Serre Duality

Yordanka Kovacheva April 9, 2012

Submitted to the Department of Mathematics of Amherst College in partial fulﬁllment of the requirements for the degree of Bachelor of Arts with honors

Faculty Advisor: Professor Robert Benedetto

We begin with a few basic definitions. After that we introduce the notions of presheaf, sheaf, stalk, morphism of sheaves, and exact sequence of sheaves. Then we proceed with defining two of the most important examples of sheaves—the structure sheaf for the spectrum of a ring with the Zariski toplogy, and the structure sheaf of a graded ring. Then we define schemes and some of the most common types of schemes. After that we introduce the notions of sheaf associated to module for the cases of affine or graded ring and quasi- coherent sheaves. We also present an operation known as twisting. In the third chapter we give some basic definitions in homological algebra—category, functor, complexes, resolutions, and cohomology groups. We then define the cohomology of sheaves by the functorial concepts. Next, we introduce Cechˇ cohomology as an alternate way of calculating the cohomology groups of a sheaf, and we prove that the two methods give the same answer in many common situations. Then we explicitly calculate the cohomology

n groups of certain coherent sheaves over projective space PA over a ring A. After that we introduce the functors Ext and Ext and prove some relations for the cohomology groups of locally free sheaves for these functors. Finally, we state and prove Serre duality for coherent sheaves on projective space and give some applications.

ii Acknowledgements

Thanks to Mom, Dad, and Pep for everything they have done for me. Thanks to my friends for the emotional support and understanding while writing this paper. Thanks to Professor Benedetto for his wonderful support, valuable advice, and enormous patience during this long and hard process. Thanks to all my Math Professors for their guidance and help about this thesis and throughout all my years at Amherst.

iii Contents

1 Basic Deﬁnitions 1

2 Background on Sheaves and Schemes 4 2.1 Introductory Deﬁnitions ...... 4 2.2 The Structure Sheaf ...... 9 2.3 Sheaves of Graded Rings ...... 12 2.4 Schemes ...... 14 2.5 The Sheaf Associated to a Module ...... 16 2.6 Twisting ...... 18

3 Cohomology 22 3.1 Derived Functors ...... 22 3.2 Sheaf Cohomology ...... 29 3.3 Cechˇ Cohomology ...... 35 3.4 Comparing Cechˇ and Derived Functor Cohomology ...... 38 3.5 The Cohomology of Projective Space ...... 41 3.6 Ext Groups and Sheaves ...... 44

4 Serre Duality 46 4.1 Statement of Serre Duality ...... 46 4.2 An Application of Serre Duality ...... 48

A Categories i

B Serre Duality for Diﬀerentials iii

iv C Additional Proofs x

Bibliography xii

Chapter 1

Basic Deﬁnitions

Throughout this thesis, “ring” will be understood to mean a commutative ring with 1. The deﬁnitions in this chapter are mostly from Chapters 1, 2, and 3 of [1] and p. 8 of [8]. A non-empty topological space X is called irreducible if every pair of non-empty open sets in X intersect. A subset Y X is irreducible if Y is irreducible in the induced topology. A topological space X is Noetherian if every chain of distinct closed subsets

¤ ¤ ¤ C1 C0 X is eventually constant; that is, there is some N ¥ 0 such that Cn CN for all n ¥ N. The dimension of X, denoted by dim X, is the supremum of all integers n such that there exists a chain Z0 Z1 ¤ ¤ ¤ Zn of distinct irreducible closed subsets of X. A Noetherian ring is a ring for which each ideal is generated by a ﬁnite number of elements. Equivalently, a ring A is Noetherian if each chain of ideals I0 I1 ¤ ¤ ¤ A is eventually constant. A ring is local if it has a unique maximal ideal. A ﬁnite sequence of n 1 prime ideals p0 p1 ¤ ¤ ¤ pn is called a prime chain of length n. The Krull dimension, or simply the dimension, of a Noetherian ring A is the length of the longest such chain, or 8 if such chains can be arbitrary long. A local Noetherian ring is called regular if the unique maximal ideal m can be generated by d dim A elements.

Deﬁnition 1.0.1. Let A be a ring. An A-module is an abelian group M with a mapping A ¢ M Ñ M such that apx yq ax ay, pa bqx ax bx, pabqx apbxq, and 1x x for all a, b P A and x, y P M. An A-algebra R is an A-module with a ring structure such that apr1r2q par1qr2 and ar1 r1a for all a P A and r1, r2 P R. The module M is said

to be generated by S ty1, y2,...u M if for every x P M, there exist y1, . . . , yn P S and

1 a1, . . . , an P A such that x a1x1 ¤ ¤ ¤ anxn. M is ﬁnitely generated if M if generated by a ﬁnite set.

Given a ring A, any ideal I A is an A-module. In particular, A itself is an A-module. If k is a ﬁeld, any k-vector space V is a k-module.

Deﬁnition 1.0.2. Let M and N be A-modules. A mapping f : M Ñ N is an A-module homomorphism if fpx yq fpxq fpyq and fpaxq afpxq for all x, y P M and a P A.

The composition of two A-module homomorphisms is again an A-module homomorphism. An A-module homomorphism f : M Ñ N that has an inverse A-module homomorphism g : N Ñ M (that is f ¥ g idM and g ¥ f idN ) is called an isomorphism, and the A-modules M and N are said to be isomorphic.A free A-module is one which is À isomorphic to an A-module of a form iPI Mi, where I is an arbitrary set and Mi A as an A-module. A map B : M ¢ N Ñ N is called A-bilinear if Bpm m1, nq Bpm, nq Bpm1, nq, Bpm, n n1q Bpm, nq Bpm, n1q, and Bpam, nq Bpm, anq aBpm, nq, for any m, m1 P M, n, n1 P N and a P A. A bilinear mapping B : V ¢ W Ñ A is said to be a perfect pairing if the mapping V Ñ HompW, Aq given by v Ñ fv, where fvpwq Bpv, wq, is an isomorphism. Equivalently, B is a perfect pairing if the analogous mapping W Ñ HompV,Aq is an isomorphism.

Deﬁnition 1.0.3. Let M and N be A-modules. The tensor product of M and N, denoted

M bA N or simply M b N, is an A-module with an A-bilinear map

φ : M ¢ N Ñ M b N satisfying the following universal property: given any A-module P and any A-bilinear mapping F : M ¢ N Ñ P , there exists a unique A-homomorphsim g : M b N Ñ P such that f φ ¥ g.

The tensor product M b N is unique up to a unique A-module homomorphism. More- over, M b N is generated as a module by the elements m b n φpm, nq subject to the restrictions pm m1q b n m b n m1 b n, m b pn n1q m b n m b n1, and pamq b n m b panq apm b nq, for all m, m1 P M, n, n P N, and a P A.

2 For example, if V and W are k-vector ﬁelds with basis teiuiPI and tfjujPJ respectively, b t b u P then V k W is a k-vector ﬁeld with basis ei fj pi,jqPI¢J . In addition, for any n Z, we b b a b a b have an isomorphism Q Q Zn 0, given by a b n bn n 0 0.

Deﬁnition 1.0.4. Let A be a ring. A multiplicatively closed subset of A is a subset S A such that 1 P S, and rs P S for all r, s P S. Given an A-module M and a multiplicatively closed subset S A, the localization of M with respect to S is the A-module 1 1 2 1 1 MS tpm, sq | m P M, s P Su, where pm, sq pm , s q if and only if s pms ¡ m sq 0 for 2 P m p q some s S. We usually write s to denote the equivalence class of m, s of MS. If B is

an A-algebra, then BS is also an A-algebra. In particular, AS is an A-algebra. Note that for any prime ideal p , its complement A ¡ p is a multiplicatively closed subset of A, as is

n the set tf | n P Z, n ¥ 0u for any f P Azt0u. The localizations of M with respect to them Ñ Ñ a are denoted respectively by Mp and Mf . The homomorphism A AS given by a 1 is called a localizing homomorphism.

Deﬁnition 1.0.5. A sequence of A-modules and A-homorphisms

fi fi 1 ¤ ¤ ¤ Ñ Mi¡1 ÝÑ Mi ÝÝÝÑ Mi 1 Ñ ¤ ¤ ¤

1 f is said to be exact if impfiq kerpfi 1q for all i. In particular, if 0 Ñ M ÝÑ M is exact, f then f is injective, and if M ÝÑ M 1 Ñ 0 is exact, then f is surjective.

A partially ordered set is a set I with a reﬂexive and transitive binary relation ¤ such that a ¤ b and b ¤ a implies a b. A partially ordered set I is said to be a directed set if for each pair i, j P I, there exists k P I such that i ¤ k and j ¤ k.

Deﬁnition 1.0.6. Let A be a ring, I a directed set, and pMiqiPI a family of A-modules indexed by I. For each i ¤ j, let fij : Mi Ñ Mj be an A-homomorphism with the properties P ¥ ¤ ¤ that fii idMi for all i I, and fik fjk fij whenever i j k. Then there exists an p q A-module M, called the direct limit of Mi iPI , denoted by limÝÑ M, and A-homomorphisms fi : M Ñ Mi, such that fi fj ¥ fij whenever i ¤ j. ( We can represent the direct limit M as rmi,Mis | mi P Mi { where the equivalence relation is given by rmi,Mis rmj,Mjs if there is some k ¥ i, j with fikpmiq fjkpmjq. For more information see p. 1 of [15].

3 Chapter 2

Background on Sheaves and Schemes

2.1 Introductory Deﬁnitions

This section follows mostly Section II.1 of [8] and Section II.1 of [3].

Deﬁnition 2.1.1. Let X be a topological space. A presheaf F of abelian groups (rings) on X consists of:

1. for every open subset U X, an abelian group (ring) FpUq, and

2. for every inclusion V U of open subsets of X, a homomorphism ρUV : FpUq Ñ FpV q of groups (rings), such that:

(a) FpHq 0,

(b) ρUU idU , and

The maps ρUV are called restriction maps. The elements of FpUq are called sections of F over U. The set of global sections, when X U, is denoted by ΓpX, Fq FpXq. For simplicity, we say that the presheaf F is given by U Ñ AU , where U X is any open set and AU is the group FpUq.

Deﬁnition 2.1.2. A presheaf F is called a sheaf if for every open covering tUαuαPI of an open set U X, and for each collection of elements fα P FpUαq having the property that

4 the restrictions of fα and fβ to Uα X Uβ are equal for α, β P I, that is

p q | X | X p q ρUαpUαXUβ q fα fα Uα Uβ fβ Uα Uβ ρUβ pUαXUβ q fβ ,

P p q | P there is a unique element f F U such that f Uα fα for each α I.

Example 2.1.3. Equip the set X tx, yu with the discrete topology. Deﬁne a presheaf

F1 of rings on X as follows: F1pHq 0, F1ptxuq Z2, F1ptyuq Z2, and F1ptx, yuq

Z2 ¢Z2 ¢Z2. The restriction map F1ptx, yuq Ñ F1ptxuq is the projection Z2 ¢Z2 ¢Z2 Ñ Z2 onto the ﬁrst coordinate, and F1ptx, yuq Ñ F1ptyuq is the projection Z2 ¢ Z2 ¢ Z2 Ñ Z2 onto the second coordinate. Consider 1 P F1ptxuq and 0 P F1ptyuq. They agree on the intersection txu X tyu H, but both elements p1, 0, 0q and p1, 0, 1q in F1ptx, yuq restricted to F1ptxuq and F1ptyuq give 1 and 0, respectively. Thus, F1 is only a presheaf and not a sheaf, because the uniqueness part of the sheaf condition fails. On the other hand, consider the presheaf F2 on X tx, yu given by F2ptx, yuq Z2, F2ptxuq Z2, F2ptyuq Z2, and

FpHq 0 with the restriction maps being the identity. Thus, F2 diﬀers from F1 only on

the open set X tx, yu. The presheaf F2 still is not a sheaf, but this time the existence

condition fails because there is no element in F2ptx, yuq Z2 that identically maps to both

1 P F2ptxuq Z2 and 0 P F2ptyuq Z2.

If F is a presheaf on X and U X is open, the restriction of F to U, denoted by F|U ,

is a presheaf on U, constructed by setting F|U pV q FpV q for every open subset V U. 1 Note that if F is a sheaf, then so is F|U .A subsheaf of a sheaf F on X is a sheaf F on X such that for each open set U X, we have F1pUq FpUq, and the restriction maps of F1 are those induced by F.

Deﬁnition 2.1.4. For any presheaf F on X and any point x P X, we deﬁne the stalk Fx to be the direct limit of the groups (rings) FpUq over all open neighborhoods U of x. That is,

( p q r s | P p q P { Fx ÝÑlim F U a, U a F U , x U , xPU

where is given by ra, Us rb, V s if there is an open set W U X V such that x P W and

a|W b|W .

5 For any open set U X, x P X, and a P FpUq, denote the image of a in the stalk Fx by ax. That is, ax denotes the equivalence class ra, Us P Fx. (See p. 2 of [9])

Remark 2.1.5. If F is a sheaf on X and U X is an open subset, consider the map ± p q Ñ P p q P F U xPU Fx that sends a section f F U to its collection of images ax Fx for any x P U. Then this map is injective. The proof is as follows. Suppose that a, b P FpUq are two elements such that ax bx for any x P U. Then by the deﬁnition of stalk, for any x P U | | there exists an open neighborhood Ux U of x such that a Ux b Ux . Note that the sets

Ux form an open covering of the set U. Hence, by the uniqueness property for the sheaf F, the elements a and b are equal. (See Proposition 2.1 of [9])

Definition 2.1.6. Sheafification is a method for transforming a presheaf into a sheaf. Given a presheaf F on X, the sheafification of F is a sheaf F˜ defined as follows. First, for any open ± ˜p q Ñ set U X, we define F U to be the set of functions h : U xPU Fx for which there exists an open covering tUiu of U and gi P FpUiq such that for all y P Ui we have hpyq gi,y, where gi,y is the image of gi in Fy. Second, for any inclusion V U of open sets, the restriction map F˜pUq Ñ F˜pV q is given by restricting the functions of F˜pUq to functions on V .

It is easy to check that F˜ is a presheaf. Moreover, it is a sheaf because the uniqueness and existence properties follow from glueing pointwise functions defined on subsets Vi V that agree on all intersections Vi X Vj to a function defined on V . Note that for any y P U and f P F˜pUq, we can associate fpyq P Fy with fy P Fy. By the sheafification construction, for any f P F˜pUq, there are an open cover tViu of U and elements gi P FpViq such that fx gi,x for all x P Vi. Hence, for any x P X we have Fx F˜x.

Example 2.1.7. Consider the sheaf F on X tx, yu with the discrete topology given by

Fptxuq Z2, Fptyuq Z2, and Fptx, yuq Z2 ¢Z2. The restriction map Fptx, yuq Ñ Fptxuq is the projection Z2 ¢ Z2 Ñ Z2 onto the first coordinate, and Fptx, yuq Ñ Fptyuq is the projection Z2¢Z2 Ñ Z2 onto the second. The sheaf F is the sheafification of both presheaves of Example 2.1.3, as all three presheaves agree on the sets txu and tyu, and each element pa, bq of Fptx, yuq Z2 ¢ Z2 Fi,x ¢ Fi,y is uniquely determined by its projections on the

ﬁrst and second coordinate to Fptxuq Z2 Fi,x and Fptyuq Z2 Fi,y, respectively.

6 Deﬁnition 2.1.8. A morphism φ : F Ñ G of sheaves on a space X is a collection of group homomorphisms φU : FpUq Ñ GpUq such that for every inclusion V U, the following diagram commutes: φ FpUq ÝÝÝÝÑU GpUq ρU,V ρU,V

φ FpV q ÝÝÝÝÑV GpV q Let φ : F Ñ G be a morphism of sheaves. We define the presheaf kernel of φ, presheaf cokernel of φ, presheaf image of φ to be respectively the presheaves given by U Ñ kerpφpUqq,U Ñ cokerpφpUqq,U Ñ impφpUqq. Note that presheaf kernel is actually a sheaf, but in general the presheaf cokernel and the presheaf image of φ are not sheaves. Hence, we define the sheaf cokernel and the sheaf image to be the sheafifications coker φ and im φ of the presheaf cokernel and the presheaf image, respectively. We say φ is injective if ker φ 0 and surjective if im φ G.

Deﬁnition 2.1.9. Let X be a topological space. We say that a sequence

φi¡1 φi ¤ ¤ ¤ Ñ Fi¡1 ÝÝÝÑ Fi ÝÑ Fi 1 Ñ ¤ ¤ ¤ of sheaves on X and morphisms is exact if at each stage ker φi im φi¡1.

By the comments after Deﬁnition 2.1.8, an exact sequence of sheaves does not necessarily induce an exact sequence of sections

¤ ¤ ¤ Fi¡1pUq Ñ FipUq Ñ Fi 1pUq Ñ ¤ ¤ ¤ for any open subset U X. In particular, a morphism of sheaves φ : F Ñ G is injective if and only if the induced maps φU are injective. However, for φ to be surjective it is not necessary for the induced maps φU to be surjective. Instead, it is only required that the sheaf image im φ coincides with G. Given sheaves F and G on X and any x P X, note that a morphism of sheaves φ : F Ñ G induces a homomorphism of groups φx : Fx Ñ Gx between the corresponding stalks.

Lemma 2.1.10. A morphism of sheaves φ : F Ñ G is surjective (injective) if and only if the homomorphism of groups φx : Fx Ñ Gx is surjective (injective) for all x P X.

7 Proof. If φ is injective, then the sheaf ker φ is the zero sheaf. Hence 0 pker φqx ker φx; that is, φx is injective for any x P X. Conversely, assume φx is injective for all x P X.

Consider t P FpUq for some open set U X such that φptq 0. Then φxptxq 0 P Gx for all x P U. Hence tx 0 for all x P U. By Remark 2.1.5, this implies that t 0. Similarly,

φ is surjective if and only if φx for all x P X. (See Exercise 1.2 of [3].)

Lemma 2.1.10 induces the following useful lemma.

Lemma 2.1.11. Let φ : F Ñ G be a morphism of sheaves on X. The morphism φ is surjective if and only if for every open set U X and any a P GpUq, there are an open t u P p q p q | covering Ui iPI of U and elements bi F Ui such that φ bi a Ui for all i.

Proof. We begin with the inverse implication. By Lemma 2.1.10, it suﬃces to show that for any x P X, the group homomorphim φx is surjective. To show this, pick c P Gx and pull it back to d P GpUq for some open set U X. By assumption, there are an open covering t u P p q p q | Ui iPI of U and elements bi F Ui such that φ bi d Ui . Then there is an index i such that x P Ui, and thus φxpbi,xq dx c; that is, φx is surjective.

Conversely, if φ is surjective, then φx is surjective for every x P X. Consider an arbitrary section a P GpUq. Since φx is surjective, there is some b P Fx such that φxpbq ax. That is, P p q p q | there is a neighborhood Ui of x and a section bi F Ui such that φ bi a Ui , with bi,x b.

The sets Ui for each x P X give the desired cover of U. (See Exercise 1.3 of [3].)

β Remark 2.1.12. The sequence F1 ÝÑα F ÝÑ F2 is exact if ker β im α. Applying Lemma 2.1.11, this means that for each open subset U X and any element g P ker βpUq, there t u P 1p q p q | are an open cover Ui of U and elements fi F Ui such that α fi g Ui .

Proposition 2.1.13. Let F be a presheaf on X, and let F˜ be its sheafification. Then there is a sheafification map ψ : F Ñ F˜ with the property that for any morphism of presheaves φ F ÝÑ G, where G is a sheaf, there is a unique morphism ψ1 : F˜ Ñ G such that φ ψ1 ¥ ψ.

Proof. Since Fx F˜x for any x P X, applying Remark 2.1.5, we have a map of presheaves F Ñ F˜ such that for any open U X and a P FpUq, a is mapped to the unique element b P F˜pUq such that ax bx for any point x P U. Indeed, for any section t P F˜pUq for

8 any open set U X, there are an open cover tUiu of U and elements ti P FpUiq such that tx ti,x for all x P Ui. Set gi φptiq P GpUiq. Note that for all x P Ui X Uj, we have gi,x φxpti,xq φxptj,xq gj,x, where we have used the fact that ti,x tx tj,x for P X | | x Ui Uj. By Remark 2.1.5, we have gi UiXUj gj UiXUj , and since G is a sheaf, there P p q | P ˜p q is some g G U such that g Ui gi. Thus, we can map t F U to the corresponding g P GpUq. Moreover, the map F˜ Ñ G is unique because t P F˜pUq can be mapped only to such g P GpUq for which at the level of stalks, ti,x ÞÑ gx for each x P Ui and for each i. Again by Remark 2.1.5, there is a unique such g P GpUq.

Example 2.1.14. Note for both presheaves F1 and F2 of Example 2.1.3, the stalks over x or y are Z2, and they equal the stalks of the sheaﬁﬁcation F. Also, we have a presheaf morphism φ1 : F1 Ñ F given by φ1 : F1ptx, yuq Ñ Fptx, yuq via pa, b, cq ÞÑ pa, bq and the identity maps φ1 : F1ptxuq Ñ Fptxuq and φ1 : F1ptyuq Ñ Fptyuq. Similarly, we have a presheaf morphism φ2 : F2 Ñ F given by φ2 : F2ptx, yuq Ñ Fptx, yuq via a ÞÑ pa, aq and the identity map on φ2 : F2ptxuq Ñ Fptxuq and φ2 : F2ptyuq Ñ Fptyuq.

Let f : X Ñ Y be a continuous map of topological spaces. For any sheaf F on X, the

¡1 direct image sheaf f¦F on Y is the presheaf given by pf¦FqpV q Fpf pV qq for any open set

V Y . This presheaf is a sheaf. The restriction maps on f¦F are given by the corresponding f¦F F restriction maps on F, i.e. ρUV ρf ¡1pUqf ¡1pV q for open subsets V U Y .

2.2 The Structure Sheaf

This discussion follows mostly [10], [11], [12] and Section II.2 of [8]. One very important example of a sheaf is the structure sheaf OX , which we will use throughout this thesis.

Before deﬁning OX , however, we need some further terminology.

Deﬁnition 2.2.1. Let A be a commutative ring with 1. Then X Spec A denotes the set of all proper prime ideals of A. X can be made into a topological space using the Zariski topology, where the closed sets are deﬁned to be the sets of the form

V pSq tp P Spec A|S pu,

9 for each subset S A. The open sets of the form Dpfq X ¡ V pfq, for f P A, are called the distinguished open sets.

Any ring homomorphism φ : A Ñ B induces a continuous map φ1 : Spec B Ñ Spec A given by φ1ppq φ¡1ppq. Furthermore, the distinguished open sets form a basis of open sets for the Zariski topology. Using the fact that each open set is a union of distinguished open sets, a sheaf F on X can be described uniquely up to an isomorphism by specifying the p q t u groups F Bα and the restriction maps ρBαBβ for any family Bα αPA of distinguished open subsets of X. The next deﬁnition uses this fact to build the structure sheaf OX by assigning

OX pDpfqq Af for all distinguished open sets, in a manner similar to sheaﬁﬁcation.

Definition 2.2.2. The structure sheaf OX on X Spec A is constructed as follows. For P Bp p qq p q p q P P any element f A, define OX D f Af . If D g D f , there exist b A and k N k such that g bf. Hence, we can define the restriction maps ρDpfqDpgq : Af Ñ Ag by ¡ © a abn ρ p q p q . D f D g f n gkn ± For any open set U X define O pUq to be the subset of elements of OBpV q with X V PBU X the following property: ! § ) p q p q § p q P OX U aV V PBU ρVW aV aW for all W, V BU with W V U , where BU is the set of all distinguished open sets contained in U. The restriction maps for open sets U2 U1 X are given by the projections ¹ ¹ Bp q Ñ Bp q OX V OX V , P P V BU1 V BU2 p q since BU2 BU1 . The pair Spec A, OX is called the spectrum of the ring A.

See Appendix C for more technical details. The next result shows that OX pDpfqq Bp p qq OX D f for distinguished open sets, and also identiﬁes the stalks of OX .

Proposition 2.2.3. Let A be a ring, and let pSpec A, OX q be its spectrum.

1. For any element f P A, the ring OX pDpfqq is isomorphic to Af . In particular,

ΓpSpec A, OX q A.

10 2. For any p P Spec A, the stalk OX,p is isomorphic to Ap.

Proof. The isomorphism Af Ñ OX pDpfqq in statement 1 is given by a ÞÑ aDpfq. This map is an isomorphism because for any V P BDpfq, we have V Dpfq, and hence aV aDpfq|V . p q That is, the element aV V PBDpfq is uniquely determined by the component aDpfq. In particular, we have OX pXq A1 A. In addition, by the deﬁnition of stalks as direct limits, we conclude that for any points p P Spec A we have ! § ) ! § ) § § OX,p ra, Us§p P U, a P OX pUq { ra, Dpfqs§p P Dpfq, a P OX pDpfqq { ! § ) a § §f R p, a P A A . f p

Example 2.2.4. Consider the ring A Z2 ¢Z2. The prime ideals of A are x t0u¢Z2 and y Z2 ¢t0u. Hence, we can identify X tx, yu from Example 2.1.3 with Spec A, where the distinguished open sets are Dpp1, 0qq txu tt0u ¢ Z2u and Dpp0, 1qq tyu tZ2 ¢ t0uu.

Then the structure sheaf OX is given by OX pXq Z2 ¢ Z2, pt uq p pp qqq p ¢ q OX x OX D 1, 0 Z2 Z2 p1,0q Z2, and pt uq p pp qqq p ¢ q OX y OX D 0, 1 Z2 Z2 p0,1q Z2.

Thus, the sheaf F from Example 2.1.3 is the structure sheaf OX , where X Spec A.

Deﬁnition 2.2.5. A sheaf of OX -modules (or simply an OX -module) is a sheaf F on X, such that for each open set U X, the group FpUq is an OX pUq-module, and for each inclusion of open sets V U, the restriction homomorphism FpUq Ñ FpV q is compatible with the ring homomorphism OX pUq Ñ OX pV q in the sense that for each a P OX pUq and m P FpUq, we have a|V ¤ m|V pa ¤ mq|V .

A morphism F Ñ G of sheaves of OX -modules is a morphism of sheaves such that for each open set U X, the map FpUq Ñ GpUq is a homomorphism of OX pUq-modules. We denote the group of morphisms from F to G, with the group operation being addition in G, p q | | by HomOX F, G . If U X is open and F is an OX -module, then F U is an OX U -module. Ñ p | | q If F and G are OX -modules, the presheaf U HomOX |U F U , G U is a sheaf and is an OX - p q b module, which we call sheaf Hom, denoted by HomOX F, G . The tensor product F OX G Ñ p qb p q of two OX -modules is defined to be the sheafification of the presheaf U F U OX G U .

11 À A direct sum of sheaves tFiuiPI is a sheaf G Fi such that for every open set U X À p q p q G we have G U Fi U , and the restriction maps ρUV on each coordinate are given by Fi the corresponding restiction maps ρUV for any open sets V U X. An OX -module F is called free if it is isomorphic to a direct sum of copies of OX . It is called locally free if X can be covered by open sets U for which F|U is a free OX |U -module. The rank of F on a such open set is the number of copies of the structure sheaf needed in the direct sum. Let A and B be rings, let X Spec A and Y Spec B, and let f : X Ñ Y be a continuous map of topological spaces. If F is a sheaf of OX -modules, then the direct image sheaf f¦F is an f¦OX -module. ˇ p q Given a sheaf E, we deﬁne the dual sheaf to be the OX -module E HomOX E, OX .

Proposition 2.2.6. Let E be a locally free OX -module of ﬁnite rank. q 1. pEqq E.

p q q b 2. For any OX -module F, we have HomOX E, F E OX F.

p b q p p qq 3. For any OX -modules F, G, we have HomOX E F, G HomOX F, HomOX E, G .

Proof. The statements follow from the analogous facts for modules. For more details see Exercise II.5.1 of [8].

2.3 Sheaves of Graded Rings

This discussion follows mostly Sections II.4.3 and III.2.1 of [6] and Section II.2 of [8].

Deﬁnition 2.3.1. A graded ring S is a ring S together with a family pSnqn¥0 of subgroups À 8 ¥ of the additive group of S such that S n0 Sn and SmSn Sm n for all m, n 0.

Thus, S0 is a subring of S, and each Sn is an S0-module. We say an element x P S is homogeneous of degree n if x P Sn for some n, and we write deg x n. Any element y P S ° P ¥ can be written uniquely as a ﬁnite sum n yn, where yn Sn for all n 0, and all but a

ﬁnite number of the yn’s are 0. The non-zero components yn are called the homogeneous components of y.A homogeneous ideal is an ideal of S generated by homogeneous elements. À We denote by S the ideal S n¡0 Sn.A relevant prime ideal is homogeneous prime ideal in the ring S that do not contain all of S .

12 If S is a graded ring, a graded S-module is an S-module M, together with a decompo- À sition M M , such that S ¤ M M .A homomorphism of degree d of graded nPZ n d n d n

S-modules f : M Ñ N is a collection of S0-module homomorphisms fi : Mi Ñ Ni d. Two graded S-modules are isomorphic if there are a S-module homomorphism M Ñ N and an inverse S-module homomorphism N Ñ M both of degree 0. That is, M and N are

isomorphic if Mk Nk as S0-modules for all k P Z.

Definition 2.3.2. Given a graded ring S, let | Proj S| be the set of relevant prime ideals in S. We define a topology on | Proj S|, much as in Definition 2.2.1, by taking the closed sets to be the sets of the form

V pIq tp | p is a relevant prime ideal and I pu.

The distinguished open sets of relevant primes of S not containing some homogeneous element f of S of positive degree, D pfq | Proj S|¡V pfq, form a basis for this topology.

Deﬁnition 2.3.3. The structure sheaf OX for X | Proj S|, denoted by Proj S, is constructed on the distinguished open sets and then extended to all open sets, much as in

Deﬁnition 2.2.2. We can identify D pfq with the set of relevant primes of the localization

Sf , which inherits a graded ring structure from S. To achieve that identiﬁcation, we group x ¡ the elements of Sf by degree, where deg y deg x deg y. On the other hand, these relevant primes correspond to all primes of Spfq, the ring of elements of degree 0 in Sf . Hence, we

deﬁne OX pD pfqq Spfq. Note that for any other homogeneous element g of degree 1 in the

overlap D pfgq D pfqXD pgq, we have Proj SpD pfgqq Spfgq pSpfqqg{f pSpgqqf{g. | p q Thus, Proj S D pfq OY for Y Spec Spfq . Hence we can glue the sheaf on distinguished open sets much as we did in Deﬁnition 2.2.2, and Proj S is a sheaf on | Proj S|. Furthermore, for any p P | Proj S|, the stalk Op satisﬁes Op pSpq0 Sppq

Example 2.3.4. The polynomial ring S Arx0, . . . , xrs can be graded by assigning the elements of A degree 0 and each variable degree 1. Consider the speciﬁc case when

S Zrx, ys. The distinguished sets D pxq and D pyq form a cover of | Proj S|. Also, r s { r s { Spxq Z t , where t y x, and analogously Spyq Z u where u x y. Thus, we have Proj SpD pxqq Zrts and D pxq SpecpZrtsq, with the prime ideals being p0q, ppq for

13 p P Spec Z, pfq for each irreducible polynomial f P Zrts such that the greatest common divisor of its coeﬃcients is 1, and pp, fq for p P Spec Z and f a monic polynomial irreducible over the ﬁeld Zp.

2.4 Schemes

This discussion follows mostly Sections II.2 and II.3 of [8] and Chapter 4 of [13].

Deﬁnition 2.4.1. A ringed space is a pair pX, OX q, where X is a topological space and OX is a sheaf of rings on X. The ringed space pX, OX q is a locally ringed space if for every point P P X, the stalk OX,P is a local ring. A morphism of ringed spaces # f : pX, OX q Ñ pY, OY q is a pair pf, f q, where f : X Ñ Y is continuous map and # # f : OY Ñ f¦OX is a morphism of sheaves such that f assigns to each open set U Y # p q Ñ p ¡1p q q a homomorphism fU :Γ U, OY Γ f U , OX compatible with the restriction homomorphisms. That is, for any open sets V U Y , we have the following commutative diagram: f # ΓpU, O q ÝÝÝÝÑU Γpf ¡1pUq, O q Y X Y ρX ρUV f¡1pUqf¡1pV q

f # V ¡1 ΓpV, OY q ÝÝÝÝÑ Γpf pV q, OX q # For each x P X, f then induces a homomorphism of the stalks fx : OY,fpxq Ñ OX,x by taking direct limits.

An aﬃne scheme is a locally ringed space pX, OX q which is isomorphic to the spectrum of some ring. A scheme is a locally ringed space pX, OX q for which every point x P X has an open neighborhood U such that the topological space U together with the restriction

OX |U is an aﬃne scheme. By abuse of notation, we will often write simply X for the scheme pX, OX q.A morphism of schemes is a morphism of the corresponding locally ringed spaces.

Example 2.4.2. For any ring A, the ringed space pSpec A, OSpec Aq is an aﬃne scheme. In particular, the pair pX, Fq of Example 2.1.3 is a scheme. Also, for any graded ring S, the pair p| Proj S|, Proj Sq is a scheme, which is not in general aﬃne. This scheme is denoted

Proj S. In particular, note that for the graded ring S Zrx, ys from Example 2.3.4,

p p q | q p r s q r s D x , Proj S D pxq Spec Z x, y pxq Spec Z t .

14 That is, Proj S restricted to D pxq is the structure sheaf for the topological space Spec Zrts.

Example 2.3.4 showed that the overlap D pxqXD pyq for | Proj S| is the open aﬃne subset

Dpy{xq Dptq of Spec Zrts.

Given an open subset U of a scheme X, the pair pU, OX |U q is again a scheme, called an open subscheme of X. The definition of a closed subscheme is more complicated: it is not enough to specify a closed subspace of X, because that is not enough to determine the sheaf structure. Instead, consider first the subcase of affine schemes. If the ring homomorphism φ : A Ñ B is surjective, then the map φ1 : Spec B Ñ Spec A determines a homeomorphism of Spec B onto the closed subset V pker φq Spec A. In this case, Spec B is called a closed subscheme of Spec A, and the morphism φ1 is a closed embedding. We now generalize this concept to arbitrary schemes. (See Section III.3 of [19]) A morphism of schemes f : Y Ñ X is called a closed embedding if every point x P X has an affine neighborhood U such that the scheme f ¡1pUq is affine and the homomorphism

¡1 fU : OX pUq Ñ OY pf pUqq is surjective. In that case Y is called a closed subscheme of X. We now consider some general properties of schemes. Given a morphism of schemes X Ñ S, we call X a scheme over S or simply S-scheme. Given S-schemes X and Y , the

ﬁbered product of X and Y over S, denoted by X ¢S Y , is a scheme together with morphisms p1 : X ¢S Y Ñ X and p2 : X ¢S Y Ñ Y , satisfying the commutative diagram in Figure 2.1 with the given morphisms X Ñ S and Y Ñ S.

Figure 2.1: Fibered Product

Z f θ > > g X ¢S Y > X p1

p2 > ∨ ∨ Y > S

In addition, X ¢S Y must satisfy the following universal property. Given any morphisms α : X Ñ S, β : Y Ñ S, and Z Ñ S and morphisms f : Z Ñ X and g : X Ñ Y that satisfy

β ¥ g α ¥ f, there exists a unique morphism θ : Z Ñ X ¢S Y such that f p1 ¥ θ and

15 g p2 ¥ θ; see Figure 2.1. The morphisms p1 and p2 are called the projection morphisms of the fibered product onto its factors. The fibered product X ¢S Y of any S-schemes X,Y exists and is unique up to isomorphism. In the affine case, when X Spec A, Y Spec B, and S Spec R, where A and B are R-algebras, the scheme X ¢S Y is the affine scheme

SpecpA bR Bq. (See Theorem 3.3 of [8]) A morphism of schemes f : X Ñ Y is said to be closed if the image of any closed set is closed. Given a Y -scheme X, the diagonal morphism is the unique morphism

∆ : X Ñ X ¢Y X whose composition with both projection maps p1, p2 : X ¢Y X Ñ X is the identity map on X. Let f : X Ñ Y be a morphism of schemes. Then f is separated if the diagonal morphism ∆ is closed. If f is separated, we say that X is separated over Y ; we say a scheme X is separated if it is separated over the scheme Spec Z. Note that separatedness loosely relates to the Hausdorff property, as a topological space E is Hausdorff if and only if the diagonal embedding E Ñ E ¢ E is closed, where E ¢ E has the product topology. A scheme X is called Noetherian if it can be covered by a finite number of affine sets

Ui Spec Ai such that all Ai are Noetherian rings. A scheme X is projective over a ring

A if it is isomorphic to Proj S for some graded ring S, where S0 A, and S is ﬁnitely generated by S1 as an S0-algebra. If A is a ring, the projective n-space over A is the scheme n r s Ñ PA Proj A x0, . . . , xn . A morphism of schemes i : X Y is an immersion if it gives an isomorphism from X to an open subscheme of a closed subscheme of Y .

2.5 The Sheaf Associated to a Module

This discussion follows mostly Section II.5 of [8].

Definition 2.5.1. Let A be a ring, and let M be an A-module. Define the sheaf associated to M on Spec A, denoted by M˜ , in the following way. For any element f P A, define

B M˜ pDpfqq Mf with the usual restriction maps on the basis sets. For any open set ± U X, deﬁne M˜ pUq to be the subset of elements of M˜ BpV q for which: V PBU

˜ p q tp q | p q P u M U xV V PBU ρVW xV xW for all W, V BU with W V U , where BU is the set of all distinguished open sets contained in U. The restriction maps for ± ± open sets U2 U1 are given by the projections P M˜ pV q Ñ P M˜ pV q. V BU1 V BU2

16 Here are some properties of the sheaf associated to M on X Spec A.

Proposition 2.5.2. Let A be a ring, X Spec A, and M an A-module. Then:

1. M˜ is a sheaf of OX -modules.

2. For each p P X, the stalk pM˜ qp of the sheaf M˜ at p is isomorphic to Mp.

3. For any f P A, the Af -module M˜ pDpfqq is isomorphic to Mf and ΓpX, M˜ q M.

Proof. The proof is very similar to that of Proposition 2.2.3 for the structure sheaf OX , as the two sheaves have very similar constructions.

Remark 2.5.3. The operator takes exact sequences of modules 0 Ñ M 1 Ñ M Ñ M 2 Ñ 0 to exact sequences of sheaves of modules 0 Ñ M˜ 1 Ñ M˜ Ñ M˜2 Ñ 0. This statement follows from the fact that localization preserves exactness in exact sequences (see Proposition 3.9 of [1]), and hence for each point p P X the corresponding sequence of stalks is exact.

Deﬁnition 2.5.4. Let pX, OX q be a scheme. A sheaf F of OX -modules is quasi-coherent if

X can be covered by open aﬃne subsets Ui Spec Ai in such a way that for each i there | is an Ai-module Mi with F Ui Mi. The sheaf F is coherent if furthermore each Mi is a 1 ﬁnitely generated Ai-module.

Proposition 2.5.5. Let X be a scheme and let F be a sheaf of OX -modules. Then F is quasi-coherent if and only if for each open aﬃne subset U Spec A of X, there is an A-

module M such that F|U M˜ . If X is Noetherian, then F is coherent if and only if the same condition holds, and in addition M is a ﬁnitely generated A-module.

Proof. See Theorem 5.4 of [8].

Example 2.5.6. Note that for a given A-module M, the sheaf M˜ associated to M is very similar to the structure sheaf OX for X Spec A. More precisely, if we consider a ring A as a module over itself, then OSpec A A˜. Any structure sheaf OX is a coherent sheaf, as the ring A is generated as an A-module by the identity 1 P A.

1Technically this deﬁnition is used only for local Noetherian spaces, which is case when it is most often studied. See p. 60 of [?] for the general deﬁnition.

17 Definition 2.5.7. Let S be a graded ring, and let M be a graded S-module. The sheaf associated to M on Proj S, denoted M˜ , is defined similarly to the affine case. For any

˜ B homogeneous element f P S , deﬁne M pD pfqq Mpfq with the usual restriction maps on the basis sets. The sheaf structure of M˜ can be extended to an any open set U X:

Proposition 2.5.8. Let S be a graded ring, let M be a graded S-module, let X Proj S, and let M˜ be the sheaf associated to M on X. Then

˜ 1. For any p P X, the stalk pMqp Mppq.

P ˜ | pq 2. For any homogeneous f S , we have M D pfq Mpfq via the isomorphism

D pfq Spec Spfq, where Mpfq denotes the group of elements of degree 0 in Mf .

3. M˜ is a quasi-coherent sheaf of OX -modules. If S is a Noetherian ring and M is ﬁnitely generated, then M˜ is coherent.

Proof. Similar to the aﬃne case. For more details see Proposition 5.11 of [8].

2.6 Twisting

This discussion mostly follows Section II.5 of [8].

Deﬁnition 2.6.1. Let S be a graded ring, and let X Proj S. For any n P Z, deﬁne its n-th twist to be the graded S-module Spnq given by Spnqi Sn i for any i P Z, and its n-th twisted sheaf to be OX pnq Spnq. For any sheaf of OX -modules F, denote by Fpnq b p q p q the twisted sheaf F OX OX n . Tensoring with OX n is called the twisting operation.

Example 2.6.2. The graded ring S, considered as a graded S-module, is isomorphic to Sp0q as a S-module. Analogously, the sheaf of rings Proj S is isomorphic to S˜, while Proj S, considered as sheaf of S-modules, is isomorphic to Sp0q OX p0q for X Proj S.

18 Remark 2.6.3. Let S be a graded ring, and let X Proj S. For a given f P S1, we have ! § ) s § Spmqp q §n ¥ 0, s P Spmq S pS q . f f n n m n f m

Ñ p q a ÞÑ af m P Thus, we have a homomorphism φpfq : Spfq S m pfq given by f n f n for a Sn. This p q Ñ b ÞÑ b map is actually an isomorphism, with inverse ψpfq : S m pfq Spfq given by f n f n m for P p q| | b Sm n. Hence, we also have an isomorphism OX m D pfq OX D pfq. Nevertheless, it is not true in general that OX pmq is isomorphic to OX , as they diﬀer in the way the two sheaves glue on the intersections D pfq X D pgq for another g P S . 1 y For example, consider S krx, ys. By Example 2.3.4, we have O pD pxqq k X x x and O pD pyqq k . We have isomorphisms φ : O pD pxqq Ñ O p1qpD pxqq X y x X X and φy : OX pD pyqq Ñ OX p1qpD pyqq, given by multiplication by x and y respectively.

That is, OX p1qpD pxqq krvs, with 1 and v representing x and y respectively, and

OX p1qpD pyqq krus, with 1 and u representing y and x respectively. Nevertheless, we do

not have a homomorphism φxy : OX pD pxyqq Ñ OX p1qpD pxyqq compatible with φx, φy

and the restriction maps in both OX and OX p1q. If φxy existed, then on the one hand

p q p O p qq Op1qp p qq φxy 1 φxy ρD pxqD pxyq 1 ρ φx 1 x, while on the other hand

p q p O p qq Op1qp p qq φxy 1 φxy ρD pyqD pxyq 1 ρ φy 1 y.

Proposition 2.6.4. Let S be a graded ring, let X Proj S, and suppose that S is generated by S1 as an S0-algebra. For any graded S-module M, we have M˜ pnq Mpnq and

OX pnq b OX pmq OX pm nq for all m, n P Z.

p b q ˜ b ˜ Proof. This statement follows from the fact that M S N M OX N for any graded

S-modules M and N if S is generated by S1. Indeed, for any f P S1, it is true that p b q b p b q ˜ b ˜ M S N pfq Mpfq Spfq Npfq, and hence the sheaves M N and M N agree on the

distinguished open sets D pfq.

q Proposition 2.6.5. Let S be a graded ring. Then OX pdq OX p¡dq for any d P Z.

19 Proof. For any S-graded modules N and M, the set HomgrSpM,Nq of S-graded homomorphisms has the structure of graded S-module with grading given by à HomgrSpM,Nq HomgrSpM,Nqn, nPZ where HomgrSpM,Nqn is the set of S-module homomorphisms M Ñ N of degree n. Then for X Proj S, we have an OX -module morphism λ : HomgrSpM,Nq Ñ HompM,˜ N˜q. The morphism λ is determined by the homomorphisms

p q Ñ p q λpfq : HomgrS M,N pfq HomSpfq Mpfq,Npfq , ¡ ©¡ © φ m φpmq where λp q f f t f k f t k for φ P HomgrSpM,Nqt and m P Mk.

On the other hand, we have HomgrSpSpdq,Sq Sp¡dq. Indeed, any φ P HomgrSpSpdq,Sqk is determined by φp1q P Sk¡d for 1 P Spdq¡d S0, and hence

HomgrSpSpdq,Sqk Sk¡d Sp¡dqk.

Similarly, the group HomSpSpdq,Sq of S-module homomomorphisms is isomorphic to the

S-module HomgrSpSpdq,Sq, because for any φ P HomSpSpdq,Sq, the element φp1q P S is a sum of ﬁnite number of homogeneous elements, and hence φ is a ﬁnite direct sum of graded S-module homomorphisms. This makes the morphism λ actually an isomorphism for M Spdq and N S. (See p. 75–77 of [21].) Therefore,

q OX p¡dq Sp¡dq HomSpSpdq,Sq HompSpdq, S˜q HompOX pdq, OX q OX pdq.

The twisting operation, of producing Fpnq from F, allows us to deﬁne a graded S-module associated to any sheaf of modules on X Proj A.

Deﬁnition 2.6.6. Let S be a graded ring, let X Proj S, and let F be a sheaf of OX - À p q p p qq modules. The graded S-module associated to F is Γ¦ F nPZ Γ X, F n .

Γ¦pFq is clearly a group, but it also has the structure of graded S-module. The grading comes from the direct summands in the deﬁnition; we now show it is actually a module.

20 Any s P Sd determines a global section s P Sd ΓpX, OX pdqq. Then for any t P ΓpX, Fpnqq, deﬁne the product s ¤ t in ΓpX, Fpn dqq by the tensor product s b t. By Proposition 2.6.4,

p q b p q p b p qq b p q b p q p q F n OX d F OX OX n OX OX d F OX OX n d F n d .

Proposition 2.6.7. Let A be a ring, let S Arx0, . . . , xrs, let r ¥ 1, and let X Proj S.

Then Γ¦pOX q S.

Proof. The distinguished open sets D pxiq cover X. Hence, each element t P ΓpX, OX pnqq is uniquely determined by elements ti P OX pnqpD pxiqq which agree on the intersections p q D xixj . Each ti is a homogeneous element of degree n in Sxi and its restriction to

Dxi,xj is its image in Sxixj by Proposition 2.6.4. Thus, summing over all n, we can identify À p q p p qq p q p q P Γ¦ OX nPZ Γ X, OX n with the set of r 1 -tuples t0, . . . , tr for which ti Sxi , and ti and tj have the same images in Sxixj for each i, j. Ñ Since the variables xi are not zero divisors of S, the localization maps S Sxi and Ñ 1 Sxi Sxixj are all injective, and these rings are all subrings of S Sx0...xr by Exer- p q 1 cise 3.1 of [1]. Hence Γ¦ OX is the intersection 0¤i¤r Sxi inside S . By the deﬁnition of localization, any homogeneous element of S1 can be uniquely presented as a product

n0 nr p q P x0 . . . xr f x0, . . . xr , where nj Z and f is a homogeneous polynomial not divisible by ¥ any xi. This element will be in Sxi exactly when nj 0 for all j 0. Therefore, the

n0 nr p q intersection of all the Sxi consists of homogeneous elements x0 , . . . , xr f x0, . . . , xr with all ni ¥ 0; that is, S Γ¦pOX q.

Deﬁnition 2.6.8. Let X be a scheme, and let F be a sheaf of OX -modules. We say that F is generated by global sections if F can be written as a quotient of a free module. That is, À Ñ there is a surjective morphism iPI OX F.

Proposition 2.6.9. Let X be a projective scheme over a Noetherian ring A. Then any coherent sheaf F on X can be written as a quotient of a sheaf E, where E is a ﬁnite direct

sum of twisted structure sheaves Opniq for various integers ni.

Proof. By Theorem II.5.17 of [8] there is a positive integer n such that Fpnq is generated À N Ñ p q Ñ by a ﬁnite number of global sections. Thus, we have a surjection i1 OX F n 0. À p¡ q N p¡ q Ñ Ñ Then after tensoring with OX n , we have a surjection i1 OX n F 0.

21 Chapter 3

Cohomology

3.1 Derived Functors

This discussion follows mostly Section III.1 of [8], Section 2 of [2], and Section 1 of [7].

Deﬁnition 3.1.1. A category C is a collection of objects and maps between them called morphisms, satisfying certain properties. An abelian category is a category A such that for any objects A, B P A, the set of morphisms HompA, Bq has the structure of an abelian group and in addition certain other nice properties hold, such as the existence of kernels and cokernels. In particular, modules and hence sheaves of modules each form an abelian cathegory. For a more detailed explanation of abelian categories and their properties, see Appendix A.

Deﬁnition 3.1.2. A functor from a category A to a category B is a mechanism that converts objects of A to objects of B and similarly converts morphisms to morphisms. A functor F : A Ñ B is called covariant if it maps all morphisms A Ñ A1 in A to morphisms F pAq Ñ F pA1q in B, and contravariant if it maps them to morphisms F pA1q Ñ F pAq.

Deﬁnition 3.1.3. Let A be an abelian category, and let 0 Ñ A Ñ B Ñ C Ñ 0 be an exact sequence in A. A covariant functor F : A Ñ B is exact if after applying F , the sequence 0 Ñ F pAq Ñ F pBq Ñ F pCq Ñ 0 is exact. F is only left exact if only the part 0 Ñ F pAq Ñ F pBq Ñ F pCq is necessarily exact, and right exact if the same applies to the part F pAq Ñ F pBq Ñ F pCq Ñ 0. Analogously, a contravariant functor G is exact if the sequence 0 Ñ GpCq Ñ GpBq Ñ GpAq Ñ 0 is exact, left exact if only

22 the part 0 Ñ GpCq Ñ GpBq Ñ GpAq is necessarily exact, and right exact if the part GpCq Ñ GpBq Ñ GpAq Ñ 0 is necessarily exact.

Example 3.1.4. Let A be a ring. The collection of all A-modules with A-module morphisms of A-modules forms an abelian category, as does the category of sheaves of A-modules. The correspondence M Ñ M˜ is an exact covariant functor from the category Mod of A-modules to the category ModpXq of sheaves of OX -modules by Remark 2.5.5, where X Spec A. In fact, the functor M Ñ M˜ gives an equivalence of categories between the category of

A-modules and the category of quasi-coherent OX -modules. Its inverse is the covariant functor F Ñ ΓpX, Fq. If A is Noetherian, the same functor gives an equivalence between the category of the ﬁnitely generated A-modules and the category of coherent sheaves of

OX -modules.

Deﬁnition 3.1.5. An object I of an abelian category A is injective if the contravariant functor Homp¤,Iq from A to the category Ab of abelian groups is exact.

Since Homp¤,Iq is left exact, equivalently, an object I is injective if and only if for each injective morphism g : A Ñ B and morphism f : A Ñ I, there exists a morphism h : B Ñ I such that h ¥ g f.

i Deﬁnition 3.1.6. A complex A is a collection of objects A for i P Z, and morphisms i i i 1 i 1 i i d : A Ñ A , such that d ¥ d 0 for all i P Z. The maps d are called coboundary maps.A morphism of complexes, f : A Ñ B , is a set of morphisms f i : Ai Ñ Bi for

i each i P Z, which commute with the maps d . That is, we have the following commutative diagram. i Ai ÝÝÝÝÑd Ai 1 f i f i 1

i Bi ÝÝÝÝÑd Bi 1 A short exact sequence of complexes 0 Ñ A Ñ B Ñ C Ñ 0 is a set of exact sequences for each i: 0 Ñ Ai Ñ Bi Ñ Ci Ñ 0, that commute with the coboundary maps di. The i-th cohomology of the complex A is hipA q ker di{ im di¡1.

The cohomology measures the deviation from exactness. After all, if the resulting sequence were exact, then ker di im di¡1, and hence hipA q 0 for all i.

23 Deﬁnition 3.1.7. An injective resolution of object A is a complex I for i ¥ 0 such that each Ii is an injective object, and the sequence

0 Ñ A Ñ I0 Ñ I1 Ñ ¤ ¤ ¤ is exact. We say that a category A has enough injectives if every object of A is isomorphic to a subobject of an injective object of A.

Note that if A has enough injectives, then every object has an injective resolution. To see this, ﬁrst embed a given object A in an injective I0 to produce a short exact sequence 0 Ñ A ÝÑ I0 Ñ coker Ñ 0. Next, embed the cokernel of in an injective I1, that is 0 Ñ coker Ñ I1, and ﬁnally combine the two sequences to produece an exact sequence 0 Ñ A ÝÑ I0 Ñ I1. Continue in the same way to construct an injective resolution of A.

Lemma 3.1.8 (Snake Lemma). Consider a commutative diagram of A-modules of the form: p A1 ÝÝÝÝÑ B1 ÝÝÝÝÑ C1 ÝÝÝÝÑ 0 f g h

q 0 ÝÝÝÝÑ A ÝÝÝÝÑi B ÝÝÝÝÑ C If the rows are exact, then there is a morphism δ : ker h Ñ coker f such that the sequence:

ker f ÝÝÝÝÑ ker g ÝÝÝÝÑ ker h ÝÝÝÝÑδ coker f ÝÝÝÝÑ coker g ÝÝÝÝÑ coker h is exact. Moreover, if A1 Ñ B1 is injective, then so is ker f Ñ ker g; and if B Ñ C is surjective, then so is coker f Ñ coker g.

Sketch of Proof. Given c1 P ker h, there is some b1 P B1 such that ppb1q c1 since p is surjective. Then by commutativity of the second square, we have gpppb1qq P ker q, and by the exactness of the second row, there exists a unique a P A such that ipaq gpppb1qq. Then we can deﬁne δpc1q a im f. That is, we have the formula δpc1q i¡1gp¡1pc1q, for c1 P ker h. Some more diagram chasing proves that this map is well-deﬁned, and we have exactness on each level of the long sequence. (See Lemma 11.7 of [16])

Proposition 3.1.9. Let A be an abelian category. Then

1. Each morphism of complexes f : A Ñ B in the category A induces a natural map hipfq : hipA q Ñ hipB q.

24 2. Given a short exact sequence of complexes in A:

f g 0 Ñ A ÝÑ B ÝÑ C Ñ 0,

we have a long exact sequence of cohomology

¤ ¤ ¤ Ñ hipA q Ñ hipB q Ñ hipC q Ñ hi 1pA q Ñ ¤ ¤ ¤

i i i i i Proof. (1) For each i P Z, the map h pfq, given by h pfqpa im d q f paq im d , is well-deﬁned because the morphisms f i commute with the coboundary maps di. (2) Follows from the Snake Lemma. (See Proposition 1.4 of [2]).

Definition 3.1.10. Let A be an abelian category with enough injectives, and let F : A Ñ B be a covariant left exact functor. Then for each i ¥ 0, the right derived functor RiF is defined by fixing, for each object A, an injective resolution 0 Ñ A Ñ I , and then setting RiF pAq hipF pI qq for the complex 0 Ñ F pI0q Ñ F pI1q Ñ ¤ ¤ ¤ .

The sequence 0 Ñ F pI0q Ñ F pI1q Ñ ¤ ¤ ¤ is indeed a complex, because after we apply F to the resolution, we get maps F pdkq : F pIkq Ñ F pIk 1q which satisfy

F pdkq ¥ F pdk 1q F pdk ¥ dk 1q F p0q 0.

Note that the requirement that F is left exact is important, because if F is completely exact, then RiF pAq hipF pI qq 0. On the other hand, RiF pAq is not generally trivial if F is only left exact, because then the exact sequence 0 Ñ A Ñ I0 Ñ im I0 Ñ 0, where im I0 I1, is transformed into an exact sequence 0 Ñ F pAq Ñ F pI0q Ñ F pim I0q, and in general the last map is not surjective, so that F pim I0q F pI1q. The following lemma shows that the cohomology of F is independent of the choice of injective resolution.

Lemma 3.1.11. Let A and B be objects in a abelian category C. Let 0 Ñ B Ñ J be an injective resolution of an object B and 0 Ñ A Ñ I any resolution of A. Then any morphism f : A Ñ B induces a morphism of complexes f : I Ñ J .

25 Proof. The result follows by induction. For the base case k 0, the existence of the morphism f 0 : I0 Ñ J 0 follows directly from the facts that A Ñ I0 is an injective morphism, that A Ñ B Ñ J 0 is a morphism, and that J 0 is an injective object. For k ¥ 1, by the exactness of the resolutions Ik¡2 Ñ Ik¡1 Ñ Ik and J k¡2 Ñ J k¡1 Ñ J k, we have injective morphisms Ik¡1{Ik¡2 Ñ Ik and J k¡1{J k¡2 Ñ J k. Hence, by chasing the diagram:

k¡2 Ik¡2 ÝÝÝÝÑd Ik¡1 f k¡2 f k¡1

k¡2 J k¡2 ÝÝÝÝÑd J k¡1 we have a morphism Ik¡1{Ik¡2 Ñ J k¡1{J k¡2. Then from the diagram

Ik¡1{Ik¡2 ÝÝÝÝÑ J k¡1{J k¡2

Ik J k using the morphisms 0 Ñ Ik¡1{Ik¡2 Ñ Ik and Ik¡1{Ik¡2 Ñ J k¡1{J k¡2 Ñ J k, and the fact that J k is injective, we construct the morphism Ik Ñ J k. Hence, we produce the desired map of complexes f k : Ik Ñ J k.

If A B and both A Ñ I and A Ñ J are injective resolutions, then applying Lemma 3.1.11 from both sides, we get an isomorphism between the complexes F pJ q and F pI q, and hence hipF pI qq hipF pJ qq. Moreover, any morphism f : A Ñ B induces a morphism of complexes f : I Ñ J between the injective resolutions, which induces a morphism RiF pfq : RiF pAq Ñ RiF pBq.

Theorem 3.1.12. Let A be an abelian category with enough injectives, let B be another abelian category, and let F : A Ñ B be a covariant left exact functor. Then:

1. There is an isomorphism F R0F .

2. For each short exact sequence 0 Ñ A1 Ñ A Ñ A2 Ñ 0 in A and for each i ¥ 0, there is a morphism δi : RiF pA2q Ñ Ri 1F pA1q, ﬁtting into a long exact sequence

i ¤ ¤ ¤ Ñ RiF pA1q Ñ RiF pAq Ñ RiF pA2q ÝÑδ Ri 1F pA1q Ñ Ri 1F pAq Ñ ¤ ¤ ¤ .

26 3. Given a morphism of the exact sequence of (2) to another 0 Ñ B1 Ñ B Ñ B2 Ñ 0, the resulting morphisms δi’s ﬁt into a commutative diagram

i RiF pA2q ÝÝÝÝÑδ Ri 1F pA1q

i RiF pB2q ÝÝÝÝÑδ Ri 1F pB1q

Proof. To verify that R0F F , choose an injective resolution

0 0 Ñ A Ñ I0 Ñd I1 Ñ ¤ ¤ ¤ of A. Since F is left exact, the sequence

F pq F pd0q 0 Ñ F pAq Ñ F pI0q Ñ F pI1q is still exact. Thus,

R0F pAq ker F pd0q{ im 0 ker F pd0q im F pq F pAq.

By a similar argument, if f is a morphism in A, then R0F pfq corresponds to F pfq. Let 0 Ñ A Ñ B Ñ C Ñ 0 be any short exact sequence in A. Fix injective resolutions 0 Ñ A Ñ I and 0 Ñ C Ñ K . Then by a diagram chase similar to the one in the proof of the Snake Lemma (see the Horseshoe Lemma, Theorem 2.2.8 of [24]), there exists an injective resolution 0 Ñ B Ñ J , so that there is a short exact sequence of complexes

0 Ñ I Ñ J Ñ K Ñ 0.

Using injectivity of the objects Ii’s, we have a short exact sequence of complexes

0 Ñ F pI q Ñ F pJ q Ñ F pK q Ñ 0.

Therefore, by Lemma 3.1.8, there is a long exact sequence

¤ ¤ ¤ Ñ RiF pA1q Ñ RiF pAq Ñ RiF pA2q Ñ Ri 1F pA1q Ñ Ri 1F pAq Ñ ¤ ¤ ¤ .

Statement (3) may be proven by another diagram chase.

Proposition 3.1.13. If F : A Ñ B is any left exact functor between abelian categories with enough injectives, and I is an injective object of A, then RiF pIq 0 for all i ¡ 0.

27 Proof. Immediate by computing RiF pIq using the injective resolution 0 Ñ I Ñ I Ñ 0.

Definition 3.1.14. A functor that satisfies the properties of the functor F of Theorem 3.1.12 is called a δ-functor. A functor F : A Ñ B is effaceable if for each object A, there is monomorphism 0 Ñ A ÝÑu M, for some M P A, such that F puq 0. It is coeffaceable if for each A, there is an epimorphism R ÝÑu A Ñ 0 such that F puq 0. Similarly, the δ-functor T pT iq : A Ñ B is said to be universal, if given any other δ-functor T 1 pT 1iq : A Ñ B and a morphism of functors f 0 : T 0 Ñ T 10, there exists a unique sequence of morphisms f i : T i Ñ T 1i for each i ¥ 0, starting with the given f 0, which commute with the δi for each exact sequence.

i i Theorem 3.1.15. Let T pT qi¥0 be a covariant δ-functor from A to B. If T is eﬀaceable for each i ¡ 0, then T is universal.

Proof. See Theorem II.2.2.1 of [5].

Remark 3.1.16. Assume that A has enough injectives. Then for any left exact functor

i 0 F : A Ñ B, the derived functors pR F qi¥0 form a universal δ-functor with F R F . Indeed, any object A P A can be embedded into an injective object by some morphism u : A Ñ I. By Proposition 3.1.13, RiF pIq 0 for i ¡ 0, and thus RiF puq 0. Hence RiF is eﬀaceable and therefore universal by Theorem 3.1.15. We can use this fact to show that two derived functors are the same by showing that they coincide for i 0, and that they are both 0 for i ¡ 0.

Deﬁnition 3.1.17. Given a left exact functor F : A Ñ B between abelian categories with enough injectives, an object I is acyclic for F if RiF pIq 0 for i ¡ 0.

Thus, according to Proposition 3.1.13, each injective object is acyclic.

Proposition 3.1.18. Let the categories A, B, and the functor F be as in Deﬁnition 3.1.17, and let 0 Ñ A Ñ J 0 Ñ J 1 Ñ ¤ ¤ ¤ be an exact sequence in A, where each J i is acyclic for F for all i ¥ 0. Then for each i Ñ 0, there is a natural isomorphism RiF pAq hipF pJ qq.

Proof. For i 0, by the left exactness of F and Theorem 3.1.12, we have ¡ © ¡ © h0pF pJ qq ker F pJ 0q Ñ F pJ 1q im F pAq Ñ F pJ 0q F pAq R0F pAq.

28 We will prove the statement for i ¡ 0 using induction. The long exact sequence of the resolution 0 Ñ A Ñ J can be decomposed into a short exact sequence

0 Ñ A ÝÑ J 0 Ñ B Ñ 0, where B im d0{ ker , and an acyclic resolution 0 Ñ B Ñ J 1 Ñ J 2 Ñ ¤ ¤ ¤ , which we denote by 0 Ñ B Ñ K . For the base case i 1 we have an exact sequence

0 Ñ F pAq Ñ F pJ 0q Ñ F pBq Ñ R1F pAq, (3.1)

and a resolution 0 Ñ F pBq Ñ F pJ 1q Ñ F pJ 2q Ñ ¤ ¤ ¤ . Hence, ¡ © ¡ © ker F pJ 1q Ñ F pJ 2q im F pBq Ñ F pJ 1q F pBq.

Therefore, ¡ © ¡ © h1pJ q ker F pJ 1q Ñ F pJ 2q { im F pJ 0q Ñ F pJ 1q ¡ © ¡ © F pBq{ im F pJ 0q Ñ F pBq F pBq{ ker F pBq Ñ R1F pAq R1F pAq,

where the last isomorphism follows by the ﬁrst isomorphism theorem. Writing out the long exact sequence

¤ ¤ ¤ Ñ Ri¡1F pJ 0q Ñ Ri¡1F pBq Ñ RiF pAq Ñ RiF pJ 0q Ñ ¤ ¤ ¤ corresponding to (3.1), and using the acyclicity of each J i, we obtain Ri¡1F pBq RiF pAq for all i ¥ 2. Hence,

RiF pAq Ri¡1F pBq hi¡1pF pK qq hipF pJ qq, where we used the inductive hypothesis for B in the middle isomorphism.

3.2 Sheaf Cohomology

This discussion follows mostly Section 2 of [2]. We want to deﬁne the cohomology groups of a sheaf by taking the derived functors of the global sections functor. In order to be able to do this, we must ﬁrst ensure that the global sections functor Γ is left exact, and that the category of sheaves of abelian groups has enough injectives.

29 Proposition 3.2.1. The functor of global sections ΓpX, ¤q : ModpXq Ñ Mod is left exact.

1 f g 2 Proof. Let 0 Ñ F ÝÑ F ÝÑ F Ñ 0 be a short exact sequence of OX -modules. To show that f g ΓpX, ¤q is left exact, we will prove that 0 Ñ F1pXq ÝÝÑX FpXq ÝÝÑX F2pXq is also exact. First, we have ker fX pker fqpXq 0, since ker f 0, and the sheaf kernel coincides with the presheaf kernel and hence consists of kernels of the individual morphisms.

To show that ker gX im fX , we take an arbitrary a P FpXq such that gpaq gX paq 0. 1 Then by Remark 2.1.12, there exists an open cover tUiuiPI of X and bi P F pUiq such that p q | f bi a Ui . Then

p | q p q| p | q | f bi UiXUj f bi UiXUj a Ui UiXUj a UiXUj .

p | q p | q By symmetry, therefore we have f bi UiXUj f bj UiXUj . Hence, since f is injective, we | | P p q have bi UiXUj bj UiXUj . By the sheaf axioms for the sheaf F, there is a unique b F X | p q such that b Ui bi. It remains to check that f b a. This is true because on each Ui,

p q| p | q p q | f b Ui f b Ui f bi a Ui ,

and by the uniqueness property for the sheaf F, we have fpbq a. 1 To show im fX ker gx note that for any b P F pXq, we have fpbq P pim fqpXq pker gqpXq, since the sequence of sheaves is exact, and hence gpfpbqq gX pfX pbqq 0.

Proposition 3.2.2. If A is a ring, then every A-module is isomorphic to a submodule of an injective A-module. In particular, the category of A-modules has enough injectives.

Proof. See Theorem I.1.2.2 of [4].

The following proposition is the corresponding fact for the category of OX -modules.

Proposition 3.2.3. Let pX, OX q be a ringed space. Then the category of OX -modules has enough injectives.

Proof. Let F be an OX -module. For any point x P X, by Proposition 3.2.2 there is an inclusion Fx Ñ Ix of the stalk of F at x into an injective OX,x-module. Let I be the sheaf ² p q given by I U xPU Ix. Note that the stalks of the sheaf I are the injective modules

30 Ix. To show that I is injective, given an injective morphism of sheaves G Ñ L and any

morphism of sheaves f : G Ñ I, we want to extend f to a morphism g : L Ñ I. Since Ix

is an injective module for each x P X, we can extend every fx : Gx Ñ Ix to a morphism

gx : Lx Ñ Ix. The collection of morphisms gx induces a morphism g : L Ñ I which extends f. Hence, I is injective.

Definition 3.2.4. Let X be a topological space. Let ΓpX, ¤q be the global section functor from the category AbpXq of sheaves of abelian groups on X to the category Ab of abelian groups. We define the cohomology functors HipX, ¤q to be the right derived functors of ΓpX, ¤q. We also define HipX, ¤q to be the analogous functors for the category ModpXq of sheaves of OX -modules.

The following statements will imply that computing cohomology groups by injective resolutions either of the category AbpXq or of the category ModpXq give the same group.

Deﬁnition 3.2.5. A sheaf F on a topological space X is ﬂasque if for every inclusion of open sets V U, the restriction map FpUq Ñ FpV q is surjective.

Note that to show that F is ﬂasque, it is enough to show that for any open set U X, the restriction map FpXq Ñ FpUq is surjective. Indeed, if this is true, then for any open sets V U, we have ρX,V ρX,U ¥ ρU,V , and since ρX,V is surjective, the restriction map

ρU,V : FpUq Ñ FpV q is surjective as well.

Example 3.2.6. The sheaf F on X tx, yu from Example 2.1.3 is ﬂasque because the ¢ Ñ ¢ Ñ restriction maps ρtx,yu,txu : Z2 Z2 Z2 and ρtx,yu,tyu : Z2 Z2 Z2 are surjective as projections on the ﬁrst and second coordinate, respectively.

f f Lemma 3.2.7. If 0 Ñ F1 ÝÑ F ÝÑ F2 Ñ 0 is an exact sequence of sheaves and F1 is ﬂasque, f g then for each open U X, the sequence 0 Ñ F1pUq ÝÑU FpUq ÝÑU F2pUq Ñ 0 is exact too.

Proof. The partial sequence 0 Ñ F1pUq Ñ FpUq Ñ F2pUq is exact by Proposition 3.2.1 applied to the functor ΓpU, ¤q. Hence, we need to show only the surjectivity part. Given s2 P F2pXq, we want to ﬁnd s P FpXq such that gpsq s2. Consider the set 2 all pairs pV, sq of an open subset V U and s P FpV q such that gpsq s |V . This set is

31 not empty, since by Remark 2.1.12 the surjectivity of g implies that there exist an open t u P p q p q 2| covering U Ui iPI of U and elements si F Ui such that g si s Ui . We can order p q p q p q | the pairs V, s by declaring Vi, si Vj, sj when Vi Vj and sj Vi si. We will prove by contradiction that there exists a maximal pair pV, sq. If there were no maximal element, starting with any pair pV0, s0q we can construct a chain pV0, s0q pV1, s1q ¤ ¤ ¤ . Deﬁne V Vi. Then tViui¥0 is an open covering of V , and for any si P FpViq and sj P FpVjq, | | P p q p q tp q p qu we have si ViXVj sj ViXVj sk F Vk , where sk,Vk min si,Vi , sj,Vj . By the P p q | existence property for the sheaf F, there exists s F V such that s Vi si for any Vi. p q| p | q p q 2| 1 Then g s Vi g s Vi g si s Vi . Now by the uniqueness property of sheaf F , we 2 have gpsq s |V . Thus pVi, siq pV, sq for all i, and hence pV, sq is an upper bound for the pVi, siq’s. By Zorn’s Lemma, there is a maximal element pV, sq among the whole set. If V is not the whole set U, then using the covering U of U, there exist a non-empty 2 set W P U and sW P FpW q such that gpswq s |W and W V . Then

1 P 1p X q p 1 q | ¡ | 1 By exactness, there is some s0 F V W such that f s0 sW V XW s V XW . Because F 1 P p q 1| 1 p p 1q q| | is ﬂasque, there is some s F V such that s V XW s0. Since f s s V XW sW V XW 1 by the sheaf existence property, there is some t P FpV Y W q such that t|V fps q s and t|W sW . Moreover,

1 2 gptq|V gpt|V q gpfps q sq gpsq s |V ,

2 and gptq|W gpt|W q gpsW q s |W .

2 Hence, by the sheaf uniqueness property, we have gptq s |V YW . Thus, pV, sq pV YW, tq, which contradicts the maximality of the pair pV, sq. Therefore, V U, and gpsq s2.

f Lemma 3.2.8. If 0 Ñ F1 Ñ F ÝÑ F2 Ñ 0 is an exact sequence of sheaves and F1 and F are ﬂasque, then so is F2.

Proof. By the discussion following Deﬁnition 3.2.5, to show that F2 is ﬂasque, it is enough 2 P 2p q 2 P 2p q to show that for any open subset U X and any s0 F U , there exists s F X such 2| 2 P p q p q 2 that s U s0. By Lemma 3.2.7, there exists s0 F U such that f s0 s0. Since F itself

32 P p q | p q| p | q p q 2 is ﬂasque, there exists s F X such that s U s0. Then f s U f s U f s0 s0; that is, fpsq is the desired s2.

Lemma 3.2.9. Let pX, OX q be a ringed space. Then any injective OX -module is ﬂasque.

Proof. Let I be an injective OX -module. We will prove that for any open subsets V U, the map IpUq Ñ IpV q is surjective. Let OU denote the subsheaf of OX formed by sheaﬁfying pre Ñ p q Ñ the presheaf OU given by W OX W for any open subset W U, and W 0 otherwise. pre pre pre Deﬁne OV and OV similarly. Note that OV is subsheaf of OU , and hence by Proposition

2.1.13, there is a sheaf morphism OV Ñ OU . To show that this morphism of sheaves is injective, by Proposition 2.1.10 it is suﬃcient to show that for any x P X, the morphism of stalks OV,x Ñ OU,x is injective. Note that for x P V , we have

pre p pre| q p | q OV,x OV,x OV V x OX V x OX,x, and similarly OU,x OX,x. That is, for x P V , OV,x OU,x. If x R V , then for any open

neighborhood W of x, we have OV pW q 0, and hence OV,x 0. Thus, for all x P X, the

morphism of stalks OV,x Ñ OU,x is injective, and hence the morphism of sheaves OV Ñ OU

is injective as well. Hence, we have an exact sequence of sheaves 0 Ñ OV Ñ OU . Since I

is injective, we get a surjective group homomorpshim HompOU , Iq Ñ HompOV , Iq Ñ 0.

We will prove that HompOU , Iq IpUq by showing that each morphism f P HompOU , Iq

is determined uniquely by the image of 1 P OU pUq. Note by Proposition 2.1.13 and Def- pre Ñ Ñ inition 3.1.5 that each morphism OU I factors through a unique morphism OU I, Ñ pre Ñ and each morphism OU I composed with the sheaﬁﬁcation map OU OU gives a pre Ñ p q Ñ p pre q unique morphism OU I. Thus, we get homomorphisms Hom OU , I Hom OU , I p pre q Ñ p q p pre q and Hom OU , I Hom OU , I which are mutually inverses. Hence, Hom OU , I and p q P p pre q p q p q Hom OU , I are isomorphic. But then for any f Hom OU , I , we have fW a afW 1 for any a P OU pW q, since fW : OU pW q Ñ IpW q is OU pW q-module homomorphism. Thus, for W U, we have fW p1q fU p1q|W , as 1 P OU pUq restricts to 1 P OU pW q. For any prep q other open W X, the morphism fW is the zero morphism, since OU W 0. Hence, P prep q the morphism f is indeed determined uniquely by the image of 1 OU U , and thus

HompOU , Iq IpUq. Similarly, we have HompOV , Iq IpV q. Therefore, the surjection

HompOU , Iq Ñ HompOV , Iq Ñ 0 gives the desired surjection IpUq Ñ IpV q Ñ 0.

33 Proposition 3.2.10. If F is a ﬂasque sheaf on a topological space X, then HipX, Fq 0 for all i ¡ 0.

Proof. We embed F into an injective sheaf I on X and let G be the quotient. Then we have f g a short exact sequence 0 Ñ F ÝÑ I ÝÑ G Ñ 0. By Lemma 3.2.9, I is flasque, and since F is also flasque, by Lemma 3.2.8, it follows that G is flasque as well. By Proposition 3.2.7, the sequence IpXq Ñ GpXq Ñ 0 is exact. Since I is injective, we have HipX, Iq 0 for all i ¡ 0. Then the sequence

0 H0pX, Iq Ñ H0pX, Gq ÝÑδ H1pX, Fq Ñ H1pX, Iq becomes IpXq Ñ GpXq Ñ H1pX, Fq Ñ 0. Therefore, since ¢ ker δ0 im IpXq Ñ GpXq GpXq H0pX, Gq, it follows that H0pX, Gq Ñ H1pX, Fq is surjective, and the kernel is the whole of H0pX, Gq. Thus, H1pX, Fq 0. In addition, the exact sequence

HipX, Iq Ñ HipX, Gq Ñ Hi 1pX, Fq Ñ Hi 1pX, Iq

0 becomes 0 Ñ HipX, Gq ÝÑδ Hi 1pX, Fq Ñ 0, which implies that HipX, Gq Hi 1pX, Fq for i ¥ 1. Since G is ﬂasque as well, by induction obtain that HipX, Fq 0 for all i ¡ 0.

Remark 3.2.11. Let pX, OX q be a ringed space. Then the derived functors of the functor ΓpX, ¤q coincide with the cohomology functors HipX, ¤q. Indeed, to calculate the derived functors ΓpX, ¤q on ModpXq, we use a resolution of injective OX -modules, which we proved are ﬂasque sheaves and hence acyclic for the functor ΓpX, ¤q : AbpXq Ñ Ab. Proposition 3.2.10 implies that ﬂasque sheaves are acyclic for the functor ΓpX, ¤q : AbpXq Ñ Ab and by Proposition 3.1.18 will give us the usual cohomoglogy. That is, the following diagram commutes: i ModpXq ÝÝÝÝÑR Γ Mod forget forget

i AbpXq ÝÝÝÝÑR Γ Ab where forget is the forgetful functor, which means to consider a sheaf of OX -modules as a sheaf of groups on X and an A-module as a group, respectively; the upper RiΓ is calculating

34 i the derived functor using resolutions of ﬂasque OX -modules, and the lower R Γ is calculating the derived functor using resolutions of injective sheaves of rings on X.

Theorem 3.2.12. Let X Spec A be the spectrum of a Noetherian ring A. Then for all quasi-coherent sheaves F on X and for all i ¡ 0, we have HipX, Fq 0.

Proof. Given a quasi-coherent sheaf F, deﬁne M ΓpX, Fq. Then F M˜ by Proposition 2.5.2. Choose an injective resolution 0 Ñ M Ñ I of M in the category of A-modules. By Remark 2.5.3, the sequence 0 Ñ M˜ Ñ I˜ of sheaves on X is exact. By Proposition r III.3.4 of [8], Ik is ﬂasque for all k ¥ 0, since X Spec A is the spectrum of a Noetherian ring. Hence, by Proposition 3.2.10, we can use this resolution to calculate the cohomology groups of F. Then by Theorem 3.1.12, H0pX, Fq ΓpX, Fq M, and by Proposition 3.2.10, HipX, Fq 0 for i ¡ 0. (See Theorem II.3.5 of [8])

3.3 Cechˇ Cohomology

This discussion follows Section III.4 of [8] and Section 3 of [2]. In general, it is diﬃcult to compute cohomology groups using injective objects in a given category A. The following construction allows us to calculate the cohomology groups of a sheaf on a scheme X more easily, in the special case when the sheaf is quasi-coherent and X is a Noetherian separated scheme.

Let X be a topological space, and let U tUiuiPI be an open covering of X. Fix an arbitrary well-ordering ¤ of the index set I. Then for any ﬁnite subset of indices t u σ i0, . . . , ip I, we denote the intersection by Uσ jPσ Uj. For a given sheaf F of abelian groups on X, deﬁne the Cechˇ complex of abelian groups, denoted by C pU , Fq, as follows. For each integer p ¥ 0, let ¹ p C pU , Fq FpUσq. |σ|p 1

Hence, an element α P CppU , Fq is determined by all elements αpσq, where |σ| p 1, and

αpσq P FpUσq denotes the σ-th component of α. If σ is a ﬁnite subset of I of p 1 elements

j0 j1 ¤ ¤ ¤ jp, we deﬁne σk σztjku for 0 ¤ k ¤ p. Now we deﬁne the coboundary

35 maps dp : Cp Ñ Cp 1 by p¸ 1 § p k § pd αqpσq p¡1q αpσkq§ . U k0 σ We sometimes abbreviate notation and write d instead of dp.

Lemma 3.3.1. dp 1 ¥ dp 0.

Proof. Let σ P CppU , Fq. We want to show that d ¥ dα 0 P Cp 2pU , Fq. That is, we must check that pd ¥ dαqpσq 0 for an arbitrary subset σ of p 3 elements of I. Given such

σ I, write σ tj0, . . . , jp 2u with j0 j1 ¤ ¤ ¤ jp 2. Then

p¸ 2 § p¸ 2 p¸ 1 § § k § k t § § pd ¥ dαqpσq p¡1q pdαqpσkq § p¡1q p¡1q αpσk,tq§ § U U U k0 σ k0 t0 σk σ p¸ 2 p¸ 1 § k t § p¡1q αpσk,tq§ . U k0 t0 σ For any pair of indices j j , consider the subset τ σztj , j u, and note that αpτq| k t k t Uσk appears exactly twice in the last sum, as τ σk,t¡1 and τ σt,k, but with opposite signs. Hence, the two terms cancel each other, giving pd ¥ dαqpσq 0.

Lemma 3.3.1 says that C pU , Fq is indeed a complex for any sheaf F on X and open cover U . Thus, we can deﬁne Hˇ ipU , Fq hipC pU , Fqq to be the i-th Cechˇ cohomology group of F with respect to the covering U . However, although any morphism of sheaves

F Ñ G induces a morphism of complexes Hˇ pU , Fq Ñ Hˇ pU , Gq, there is no δ-functor, and thus Cechˇ cohomology does not take short exact sequences of sheaves to long exact sequences of cohomology groups. For example, if U consists of the single set X, then this statement follows from the fact that the global section functor ΓpX, ¤q is not always exact. To rectify this problem, we need a sheafified version of the Cechˇ complex. Thus, for any open set V X, let f : V Ñ X be the inclusion map. Then given X, U , and F, we can construct a complex C pU , Fq by defining ¹ p C p , Fq f¦pF| q U Ui0...ip i0 ¤¤¤ ip and dp : Cp Ñ Cp 1 as before. It is helpful to note that

f¦pF| qpV q FpV X U q. Ui0...ip i0...ip

36 Note that by construction we have ΓpX, CppU , Fqq CppU , Fq. Also, Lemma 3.3.1 is

still true for the complex of groups C pU , FqpV q on any open V X, and hence d ¥ d 0

is true for the complex of sheaves C pU , Fq.

Example 3.3.2. We will calculate Cechˇ cohomology for the sheaf F of Example 2.1.3

with respect to the open cover U ttx, yu, txu, tyuu. Denote these sets U1,U2 and U3, respectively. Then

0 C pU , Fq FpU1q ¢ FpU2q ¢ FpU3q pZ2 ¢ Z2q ¢ Z2 ¢ Z2.

Similarly,

1 C pU , Fq FpU12q ¢ FpU13q ¢ FpU23q FpU2q ¢ FpU3q ¢ FpHq Z2 ¢ Z2.

i 0 0 1 For i ¥ 2 we have C pU , Fq FpU123q FpHq 0. The map d : C pU , Fq Ñ C pU , Fq is given by 0 1 dppa, bq, c, dq12 p¡1q c|txu p¡1q pa, bq|txu c ¡ a,

0 1 dppa, bq, c, dq13 p¡1q d|tyu p¡1q pa, bq|tyu d ¡ b,

0 1 dppa, bq, c, dq23 p¡1q d|txuXtyu p¡1q c|txuXtyu 0.

Thus, ˇ 0 0 H pU , Fq ker d tppa, bq, dq | a c, b du Z2 ¢ Z2 ΓpX, Fq

ˇ 1 1 0 and H pU , Fq ker d { im d pZ2 ¢ Z2q{pZ2 ¢ Z2q 0.

Now consider the sheaﬁﬁed Cechˇ complex. For i 0, we have

0 C pU , FqpXq FpU1 X Xq ¢ FpU2 X Xq ¢ FpU3 X Xq pZ2 ¢ Z2q ¢ Z2 ¢ Z2,

0 C pU , Fqptxuq FpU1Xtxuq¢FpU2Xtxuq¢FpU3Xtxuq Fptxuq¢Fptxuq¢FpHq Z2¢Z2,

0 and similarly C pU , Fqptyuq Z2 ¢ Z2. For i 1 we have

1 C pU , FqpXq FpU12 XXq¢FpU13 XXq¢FpU23 XXq Fptxuq¢Fptyuq¢FpHq Z2 ¢Z2,

1 C pU , Fqptxuq FpU12 X txuq ¢ FpU13 X txuq ¢ FpU23 X txuq Fptxuq ¢ FpHq ¢ FpHq Z2,

1 and similarly, C pU , Fqptyuq Z2.

37 3.4 Comparing Cechˇ and Derived Functor Cohomology

Through this section, let X, U , and F be as in Section 3.3. This discussion follows Section 4 of [2].

Lemma 3.4.1. Hˇ 0pU , Fq ΓpX, Fq.

d0 Proof. Recall that Hˇ 0pU , Fq ker d0 from the exact sequence 0 Ñ C0pU , Fq ÝÑ C1pU , Fq. 0 1 0 For any α pαjqjPJ P C pU , Fq, the σ tj, j u component of d α is

0 pd αq 1 α | ¡ α 1 | , j,j j UJ Uj1 j Uj Uj1 0 so that d α 0 if and only if αj P FpUjq and αj1 P FpUj1 q coincide on Uj Uj1 for all pairs 1 of indices j, j P I. Since F is a sheaf, the αj lift in that case to a global section α P ΓpX, Fq. Thus, the map Hˇ 0pU , Fq Ñ ΓpX, Fq is a well-deﬁned injection. It is surjective by applying Ñ | the restriction maps α α Ui .

Note that we already showed explicitly in Example 3.3.2 that for the sheaf F of that example, we have Hˇ 0pU , Fq ΓpX, Fq.

Proposition 3.4.2. 0 Ñ F ÝÑ C pU , Fq is a resolution of F, where the morphism of sheaves ± Ñ 0p q P p q p q | : F C U , F is given on any t F V by t jPJ t V XUj for any open V X.

Proof. For every i ¥ 0, the sheaf im di is subsheaf of the sheaf ker di 1, since di 1 ¥ di 0. We need to check only that ker di 1 is a subsheaf of im di, that ker d0 im , and that is injective. The latter two facts are immediate from the uniqueness property of the sheaf F. We will show the exactness of the sequence for i ¡ 0 at the level of stalks. Consider

i i 1 i dx i 1 dx i 2 C pU , Fqx ÝÑ C pU , Fqx ÝÝÝÑ C pU , Fqx.

i 1 i By Proposition 2.1.10, we need to prove that ker dx im dx, as we already know the opposite inclusion from the analogous statement at the level of sheaves. P i P i 1p qp q Any germ αx ker dx can be represented by an element α C U , F V , where the open set V can be chosen small enough to lie entirely inside one of the open sets Uj of the cover U . If σ I is a ﬁnite subset such that |σ| i 1, we have V XUσ V XUσYtju, since

38 i pσ V Uj. Thus, we can deﬁne β P C pU , FqpV q by βpσq p¡1q αpσ X tjuq, where we set

Given a sheaf F, consider the resolution 0 Ñ F Ñ C pU , Fq. By deﬁnition we have

Hˇ ipU , Fq hipC pU , Fqq. But then Lemma 3.4.1 tells us that Hˇ ipU , Fq is the cohomology of F for the right derived functor ΓpX, ¤q applied to the resolution 0 Ñ F Ñ C pU , Fq. Moreover, it follows from Lemma 3.1.11 that if 0 Ñ F Ñ I is any injective resolution of F, then the identity map on F lifts to a morphism of complexes C pU , Fq Ñ I , which in turn induces a morphism H˜ pU , Fq Ñ H pX, Fq from the Cechˇ cohomology to sheaf cohomology, which enables us to compare the two cohomologies. The following results will show that under some additional conditions this morphism becomes an isomorphism, thus enabling us to calculate sheaf cohomology via Cechˇ cohomology.

Lemma 3.4.3. If a sheaf F on X is ﬂasque, then Hˇ ipU , Fq HipX, Fq 0 for i ¡ 0.

Proof. By Lemma 3.2.10, we have HipX, Fq 0; we need only to show the same statement for Hˇ ipU , Fq. Since F is flasque, CipU , Fq is a product of flasque sheaves, because the maps i FpV X Uσq Ñ FpW X Uσq are surjective for any open subsets W V X. Hence C pU , Fq itself is flasque. Then 0 Ñ F Ñ C pU , Fq is a flasque resolution of F, and therefore by

Proposition 3.2.10, we have Hˇ ipU , Fq HipX, C q 0 for all i ¡ 0.

Note that we already showed in Example 3.3.2 that for the ﬂasque sheaf F of that

example, we have Hˇ 1pU , Fq 0.

Theorem 3.4.4 (Leray). Let X be a topological space, let F be a sheaf of abelian groups on

X, and let U be an open cover of X. Assume that for any ﬁnite intersection of open sets ¤ ¤ ¤ P ¡ ip | q V Ui0 Uip with Uij U , and for any positive index i 0, we have H V, F V 0. Then the natural maps Hˇ ipU , Fq Ñ HipX, Fq are isomorphisms for all integers i ¥ 0.

Proof. We proceed by induction on the degree i. For i 0, the result is Lemma 3.4.1.

For i ¡ 0, embed F into an injective sheaf G and let H G{F, so that we have an exact

39 sequence 0 Ñ F Ñ G Ñ H Ñ 0. The base case i 1 follows by applying Lemma 3.4.1 and the Five Lemma (see [23]) to the diagram

0 ÝÝÝÝÑ Hˇ 0pX, Fq ÝÝÝÝÑ Hˇ 0pX, Gq ÝÝÝÝÑ Hˇ 0pX, q ÝÝÝÝÑ Hˇ 1pX, Fq ÝÝÝÝÑ 0 H

0 ÝÝÝÝÑ H0pX, Fq ÝÝÝÝÑ H0pX, Gq ÝÝÝÝÑ H0pX, H q ÝÝÝÝÑ H1pX, Fq ÝÝÝÝÑ 0 Now consider i ¡ 0. For each ﬁnite subset σ I, the sequence

0 Ñ FpUσq Ñ GpUσq Ñ H pUσq Ñ 0 (3.2)

is exact, using the long exact sequence for H pUσ, ¤q, the fact that, by Theorem 3.1.12 Ñ p q 0p | q p | q p q applied to the functor F F Uσ , we have H Uσ, F Uσ Γ Uσ, F Uσ F Uσ , and the 1p | q hypothesis that H Uσ, F Uσ 0. Taking products, the corresponding short sequence of Cechˇ complexes

0 Ñ C pU , Fq Ñ C pU , Gq Ñ C pU , H q Ñ 0 is exact, so that we get a long exact sequence in Cechˇ cohomology. The fragment

¤ ¤ ¤ Ñ Hˇ ipU , Gq Ñ Hˇ ipU , H q Ñ Hˇ i 1pU , Fq Ñ Hˇ i 1pU , Gq Ñ ¤ ¤ ¤ , together with the fact that Hˇ ipU , Gq 0 for all i ¡ 0, since G is ﬂasque, implies that

Hˇ ipU , H q Hˇ i 1pU , Fq for each i ¡ 0. Consider the long exact sequence of the derived functor cohomology corresponding to (3.2):

¤ ¤ ¤ Ñ ip | q Ñ ip | q Ñ i 1p | q Ñ ¤ ¤ ¤ H Uσ, G Uσ H Uσ, H Uσ H Uσ, F Uσ .

| kp | q Applying Proposition 3.2.10 to the ﬂasque sheaf G Uσ and the assumption H Uσ, F Uσ 0 ¡ ip | q ¡ for k 0, we have H Uσ, H Uσ 0 for each ﬁnite intersection Uσ and each i 0. Thus, the inductive hypothesis applies to H as well. Then by the following commutative diagram

Hˇ ip , q ÝÝÝÝÑ Hˇ i 1p , Fq U H U

HipX, H q ÝÝÝÝÑ Hi 1pX, Fq and the inductive hypothesis for H , we get that the natural maps for F are isomorphisms for1 ¤ i ¤ n.

40 Corollary 3.4.5. Let X be a Noetherian separated scheme, let U be an open aﬃne cover of X, and let F be a quasi-coherent sheaf on X. Then for all i ¡ 0, the natural maps

Hˇ ipU , Fq Ñ HipX, Fq are isomorphisms. ¤ ¤ ¤ Proof. Since X is a separated scheme, all ﬁnite intersection V Ui0 Uip of open sets in the covering U are aﬃne by Exercise II.4.3 of [8]. Moreover since X is Noetherian, i they are the spectra of Noetherian rings. Then because F is quasi-coherent, H pV, F|V q 0 for i ¡ 0 by Theorem 3.2.12. The conclusion now follows from Theorem 3.4.4.

3.5 The Cohomology of Projective Space

In this section we explicitly compute of the cohomology of the sheaves Opnq on projective space, by using Cechˇ cohomology for a suitable open aﬃne covering. These explicit calcu- lations form the basis for various general results about the cohomology of coherent sheaves on projective space, such as Serre duality.

¥ r ¥ Theorem 3.5.1. Let A be a Noetherian ring, let r 1, and let X PA with r 1. Then: À 0 1. the natural map S Ñ Γ¦pO q H pX, O pnqq is an isomorphism of graded X nPZ X S-modules.

i 2. H pX, OX pnqq 0 for 0 i r and all n P Z.

r 3. H pX, OX p¡r ¡ 1qq A.

0 r r 4. The natural map H pX, OX pnqq¢H pX, OX p¡n¡r ¡1qq Ñ H pX, OX p¡r ¡1qq A

is a perfect pairing of ﬁnitely generated free A-modules, for each n P Z.

Proof. We will give a proof only of statements (1), (3), and (4). For the rest of this rather technical proof, see Theorem III.5.1. of [8]. À À The sheaf F O pnq Spnq is quasi-coherent. Since cohomology on a nPZ X nPZ Noetherian topological space commutes with arbitrary direct sums by Remark III.2.9.1 of

[8], the cohomology on F is the direct sum of the cohomologies of the sheaves OX pnq. p q For each i 0, . . . , r, let Ui D xi . Then each Ui is an open aﬃne subset Spxiq, r and together they cover X. Since PA is Noetherian separated scheme by Hilbert’s Theorem

41 and Exercise 8, p. 262 of [19], we can use Cechˇ cohomology for this cover. As before, for t u p q any set of indices i0, . . . , ip I, we write D xi0 , . . . , xip Ui0...ip . By the deﬁnition of quasi-coherent sheaf, for the quasi-coherent sheaf F we have FpU q S , and the i0...ip xi0 ...xip grading on F corresponds to the grading of S . Hence, the Cechˇ complex of F is given xi0 ...xip by ¹ ¹ C p , Fq : S Ñ S Ñ ... Ñ S . U xi0 xi0 xi1 xi0 ...xip By Theorem 3.4.4, then, H0pX, Fq is the kernel of the ﬁrst map, which is just S S xik by Proposition 2.6.7. This proves statement (1).

As for statement (3), HrpX, Fq is the cokernel of the last map in the Cechˇ complex, ± r¡1 Ñ namely d : k Sx0...xˆk...xr Sx0...xr , wherex ˆk means omit the index xk. We can

t t0 tr | P u consider the local ring Sx0...xr as a free A-module with basis x0 . . . xr t Z . Then as the image of a localizing homomorphism, the image of dr¡1 is the free submodule generated

r by those basis elements for which at least one ti is nonnegative. Thus, H pX, Fq is a free

t0 ¤ ¤ ¤ tr A-module with basis consisting of negative monomials, i.e. monomials of the form x0 xr ° with all ti 0. Since the degree of each element is ti, the only such element of degree ¡ ¡ ¡1 ¡1 rp p¡ ¡ qq r 1 is x0 . . . xr . Thus, H X, OX r 1 is a free A-module of rank 1. That is, r H pX, OX p¡r ¡ 1qq A.

0 r r To prove that the map H pX, OX pnqq ¢ H pX, OX p¡n ¡ r ¡ 1qq Ñ H pX, OX p¡r ¡ 1qq 0 is a perfect pairing, it suﬃces to show that for any nonzero element v P H pX, OX pnqq, r there is a nonzero element w P H pX, OX p¡n ¡ r ¡ 1qq such that pv, wq does not map to r 0 P H pX, OX p¡r ¡1qq. First, consider the case n 0. Then by the proof of statement (1), 0 we have H pX, OX pnqq Sn. However, by the deﬁned grading on S Arx0, . . . , xrs, all 0 elements have non-negative degree, and hence H pX, OX pnqq Sn 0 for n 0. Similarly, r by the proof of statement (3), H pX, OX p¡n ¡ r ¡ 1qq has basis ¸ t t0 tr | ¡ ¡ ¡ u x0 , . . . , xr ti 0, ti n r 1 .

r There are no such elements for n 0, and hence H pX, OX p¡n ¡ r ¡ 1qq 0. Therefore, the map of statement (4) is trivially a perfect pairing. ° ¥ 0p p qq t m0 mr | ¥ u For n 0, H X, OX n has basis x0 , . . . , xr mi 0, mi n . The pairing

42 r r with H pX, OX p¡r ¡ 1qq into H pX, OX p¡r ¡ 1qq is given by

p m0 mr q ¤ p l0 lr q p m0 t0 mr tr q x0 , . . . , xr x0 , . . . , xr x0 , . . . , xr , ° where ti ¡n¡r¡1. The image is 0 if any mi ti ¥ 0, because by the proof of statement ° p q rp p qq t t0 tr | u (3), for any sheaf OX q , the module H X, OX q has basis x0 , . . . , xr ti 0, ti q .

m0 mr P 0p p qq Thus, the dual basis element for x0 , . . . , xr H X, OX n is

¡ ¡ ¡ ¡ m0 1 mr 1 P rp p¡ ¡ ¡ qq x0 , . . . , xr H X, OX n r 1 .

We can summarize these results in the following statement.

Proposition 3.5.2. Consider the coherent sheaf OX pmq on an n-dimensional projective n pp p qq space X Pk over a ﬁeld k. Then H X, OX m does not equal 0 if and only if either p 0 and m ¥ 0, or p n and m ¤ ¡n ¡ 1. Moreover, for m ¥ 0, ¢ m n dim H0pX, O pmqq , k X n ¢ m n dim HnpX, O p¡m ¡ n ¡ 1qq . k X n

Proof. Consider the Cechˇ complex for OX pmq from the proof of Theorem 3.5.1 ¹ ¹ C p , Fq : S Ñ S Ñ ¤ ¤ ¤ Ñ S U xi0 xi0 xi1 xi0 ...xin

p for U tUiu, where we set Ui D pxiq. Note that C pU , OX pmqq 0 for p 0 p p or p ¡ n. Then by Corollary 3.4.5, we have H pX, OX pmqq Hˇ pU , OX pmqq 0 for p p 0 or p ¡ n. By statement (2) of Theorem 3.5.1, we have H pX, OX pmqq 0 for ° 0p p qq t m0 mn | ¥ u 0 p m. Similarly, H X, OX m has basis x0 , . . . , xn mi 0, mi m , ¨ 0p p qq m n ¥ 0p p qq and hence dimk H X, OX m n for m 0, and H X, OX m 0 for m 0. ° np p¡ ¡ ¡ qq t m0 mn | ¡ ¡ ¡ u Analogously, H X, OX m n 1 has basis x0 , . . . , xn mi 0, mi m n 1 , ¨ np p¡ ¡ ¡ qq m n ¥ np p¡ ¡ ¡ qq and hence dimk H X, OX m n 1 n for m 0, and H X, OX m n 1 0 for m 0.

1 1 r s Example 3.5.3. Consider X Pk over a ﬁeld k. That is, Pk Proj k x0, x1 . We will now calculate the cohomology of the structure sheaf using Cechˇ cohomology for the cover

U consisting of U0 D px0q and U1 D px1q. As we showed in Example 2.3.4, p q r s x1 OX U0 k x0, x1 px0q k , x0

43 x0 O pU q krx , x sp q k , and X 1 0 1 x1 x 1 x0 x1 OX pU01q k , . x1 x1 p q r s p q r s p q r 1 s Denote OX U0 k x , OX U1 k y , and OX U01 k t, t . Thus,

0 C pU , OX q OX pU0q ¢ OX pU1q krxs ¢ krys, 1 1 C pU , OX q OX pU01q k t, , t p 0 and C pU , OX q 0 for all p ¡ 1, by Lemma 4.1.4. For any α P C pU , OX q, α is given by its two components α0 f0pxq P krxs and α1 f0pyq P krys. Then ¡ © 1 d0pαq α ¡ α f ¡ f ptq. 1 0 1 t 0 0 p 1 q p q P For α to be in ker d , we need f1 t f0 t , that is, f1 f0 a k. Hence,

0 0 0 H pX, OX q Hˇ pU , OX q ker d k.

P r 1 s p q p 1 q P r s Any element β k t, t can be represented as β f0 t f1 t for f0, f1 k s . Hence, for 0 0 1 0 p¡f0, f1q P C pU , OX q, we have d pp¡f0, f1qq f1 f0 β, and thus C pU , OX q im d . 2 Because C pU , OX q 0, we obtain

1 1 0 1 1 1 H pX, OX q Hˇ pU , OX q im d { ker d C pU , OX q{C pU , OX q 0.

Note that these results agree with the results of Theorem 3.5.1 and Proposition 3.5.2.

3.6 Ext Groups and Sheaves

This discussion follows Section III.6 of [8].

Deﬁnition 3.6.1. Let X be a scheme, and let F and G be sheaves of OX -modules. We denote by HompF, Gq the group of sheaves of OX -modules homomorphisms, and by HompF, Gq the sheaf Hom. (See p. 12)

For a ﬁxed F, HompF, ¤q is a left exact covariant functor from the category of sheaves of modules on X to the category of abelian groups, and HompF, ¤q is left covariant functor from the category of sheaves of modules on X to the same category. By Proposition 3.2.3, the following deﬁnition is valid.

44 Deﬁnition 3.6.2. Let pX, OX q be a ringed space, and let F be a sheaf of OX -modules. The functors ExtipF, ¤q are the right derived functors of HompF, ¤q, and the functors ExtipF, ¤q are the right derived functors of HompF, ¤q.

Then using the general properties of derived functors, we have Ext0 Hom, Ext pF, ¤q forms a long exact sequence for a given short exact sequence, and ExtipF, Iq 0 for any injective sheaf I and positive interger i ¡ 0. The analogous statements also hold for Ext.

Proposition 3.6.3. Let X be a scheme. For any sheaf G of OX -modules, we have:

0 1. Ext pOX , Gq G;

i 2. Ext pOX , Gq 0 for i ¡ 0; and

i i 3. Ext pOX , Gq H pX, Gq for all i ¥ 0.

Proof. Note that HompOX , Gq G, because G is OX -module. This implies statements (1) and (2). Similarly, HompOX , Gq ΓpX, Gq, and hence we have statement (3).

Proposition 3.6.4. Let L be a locally free sheaf of ﬁnite rank, and let Lˇ HompL, OX q be its dual. Then for any sheave F and G of OX -modules, we have

ExtipF b L, Gq ExtipF, Lˇ b Gq, and ExtipF b L, Gq ExtipF, Lˇ b Gq ExtipF, Gq b Lˇ.

Proof. First, note that each Ext and Ext is δ-functors in G, since for a locally free sheaf L of ﬁnite rank, the functor R Ñ R b Lˇ is exact. The case i 0 follows from Proposition 2.2.6. Applying Lemma III.6.6 of [8] to the injective sheaf G implies that Lˇ b G is injective. Thus, for all the Exti groups and Exti sheaves in the two expressions, the second component is one of the two injective sheaves G and Lˇ b G, and hence all functors Exti and Exti vanish for i ¡ 0. That is, each of the functors Ext and Ext is eﬀaceable, and by Theorem 3.1.15 they are equal.

45 Chapter 4

Serre Duality

4.1 Statement of Serre Duality

This discussion follows Section III.7 of [8].

n n Theorem 4.1.1 (Duality for Pk ). Let k be a ﬁeld, and let X Pk . Then:

n 1. H pX, OX p¡n ¡ 1qq k.

2. For a ﬁxed such isomorphism and any coherent sheaf F on X, the natural pairing

n n HompF, OX p¡n ¡ 1qq ¢ H pX, Fq Ñ H pX, OX p¡n ¡ 1qq k

is a perfect pairing of ﬁnite-dimensional vector spaces over k.

3. For every i ¥ 0, there is an isomorphism

i n¡i q Ext pF, OX p¡n ¡ 1qq H pX, Fq q where denotes the dual vector space. In the special case i 0, the isomorphism is the one induced by the pairing of statement (2).

Proof. Statement (1) follows from statement (3) of Theorem 3.5.1.

A morphism of sheaves F Ñ OX p¡n ¡ 1q induces a morphism of cohomology groups n n H pX, Fq Ñ H pX, OX p¡n ¡ 1qq. This gives the pairing. If F OX pqq for some q, then

0 HompF, OX p¡n ¡ 1qq ΓpX, OX p¡q ¡ n ¡ 1qq H pX, OX p¡q ¡ n ¡ 1qq, and therefore the result follows from statement (4) of Theorem 3.5.1. Hence, the result is also true for a ﬁnite direct sum of sheaves of the form OX pqiq.

46 If F is an arbitrary coherent sheaf, we can write it as the cokernel E1 Ñ E0 Ñ F Ñ 0 of a

map of sheaves Ej, each Ej being a direct sum of sheaves OX pqiq. Since Homp¤, OX p¡n¡1qq q and HnpX, ¤q are both left exact contravariant functors, by the Five Lemma (see [23]) we

n q get an isomorphism HompF, OX p¡n ¡ 1qq H pX, Fq. Both sides of the expression for statement (3) are contravariant δ-functors for coherent sheaves F. For i 0 we have an isomorphism by statement (2). Thus, to show that they are isomorphic, by Theorem 3.1.15, it will be sufficient to show that both sides are coeffaceable for i ¡ 0. Given that F is coherent, by Proposition 2.6.9 we can write F as a quotient of a À N p¡ q ¡ sheaf E j1 OX q , for some large enough q 0. Then à i i Ext pE , OX p¡n ¡ 1qq H pX, OX p¡n ¡ 1 ¡ qqq 0 À n¡i q n¡i q for i ¡ 0, by Proposition 3.6.3. On the other hand, H pX, E q H pX, OX p¡qqq, which is 0 for i ¡ 0 by Theorem 3.5.1. Then both sides are coeffaceable for i ¡ 0. Therefore, the δ-functors are universal, and hence they are isomorphic.

Remark 4.1.2. The usual statement of the result of Theorem 4.1.1 is phrased not for n OX p¡n ¡ 1q but for the canonical sheaf ωX ΩX{K . Nevertheless, Example II.8.20.1 n of [8] proves that for X Pk , the two sheaves are isomorphic. For the usual statement, see Theorem B.0.13 of Appendix B.

n Corollary 4.1.3. Let X be the n-dimensional projective space Pk over a ﬁeld k. Then for any locally free sheaf of ﬁnite rank F on X, there are isomorphisms

i n¡i q H pX, Fq H pX, Fˇ b OX p¡n ¡ 1qq for all i ¥ 0.

Proof. n¡i q n¡i q H pX, Fˇ b OX p¡n ¡ 1qq Ext pOX , Fˇ b OX p¡n ¡ 1qq

n¡i q n¡i q Ext pOX b F, OX p¡n ¡ 1qq Ext pF, OX p¡n ¡ 1qq qq pHipX, Fqq HipX, Fq, using Proposition 3.6.3 for the ﬁrst isomorphism, Proposition 3.6.4 for the second, Theorem 4.1.1 for fourth, and Statement (1) of Theorem III.5.2 of [8] for the ﬁfth.

47 n Proposition 4.1.4. Let X Pk over a ﬁeld k. Then for any locally ﬁnitely free sheaves F and G on X, there is an isomorphism of k-modules ExtipG, Fq Ñ Extn¡ipF, Gp¡n ¡ 1qq.

Proof. Applying Proposition 3.6.4, Proposition 3.6.3, Corollary 4.1.3, Propostion 2.2.6, and then Propositions 3.6.3 and 3.6.4 again, we have

i i i n¡i q q Ext pG, Fq Ext pOX , Gˇ b Fq H pX, Gˇ b Fq H pX, pGˇ b Fq b OX p¡n ¡ 1qq

n¡i q n¡i q H pX, G b Fˇ b OX p¡n ¡ 1qq Ext pOX , G b Fˇ b OX p¡n ¡ 1qq

n¡i n¡i Ext pF, G b OX p¡n ¡ 1qq Ext pF, Gp¡n ¡ 1qq. (See Corollary 9 of [17]).

n Example 4.1.5. Consider the 1-dimensional projective space X Pk over a ﬁeld k and the coherent sheaf OX pmq for m P Z. Applying Corollary 4.1.3 and Proposition 2.6.5, we have i n¡i q q H pX, OX pmqq H pX, OX pmq b OX p¡n ¡ 1qq

n¡i q n¡i q H pX, OX p¡mq b OX p¡n ¡ 1qq H pX, OX p¡m ¡ n ¡ 1qq. Note that this result agrees with Proposition 3.5.2, which implies that both sides are 0 for

0 i n and m P Z or for i 0, n if m 0, and otherwise both sides have dimension ¨ m n ¥ n over k for i 0, n and m 0. 1 0p q In particular, in Example 3.5.3 we showed that for X Pk, we have H X, OX k 1 and H pX, OX q 0. Thus, applying Corollary 4.1.3, we have

0 1 q H pX, OX p¡2qq H pX, OX q 0

1 0 q q and H pX, OX p¡2qq H pX, OX q k k.

4.2 An Application of Serre Duality

Serre Duality can be generalized to projective schemes and in this form is the most important ingredient in the proof of the Riemann-Roch theorem about the degree of divisors in algebraic geometry. We will explain the idea of the proof after providing first some (rather brief) prelimiary definitions. For more information on divisors see Section II.6 of [8]. Let k be an algebraically closed field. A curve over k is an integral separated scheme X of finite type over k, of dimension 1. (See Definition B.0.20.) The genus of a curve X is

48 1 g dimk H pX, OX q, where OX is the structure sheaf for the scheme X.A divisor D on X is an element of the free abelian group generated by the irreducible closed subschemes ° of codimension 1 in X. We write a divisor D as D nP P , where the P ’s are irreducible

subschemes of codimension 1, called points, each nP P Z, and only a ﬁnite number of nP ’s

are different from 0. The divisor D is positive if each nP ¥ 0. We denote the degree of ° the divisor D by D nP . Two divisors can be added or substracted by adding the corresponding coefficients. For a sheaf F on X, we define the Euler characteristic to be

0 1 χpFq dimk H pX, Fq ¡ dimk H pX, Fq.

Every divisor D has an associated coherent sheaf L pDq. A divisor K that corresponds

to the sheaf ΩX{k of relative diﬀerentials of X over K is called a canonical divisor. (See

Deﬁnition B.0.9 of Appendix B for more information on the sheaf ΩX{k.) It is also true that ˇ the sheaf associated to the divisor D1 ¡D2 for given divisors D1 and D2 is L pD1qbL pD2q, 0 and hence, L p0q OX . We denote dimk H pX, L pDqq by `pDq. (See p. 294–295 of [8]).

Theorem 4.2.1 (Riemann-Roch). Let D be a divisor on a curve X of genus g. Then

`pDq ¡ `pK ¡ Dq deg D 1 ¡ g.

q Proof. Serre Duality tells us that H1pX, L pDqq H0pX, LˇpDq b L pKqq. Thus,

`pDq ¡ `pK ¡ Dq dim H0pX, L pDqq ¡ dim H0pX, LˇpDq b L pKqq

dim H0pX, L pDqq ¡ dim H1pX, L pDqq χpL pDqq.

Then the statement of the theorem becomes χpL pDqq deg D 1 ¡ g. For the zero divisor we have

0 1 χpL pDqq dim H pX, OX q ¡ dim H pX, OX q 1 ¡ g. ° Now consider a positive divisor D nP P and a point P such that the corresponding

coeﬃcient satisﬁes nP ¥ 1. For the divisor E D ¡ P , we have χpL pDqq χpL pEqq 1. Thus, by induction on deg D, we have

χpL pDqq χpL p0qq deg D deg D 1 ¡ g.

49 To generalize to an arbitrary divisor D, write D D ¡ D¡, where D ,D¡ ¥ 0. We will prove the theorem by induction on deg D . If deg D 0, then D D ¥ 0, and the theorem holds. We now proceed by induction on deg D¡. If there is a point P such that the coeﬃcient for P in D¡ is positive, set E D P . We have E D and E¡ D¡ ¡ P , and therefore deg E¡ deg D¡ ¡ 1. Then we can apply the inductive hypothesis for E to deduce that χpL pEqq deg E 1 ¡ g. But we also have χpL pEqq χpL pDqq 1 and deg E deg D 1, and hence the statement holds for D as well. (See Theorem 8.2 of [22].)

50 Appendix A

Categories

This discussion follows Chapter 1 of [18].

Definition A.0.2. A category C is an ordered tuple C pO, M , d, c, ¥q consisting of a collection O of objects, a collection M of morphisms, and two functions d, c : M Ñ O called the domain and codomain functions respectively. If f is a morphism with dpfq A and cpfq B, then we write f : A Ñ B. If we define D tpf, gq | f, g P M , dpfq cpgqu, then ¥ is a function ¥ : D Ñ M , called the composition law. We requiere the following conditions to be satisfied:

1. Matching Condition: If f ¥ g is deﬁned, then dpf ¥ gq dpgq and cpf ¥ gq cpfq.

2. Associativity condition: If f ¥ g and h ¥ f are deﬁned, then h ¥ pf ¥ gq ph ¥ fq ¥ g.

3. Identity Existence Condition: For each object A there exists a morphism 1A such that

dp1Aq A cp1Aq and

(a) f ¥ 1A f whenever f ¥ 1A is deﬁned

(b) 1A ¥ g g whenever 1A ¥ g is deﬁend.

1 1 1 1 1 Deﬁnition A.0.3. Let A pO, M , d, c, ¥q and B pO , M , d , c , ¥ q be two categories. A covariant functor F : A Ñ B is an ordered tuple F pφ, ψq consisting of mappings 1 1 φ : O Ñ O and ψ : M Ñ M satisfying the following conditions:

1. If g : A Ñ A1, then ψpgq : φpAq Ñ φpA1q.

2. If f ¥ g is deﬁned in A, then ψpf ¥ gq ψpfq ¥1 ψpgq.

i 3. For each A P O, we have ψp1Aq 1φpAq.

A natural transformation ξ : F Ñ G between covariant functors F,G : A Ñ B is a mapping 1 ξ : O Ñ M which assigns to every object A P O a morphism hA : F pAq Ñ GpAq such that 1 for a morphism f : A Ñ A of M we have GpfqhA hA1 F pfq.

1 1 1 1 1 Deﬁnition A.0.4. Let A pO, M , d, c, ¥q and B pO , M , d , c , ¥ q be two categories. A contravariant functor F : A Ñ B is an ordered tuple F pφ, ψq consisting of function 1 1 φ : O Ñ O and ψ : M Ñ M satisfying the following conditions:

1. If g : A Ñ A1, then ψpgq : φpA1q Ñ φpAq.

2. If f ¥ g is deﬁned in A, then ψpf ¥ gq ψpgq ¥1 ψpfq.

3. For each A P O, we have ψp1Aq 1φpAq.

A natural transformation ξ : F Ñ G between contravariant functors F,G : A Ñ B is a 1 mapping ξ : O Ñ M which assigns to every object A P O a morphism hA : F pAq Ñ GpAq Ñ 1 p q 1 p q such that for a morphism f : A A of M we have G f hA hAF f .

Deﬁnition A.0.5. An abelian category is a category C pO, M , d, c, ¥q such that the following properties hold. a) For each A, B P O, the collection of morphisms

HompA, Bq tf P M | dpfq A, cpfq Bu has a structure of an abelian group, and the composition law is linear with respect to the group law; b) ﬁnite direct sums exist; c) every morphism has a kernel and a cokernel; d) every monomorphism is the kernel of its cokernel and every epimorphism is the cokernel of its kernel; and ﬁnally, e) every morphism can be factored into an epimorphism followed by a monomorphism.

See p. 202 of [8].

ii Appendix B

Serre Duality for Diﬀerentials

This section follows mostly Sections II.8 and III.7 of [8] and Sections 10.1 and 10.2 of [14].

Deﬁnition B.0.6. Let A be a ring, let B be an A-algebra, and let M be a B-module. An A-derivation of B into M is a map d : B Ñ M such that for all a P A and b, b1 P B,

1. dpb b1q dpbq dpb1q,

2. dpbb1q bdpb1q b1dpbq, and

3. dpaq 0.

The module of relative diﬀerential forms of B over A is a B-module ΩB{A, together with an

A-derivation d : B Ñ ΩB{A, with the universal property that for any B-module M and any 1 A-derivation d : B Ñ M, there exists a unique B-module homomorphism f :ΩB{A Ñ M such that d1 f ¥ d, as in the following diagram.

d B > ΩB{A f d1 ∨ < M

One way to construct such a module ΩB{A is to take the free B-module F generated by the symbols tdb | b P Bu and to divide out by the relations (1),(2),(3) of Deﬁnition B.0.6.

Then the derivation d : B Ñ ΩB{A is deﬁned by b Ñ db for b P B. Thus, ΩB{A is generated as B-module by tdb | b P Bu. Moreover, it follows from the universal property that the pair

pΩB{A, dq is unique up to a unique isomorphism.

iii Example B.0.7. If B Arx1, . . . , xns is a polynomial ring over A, then ΩB{A is the free

B-module of rank n generated by dx1, . . . , dxn.

Proposition B.0.8. Let A1 and B be A-algebras, and let B1 B b A1. Then

1 ΩB1{A1 ΩB{A bB B .

p q Furthermore, if S is a multiplicative system in B, then ΩBS {A ΩB{A S.

Proof. See p. 182 of [14].

The sheaf-theoretic version of modules of differentials is a sheaf of relative differentials. But first we need some preliminary definitions. Consider a continuos map between topological spaces f : X Ñ Y . For any sheaf G on Y , the inverse image sheaf f ¡1G on X is the sheafification of the presheaf given by U Ñ lim GpV q, where the direct limit is taken over all open sets U X with ÝÑfpUqV fpUq V Y . The restriction maps are the maps between the direct limits via the universal property of direct limits (Definition 1.0.6), i.e.

¡ f 1G p q Ñ p 1q ρUV φ : limÝÑ G W ÝÑlim G W fpUqW fpV qW 1

This map exists because given V U, each element of the directed system tGpW qufpUqW is 1 1 also an element of tGpW qu p q 1 and hence has a homomorphism into lim GpW q, f V W ÝÑfpV qW 1 which factors uniquely through lim GpW q. Similarly when f : X Ñ Y is a morphism ÝÑfpUqW ¡1 ¡1 of schemes, for a given sheaf G of OY -modules, f G is an f OY -module, and we have ¡1 morphism f OY Ñ OX of sheaves of rings on X.

Definition B.0.9. Let F Ñ G be a morphism of sheaves of rings on X. Consider G as a pre Ñ F-module. Define ΩG{F to be the presheaf U ΩGpUq{FpUq. The restriction maps for open sets V U are given by GpUq Ñ GpV q Ñ ΩprepV q, which is an FpUq-derivation, and so p qpre pre factors through Ω U , giving the desired restriction map ρUV on ΩG{F. Finally, the sheaf pre of relative differentials ΩG{F is defined to be the sheafification of ΩG{F. We now generalize this notion for schemes. Let f : X Ñ Y be morphism of schemes. By

¡1 ¡1 deﬁnition we have maps OY Ñ f¦OX and f OY Ñ OX , which makes OX into a f OY -

¡1 module. Then we can deﬁne the sheaf of diﬀerentials ΩX{Y to be ΩX{Y ΩOX {f OY .

iv Note that the sheaf of differentials ΩX{Y has the structure of an OX -module. Also, when both X and Y are affine schemes, that is X Spec A and Y Spec B for rings A and B, then the sheaf of differentials is ΩX{Y ΩA{B by Proposition B.0.8. Hence, ΩX{Y is always a quasi-coherent sheaf. The sheaf ΩX,Y is coherent if X and Y are Noetherian. Then an alternative way to define to define ΩX{Y is by covering X and Y with open affine subsets V and U and glueing the corresponding sheaves ΩX{Y . The derivations d : B Ñ ΩB{A glue together to give a map d : OX Ñ ΩX{Y of sheaves of abelian groups on X. (See [20]) Let A be a ring, and let M be an A-module. Define T npMq to be the tensor product M b ¤ ¤ ¤ b M of M with itself n times, for n ¥ 1. For n 0 we define T 0pMq A. Then À p q np q T M n¥0 T M is noncommutative A-algebra, in which multiplication is given by ° ° n m P kp q P tp q tensoring up: for all x k1 xik and y t1 ytk , such that xik T M , yit T M , ° ¤ b b p q we define x y x y k,t xik yit . We call T M the tensor algebra of M. We define the n-th exterior power of M to be the subset npMq T npMq of all alternating elements α P T npMq for which

1 ¸ α Altpαq sgnpσqpσαq, n! σPΣn where σα is deﬁned additively from the action on elementary tensors as

σ pm1 b ¤ ¤ ¤ b mnq mσp1q b ¤ ¤ ¤ b mσpnq.

À p q np q ^ We deﬁne the exterior algebra M n¥0 M of M. Denote by the multiplication in this algebra; in particular, the image of x ^ y for x P kpMq and y P tpMq is deﬁned to be pk tq! x ^ y Altpx b yq P T k tpMq. k!t! Thus, pMq is a skew-commutative graded A-algebra, meaning that if x P rpMq and y P spMq, then x ^ y p¡1qrsy ^ x.

Let pX, OX q be a ringed space, and let F be a sheaf of OX -modules. We define the exterior algebra on F by the sheafification of the presheaf, given by U Ñ pFpUqq, where we view FpUq as an OX pUq-module.

v n Theorem B.0.10. Let A be a ring, let Y Spec A, and let X PA. Then there is an exact sequence of sheaves on X,

n 1 0 Ñ ΩX{Y Ñ OX p¡1q Ñ OX Ñ 0.

The exponent n 1 in the middle term means a direct sum of n 1 copies of OX p¡1q.

Proof. Let S Arx0, . . . , xns be the homogeneous coordinate ring of X. Let S be the n 1 graded S-module Sp¡1q , with basis e0, . . . , en in degree 1. Deﬁne a homomorphism of graded S-modules E Ñ S by sending ei Ñ xi, and let M be the kernel. Then the exact sequence 0 Ñ M Ñ E Ñ S of graded S-modules gives rise to an exact sequence of sheaves on X via the functor,

n 1 0 Ñ M˜ Ñ OX p¡1q Ñ OX Ñ 0.

n 1 We have a surjection OX p¡1q Ñ OX , even though E Ñ S is not surjective, because the S-module homomorphism is surjective in all degrees ¥ 1, and hence we have surjection on

n 1 the distinguished open sets for OX p¡1q . ˜ Ñ We will now prove that M ΩX{Y . First note that if we localize at xi, then Exi Sxi is a surjective homomorphism of free Sxi -modules, and hence the module Mxi is free of rank n, generated by tej ¡ pxj{xiqei | j iu. Thus, if we denote by Ui the distinguished ˜ | open sets of X generated by xi, then M Ui is a free OUi -module generated by the sections p { q ¡ p { 2q { 1 xi ej xj xi ei for j i. (We multiply by an additional factor 1 xi to reduce the degree of the elements in the module Mxi to 0.) | Ñ ˜ | We deﬁne a morphism φi :ΩX{Y Ui M Ui as follows. Recall that we have Ui r { { s | Spec A x0 xi, . . . , xn xi , and hence ΩX{Y Ui is a free OUi -module generated by the elements dpx0{xiq, . . . , dpxn{xiq by Example B.0.7. Therefore we may deﬁne φi by

p p { qq p { 2qp ¡ q φi d xj xi 1 xi xiej xjei .

Clearly φi is an isomorphism. The isomorphisms φi glue together to give an isomorphism ˜ φ :ΩX{Y Ñ M as follows. On Ui XUj, we have pxk{xiq pxk{xjq¤pxj{xiq for any k. Hence, in Ω | X we have X{Y Ui Uj ¡ © ¡ © ¡ © x x x x x d k k d j j d k . xi xj xi xi xj

vi The applying φi to the left-hand side and φj to the right-hand side produces the same expression both times, that is p1{xixjqpxjek ¡ xkejq. Thus, the isomorphisms φi glue to an ˜ isomorphism φ :ΩX{Y Ñ M.

n Definition B.0.11. Let k be a field, and let X Pk be the n-dimensional projective space n over a field k. We define the canonical sheaf of X to be ωX ΩX{k, the n-th exterior power of the sheaf of differentials, where n dim X. It is a locally free sheaf of rank 1 on X.

n Proposition B.0.12. Let X Pk be then n-dimensional projective space over a ﬁeld k. n To obtain the canonical sheaf of Pk , take the n-th exterior power of the exact sequence of Proposition B.0.10 to deduce that

ωX OX p¡n ¡ 1q.

n Then Theorem 4.1.1 stated in terms of the canonical sheaf for Pk is as follows.

n n Theorem B.0.13 (Duality for Pk ). Let k be a ﬁeld, let X Pk , and let ωX be the canonical n sheaf for Pk . Then:

n 1. H pX, ωX q k.

2. For a ﬁxed such isomorphism and any coherent sheaf F on X, the natural pairing

n n HompF, ωX q ¢ H pX, Fq Ñ H pX, ωX q k

is a perfect pairing of ﬁnite-dimensional vector spaces over k.

3. For every i ¥ 0, there is an isomorphism

i n¡i q Ext pF, ωX q H pX, Fq

q where denotes the dual vector space. In the special case i 0, the isomorphism is the one induced by the pairing of statement (2).

We conclude this section with stating some generalizations of Serre duality for projective schemes using the dualizing sheaf from Section III.7 of [8].

vii Deﬁnition B.0.14. Let X be a proper scheme of dimension n over a ﬁeld k.A dualizing ¥ np ¥ q Ñ sheaf for X is a coherent sheaf ωX on X, together with a trace morphism t : H X, ωX k, such that for all coherent sheaves F on X, the natural pairing

p ¥ q ¢ np q Ñ np ¥ q Hom F, ωX H X, F H X, ωX followed by t gives an isomorphism

p ¥ q np qq Hom F, ωX H X, F .

Proposition B.0.15. Let X be a proper scheme over k. Then a dualizing sheaf for X, if it exists, is unique. That is, if ω¥ is a dualizing sheaf with trace map t, and if ω1, t1 is another, then there is a unique isomorphism φ : ω¥ Ñ ω1 such that t t1 ¥ Hnpφq.

Note that Proposition B.0.15 says only that if a dualizing sheaf exists, it is unique. In fact, a unique dualizing sheaf exists for any proper scheme X over a ﬁeld k, but here we will provide the necessary results to prove this statement only for projective schemes.

n Lemma B.0.16. Let X be a closed subscheme of P Pk of codimension r. Then i p q ExtP OX , ωP 0 for all i r.

¥ r p q Lemma B.0.17. With the same hypothesis as Lemma B.0.16, let ωX ExtP OX , ω .

Then for any sheaf of OX -modules F, there is an isomorphism

p ¥ q r p q HomX F, ωX ExtP F, ωP

Proposition B.0.18. Let X be a projective scheme over a ﬁeld k. Then X has a dualizing sheaf.

Deﬁnition B.0.19. Given a ring A, and A-module M, a sequence of elements x1, . . . , xr P A is called a regular sequence for M if x1 is not a zero-divisor in M (i.e. ax1 0 ô a 0), and for each i ¥ 2, xi is not a zero-divisor in M{px1, . . . , xi¡1qM. If A is a local ring with a maximal ideal m, then the depth of M is the maximal length of a regular sequence for

M with all xi P m. A local Noetherian ring A is Cohen-Macaulay if depth A dim A.A scheme X is Cohen-Macaulay if for all aﬃne subschemes Ui Spec Ai, the rings Ai are Cohen-Macaulay.

viii Deﬁnition B.0.20. A morphism of schemes f : X Ñ Y is of ﬁnite type if there exists a

¡1 covering of Y by open affine subsets Vi Spec Bi, such that for each i, f pBiq can be covered by a finite number open affine subsets Uij Spec Aij, where the Aij are finitely generated Bi-algebras. A scheme X is conected if it is connected as a topological space. A connected scheme X is integral if it can be covered by the spectra of integral domains. The dimension of a scheme X is its dimension as a topological space.

Now we have the necessary background to state the Serre duality for projective schemes.

Theorem B.0.21 (Duality for a Projective Scheme). Let X be a projective scheme of ¥ dimension n over an algebraically closed ﬁeld k. Let ωk be a dualizing sheaf on X, and let Ñ r P there be an immersion X Pk for some r Z. Then:

1. For all i ¥ 0 and F coherent on X, there are maps

i p ¥ q Ñ n¡ip qq θ : Ext F, ωX H X, F ,

such that θ0 is the trace map of Deﬁnition B.0.14.

2. The following conditions are equivalent:

(a) X is Cohen-Macaulay, and all of its irreducible components have the same dimension.

(b) For any locally free sheaf F on X, we have HipX, Fp¡qqq 0 for i n and q " 0.

Corollary B.0.22. Let X be a projective Cohen-Macaulay scheme over k of dimension n such that its irreducible components have equal dimensions. Then for any locally free sheaf F on X, there are isomorphisms

ip q n¡ip ˇ b ¥ qq H X, F H X, F ωX for all i ¥ 0.

ix Appendix C

Additional Proofs

This discussion follows mostly [12].

Let A be a ring, I a directed set, and pMiqiPI a family of A-modules indexed by I. For ¤ Ñ each i j, let fij : Mi Mj be an A-homomorphism with the properties that fii idMi for all i P I, and fik fjk ¥ fij whenever i ¤ j ¤ k. Let the A-module M together with the A-homomorphisms fi : M Ñ Mi, such that fi fj ¥ fij whenever i ¤ j be the direct limit. Then the direct limit has the universal property that if for each i P I there is a homomorphism φi : Mi Ñ N such that φi fij ¥ φj when i ¤ j, then there exists a unique homomorphism φ : lim M Ñ N such that f ¥ φ φ for all i P I. ÝÑiPI i i i

Proposition C.0.23. Let A be a ring and f and g be elements in A. Then Dpgq Dpfq if and only if there exists an element a P A and an integer k ¥ 0 such that gk af.

Proof. We give the proof in a series of equivalent statements. The condition Dpgq Dpfq is equivalent to the condition pg R p ñ f R p, @p P Spec Aq ô pf P p ñ g P p, @p P Spec Aq. Since the intersection of all prime ideals containing f is the radical of f, denoted by radpfq. The previous statement is equivalent to g P radpfq ô Dk P Z : gk P pfq. That is, Dk P Z and a P A : gk af.

The next two propositions will prove that given a ring A, the restriction map ρDpfqDpgq a abn k of the structure sheaf OSpec A, which maps f n to gkn where g bf, is well-deﬁned.

k l abn acn Proposition C.0.24. If g bf and g cf then gnk gnl .

x Proof. We need to verify that there is a power of g mutliplication with which transforms the expression abngnl ¡ acngnk into 0. In fact, we are going to show that this expression is already 0. To show that note:

abngnl ¡ acngnk abnpglqn ¡ acnpgkqn abnpcfqn ¡ acnpbfqn 0.

a b k acn bcm Proposition C.0.25. If f n f m in Af and g cf in A, then gkn gkm .

a b sp m ¡ nq Proof. If f n f m in Af , then there is a number s such that: f af bf 0. Then, to acn bcm n km ¡ m kn ks prove that gkn gkm , we will show that multiplication of ac g bc g with g gives 0. Indeed,

gkspacngkm ¡ bcmgknq f scspacnpcfqm ¡ bcmpcfqnq cscn mrf spaf m ¡ bf nqs 0.

The last property we need to check in order for the structure sheaf OSpec A to satisfy all requirements for presheaf is the cocycle condition.

Proposition C.0.26. If Dphq Dpgq Dphq, then the restriction maps satisfy the cocycle

condition: ρDpgqDphq ¥ ρDpfqDpgq ρDpfqDphq.

Proof. The condition Dpgq Dpfq implies that there exist an element c P A and an integer k ¥ 0 such that gk cf. Also, the condition Dphq Dpgq implies that there exist an element d P A and an integer t ¥ 0 such that ht dg. From the second condition we get hkt dkgk and then the ﬁrst implies that: hkt dkcf. This means that we ¨ ¨ a adkncn a acn have ρDpfqDphq n ktn . On the other hand, we have: ρDpfqDpgq n nk and ¨ f h f g b bdm ρDpgqDphq gm htm . We can combine all these to obtain: ¡ © ¡ © ¡ © a acn acndnk a ρ p q p q ¥ ρ p q p q ρ p q p q ρ p q p q . D g D h D f D g f n D g D h gnk hnkt D f D h f n

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xiv Corrections

When originally submitted, this honors thesis contained some errors which have been corrected in the current version. Here is a list of the errors that were corrected.

Various places in the thesis. Approximately 20 spelling errors were corrected, 10 miss- ing periods or commas were added in mathematical formulae, and approximately 30

n spacing and sizing changes were made to mathematical formulae. The notation PA n was changed to PA in approximately 15 places.

Other changes:

p. 5, Example 2.1.3. The clarifications F1pHq 0 and F2pHq 0 were added. p. 6, -8. The sentence “Hence, for any x P X we have Fx F˜x.” was added. p. 7, Example 2.1.8. The phrase “a collection of maps” was changed to “a collection of group homomorphisms.” p. 8, -3. The begining of the sentence “By the construction of sheafification, for any x P X

we have Fx F˜x, applying...” was changed to “Since Fx F˜x for any x P X, applying...”. p. 9. In the discussion after Example 2.1.14, the phrase “f¦F on Y is the sheaf” was changed

to “f¦F on Y is the presheaf” and the sentence “This presheaf is a sheaf.” was added after that. p. 10. In the discussion after Deﬁnition 2.2.2 the reference “See Appendix C for more technical details.” was added.

xv p. 12, 1. The phrase “A direct sum of sheaves F1,... F2” was changed to “A direct sum

of sheaves tFiuiPI .”.

# p. 12. The last sentence of the second paragraph “Furthermore, the morphism f : OY Ñ

f¦OX of sheaves of rings on Y equips f¦F with the structure of an OY -module.” was deleted. p. 13, Deﬁnition 2.3.2. In the next to the last sentence, the phrase “element f of S1 of

degree 1” was changed to “element f of S of positive degree”. p. 13. In the end of Deﬁnition 2.3.3, the formula “Op Sppq” was changed to “Op

pSpq0 Sppq”.

# p. 14, Definition 2.4.1. The formulae “f : OY Ñ OX ”, “U X”, and “V U X” # were changed to “f : OY Ñ f¦OX ”, “U Y ”, and “V U Y ” respectively. p. 15, -3. The beginning of the sentence “Given any scheme Z over S and any morphisms f : Z Ñ X and g : Z Ñ Y ” was changed to “Given any morphisms α : X Ñ S, β : Y Ñ S, and Z Ñ S and morphisms f : Z Ñ Y and g : Z Ñ Y that satisfy β ¥ g α ¥ f”. p. 16. In the last sentence of the second paragraph, the clause “where E ¢ E has the product topology” was added. p. 17, Proposition 2.5.3. The fragment “See Proposition3.9” was changed to “see Propo- sition 3.9”. p. 17, Definition 2.5.4 The footnote “Technically this definition is used only for local Noetherian spaces, which is case when it is most often studied. See p. 60 of [13] for the general definition.” was added. p. 19. In the first display of Remark 2.6.2, the formula “Sf pmq’ was changed to “pSf qm”. p. 26. The next to the last sentence of Lemma 3.1.11 “first using that Ik is injective we consturct the morphism J k¡1{J k¡2 Ñ Ik, and then using that J k is injective, we construct the morphism Ik Ñ J k” was changed to “using the morphisms 0 Ñ

xvi Ik¡1{Ik¡2 Ñ Ik and Ik¡1{Ik¡2 Ñ J k¡1{J k¡2 Ñ J k, and the fact that J k is injective, we construct the morphism Ik Ñ J k”.

p. 41. In the proof of Therorem 3.5.1 the sentence “Thus, we can use Cechˇ cohomology”

r was changed to “Since PA is Noetherian separated scheme by Hilbert’s Theorem and Exercise 8, p. 262 of [19], we can use Cechˇ cohomology for this cover.”

Appendix C By idea of the thesis readers, a third appendix was added to provide some relevent facts from the used sources.

xvii