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Home , Constant sheaf, Derived functor, Exact functor, Injective sheaf, Resolution (algebra), Section (fiber bundle)

An Introduction to in Algebraic

Ryan Mike

University of Colorado Boulder Department of Defended April 7, 2020

Thesis Advisor: Dr. Sebastian Casalaina-Martin, Department of Mathematics

Defense Committee: Dr. Sebastian Casalaina-Martin, Department of Mathematics Dr. Nathaniel Thiem, Department of Mathematics Dr. Rafael Frongillo, Department of Computer Science Contents

0 Introduction and Preliminaries 1

1 in R-Mod 2 1.1 (Co)chain Complexes ...... 2 1.2 Injective and Projective Modules ...... 5 1.3 Resolutions ...... 7 1.4 Derived ...... 10

2 Sheaves 12 2.1 Complex ...... 13 2.2 Basic Definitions and Examples ...... 14 2.3 Homological Algebra in Sh(X)...... 17 2.4 Cohomology of Sheaves ...... 19

3 Applications 20 3.1 Relation to Other Cohomology Theories ...... 20 3.2 The Exponential Sequence and the ...... 22 3.3 The Riemann-Roch Theorem for Line Bundles ...... 23

0 Introduction and Preliminaries

Given a topological X and an abelian A, the singular cohomology groups Hn(X; A) provide important algebraic invariants of X which form obstructions to X being contractible. may be thought of as a generalization of singular cohomology (and other cohomology theories) in the sense that it accommodates more general “coefficients” than simply an A. The abelian group A is replaced by a sheaf of abelian groups F which in turn gives rise to the sheaf cohomology groups Hn(X, F). Broadly speaking, these groups constitute obstructions to solving geometric problems globally, given that they can be solved locally. In this paper, we intend to give an introduction to sheaf cohomology via the derived functors approach of Grothendieck. This paper is expository in nature, and the interested reader can find more information about all of the topics we in the list of references we provide at the end. 1 covers all of the homological algebra needed to define sheaf cohomology. Assuming familiarity with R-modules, we begin with the basics of cochain complexes, exactness, and coho- mology. This brings us to a discussion of exact functors and injective resolutions which then lets us define derived functors. Section 2 is devoted to developing the theory of sheaves on a . We begin with an informal review of complex manifolds and line bundles. This leads to some examples of sheaves later on, and gives an impression of the types of geometric objects which can be studied using sheaves and cohomology. We then provide basic definitions and examples of sheaves and address how the tools from homological algebra which we developed in section 1 can be transferred to the of sheaves on a topological space. Finally, we define the sheaf cohomology functors as the derived functors of the global sections from the category of sheaves on a topological space to the category of abelian groups. In section 3, we attempt to give the reader a sense for how sheaf cohomology is used in and more broadly. First, we examine how sheaf cohomology fits within the larger picture

1 of cohomology theories, and in particular, its relation to singular cohomology and de Rham coho- mology. We then introduce the exponential sequence and explore its cohomological consequences to find some interesting properties of the complex logarithm from complex analysis. Finally, we discuss Chern classes and the Euler-Poincar´echaracteristic of a sheaf which lets state the Riemann-Roch theorem for line bundles on curves. Before we begin, we will review a couple preliminaries to -up the category of R-modules. For our purposes, R will be a commutative so we will not distinguish between left and right R-modules.

Definition 0.1. Let R be a with 1. An R- is an abelian group M together with an action of R on M (that is, a R × M → M) denoted rm for all r ∈ R and m ∈ M, such that

• r(m + n) = rm + rn for all r ∈ R and m, n ∈ M.

• (r + s)m = rm + sm for all r, s ∈ R and m ∈ M,

• r(sm) = (rs)m for all r, s ∈ R and m ∈ M, and

• 1m = m for all m ∈ M.

Given two R-modules M and N, we say a φ : M → N is an R-module homomorphism if it is a homomorphism of the underlying abelian groups which additionally satisfies φ(rm) = rφ(m) for all r ∈ R and m ∈ M.

Note that if R = Z, the ring of , then the definition of R-module collapses to the definition of abelian group. Thus Z-modules are abelian groups. There is a category whose objects are R-modules and whose are R-module homo- morphisms. This is the category of R-modules, denoted R-Mod. For convenience, we will often call a homomorphism of R-modules an R-module map, or just a map when the context is understood.

1 Homological Algebra in R-Mod

The purpose of this section is to establish the algebraic preliminaries needed to develop sheaf cohomology. Homological algebra unfolds in the abstract setting of abelian categories. Loosely speaking, these are categories equipped with sensible notions of product, coproduct, zero object, , cokernel, and image, in which the sets of morphisms are enriched with an abelian group structure. General abelian categories can look very foreign at first glance, making results more difficult to state and prove. For this reason, we will restrict attention to the category R-Mod, where we assume our reader will have more familiarity. This is the prototypical example of an , and a good place to develop intuition and understanding for homological algebra. We emphasize, however, that all of the key results we prove extend to the more general setting of abelian categories. Indeed, this will be an important point when we move to the category of sheaves on a topological space in the next section. For the remainder of this section, we will fix a commutative ring R with 1.

1.1 (Co)chain Complexes In many areas of mathematics, objects often appear not by themselves, but in sequences with other objects of the same type. Given a collection of R-modules M1,M2,...,Mn and R-module maps

2 fi : Mi → Mi+1 for 1 ≤ i < n, we can form the sequence

f1 f2 f3 fn−1 M1 M2 M3 ··· Mn.

One curiosity about such a sequence is how the image of the incoming map fi−1 interacts with the kernel of the outgoing map fi in the module Mi. We say the sequence is exact at Mi if these submodules are equal, that is, if ker fi = im fi−1. The sequence is called exact if it is exact at Mi for 1 < i < n. Exactness also tells us something about the composite maps fi ◦ fi−1. In particular, exactness at Mi implies fi ◦ fi−1 = 0. Our next definition singles out precisely this case.

Definition 1.1. A C• of R-modules is a family {Cn}n∈Z of R-modules together with R-module maps {dn : Cn → Cn−1}n∈Z called differentials satisfying the property that dn ◦dn+1 = 0 for all n ∈ Z. Because dn ◦ dn+1 = 0 we have

0 ⊆ im dn+1 ⊆ ker dn ⊆ Cn

th and we define the n module of Cn to be the quotient Hn(C•) := ker dn/ im dn+1. It is common to depict a chain complex C• by the diagram

d d d d ··· Cn+1 Cn Cn−1 ··· and omit indices on the differentials for convenience.

A chain map f : C• → D• is family of R-module maps {fn : Cn → Dn}n∈Z which commute with the differentials, that is, so that following diagram commutes

d d d d ··· Cn+1 Cn Cn−1 ···

fn+1 fn fn−1 d d d d ··· Dn+1 Dn Dn−1 ··· .

Note that if C• is exact at Cn, then Hn(C•) = 0. In this sense, we can view the homology modules Hn(C•) as obstructions to C• being exact. Examples 1.1.

• The sequence consisting of all zero modules and zero differentials

··· 0 0 0 ···

is a chain complex which we will call the zero complex and often simply denote by 0.

• More generally, any arbitrary family of R-modules {Cn} with zero differentials is a chain complex.

• The sequence ·2 ·2 ·2 ·2 ··· Z/4 Z/4 Z/4 ··· with all differentials as ‘times-two’ maps x 7→ 2x is a chain complex which is exact.

Anyone familiar with some will be well-acquainted with the ‘co-’ prefix, indi- n n n n+1 cating arrow reversals. This gives a variant in notation: C = C−n and d = d−n : C → C .

3 • n Definition 1.2. A cochain complex C of R-modules is a family {C }n∈Z of R-modules together n n n+1 n n−1 th with R-module maps {d : C → C }n∈Z such that d ◦ d = 0 for all n ∈ Z. The n cohomology module of Cn is the quotient Hn(C•) := ker dn/ im dn−1. Maps of cochain complexes are defined in the same way as chain maps and are called cochain maps. Elements of Hn(C•) are called cohomology classes and for a representative c ∈ ker dn, we use square brackets to denote its cohomology class [c] ∈ Hn(C•). Note on conventions. When we develop sheaf cohomology later in this paper, we will be chiefly concerned with cochain complexes and, as the reader may have guessed, cohomology. Therefore, to ease the transition in section 2, we will adopt the ‘co-’ notational variant as our convention, and emphasize the ‘co-’ half of the story throughout this section. The attentive reader may notice that our definitions for cochain complex and cochain map actu- ally give rise to a category, the category of cochain complexes of R-modules, denoted Ch•(R-Mod). Further, a cochain map f : C• → D•, induces a map Hn(f): Hn(C•) → Hn(D•) in cohomology • given by [c] 7→ [f(c)]. To see that this is well-defined, let dC denote the differentials in C , let dD • 0 n+1 denote the differentials in D , let k = d(c ) ∈ im dC and compute n 0 0 n • H (f)([k]) = [f(k)] = [f(dC (c ))] = [dD(f(c ))] = 0 ∈ H (D )

0 n+1 since [dD(f(c ))] ∈ im dD . This leads us to the following proposition. • n • n Proposition 1.1. For each n ∈ Z, the assignment C 7→ H (C ) and f 7→ H (f) for any cochain map f : C• → D•, defines a functor from Hn : Ch•(R-Mod) → R-Mod. Proof. We have already seen that Hn(f) is well-defined. The other necessary verifications are routine.

Another important type of sequence is the short . Definition 1.3. A short exact sequence is an exact sequence of the form

f g 0 A B C 0.

A map of short exact sequences from 0 → A → B → C → 0 to 0 → A0 → B0 → C0 → 0 is a 0 A B C 0 α β γ 0 A0 B0 C0 0. Now is a good time to make a few observations highlighting how exact sequences can efficiently present concepts already familiar to the reader:

f • A map f : A → B is injective if and only if the sequence 0 A B is exact.

f • A map f : A → B is surjective if and only if the sequence A B 0 is exact.

f • A map f : A → B is an if and only if the sequence 0 A B 0 is exact.

f g • The sequence 0 A B C 0 is exact if and only if C =∼ B/A, where we have identified A as a submodule of B via f.

4 The first three observations are clear. The last observation holds because exactness at B implies ker g = im f; exactness at A implies that f is injective, whence im f is naturally identified with A; and exactness at C implies that g is surjective. Putting this all together with the first isomorphism theorem, we get the natural identification C =∼ B/A. Now, we come to the theorem which brings all of these definitions to life: the existence of long exact sequences.

f g Theorem 1.1. Let 0 A• B• C• 0 (∗) be a short exact sequence of cochain complexes, that is, a commutative diagram

......

d d d f n gn 0 An Bn Cn 0

d d d f n+1 gn+1 0 An+1 Bn+1 Cn+1 0

d d d ......

Then there exist natural maps δn : Hn(C•) → Hn+1(A•) such that the sequence

. .

δn−1 Hn(f) Hn(g) Hn(A•) Hn(B•) Hn(C•)

δn Hn+1(f) Hn+1(g) Hn+1(A•) Hn+1(B•) Hn+1(C•)

δn+1 . . is exact. We call this sequence the long exact sequence associated to the short exact sequence (∗).

Proof. The map δn is defined by taking a representative c ∈ Cn for a cohomology class [c] ∈ Hn(C•), lifting it via g, then applying d (in B•) to obtain an element in Bn+1 which is the image under f of some element a ∈ An+1, and finally assigning δn([c]) = [a] ∈ Hn+1(A•). Checking that δn is well-defined and that the long exact sequence is, in fact, exact at Hn(A•), Hn(B•), and Hn(C•) for all n involves a laborious diagram chase which is rather unenlightening. More details can be found in [Alu09, §3.3].

1.2 Injective and Projective Modules We will now turn our attention to special types of R-modules with desirable properties in relation to the definitions made in the previous section. These properties will come to light particularly in §1.3 and §1.4 when we define resolutions and derived functors.

5 To motivate the following discussion, consider the image of a short exact sequence of R-modules f g 0 A B C 0 under a functor F : R-Mod → R-Mod. Further, suppose that F is additive meaning it induces abelian group homomorphisms F : Hom(M,N) → Hom(F (M),F (N)) for any two R-modules M and N. Applying F , we obtain the sequence

F (f) F (g) 0 F (A) F (B) F (C) 0.

But is this sequence still exact? If not, is it exact anywhere? The answers to these questions depend on the functor F and prompt the following definition. Definition 1.4. A covariant additive functor F : R-Mod → R-Mod is called left exact (resp. right exact) if for any short exact sequence 0 → A → B → C → 0, the sequence

0 F (A) F (B) F (C) (resp. F (A) F (B) F (C) 0 ) is exact. We say F is exact if it is left exact and right exact. Thus exact functors take short exact sequences to short exact sequences. We make analogous definitions if F is contravariant. Of particular relevance to us are the Hom functors Hom(M, −): R-Mod → R-Mod and Hom(−,M): R-Mod → R-Mod. For general M, they turn out not to be exact. We do, however, have the following proposition. Proposition 1.2. For a fixed R-module M, the functors Hom(M, −) and Hom(−,M) are left exact. Proof. Verifying that the relevant sequences (0 → Hom(C,M) → Hom(B,M) → Hom(A, M) and 0 → Hom(M,A) → Hom(M,B) → Hom(M,C)) are exact follows immediately from the definitions. We invite the reader to work out the details.

However, for some special R-modules, the Hom functors are exact. Definition 1.5. An R-module P is called projective if the functor Hom(P, −) is exact. An R- module I is injective if the functor Hom(−,I) is exact. Since the Hom functors are already left exact as mentioned, the content of projectives and injectives is that they make the Hom functors right exact. Thus we get the following, somewhat more concrete, characterization of projective and injective modules. Lemma 1.1. An R-module P is projective if and only if it satisfies the following lifting property: For all surjections g : B → C, every map γ : P → C lifts to a map β : P → B:

P β γ g B C 0. An R-module I is injective if and only if satisfies the following extension property: For all injections f : A → B, every map α : A → I extends to a map β0 : B → I:

f 0 A B α . β0 I

6 Proof. This is immediate from the definition of projectives and injectives and the fact that the Hom functors are left exact in general. These definitions may seem abstract but they distinguish modules with very tangible properties. For example, any free R-module F is projective because it enjoys the lifting property in Lemma 1.1 – define β by lifting the image of a of F and invoking the of free modules. Furthermore, if we restrict our attention to abelian groups (that is, take R = Z), the property of being free is actually equivalent to the property of being projective. We also have a tangible characterization of injective R-modules when R is a . In this case, it turns out that the injective R-modules are exactly the divisible R-modules – those modules M satisfying the property that aM = M for all elements a ∈ R which are not zero divisors. When R = Z, the additive group Q and Q/Z are good examples of this. Proofs of these characterizations can be found in [Alu09, §6.2, §6.3]. We conclude this section with a statement that hints at the “abundance” or “density” of projec- tive and injective modules in the category R-Mod. Loosely speaking, it says that every R-module mimics the structure of some and some in a certain way. Theorem 1.2. Every R-module is a quotient of some projective module and is a submodule of some injective module. More precisely, for all R-modules M, • there exists a projective module P and a surjection P → M; • there exists an injective module I and an injection M → I. Proof. The statement about projectives is easy to prove. Any R-module M is surjected upon by the free (and hence projective) module R⊕M generated by the elements of M. The statement about injectives is hard to prove. One shows that result holds for abelian groups and bootstraps up to R-Mod using general categorical techniques. More details can be found in [Alu09, §6.3, Cor. 6.12]. Our comments from above are made slightly more precise in the next section and particularly in light of Proposition 1.3. To truly do justice to these statements, however, we must pass to the which is beyond the scope of this paper. The interested reader should look in [Wei95, Chap. 10].

1.3 Resolutions In many areas of mathematics, R-modules naturally arise and they can often be quite complicated. Loosely speaking, resolutions provide a way to “spread out” this complexity into a chain of simpler objects which has similar properties to the original module. In some sense, resolutions function like generalized presentations.

Definition 1.6. A left of an R-module M is a chain complex P• with Pi = 0 for i < 0 together with a map  : P0 → M making the augmented complex d d d  ··· P2 P1 P0 M 0 exact. It is a projective resolution if each Pi is projective. Analogously, a right resolution of an R-module M is a cochain complex I• with Ii = 0 for i < 0 together with a map  : M → I0 making the augmented complex

0 M  I0 d I1 d I2 ··· exact. It is an injective resolution if each Ii is injective.

7 Again, we will primarily present the ‘co-’ conventions. In the context of resolutions, this means emphasizing the story in the case of left resolutions and injective resolutions over right resolutions and projective resolutions. Another way to think about resolutions is via the following definition and lemma.

Definition 1.7. A cochain map f : C• → D• is called a quasi-isomorphism if it induces isomor- phisms in cohomology, that is, if Hn(f): Hn(C•) → Hn(D•) is an isomorphism for all n.

Lemma 1.2. Suppose I• is a cochain complex of injectives with Ii = 0 for i < 0. Then a map  : M → I0 giving an injective resolution for M is the same thing as a quasi-isomorphism  : M → I•, where M is considered as a cochain complex

··· 0 M 0 ··· concentrated in degree 0. An analogous statement holds for projective resolutions.

Proof. The map  defines a cochain map M → I•. Since the augmented complex in Definition 1.6 is exact, we have Hi(M) =∼ Hi(I•) for all i. Moreover, this isomorphism is induced by . Hence  is a quasi-isomorphism. Conversely, if  : I• → M is a quasi-isomorphism, then Hi(I•) = 0 for i 6= 0 and H0(): H0(M) = M → H0(I•) is an isomorphism implying  must have been injective (since ker  ⊆ ker(H0()) = 0). All of this means that the augmented complex is exact whence I• and  form an injective resolution for M.

Another interesting note related to quasi- is that a cochain complex is exact if and only if it is quasi-isomorphic to the zero complex. The reader should verify this!

Proposition 1.3. Every R-module has an injective (and a projective) resolution.

Proof. Let M be an R-module and invoke Theorem 1.2 to choose an injective R-module I0 and an injection 0 : M → I0. Setting M 0 = coker , this gives the short exact sequence

0 M  I0 M 0 0.

Now, choose an injective R-module I1 and an injection 1 : M 0 → I1. Setting M 1 = coker 1, and d0 to be the composition I0 → M 0 → I1, we get the exact sequence

0 0 M I0 d I1 M 1 0 0 . M 0

0 0

Inductively, given M n−1, we choose an injective R-module In and an injection n : M n−1 → In. Setting M n = coker n, and dn to be the composition In−1 → M n−1 → In we get the sequence

0 0 1 2 0 M  I0 d I1 d I2 d ··· which is exact since ker dn = M n = im 1 = im dn−1. Hence, we are left with M → I•, our desired injective resolution.

8 Now that we know that every R-module M has an injective resolution, it is natural to ask questions like: How many injective resolutions does M have? How do they relate to each other? Can we classify them? The answers turn out to be quite simple if we coarsen our notion of equivalence of cochain complexes. The next definition introduces this new notion of equivalence and the remainder of this section builds up to some answers to these questions. Definition 1.8. A cochain map h : C• → D• is called null homotopic if there exists maps sn : Cn → Dn−1 depicted below in the (not commutative!) diagram

··· d Cn−1 d Cn d Cn+1 d ···

h h h s s s s ··· d Dn−1 d Dn d Dn+1 d ··· such that h = d ◦ s + s ◦ d. Two cochain maps f, g : C• → D• are called chain homotopic if their difference is null homotopic, that is, if f − g = d ◦ s + s ◦ d. In this case, we call s a chain from f to g. Finally, we say that two cochains C• and D• are chain homotopy equivalent if there exists cochain maps f : C• → D• and g : D• → C• such that the compositions f ◦ g and g ◦ f are chain homotopic to the respective identities on C• and D•. In this case we call f a chain homotopy equivalence. Lemma 1.3. Chain homotopy equivalences are quasi-isomorphisms. Proof. First note that if h : C• → D• is null homotopic, then Hi(h) = 0 for all i. Indeed, for any class [c] ∈ Hi(C•) we have Hi(h)([c]) = [h(c)] = [d(s(c))] + [s(d(c))] = [0] + [s(0)] = 0 ∈ Hi(D•).

• • • • So if f : C → D and g : D → C are maps such that f ◦ g − idC and g ◦ f − idD are null homotopic, then

i i i i i i H (f) ◦ H (g) = H (idC ) = idHi(C•) and H (g) ◦ H (f) = H (idD) = idHi(D•) whence f (and g) are quasi-isomorphisms.

Theorem 1.3. Let M and N be R-modules, let  : M → I• be an injective resolution of N, and let f 0 : M → N be any map. Then for any right resolution η : N → J • of M, there exists a cochain map f : I• → J • 0 M  I0 I1 I2 ···

f 0 f 0 f 1 f 2 η 0 N J 0 J 1 J 2 ··· extending f 0 in the sense that η ◦ f 0 = f 0 ◦ , which is unique up to chain homotopy. An analogous result holds for projective and left resolutions. Proof. The idea here is to use the extension property of injective modules in Lemma 1.1 to induc- • • tively extend fi to fi+1, hence building the cochain map f : I → J . This is omitting a number of technicalities, but the interested reader can find more details in [Wei95, Thm. 2.2.6].

Note that this holds only when the top row is an injective resolution, and not for any arbitrary right resolution. In large part, it is precisely this theorem which motivates us to define injective and projective modules in the first place. They are the modules giving rise to resolutions with such functorial properties.

9 Corollary 1.1. Injective resolutions are unique up to cochain homotopy equivalence.

Proof. Invoke Theorem 1.3 twice to extend the identity map M → M and obtain a cochain homo- topy equivalence and its inverse up to cochain homotopy.

Therefore, in response to the questions we posed below Proposition 1.3, there may be many injective resolutions of a given R-module M, but there is only one up to chain homotopy equivalence!

1.4 Derived Functors In §1.2, we asked, “what happens when we apply a functor F to a short exact sequence?” As we saw, the resultant sequence may or may not be exact, and the answer depends on F . In this section, we will develop tools which measure the extent to which functors fail to be exact. Just as the cohomology modules of a cochain complex C• can be viewed as obstructions to C• being exact, the derived functors of F will constitute obstructions to F being exact.

Definition 1.9. Let F : R-Mod → R-Mod be a left . Then we can construct the right derived functors RiF (for i ≥ 0) of F as follows. Fix, once and for all, an injective resolution A → I• for all R-modules A and define

RiF (A) = Hi(F (I•)).

Note that left exactness of F implies that 0 → F (A) → F (I0) → F (I1) is exact. This means

R0F (A) = H0(F (I•)) = ker(F (I0) → F (I1)) = F (A).

We define left derived functors of a right exact functor in an analogous fashion using projective resolutions.

There is an immediate issue of well-definedness stemming from our choice of injective resolutions. This is where our work from section §1.3 comes into play.

Lemma 1.4. The modules RiF (A) are well-defined up to natural isomorphism. That is, if A → I• and A → J • are two injective resolutions for A, there is a canonical isomorphism Hi(F (I•)) → Hi(F (J •)). In particular, if RˆiF is the right of F arising from a different choice of injective resolutions, then RˆiF is naturally isomorphic to RiF .

Proof. Given a map f 0 : A → B of R-modules, injective resolutions A → I• and A → Iˆ• for A, and injective resolutions B → J • and B → Jˆ• for B, we get following (not necessarily commutative) diagram φ I• A Iˆ•

f fˆ φ J • B Jˆ• ˆ 0 where f and f extend f to maps of the respective resolutions, φA extends idA, and φB extends ˆ idB. Since φA and φB extend the identities, the compositions f ◦ φA and φB ◦ f are both maps I• → Jˆ• which extend f 0, meaning they are cochain homotopic by Theorem 1.3. Now since additive

10 ˆ functors preserve cochain (the reader should check this), we have that F (f ◦ φA) is cochain homotopic to F (φB ◦ f). This implies that the following diagram commutes

Hi(F (φ )) Hi(F (I•)) A Hi(F (Iˆ•)) =∼ Hi(F (f)) Hi(F (fˆ)) Hi(F (φ )) Hi(F (J •)) B Hi(F (Jˆ•)) =∼ where the maps on the top and bottom rows are isomorphisms by Corollary 1.1.

Corollary 1.2. If I is injective, then ( F (I) if i = 0 RiF (I) = 0 if i > 0.

Proof. If I is injective, then we can take I (as a complex concentrated in degree zero with 0’s in every other degree) to be its own injective resolution (0 is an injective R-module!). In this case, RiF (I) is computed by cohomology of the cochain complex concentrated in degree zero with F (I).

Without too much effort, just a few quick arguments primarily working the definitions and invoking Theorem 1.3, one shows that the derived functors RiF , are indeed additive functors from R-Mod to R-Mod. More details can be found in [Wei95, Thm. 2.4.5]. Finally, we have a theorem highly analogous to Theorem 1.1 which makes precise our comments at the beginning of this section: that the derived functors RiF are obstructions to F being exact.

Theorem 1.4. For any left exact functor F : R-Mod → R-Mod and short exact sequence

f g 0 A B C 0 of R-modules, there exist natural maps δi : RiF (C) → Ri+1F (A) such that the sequence

. .

δi−1 RiF (f) RiF (g) RiF (A) RiF (B) RiF (C)

δi Ri+1F (f) Ri+1F (g) Ri+1F (A) Ri+1F (B) Ri+1F (C)

δi+1 . . is exact.

Proof. The proof of this theorem is very long, technical, and not particularly enlightening. For more details, see [Wei95, Thm. 2.4.6].

11 In practice, it is generally difficult to compute right derived functors and part of the difficulty arises in the first step: finding injective resolutions. Luckily, it turns out that we do not always need to find injective resolutions. Instead, we can compute using F -acyclic resolutions. Definition 1.10. An R-module Q is called F -acyclic if RiF (Q) = 0 for i 6= 0, that is, if the higher derived functors of F vanish on Q. An F -acyclic resolution of an R-module A is a right resolution A → Q• such that Qi is F -acyclic for all i. Corollary 1.2 shows injectives are F -acyclic for every left exact functor F . This leads to the next result, which will become particularly important when we discuss in §3.1. Proposition 1.4. Let A → Q• be an F -acyclic resolution of A. Then RiF (A) =∼ Hi(F (Q•)) for all i. Proof. Let Zn = ker(Qn → Qn+1). Then we have a short exact sequence

0 Zn Qn Zn+1 0 which gives rise the long exact sequence

0 F (Zn) F (Qn) F (Zn+1) R1F (Zn) ··· by Theorem 1.4. But the Qi are F -acyclic, hence R1F (Zn) =∼ coker(F (Qn) → F (Zn+1)) and for i > 0 we have RiF (Zn+1) =∼ Ri+1(Zn). By induction, this implies that

Ri+1F (Z0) =∼ Ri(Z1) =∼ ... =∼ R1F (Zi). Now observe that F (Zi+1) = F (ker(Qi+1 → Qi+2)) = ker(F (Qi+1) → F (Qi+2)) because F is left exact, so since Z0 = A, we get

Ri+1F (A) =∼ coker(F (Qi) → F (Zi+1)) =∼ Hi+1(F (Q•)) for i > 0. Finally, to show RiF (A) =∼ Hi(F (Q•)) for i = 0, 1, simply observe that

0 F (Z0) F (Q0) F (Z1) R1F (Z0) 0 is exact and again note Z0 = A.

2 Sheaves

The purpose of this section is to develop the theory of sheaves so that we may explore the cohomol- ogy of sheaves. Sheaves provide a way to systematically keep track of local data on a topological space. As such, they give a means to formalize the notion of a “local condition.” Sheaves appear in many areas of geometry and have historically played an important role in , differential geometry, , and algebraic geometry. We will be making the leap from the familiar category R-Mod of R-modules, to the less familiar category Sh(X) of sheaves on a topological space X. As alluded to at the beginning of section 1, the tools and results that we developed in that section actually extend to the more general setting of abelian categories, of which Sh(X) is an example. Indeed, the attentive reader will note that we rarely invoked specific properties of R-modules throughout the entire discussion in section 1. We will revisit this point in §2.3, but to begin we will get our bearings with some geometry.

12 2.1 Complex Manifolds With this subsection, we would like to give an informal introduction to the geometric objects that we will use to highlight the applications of sheaf cohomology: complex manifolds. As mentioned before, this section is primarily devoted to sheaf theory. Nevertheless, we place this subsection at the beginning of this section so as to give the reader something geometric to latch onto while treading the abstract landscape of sheaves. In particular, our work here will lend itself to some examples of sheaves later on. To begin, we recall some basics from topology (the reader seeking more background can look in any standard topology text such as [Mun08]). An n-dimensional (real) topological is a second countable, Hausdorff, topological space M that admits a covering by open sets which are n all homeomorphic to R . Now suppose U and V are two sets in such a covering and that U ∩ V is n n nonempty. Then there are homeomorphisms φ : U → R and ψ : V → R called chart maps, and we can form the composite ψ ◦ φ−1 : φ(U ∩ V ) → ψ(U ∩ V ). We call this composite a transition n function between the charts U and V . Since φ(U ∩V ) and ψ(U ∩V ) are both of R , we can ask that they satisfy certain properties such as differentiability or . If a manifold has differentiable transition functions, we call it a differentiable manifold; if it has smooth transition functions, we call it a smooth manifold; etc. An n-dimensional is a 2n-dimensional (real) manifold X such that its tran- 2n ∼ n sition functions are holomorphic upon identifying R = C . We think of X as locally “looking n like” C , and since holomorphicity is checked locally, we obtain a notion for a holomorphic map f : X → Y between two complex manifolds X and Y . Examples 2.1. 1 1 2 • Complex projective line P : Let P be the set of complex lines through the origin in C . More precisely, 1 2 0 0 0 0 ∗ P = C \{(0, 0)}/ ∼ (x, y) ∼ (x , y ) ⇐⇒ (x, y) = λ(x , y ) for some λ ∈ C . 1 Letting (x : y) denote the equivalence class of (x, y), the sets U = {(x : y) ∈ P | x 6= 0} and 1 1 V = {(x : y) ∈ P | y 6= 0} form a cover of P and admit homeomorphisms φU : U → C given 1 by (x : y) 7→ y/x and φV : U → C given by (x : y) 7→ x/y, where P carries the quotient topology. The transition function for these charts is given by z 7→ 1/z which is holomorphic ∗ 1 on φU (U ∩ V ) = C and thus P is a 1-dimensional complex manifold, the complex projective line. In fact, one can show that it is homeomorphic to S2, the 2-dimensional sphere.

n • Complex P : One can generalize this construction to complex lines through n+1 n+1 the origin in C . Here we let X = C \{0}/ ∼ where (x0, . . . , xn) ∼ λ(x0, . . . , xn) for ∗ n any λ ∈ C . This time, there are n + 1 charts to C corresponding to the nonvanishing of each coordinate xi. The transition functions are analogous to those in the case n = 1 and are also holomorphic. Hence X is an n-dimensional complex manifold.

n 2n 2n 2n n • Complex tori: Let X be the quotient space C /Z where Z ⊂ R = C is the natural inclusion. Then X is a complex manifold whose transition functions are given by translation 2n n 1 1 by an element of Z in C . In the case n = 1, this gives rise to the torus S × S familiar from topology.

n • Affine hypersurfaces: If f : C → C is a for which 0 ∈ C is a regular value (meaning the differential of f doesn’t vanish anywhere in f −1(0)). Then X = f −1(0) is a complex manifold by the implicit function theorem. For more details on any of these examples, we refer the reader to [Huy06, §2.1].

13 If one wishes to understand the geometry of a complex manifold X, one must study, in particular, holomorphic vector bundles on X.

Definition 2.1. Let X be a complex manifold. A holomorphic vector of rank r on X is a complex manifold E together with a holomorphic map π : E → X such that each fiber π−1(x) has the structure of an r-dimensional complex and satisfies the following condition: There −1 r exists an open cover {Ui} of X and biholomorphic maps ψi : π (Ui) → Ui × C commuting with the projections to Ui −1 ψi r π (Ui) Ui × C

π Ui

−1 r such that the restriction ψi|π−1(x) : π (x) → C is C-linear. A holomorphic on X is a holomorphic of rank 1.

Definition 2.2. Let E be a holomorphic vector bundle on X with projection map π : E → X.A section of E is a holomorphic map s : X → E such that π ◦ s = idX . We can also speak of sections of E over some U ⊆ X. They are simply ordinary sections of E composed with the inclusion U,→ X. Since each fiber of π carries a vector space structure, any section s of E may be scaled point-wise by a holomorphic function f : X → C to obtain another section f · s of E. Thus, the set of sections of E inherits the structure of a module over the ring of holomorphic functions on X. In particular, we can consider only scaling by the constant functions, in which case the sections form a C-vector space. Interesting results arise from asking questions such as: Which vector bundles admit non-trivial global sections? If a vector bundle E admits non-trivial sections, what is the of the C-vector space of sections of E? etc.

2.2 Basic Definitions and Examples The reader should now have picture of the type of geometric objects we would like to study and some questions we can ask about them. In the remainder of this section, we will just need to consider the topological space underlying these objects. Accordingly, we fix a topological space X for the rest of the section.

Definition 2.3. Let Top(X) be the category whose objects are the open subsets of X and whose morphisms are inclusions of open sets, so that for every pair of open subsets U, V ⊆ X, the set HomTop(X)(U, V ) is either a (if U ⊆ V ) or the empty set (if U 6⊆ V ). A presheaf of abelian groups on X is a contravariant functor F : Top(X) → Ab, where Ab is the category of abelian groups.

As a matter of terminology, given a presheaf F on X, we refer to elements of F(U) as sections of F over the open set U ⊆ X. Further, for open subsets V ⊆ U ⊆ X, we refer to the maps F(V → U): F(U) → F(V ) as restriction maps and for a section s ∈ F(U), we will use the shorthand s|V to denote the restricted section F (V → U)(s). Definition 2.4. A presheaf F of X is called a sheaf if it satisfies the following additional condition:

(?) for any open cover {Vi} of any open set U ⊆ X and any elements si ∈ F(Vi) satisfying

si|Vi∩Vj = sj|Vi∩Vj for all i and j, there exists a unique element s ∈ F(U) such that s|Vi = si.

14 These definitions can be easily modified to yield presheaves and sheaves valued in other cate- gories. For example a sheaf of rings on X would be a contravariant functor F : Top(X) → Ring subject to the same condition (?). Other common examples are sheaves valued in sets or R-modules. From the terminology we set above, one might get the impression that sections of a presheaf somehow behave like functions on the open sets of X. This is not a bad intuition to adopt and leads us to some examples. Examples 2.2. • Given a topological space X, the assignment of any open set U ⊆ X to the ring of continuous functions U → R where the restriction maps are ordinary restriction of functions gives rise to a sheaf of rings on X. Here, the sheaf condition is precisely the pasting lemma for continuous functions from topology.

• In the same way, smooth real-valued functions on the open sets of a smooth manifold form a sheaf, commonly denoted C∞.

• Also, complex-valued holomorphic functions on the open sets of a complex manifold form a sheaf. This is often denoted OX , the sheaf of holomorphic functions on X. • More generally, if X is a complex manifold and E is a holomorphic vector bundle, then the vector space of sections of E over an open set U ⊆ X gives rise to a sheaf of complex vector spaces.

• Give Z the discrete topology and let Z(U) be the group of continuous functions U → Z. Then together with restriction maps which are, again, ordinary restriction of functions, we obtain a sheaf Z called the on X. If U is connected, a section of Z over U is just a constant , hence the name “constant sheaf.” Generally, we have ∼ M Z(U) = Z i∈π0(U)

where π0(U) is the set of connected component of U. One defines the constant sheaf R or more generally A for any abelian group A, in a similar fashion.

• For any point x ∈ X and abelian group A, we define the skyscraper sheaf x∗A at the point x to be the presheaf given by  ( idA if x ∈ V ⊆ U A if x ∈ U  (x∗A)(U) = , and (x∗A)(V → U) = A → 0 if x∈ / V, x ∈ U 0 if x∈ / U 0 → 0 if x∈ / U ⊇ V.

The reader should verify that this is, in fact, a sheaf.

• The zero sheaf, denoted 0 is the sheaf given by U 7→ 0 for all open sets U ⊆ X. From these examples, the reader might get the impression that any class of functions on a space X gives rise to a sheaf on X. This is not the case and it is instructive to see a counterexample: bounded real-valued functions on subsets of R form a presheaf but not a sheaf on R. To see this, consider the identity map idR : x 7→ x restricted to all sets in the open cover {Vn = (−n, n) ⊆ R} of R, where n ranges in the positive integers. It is clear that idR |Vn is bounded for all n, yet idR : R → R is not bounded. Thus the sheaf condition is violated!

15 We might say that this is due to the fact that boundedness is a global condition, whereas the properties of functions we considered in Examples 2.2 – continuity, differentiability, and holomor- phicity – are local conditions. It is in this sense that sheaves help formalize our notions of local and global properties. The sheaf condition dictates that in order for some data to form a sheaf, we must be able to “glue together” local data to obtain global data. Our next definition uses the categorical notion of a direct to capture the behavior of a sheaf near a single point p of X.

Definition 2.5. If F is a presheaf on X and p ∈ X is any point, we define the of F at p, denoted Fp, to be the F = lim F(U). p −→ U3p To illustrate this definition further for the reader without as much familiarity with categorical Q limits, we think about the direct limit above as some quotient of the coproduct U3p F(U) where we have quotiented out by exactly enough, but no more than is needed, for the following diagram to commute:

··· F(U) ···

F(X) ··· ··· F(W ) ···

··· F(V ) ···

Fp.

Here, the top three rows represent the (massive!) directed system of F(U)’s for all open sets U 3 p. Any sections in F(U) and F(V ) which restrict to the same section in F(W ) must get sent to the same place in Fp by commutativity of the diagram. Thus elements of Fp are represented by pairs hU, si where U ⊆ X is an open set and s ∈ F(U) is a section over U, and two representatives hU, si and hV, ti represent the same element in Fp if and only if there exists an open set W contained in U ∩ V such that s|W = t|W . In this case we write hU, si ∼ hV, ti. Formally, this is saying that elements of the stalk Fp are germs of sections of F at the point p. Definition 2.6. A of two presheaves F and G is a F → G considered as functors. We use the same definition for a morphism of sheaves.

Now that we have a notion of maps of sheaves, we can talk about the category of sheaves of abelian groups on X denoted Sh(X), whose objects, as the reader no doubt can guess, are sheaves on X and whose morphisms are morphisms of sheaves. A morphisms φ : F → G of presheaves induces a map φp : Fp → Gp on the stalks given by φp : hU, si 7→ hU, φ(U)(s)i. To check well-definedness, suppose hU, si ∼ hV, ti and choose W ⊆ U ∩V with s|W = t|W . Then

φ(U)(s)|W = φ(W )(s|W ) = φ(W )(t|W ) = φ(V )(t)|W

16 by naturality of φ whence hU, φ(U)(s)i ∼ hV, φ(V )(t)i. Our next proposition (which does not hold for presheaves) further illustrates the local nature of sheaves.

Proposition 2.1. A morphism φ : F → G of presheaves on X is an isomorphism if and only if the induced map φp : Fp → Gp is an isomorphism for every p ∈ X.

Proof. The forward direction is clear, if φ is an isomorphism, then φp is an isomorphism for all p ∈ X. The proof of the converse, though a little lengthy, involves a straightforward verification that φ(U) is injective and surjective for all open sets U ⊆ X. All one needs is the sheaf condition and definition of stalks. For more details, we refer the reader to the very clear proof of this result in [Har13, Chap. II, Prop. 1.1].

2.3 Homological Algebra in Sh(X) Section 1 was devoted to developing tools from homological algebra in the category R-Mod. We would like to “port” this tool-kit to category Sh(X). We can do this but the process of “porting” is not immediate. For example, what does it mean for a sequence of sheaves to be exact? How do we make sense of quotients of sheaves? Questions like this demand clarification of the blurry comparison between R-Mod and Sh(X) that we have thus far impressed upon the reader. The purpose of this section is to address these issues and provide the clarification.

Definition 2.7. Let φ : F → G be a morphism of presheaves. We define

• the presheaf kernel of φ as the presheaf U 7→ ker(φ(U)),

• the presheaf cokernel of φ as the presheaf U 7→ coker(φ(U)), and

• the presheaf image of φ as the presheaf U 7→ im(φ(U)).

In fact, if φ : F → G is a morphism of sheaves, the presheaf kernel actually satisfies the sheaf condition (exercise for the reader) and we define the sheaf kernel of φ accordingly (Definition 2.9). We run into a slight technical difficulty, however, if we try to define sheaf cokernels and sheaf images the same way: the presheaf cokernel and presheaf image of φ turn out not to be sheaves in general. We remedy this issue with a technical construction allowing us to “upgrade” a presheaf to a sheaf.

Definition 2.8. Given a presheaf F on X, we define the sheafification of F to be the sheaf F + on X which assigns to each open set U ⊆ X, the set of functions s from U to the disjoint union F p∈U Fp satisfying the property that for all p ∈ X

(1) s(p) ∈ Fp, and (2) there exists a neighborhood V with p ∈ V ⊆ U and a section t ∈ F(V ) such that for all points q ∈ V , we have hV, ti = s(q) in Fq. The restriction maps are given by ordinary restriction of functions.

We leave it to the reader to check that F + is, in fact, a sheaf. Loosely speaking, conditions (1) and (2) endow the sections of F + with local properties. The sheafification F + also comes with a natural morphism of presheaves θ : F → F + which sends sections s ∈ F(U) to functions + F + + s : U → p∈U Fp ∈ F (U) given by s (q) = hU, si ∈ Fq. This gives rise to a universal property of the sheafification.

17 Proposition 2.2. Given a presheaf F, a sheaf G, and any morphism of presheaves φ : F → G, there exists a unique morphism of sheaves φ+ : F + → G such that φ = φ+ ◦ θ:

φ F G

θ . φ+ F +

Proof. [Har13, Chap. 2, Prop. 1.2]

Definition 2.9. A subsheaf of F is a sheaf F 0 such that for each open set U ⊆ X, the group F 0(U) is a subgroup of F(U), and the restriction maps of F 0 are induced by those of F so to make the natural map F 0 → F a morphism of sheaves. If F 0 is a subsheaf of F, we define the quotient sheaf F/F 0 to be the sheafification of the presheaf given by U 7→ F(U)/F 0(U). Finally for any morphisms of sheaves φ : F → G, • we define the kernel of φ denoted ker φ to be the presheaf kernel of φ (which is a sheaf);

• we define the image of φ denoted im φ to the be sheafification of the presheaf image of φ;

• we say φ is injective if ker φ = 0;

• we say φ is surjective if the induced map φ+ : im φ → G from Proposition 2.2 is an isomor- phism. This gives a means to define exactness for a sequence of sheaves in the same way we defined exactness for a sequence of R-modules. Namely, a sequence of sheaves F i and morphisms φi

i−2 i−1 i i+1 φ i−1 φ i φ i+1 φ ··· Fi F F ··· is called exact if ker φi = im φi−1 for all i. Continuing as one would expect by replacing R-modules with sheaves, R-module homomorphisms with morphisms of sheaves, submodules with subsheaves, etc., we can essentially transfer all of our algebraic technology developed in section 1 to our current setting. We leave it to the reader to write out definitions for: cochain complexes of sheaves, coho- mology of cochain complexes of sheaves, short exact sequences of sheaves, (left/right) exact func- tors F : Sh(X) → R-Mod, injective sheaves, projective sheaves, (left/right/injective/projective/F - acyclic) resolutions of sheaves, and (left/right) derived functors of a (right/left) exact functor F : Sh(X) → R-Mod. The only difficulty with the above task that we have set for the reader might be in providing the final two definitions: resolutions and derived functors. Indeed, it is not yet clear that, for instance, injective resolutions of sheaves even exist. Luckily, they do and we will spend the rest of this section building up to a theorem for Sh(X) which is directly analogous to Theorem 1.2 which held for R-Mod. In fact, for the purposes of cohomology, we will only need half of the theorem – the statement about injectives. To begin, we will introduce an important class of sheaves, sheaves of OX -modules.

Definition 2.10. A (X, OX ) is a topological space X together with a sheaf of rings OX on X.

In many contexts, the sheaf OX of a ringed space is referred to as the structure sheaf. The first three bullet points in Examples 2.2 gives sheaves which, together with X, form a ringed space.

18 Definition 2.11. Let (X, OX ) be ringed space. A sheaf of OX -modules is a sheaf F on X such that for each U, the sections F(U) carry an OX (U)-module structure, and for each inclusion V ⊆ U, the restriction map F(U) → F(V ) is an OX (U)-module homomorphism where F(V ) is viewed as an OX (U)-module via the action induced by restriction: s · m = s|V · m for any s ∈ OX (U) and m ∈ F(V ). Thus, we have (s · m)|V = s|V · m|V for any module element m ∈ F(U) and ring element s ∈ OX (U). The fourth bullet of Examples 2.2, sections of a vector bundle E over a complex manifold X, can actually be viewed as giving rise to a sheaf of OX -modules, more than just a sheaf of vector spaces. Indeed, sections of E can be scaled by any holomorphic function, not just constant functions.

Proposition 2.3. Let (X, OX ) be a ringed space. Then any sheaf of OX -modules is a subsheaf of some of OX -modules. That is, for any sheaf of OX -modules F, there exists an injective sheaf of OX -modules I and an injective morphism of OX -modules F → I. Proof. This proof involves a technical construction of I. Informally, one builds I from stalks which are injective modules and then invokes Theorem 1.2 to get the map F → I. We refer the reader to [Har13, Prop. 2.2] for more details.

Generally, any abelian category in which this result holds, that is, in which every object is a of some , is said to have enough injectives.

Corollary 2.1. The category Sh(X) of sheaves of abelian groups on X has enough injectives.

Proof. Let OX be the constant sheaf of rings Z (Example 2.2). Then a sheaf of OX -modules is the same as a sheaf of abelian groups and we may apply Proposition 2.3 directly.

Corollary 2.2. Every sheaf on X has an injective resolution.

Proof. This is the same proof as for Proposition 1.3.

2.4 Cohomology of Sheaves It is now that all of our hard work developing homological algebra and sheaf theory finally pays off. We begin by introducing, for any topological space X, the global sections functor

Γ(X, −): Sh(X) → Ab which takes any sheaf F and returns the global sections Γ(X, F) := F(X) of F. This functor turns out to be left exact.

Lemma 2.1. The global sections functor is left exact.

Proof. This follows from general categorical properties of . One can show that the global section functor is a right-adjoint, and hence is left exact.

In light of this lemma and Corollary 2.2, we are finally ready to define sheaf cohomology.

Definition 2.12. We define the cohomology functor Hi(X, −) to be the ith right derived functor of Γ(X, −), for i ≥ 0. For any sheaf F, the groups Hi(X, F) are the cohomology groups of F.

19 Note that even if X and F have some additional structure, for instance if X is a ringed space and F an OX -module, we always take cohomology in this sense, viewing F as a sheaf of abelian groups on the underlying topological space X. Defining the cohomology functors as derived functors automatically affords us the long exact sequence in cohomology. That is, if

0 F G H 0 is a short exact sequence of sheaves on X, then by Theorem 1.4, there exist natural connecting homomorphisms δi : Hi(X, H) → Hi+1(X, F) yielding the long exact sequence:

. .

δn−1 Hn(X, F) Hn(X, G) Hn(X, H)

δn Hn+1(X, F) Hn+1(X, G) Hn+1(X, H)

δn+1 . . .

Further, by general properties of derived functors, we have H0(X, F) = Γ(X, F).

Proposition 2.4. Let F be a sheaf on X and let Q• be a Γ(X, −)-acyclic resolution of F. Then Hi(X, F) =∼ Hi(Γ(X, Q•)) for all i, meaning acyclic resolutions can be used to compute sheaf cohomology.

Proof. This is precisely Proposition 1.4 in the context of sheaves.

3 Applications

Up until this point, the entire paper has been devoted to theory. Now we will attempt to give the reader a sense of how sheaf cohomology is used. In particular, we will highlight some connections to topology, differential geometry, complex analysis, and algebraic geometry.

3.1 Relation to Other Cohomology Theories Sheaf cohomology can be viewed as unifying other cohomology theories under one coherent frame- work. Two of the most widely known such cohomology theories are singular cohomology coming from , and de Rham cohomology coming from differential geometry. One major contrast between sheaf cohomology and other cohomology theories, is that the latter views cohomology as a functor out of the category of topological spaces. Meanwhile, sheaf cohomology fixes a topological space X, and views cohomology as a functor out of the category of sheaves on X. This change in perspective is exactly what allows sheaf cohomology to accommodate more general “coefficients” than simply an abelian group or ring. Hence, from the perspective of sheaf cohomology, the emphasis is on the sheaf more than the space.

20 Singular Cohomlogy

Let X be a topological space and let ∆k(X) denote the on all continuous maps k+1 σ : ∆k → X where ∆k ⊂ R is the standard k-simplex. Precomposing σ with the inclusion th of the i face ∆k−1 ,→ ∆k yields a map ∆k−1 → X (for 1 ≤ i ≤ k), which in turn induces a homomorphism ∂i : ∆k−1(X) → ∆k(X). We can then define d : ∆k−1(X) → ∆k(X) to be the P i alternating sum dk = i(−1) ∂i, and we get a chain complex

dk+1 dk ··· ∆k+1(X) ∆k(X) ∆k−1(X) ··· .

Dualizing this complex by mapping into Z, we obtain the cochain complex k−1 k ··· ∆k−1 d ∆k(X) d ∆k+1(X) ···

k k where ∆ (X) = Hom(∆k(X), Z) and d = Hom(dk, Z). The cohomology of this complex is called n n • singular cohomology of X (with coefficients in Z) and we write H (X; Z) := H (∆ (X)). Theorem 3.1. Let X be a topological space which admits a triangulation. Then for all n ∈ Z, n ∼ n H (X; Z) = H (X, Z). That is, singular cohomology (with coefficients in Z) coincides with cohomology of the constant sheaf Z on X. Proof. [GH14, Chap. 0, §3, pg. 42]

De Rham Cohomology Let M be an n-dimensional smooth (real) manifold and let T ∗M denote the to M. Then for 0 ≤ k ≤ n we can form the kth wedge of T ∗M denoted Vk T ∗M, the sections of which are k called k-forms. Let Ω (M) be the R-vector space of all such k-forms. The together with the d :Ωk(M) → Ωk+1(M) (which acts by ordinary differentiation on Ω0(M) = C∞(M)), we obtain the cochain complex 0 Ω0(M) Ω1(M) Ω2(M) ··· which is called the de Rham complex. The nth cohomology groups of this complex define the de n n • Rham cohomology: HdR(M) := H (Ω (M)). Theorem 3.2 (The de Rham Theorem). Let M be any smooth manifold. Then for all n ∈ Z, n ∼ n HdR(M) = H (M, R). That is, de Rham cohomology coincides with cohomology of the constant sheaf R on M. Proof. Let Ωk denote the sheaf on M arising from sections of Vk T ∗M over any open set U ⊆ M. Then by the Poincar´elemma [BT13, §4, pp.33-35],

0 Ω0 d Ω1 d ··· d Ωn 0 is a cochain complex which is exact at Ωi for i > 0. Moreover, since d : C∞(M) =∼ Ω0(M) → Ω1(M) 0 1 acts by the ordinary differential, we have R = ker(Ω → Ω ) meaning the above cochain complex is a right resolution for R. Furthermore, one can show [GH14, Chap. 0, §3, pg. 42] that the sheaves Ωk are Γ(M, −)-acyclic by virtue of the existence partitions of unity on smooth manifolds. Hence • they form an acyclic resolution of R. But Γ(M, Ω ) is the de Rham complex, hence by Proposition 2.4 we find n ∼ n HdR(M) = H (M, R).

21 3.2 The Exponential Sequence and the Complex Logarithm We use this subsection to give an extended example showing how one can use the long exact se- quence in sheaf cohomology to understand global properties of the complex logarithm from complex analysis.

Example 3.1. For any complex manifold X, we have the following short exact sequence of sheaves on X ι exp ∗ 0 Z OX OX 0 called the exponential sequence. Here, Z is the constant sheaf, OX is the structure sheaf (the sheaf ∗ of holomorphic functions) on X, and OX is the sheaf of nonvanishing holomorphic function on X: ∗ for any open set U ⊆ X, the sections OX (U) are nonvanishing holomorphic functions U → C and they form a multiplicative group. The morphism ι identifies Z with a subsheaf of OX by mapping n 7→ 2πin on each connected component of any open set. The morphism exp sends any section U → C ∈ OX (U) to the section obtained by exponentiating:

z7→ez ∗ U C C ∈ OX (U).

This is nonvanishing since the complex exponential is nonvanishing. f(z) It is clear that Z = ker(exp) because e = 1 on any open set U ⊆ X if and only if f(z) = 2πin for some n ∈ Z and all z in any connected component of U. The fact that exp is surjective can be checked at the level at stalks. For any p ∈ X, let V be an open neighborhood of p and let ∗ ∗ s ∈ OX (V ) be a section giving the hV, si ∈ (OX )p. Since s is nonvanishing, we can find a sufficiently small open disk D ⊆ s(V ) ⊆ C containing p such that 0 ∈/ D and choose a ray extending outward from origin in C which does not intersect D. A choice of such a ray determines a branch of the complex logarithm on its compliment and hence on D, call it logD,s : D → C. Letting V 0 = s−1(D), we find 0 0 expp(hV , logD,s ◦ s|V 0 i) = hV , s|V 0 i = hV, si meaning exp is surjective on stalks, and hence is surjective as a morphism of sheaves. Thus, we can write out the corresponding long exact sequence:

n−1 ∗ δn−1 n n n ∗ δn n+1 ··· H (X, OX ) H (X, Z) H (X, OX ) H (X, OX ) H (X, Z) ··· .

Now consider the situation where X is an open submanifold of the complex plane. There is a natural question arising from complex analysis: given any nonvanishing holomorphic function f g : X → C, can we write g = e for some holomorphic function f : X → C? The answer depends on X, and more specifically, on the topology of X. To illustrate this dependence, we consider two ∗ scenarios: X = C, and X = C = C \{0}. In either case, the beginning of the long exact sequence in cohomology associated to the expo- nential sequence on X is

∗ δ0 1 0 Γ(X, Z) Γ(X, OX ) Γ(X, OX ) H (X, Z) ··· .

1 ∼ 1 If X = C, then X is contractible so by Theorem 3.1 we have H (X, Z) = H (X, Z) = 0, meaning ∗ Γ(X, OX ) → Γ(X, OX ) is actually surjective. So in this case, the answer to our question is yes: f any nonvanishing holomorphic function g : C → C can be written as g = e for some holomorphic function f : C → C!

22 ∗ 1 ∼ On the other hand if X = C , then X is homotopy equivalent to the circle meaning H (X, Z) = 1 ∗ H (X, Z) = Z by Theorem 3.1. Therefore, there is no reason for Γ(X, OX ) → Γ(X, OX ) to be ∗ ∗ surjective and indeed it is not! If g : C → C is the identity map, then there exists no nonvanishing ∗ f holomorphic function f : C → C for which g = idC∗ = e . In particular, this implies that we ∗ cannot define a branch of the complex logarithm on all of C .

3.3 The Riemann-Roch Theorem for Line Bundles The Atiyah-Singer Index Theorem and the Grothendieck-Riemann-Roch Theorem are widely re- garded as two of the major achievements of 20th-century mathematics. They are both generalization of the Riemann-Roch Theorem which gives a formula relating the sheaf cohomology of a complex manifold X (or more generally a ), to invariants relating to vector bundles on X (or more generally, locally free sheaves on X), thus providing an algebraic insight into geometric questions. Specifically, the invariants we will need to consider are the Chern classes of a vector bundle. We introduce these now. Chern classes arise in a number of different contexts (topology, differential geometry, complex geometry, algebraic geometry, etc.) and may be introduced in various ways accordingly. The general idea is that for some space X (depending on context this could be a manifold, variety, th scheme, etc.) and vector bundle E on X, the i Chern class ci(X) is an element of the cohomology ring associated to X (where the cohomology theory also depends on the context), which somehow measures the non-triviality E. Here i is an integer ranging from 0 to the rank of E. For our treatment of Chern classes, we will restrict our attention to line bundles on complex manifolds X. Since line bundles are just vector bundles of rank 1, this means we will only need to consider c0(E) and c1(E) for any line bundle E on X. Recall the exponential sequence

ι exp ∗ 0 Z OX OX 0. from Example 3.1. This is a short exact sequence in Sh(X), so by Theorem 1.4 we get a long exact sequence

0 ∗ δ0 1 1 1 ∗ δ1 2 ··· H (X, OX ) H (X, Z) H (X, OX ) H (X, OX ) H (X, Z) ··· in cohomology. Now, borrowing a result form complex geometry [Huy06, Cor. 2.2.10], there is a 1 ∗ natural isomophism beween the set of line bundles of X and the cohomology group H (X, OX ), letting us identify line bundles with cohomology classes.

1 ∗ Definition 3.1. We can then define the first Chern class c1(E) of any line bundle E ∈ H (X, OX ) to be the image of E under the connecting homomorphism

1 1 ∗ 2 c1 := δ : H (X, OX ) → H (X, Z).

2 0 Hence c1(E) is an element of H (X, Z). Finally, we put c0(E) = 1 ∈ H (X, Z) under the identifi- 0 ∼ ∼ cation H (X, Z) = Γ(X, Z) = Z (X is connected). Definition 3.2. Let F be a sheaf of vector spaces on X. Then Hi(X, F) inherits a natural vector space structure and we define the Euler-Poincar´echaracteristic of F, denoted χ(X, F), by X χ(X, F) = (−1)i dim Hi(X, F). i

23 We will need the Euler-Poincar´echaracteristic for sheaves arising from line bundles. As we saw in Examples 2.2, the section of a holomorphic line bundle E on X gives rise a sheaf that we will denote E. This is a sheaf of OX -modules so, in particular, it is a sheaf of C-vector spaces. Hence we can actually make sense of the Euler-Poincar´echaracteristic of a line bundle by setting χ(X,E) := χ(X, E). The final technicalities to introduce are the exponential Chern character of a line bundle E, L i denoted ch(E), and the Todd class of E, denoted td(E). They are elements of i H (X, Z) ⊗ Q viewed as a under the aforementioned correspondence between sheaf cohomology of the constant sheaf and singular cohomology, and are given formally by 1 1 ch(E) = ec1(E) = 1 + c (E) + c (E)2 + c (E)3 + ··· 1 2 1 6 1 and c1(E) 1 1 2 1 4 td(E) = = 1 + c1(E) + c1(E) − c1(E) + ··· . 1 − e−c1(E) 2 12 720 This lets us finally state the Riemann-Roch Theorem for line bundles on one dimensional com- plex manifolds, also called Riemann surfaces [Huy06, §5.1].

Theorem 3.3 (Riemann-Roch). Let X be a compact Riemann surface, let E be a line bundle on X, and let TX be the on X. Then

χ(X,E) = deg(ch(E)td(TX ))1

L i where deg( )1 denotes the component in the degree 1 grading of i H (X, Z) ⊗ Q.

References

[Alu09] Paolo Aluffi, Algebra: Chapter 0: Chapter 0, vol. 104, American Mathematical Soc., 2009.

[BT13] and Loring W Tu, Differential forms in algebraic topology, vol. 82, Springer Science & Business Media, 2013.

[GH14] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, John Wiley & Sons, 2014.

[Har13] Robin Hartshorne, Algebraic geometry, vol. 52, Springer Science & Business Media, 2013.

[Huy06] Daniel Huybrechts, Complex geometry: an introduction, Springer Science & Business Me- dia, 2006.

[Mun08] James R Munkres, Topology: A first course. 1975, Received: September (2008).

[Wei95] Charles A Weibel, An introduction to homological algebra, no. 38, Cambridge university press, 1995.

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