f(P ) O X,P. Let (X, O X ) and (Y, O Y ) be two locally ringed spaces. A morphism of locally ringed spaces (f, f ) : (X, O X ) (Y, O Y ) is a morphism of ringed spaces such that for any P X, f P : O Y,f(P ) O X,P is a local homomorphism. (Recall that a local homomorphism of local rings f : A B is a ring homomorphism such that f 1 (m B ) = m A, or equivalently, f(m A ) m B, where m A and m B are the maximal ideals of A and B, respectively.) We can define the composition of morphisms of locally ringed spaces in the obvious way. An isomorphism of locally ringed spaces (f, f ) : (X, O X ) (Y, O Y ) is a morphism with a twosided inverse. This is equivalent to saying that f : X Y is a homeomorphism of topological spaces and f : O Y f O X is an isomorphism of sheaves. Any locally ringed space that is isomorphic to (SpecA, O SpecA ) for some ring A is called an affine scheme. A scheme (X, O X ) is a locally ringed space such that there exists an open covering {U i } i I of X such that each (U i, O X Ui ) is an affine scheme. We call X the underlying topological space and O X the structure sheaf. We often denote (X, O X ) by X. We define morphisms of schemes as morphisms of locally ringed spaces. We often denote a morphism of schemes (f, f ) : (X, O X ) (Y, O Y ) by f : X Y. Proposition 2.5. (i) Let φ : A B be a homomorphism of rings. Then φ induces canonically a morphism of locally ringed spaces (f, f ) : (SpecB, O SpecB ) (SpecA, O SpecA ). (ii) Any morphism (f, f ) : (SpecB, O SpecB ) (SpecA, O SpecA ) of locally ringed spaces is obtained this way. Proof. (i) Given a homomorphism φ : A B, define f : SpecB SpecA by f(q) = φ 1 (q)
25 1.2. SCHEMES AND MORPHISMS 19 for any q SpecB. For any ideal a of A, we have f 1 (V (a)) = V (ab), where ab is the ideal of B generated by φ(a). Hence f is continuous. For any open subset V of SpecA and any section s : V p SpecA A p of O SpecA (V ), define a section f (s) : f 1 (V ) q SpecB B q in O SpecB (f 1 (V )) by (f (s))(q) = φq(s(f(q))) for any q f 1 (V ), where φq : A φ 1 (q) Bq is the homomorphism induced by φ. In this way we get a morphism of sheaves f : O SpecA f (O SpecB ). One can verify that fq : O SpecA,q O SpecB,f(q) coincides with the homomorphism φq : A φ 1 (q) Bq through the identifications defined in Proposition 2.4 (i). So fq is a local homomorphism and hence (f, f ) is a morphism of schemes. (ii) Suppose (f, f ) : (SpecB, O SpecB ) (SpecA, O SpecA ) is a morphism of locally ringed spaces. Define φ : A B so that the following diagram commutes: O SpecA (SpecA) A φ B = = f (f O SpecB )(SpecA) = O SpecB (SpecB), where the vertical arrows are defined by Proposition 2.4 (ii). For every q SpecB, define φ q : A f(q) Bq so that the following diagram commutes: φ q B q = = A f(q) O SpecA,f(q) f q O SpecB,q, where the vertical arrows are defined in Proposition 2.4 (i). Since fq is a local homomorphism, φ q is also local. The following diagram commutes: O SpecA (SpecA) O SpecA,f(q) f f q O SpecB (SpecB) O SpecB,q. Through the identifications in Proposition 2.4, this diagram becomes A φ B p A p B A f(q) φ q B q, where the vertical arrows p A : A A f(q) and p B : B Bq are the canonical homomorphisms. So we have φ 1 (q) = φ 1 p 1 B (qb q) = p 1 A (φ q) 1 (qbq) = p 1 A (f(q)a f(q)) = f(q),
26 20 CHAPTER 1. SCHEMES AND COHERENT SHEAVES where the third equality follows from the fact that φ q is a local homomorphism. Therefore we have f(q) = φ 1 (q). Moreover the commutativity of the last diagram shows that φ q coincides with the homomorphism φ q : A φ 1 (q) Bq induced by φ. For any open subset V of SpecA and any q f 1 (V ), we have a commutative diagram O SpecA (V ) O SpecA,f(q) A φ 1 (q) f O SpecB (f 1 (V )) f q O SpecB,q φq B q. Note that the compositions of the vertical arrows in this diagram are the homomorphisms which send a section to its value at φ 1 (q) and at q, respectively. The commutativity of the above diagram shows that for any section s : V p SpecA A p in O SpecA (V ), the section f (s) : f 1 (V ) q SpecB B q in O SpecB (f 1 (V )) is given by f (s)(q) = φq(s(φ 1 (q))). Hence (f, f ) is induced by φ. Proposition 2.6. For any f A, we have a canonical isomorphism of locally ringed space (D(f), O SpecA D(f) ) = (SpecA f, O SpecAf ). Proof. The canonical homomorphism A A f induces a morphism (ϕ, ϕ ) : (SpecA f, O SpecAf ) (SpecA, O SpecA ). It is not hard to verify that the map ϕ : SpecA f SpecA is onetoone continuous and open with image D(f). So it induces a homeomorphism between SpecA f and D(f). One then show ϕ induces an isomorphism of sheaves O SpecA D(f) = (ϕ O SpecAf ) D(f) using Proposition 1.1 and the fact that Ap = (A f )p f for any p D(f). Corollary 2.7. Let (X, O X ) be a scheme and U an open subset of X. Then (U, O X U ) is a scheme. We call this scheme an open subscheme of (X, O X ). Proof. Cover X by affine open subschemes U i = SpecA i (i I). Then U can be covered by U U i (i I). Since each U U i is an open subset of U = SpecA i, we may cover each U U i by D(f ij ) (j J i ) for some f ij A i. Then D(f ij ) (i I, j J i ) form an open covering of U, and each (D(f ij ), O X D(fij)) is affine by Proposition 2.6. So (U, O X U ) is a scheme. The following proposition is a generalization of Proposition 2.5: Proposition 2.8. Let X be a scheme and A a ring. Then there is a onetoone correspondence between the family of morphisms of schemes from X to SpecA and the family of homomorphisms of rings from A to O X (X).
27 1.2. SCHEMES AND MORPHISMS 21 Proof. Define a map α : Hom(X, SpecA) Hom(A, O X (X)) as follows: For any morphism of schemes f : X SpecA, define α(f) to be the homomorphism f (SpecA) : A = O SpecA (SpecA) O X (X). Define a map β : Hom(A, O X (X)) Hom(X, SpecA) as follows: Let φ : A O X (X) be a homomorphism. Cover X by affine open subschemes U i = SpecA i (i I). The composition A φ O(X) O(U i ) = A i induces a morphism of affine schemes f i : U i SpecA for each i. We claim that f i Ui U j = f j Ui U j so that we can glue f i together to get a morphism f : X SpecA. We then define β(φ) = f. Indeed, cover U i U j by open affine subschemes U ijk = SpecA ijk (k K ij ). Then both f i Uijk : U ijk = SpecA ijk SpecA and f j Uijk : U ijk = SpecA ijk SpecA are induced by the composition A φ O(X) O(U ijk ) = A ijk. So f i Uijk = f j Uijk and hence f i Ui U j = f j Ui U j. One can verify α and β are inverse to each other. Recall that a graded ring is a ring S together with a decomposition S = S d of the additive group S into a direct sum of abelian groups S d (d d=0 N {0}) such that S d S e S d+e for any nonnegative integers d and e. Elements in S d are called homogeneous of degree d. Note that the identity element 1 lies in S 0. An ideal a of S is called homogeneous if it satisfies one of the following equivalent conditions: (i) a = d (a S d). (ii) If a a and a = d a d with a d S d, then each a d a. (iii) a is generated by homogeneous elements as an additive subgroup of S. (iv) a is generated by homogeneous elements as an ideal of S. For any homogeneous ideal a of S, S/a is a graded ring and the canonical homomorphism S S/a preserves the gradings. Let a be a homogeneous ideal. If for any homogeneous elements f and g in S such that fg a, we have f a or g a, then a is a prime ideal. To see this, suppose f and g are elements in S (not necessarily homogeneous) such that fg a but f, g a. Suppose f = i f i and g = i g i, where f i and g i are homogeneous of degree i for each i. Let m be the smallest integer such that f m a and n the smallest integer such that g n a. The degree m + n part in the decomposition of fg is m 1 i=0 n 1 f i g m+n i + f m g n + j=0 f m+n j g j. Since fg a, this sum lies in a. By the choice of m and n, we have f i a for i = 0,..., m 1 and g j a for j = 0,..., n 1. So we must have f m g n a. By our assumption, we then have f m a or g n a. But this contradicts to our choice of m and n.
28 22 CHAPTER 1. SCHEMES AND COHERENT SHEAVES We leave to the reader to show that sums, products, intersections, and nilpotent radicals of homogeneous ideals are homogeneous. Let S + = S d and let ProjS be the set of homogeneous prime ideals of S d=1 not containing S +. For any homogeneous ideal a of S, define V + (a) = {p ProjS a p}. We leave to the reader to prove the following proposition: Proposition 2.9. (i) V + (0) = ProjS and V + (S) =. (ii) i I V + (a i ) = V + ( i I a i) for any family of homogeneous ideals a i (i I) of S. (iii) V + (a) V + (b) = V + (ab) = V + (a b) for any homogeneous ideals a and b of S. By the above proposition, the family of subsets of ProjS of the form V + (a) is closed under the operations of intersection and finite union and includes the empty set and the total space ProjS. So we may define a topology on ProjS so that closed sets are of the form V + (a) for homogeneous ideals a of S. This topology is called the Zariski topology on ProjS. Let T be a multiplicative subset of S, that is, T is a subset of S containing 1 and is closed under multiplication. Then { a t T 1 S a S, t T, a and t are homogeneous of the same degree} is a subring of T 1 S. When T = S p for some homogeneous prime ideal p, we denote this ring by S (p). When T = {1, f, f 2,..., } for some homogeneous element f S, we denote this ring by S (f). Define a sheaf of rings O ProjS on ProjS as follows: For any open subset U of ProjS, O ProjS (U) consists of functions s : U p ProjS S (p) satisfying the following two conditions: (a) For any p U, we have s(p) S (p). (b) For any p U, there exist a neighborhood Up of p contained in U and homogeneous elements a, f S of the same degree such that for any q Up, we have f q and s(q) = a f in S (q). For any inclusion of open subsets V U, define O ProjS (U) O ProjS (V ) to be the restriction of functions. We often denote O ProjS by O if this doesn t cause any confusion. Proposition (i) For any p ProjS, we have a canonical isomorphism Op = S (p). (ii) For any homogeneous element f S + of positive degree, let D + (f) = ProjS V + ((f)) = {p ProjS f p}.
29 1.2. SCHEMES AND MORPHISMS 23 Then D + (f) is open in ProjS. Open subsets of this type form a basis for the topology of ProjS. Moreover, we have an isomorphism of locally ringed spaces (D + (f), O ProjS D+(f)) = (SpecS (f), O SpecS(f) ). In particular, (ProjS, O ProjS ) is a scheme. Proof. We leave to the reader to prove (i). Let s prove (ii). As the complement of V + ((f)), D + (f) is open. For any homogeneous ideal a of S and any point p in the open subset ProjS V + (a), we have a p and S + p and hence as + p. Let f be a homogeneous element in as + but not in p. Then f has positive degree and p D + (f) ProjS V + (a). Hence open subsets of the form D + (f) form a basis. Consider the map ϕ : D + (f) SpecS (f), p ps f S (f). We claim ϕ is bijective. Indeed, suppose p 1, p 2 D + (f) and ϕ(p 1 ) = ϕ(p 2 ). Then for any homogeneous element x p 1, we have x degf f degx p 1S f S (f) = p 2 S f S (f). (Here we use the fact that degf > 0.) So we have x degf p 2 and hence x p 2. Therefore p 1 p 2. Similarly p 2 p 1. This proves the injectivity of ϕ. For any q SpecS (f), let p be the nilpotent radical of the ideal generated by those a S lie in q for some k. (Then a is necessarily homogeneous of degree kdegf). Note that p is homogeneous. Suppose a 1 and a 2 are homogeneous such that a 1 a 2 p. Then there exists an n such that (a 1 a 2 ) n lies in the ideal generated by numerators of elements in q. Note that (a 1 a 2 ) ndegf lies in the such that a f k same ideal. This implies that a ndegf 1 f ndega 1 (a 1a 2) ndegf f n(dega 1 +dega 2 ) lies in q. Since q is prime, we have andegf q or f ndega 2 q. Hence a 2 1 p or a 2 p. Therefore p is a prime ideal. Obviously f p and hence p D + (f). One can verify that ϕ(p) = q. So ϕ is surjective. For any homogeneous ideal a of S, set b = as f S (f). One can verify ϕ(d + (f) V + (a)) = V (b). On the other hand, for any ideal b of S (f), define a to be the ideal of S generated by numerators of elements in b. Then b = as f S (f). This shows that ϕ establish a onetoone correspondence between the family of closed subsets of D + (f) and the family of closed subsets of SpecS (f). So ϕ is a homeomorphism. We have an obvious homomorphism for any p D + (f). Given any a t t degf f degt S (f) ϕ(p) and tdegf 1 a f degt (S (f) ) ϕ(p) S (p) S (p), we have t p and dega = degt. So S (f). It is easy to verify that t degf 1 a f degt t degf f degt lies in
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