2 Sheaves and Cohomology

2.1 Sheaves and Presheaves We ﬁx a topological space X. Later we will include assumptions that are satisﬁed by smooth manifolds.

2.1.1 Deﬁnitions and Examples Deﬁnition 2.1. A presheaf of abelian groups F on X assigns to each open U ⊆ X an abelian group F (U) = Γ(U, F ) and for every inclusion of open sets V ⊆ U a homomorphism of abelian groups F ρUV : F (U) → F (V ), often called the restriction map, satisfying F 1 [P1] ρUU = F(U)

F F F [P2] for W ⊆ V ⊆ U, we have ρVW ◦ ρUV = ρUW . If F and G are two presheaves (of abelian groups) on X, then a morphism ϕ : F → G consists of the data of a morphism ϕU : F (U) → G (U) for each open set U ⊆ X such that if V ⊆ U is an inclusion, then we have commutative diagrams

ϕU F (U) / G (U)

F G ρUV ρUV (V ) / (V ). F ϕV G

Remark 2.2. We may form a category TopX whose objects are open sets in X and whose morphisms are simply inclusions of open sets. Then the above deﬁnition says that a presheaf is a ◦ contravariant functor TopX → Ab, and that a morphism of presheaves is a natural transformation of the associated functors.

Deﬁnition 2.3. A sheaf F of abelian groups on X is a presheaf which, for any open set U ⊆ X and any open covering {Ui}i∈I of U, satisﬁes the two additional properties:

[S1] if s ∈ F (U) is such that s|Ui = 0 for all i ∈ I, then s = 0;

[S2] if si ∈ F (Ui) such that si|Ui∩Uj = sj|Ui∩Uj for all i, j ∈ I, then there exists s ∈ F (U) such

that s|Ui = si for each i. Note that such an element is unique by [S1]. A morphism of sheaves is a morphism of the underlying presheaves. Let A be a sheaf of rings on X, i.e. one for which A (U) is a ring for each open U ⊆ X and for which the restriction maps are ring homomorphisms. If F is a sheaf of abelian groups such that for every open U ⊆ X, F (U) is an A (U)-module, then F is called a sheaf of A -modules. Examples 2.4. (a) The constant presheaf Let X be a topological space and let A be an abelian p group. We deﬁne the constant presheaf with values in A to be the presheaf AX to be the presheaf such that

p AX (U) = A

Ap X 1 for all non-empty U ⊆ X, and ρUV = A for all V ⊆ U. (b) The constant sheaf With X and A as above, we deﬁne the constant sheaf with values in A to be the sheaf AX whose sections AX (U) over U are locally constant functions U → A; this is the same as the set of continuous functions U → A, where A is given the discrete topology.

1 0 0 0 (c) Functions Let C = CX be the sheaf for which C (U) is the set of continuous R-valued functions on U. Then C0 is a sheaf. Similarly, if X is a smooth (respectively, complex) mani- ∞ ∞ fold, then CX (respectively, OX ) is a sheaf, where CX (U) (respectively, OX (U)) is the ring of smooth (respectively, holomorphic) functions on U. These are all sheaves of rings.

(d) Skyscraper sheaves Let X and A be as above, and ﬁx x ∈ X. Then we can deﬁne a sheaf Ax by ( A if x ∈ U Ax(U) := 0 otherwise.

0 In the case that that A = C, then Cx is a sheaf of CX -modules, where a function acts on a section simply by multiplying by its value at x. (e) Vector bundles Let X be as above and let π : E → X be a vector bundle over X. Then for any U ⊆ X,

U 7→ Γ(U, E)

deﬁnes a sheaf. For a vector bundle E, we will typically also denote its sheaf of sections by E, as no confusion is likely to occur. By Remark , E is a sheaf of C0-modules; it is a sheaf of ∞ CX -modules or OX -modules if E is smooth or holomorphic, respectively.

2.1.2 Stalks We ﬁrst recall the notion of a colimit. Deﬁnition 2.5. Let I be a small category (i.e., Ob I is a set) and C any category. Let α : I → C be a functor; we thus, think of this data as a set of objects of C indexed by I , with morphisms between these objects indexed by morphisms in I . Any object X ∈ Ob C determines such a functor ∆X : I → C , namely, the constant functor, i.e., ∆X takes all objects of I to X and all morphisms to 1X . If Y ∈ Ob C is another object and f : X → Y a morphism, then there is an induced natural transformation Φ = Φf : ∆X → ∆Y which is simply f at each object of I . We say that α has a colimit X ∈ Ob C if there is a natural transformation λ : α → ∆X such that if Y ∈ Ob C and τ : α → ∆Y is a natural transformation, then there is a unique f : X → Y such that α

λ τ τ = Φf ◦ λ. Ó ∆X / ∆Y Φf commutes. [The colimit is often called the “direct limit,” but for the sake of standardizing lan- guage, “colimit” is preferable.] It is not hard to see that if a colimit exists, then it is unique up to a unique isomorphism. Lemma 2.6. If I is a small category and α : I → Ab is an functor, then colim α exists. L Proof. For i ∈ Ob I , we will write Xi for α(i). Then let Xe := i∈Ob I Xi be the direct sum and let N ⊆ Xe be the subgroup generated by elements of the form (··· , 0, xi, ··· , −α(f)(xi), ··· ) for morphisms f : i → j. It is left as an exercise to show that X := X/Ne is the colimit. L Remark 2.7. From the construction above, any element of colim α is a tuple in Xi with ﬁnitely many non-zero components, corresponding to i1, . . . , ir ∈ Ob I . Suppose that for any such i1, . . . , ir, there is j ∈ Ob I with arrows ip → j for 1 ≤ p ≤ r. Then by using the relation in the proof above, one can represent any element of colim α by one in the image of Xj → X.

2 Let X be a topological space and ﬁx a point x ∈ X. Recall that TopX was the category whose objects are open sets in X and whose morphisms are inclusions. Consider the (full) subcategory ◦ TopX,x whose objects are open neighbourhoods of x. It is easy to see that TopX,x has the property mentioned in Remark 2.7, since the intersection of any ﬁnite set of open neighbourhoods of x is also one. ◦ Now, if F is a presheaf, which we may think of as a functor F : TopX → Ab, then we can ◦ restrict it to F : TopX,x → Ab. By Lemma 2.6,

Fx := colimTopX,x F = colimx∈U F (U) exists; we call it the stalk of F at x. By properties of the colimit, for each open neighbourhood U of x, we have a natural map F (U) → Fx and these “are compatible” with the restriction maps. Remark 2.7 says that any element of Fx is the image of some s ∈ F (U) for some neighbourhood U of x. We may repeat the above deﬁnition for an arbitrary set S ⊆ X, instead of {x}, by deﬁning the subcategory TopX,S of TopX with objects open sets U ⊆ X with S ⊆ U. In this way, we can extend the presheaf F to all subsets of X by

F (S) := colimS⊆U F (U).

2.2 Sheaﬁﬁcation Proposition 2.8. Let F be a presheaf on X. Then there exists a sheaf F + together with a morphism θ : F → F + such that if ϕ : F → G is any morphism and G is a sheaf, then there is a unique morphism Φ: F + → G such that

F + θ 8

F Φ ϕ = Φ ◦ θ

ϕ & G commutes.

Definition 2.9. The sheaf F + is called the sheafification of F . Remark 2.10. If F is a sheaf, then θ : F → F + is an isomorphism. Proof. We define F + as follows. Let U ⊆ X be open. We let F +(U) be the set of elements Y σ ∈ Fx x∈U such that for each x ∈ U there is an open neighbourhood V of x contained in U and a section s ∈ F (V ) such that pry(σ) = sy for all y ∈ V , where pry is the obvious projection map. We leave it as an exercise to show that this is a sheaf. Q Given s ∈ F (U), there is a unique σ ∈ x∈U Fx, where prx(σ) = sx; clearly, we have σ ∈ F +(U) and so we set

θU (s) := σ.

It is an exercise to verify that this deﬁnes θ. Finally, we also leave as an exercise the proof of the mapping property.

3 2.3 Kernels and Images Deﬁnition 2.11. If F is a presheaf, then a subpresheaf G of F is a presheaf for which G (U) is a subgroup of F (U) for all open U ⊆ X and for which the restriction morphisms are induced from those of F .A subsheaf of a sheaf is simply a subpresheaf which is also a sheaf. Lemma 2.12. If G is a subpresheaf of a sheaf F (which is not necessarily a sheaf itself), then G + is (or may be realized as) a subsheaf of F .

Proof. It is an exercise to show that since G is a subpresheaf of F , for each x ∈ X, Gx ⊆ Fx. For each open U ⊆ X, let

p G (U) := {s ∈ F (U): ∀x ∈ X. sx ∈ Gx} ⊆ F (U).

Then G p is a subsheaf of F . There is a natural map θ : G → G p and with this map, G p satisfies the universal mapping property of G +, and so G p =∼ G +. Definition 2.13. Let ϕ : F → G be a morphism of presheaves. Then we may define the kernel presheaf kerp ϕ and the image presheaf impϕ by

p p ker ϕ(U) := ker ϕU ⊆ F (U) im ϕ(U) := im ϕU ⊆ G (U).

Lemma 2.14. If ϕ : F → G is a morphism of sheaves, then kerp ϕ is a sheaf. Proof. This is left as an exercise.

× × Example 2.15. Let X = C , and OX , OX the sheaves of holomorphic functions and non-vanishing × holomorphic functions, respectively. Consider the exponential map exp : OX → OX . × We claim the image presheaf in OX is not a sheaf. Let {Ui}i∈I be a covering of X by simply 1 × connected open sets. Then it is clear that Ui ∈ OX (Ui) for all i ∈ I. Since one can choose a 1 logarithm on each simply connected open set, each Ui lies in the image presheaf. These obvi- ously agree on overlaps Ui ∩ Uj, however, there is no holomorphic function f ∈ OX (X) with exp(f) = 1X .

Definition 2.16. If ϕ : F → G is a morphism of sheaves, then the kernel of ϕ is simply defined as the sheaf ker ϕ := kerp ϕ. We define the image of ϕ as the sheafification im ϕ := (imp ϕ)+. By Lemma 2.12, this is a subsheaf of G . We say that ϕ is injective if ker ϕ = 0 and ϕ is surjective if im ϕ = G . Proposition 2.17. Let ϕ : F → G be a morphism of sheaves. Then ϕ is injective (respectively, surjective, bijective) if and only if ϕx : Fx → Gx is injective (respectively, surjective, bijective) for all x ∈ X. Proof. Exercise.

2.4 Complexes Virtually all of the deﬁnitions we now make in the category of sheaves of abelian groups over a topological space X make sense for a general abelian category, but we will not go into this generality in the interest of saving time.

Deﬁnition 2.18. A complex F • of sheaves (of abelian groups) over X is a sequence of morphisms of sheaves of abelian groups

di−1 di · · · → F i−1 −−−→ F i −→ F i+1 → · · · such that di ◦ di−1 = 0 for all i ∈ Z; the maps di are often called differentials. Clearly, im di−1 is a subsheaf of ker di for all i ∈ Z; if they are equal, then we say that the complex is exact. If F • and

4 G • are complexes of sheaves over X, then a morphism f • : F • → G • is a collection of morphisms ϕi : F i → G i for i ∈ Z such that

ϕi F i / G i

i i+1 i i i d di F G ϕ ◦ dF = dG ◦ ϕ F i+1 / G i+1 ϕi+1 commutes. Kernels and images of morphisms of complexes can be defined in the obvious way, as can exactness of sequences. Of course, all these definitions can be made for complexes of abelian groups, as well. • i • Lemma 2.19. The complex F is exact at F if and only if for all x ∈ X, the complex Fx of i abelian groups is exact at Fx. Proof. Exercise. Definition 2.20. Let ϕ•, ψ• : F • → G • be two morphisms of complexes. A homotopy from ϕ• to ψ• is a collection of morphisms ki : F i → G i−1 such that

di F i F / F i+1 i i i i+1 i i−1 i k ϕ − ψ = k ◦ d + d ◦ k ϕi ψi F G ki+1 | | G i−1 / G i di−1 G for all i ∈ Z. It is not hard to show that the relation “there exists a homotopy from f to g” is an equivalence relation on the set of morphisms from F • to G •. We say that the morphisms ϕ• and ψ• are homotopic if there is a homotopy from ϕ• to ψ•. Proof. Exercise.

2.5 Resolutions Deﬁnition 2.21. Let F be a sheaf on X.A resolution of F is a complex of sheaves A •, non-zero only for indices in Z≥0, such that 0 → F → A 0 → A 1 → · · · is exact. Proposition 2.22 (de Rham resolution). Let M be a smooth manifold of (real) dimension r and the constant sheaf with coefﬁcients in . Then if k is the sheaf of -valued differential RM R AM,R R k-forms on M, one has a resolution

0 → → 0 −→d 1 −→·d · · −→d r → 0 RM AM,R AM,R AM,R of RM . The statement also holds with R replaced by C. Proof. 0 is the sheaf of smooth functions on M and since locally constant functions are cer- AM,R tainly smooth, we have an inclusion ,→ 0 . Furthermore, on any open set U ⊆ M, the RM AM,R kernel of d : 0 (U) → 1 (U) is precisely the subgroup of locally constant functions. To AM,R AM,R see that it is exact at k for any k ≥ 1, we may pass to the stalks: let x ∈ M; an element AM,R ν ∈ ( k ) with dν = 0 is represented by some α ∈ k (U) for some open neighbourhood AM,R x AM,R U of x with dα = 0. If U is an open ball, then the Poincare´ lemma states that there is some k−1 k−1 β ∈ A (U) with dβ = α; if ξ ∈ (A )x is the image of β in the stalk, then dξ = ν, and so we M,R M,R have exactness at stalks. The proof is exactly the same for C.

5 2.5.1 The Dolbeault Complex of a Holomorphic Bundle Let E be a holomorphic vector bundle of rank r over a complex manifold X. For an open set 0,q 0,q ∼ r U ⊆ X, let A (U, E) be the space of smooth sections of E ⊗ Ω . If ϕ : EU −→ U × C is a 0,q t trivialization, then a section b ∈ A (U, E) corresponds to a r-tuple pr r ◦ϕ◦b = β = (β1, . . . , βr) C 0,q 0,q 0,q+1 with βi ∈ A (U), 1 ≤ i ≤ r. Now, we deﬁne an operator ∂E : A (U, E) → A (U, E) by taking

−1 −1 t ∂Eb = ϕ (∂β) = ϕ (∂β1,..., ∂βr) .

Lemma 2.23. The deﬁnition of ∂E is independent of the choice of trivialization and so gives a well-deﬁned operator A 0,q(U, E) → A 0,q+1(U, E) for any open set U ⊆ X.

0,q Proof. Suppose ϕi, ϕj are trivializations of E on Ui,Uj, respectively, and suppose b ∈ A (Ui ∩ Uj) corresponds to βi, βj with respect to ϕi, ϕj. If the transition function is gij, then we know that βi = gijβj. To see that ∂E is well-deﬁned, we want to see that

−1 −1 ϕi (∂βi) = ϕj (∂βj).

But since gij is holomorphic

−1 ∂βi = ∂(gijβj) = ∂gij ∧ βj + gij∂βj = gij∂βj = ϕi ◦ ϕj (∂βj).

Proposition 2.24. In the following U ⊆ X is open.

2 (a) We have ∂E = 0. (b) We have

0,0 0,1 Γ(U, E) = ker ∂E : A (U, E) → A (U, E) .

0 0 (c) If α ∈ A 0,q(U), β ∈ A 0,q (U, E), then α ∧ β ∈ A 0,q+q (U, E) and

q ∂E(α ∧ β) = ∂α ∧ β + (−1) α ∧ ∂Eβ.

0,q (d) Let α ∈ A (U, E), q > 0 be such that ∂Eα = 0. Then given x ∈ U, there is a neighbourhood V of x and β ∈ A 0,q−1(U, E) with

α|V = ∂Eβ.

Proof. These are all clear from the local deﬁnition above and the relevant statements for (p, q)- forms: Lemma ?? for (a), Remark ?? for (b), Lemma ?? for (c) and Proposition ?? for (d). Proposition 2.25 (Dolbeault resolution). Let X be a complex manifold of dimension n and let E be a holomorphic vector bundle on X. Then the complex

∂ ∂ ∂ 0 → E → A 0(E) −−→E A 0,1(E) −−→·E · · −−→E A 0,n(E)

p gives a resolution of the sheaf of sections of E. In the particular case where E = ΩX , then one has the resolution

p p,0 ∂ p,1 ∂ ∂ p,n 0 → ΩX → A −→ A −→· · · −→ A .

Proof. The proof is as for the de Rham resolution, except that we use Proposition 2.24 (d) in the place of the Poincare´ lemma.

6 2.6 Injective Sheaves Deﬁnition 2.26. Let X be a topological space. A sheaf I over X is called injective if given an exact sequence

0 → F → G and a morphism F → I , there exists a (not necessarily unique) morphism G → I making the diagram

0 / F / G

~ I commute. (Thus, any morphism from a subsheaf to an injective can be extended to the whole sheaf.) A sheaf F is called ﬂasque or ﬂabby if for every inclusion of open sets V ⊆ U, the restriction map F (U) → F (V ) is surjective.

Lemma 2.27. A sheaf I is injective if and only if the functor HomSh(X)(·, I ): Sh(X) → Ab is exact.

Proof. Exercise. Lemma 2.28. Every injective sheaf is ﬂasque.

Proof. Let I be an injective sheaf and let V ⊆ U be an inclusion of open sets. We deﬁne a sheaf ZU,X on X by ( ZX (W ) if W ⊆ U ZU,X (W ) := 0 otherwise.

We can similarly deﬁne ZV,X and we have ZU,X (W ) = ZV,X (W ) unless W ⊆ U and W 6⊆ V in which case the right side is zero. In all cases, the sections of ZV,X are contained in those of ZU,X and hence we have an exact sequence of sheaves on X

0 → ZV,X → ZU,X .

Applying the functor HomSh(X)(·, I ), which is exact by Lemma 2.27, we get an exact sequence

HomSh(X)(ZU,X , I ) → HomSh(X)(ZV,X , I ) → 0.

But now suppose that ϕ : ZU,X → I is a sheaf homomorphism. Then if W 6⊆ U, ϕW must be zero. If W = U, then the group of sections of ZU,X (U) over any connected component is simply Z and hence ϕU on this connected component is determined by the image of 1 the space of sections of I on this connected component; thus, ϕU can be thought of an element of I (U). Now, on any proper open subset of U, ϕ is determined by the restriction maps. Hence HomSh(X)(ZU,X , I ) = I (U). Doing the same thing above for V , we get an exact sequence I (U) → I (V ) → 0, which was our claim. Lemma 2.29. Every abelian group can be embedded into an injective abelian group. Proof. Exercise: give a proof or ﬁnd a precise reference for this statement. Lemma 2.30. Let X be a topological space. Every sheaf of abelian groups on X can be embedded in an injective sheaf.

7 Remark 2.31. This statement does not hold for an arbitrary abelian category. This is the assumption that an abelian category “has enough injectives.”

Proof. For each x ∈ X, Lemma 2.29 tells us that there is an injective abelian group Ix with Fx ⊆ Ix. Let Y I (U) := Ix. x∈U

Then this deﬁnes a sheaf I on X with the restriction maps coming from the projections. We claim that for any sheaf G we have

∼ Y HomSh(X)(G , I ) = HomAb(Gx,Ix). x∈X

Given ϕ : G → I and x ∈ X, for any open neighbourhood U of x, we have maps

ϕU Y prx G (U) −−→ I (U) = Ix −−→ Ix, x∈U and since Gx is deﬁned as a colimit, this deﬁnes a unique map Gx → Ix. As we have one such for each x ∈ X, we get a map from the left to the right. In the other direction, suppose we are given Q f ∈ x∈X HomAb(Gx,Ix). Then we have a map in the opposite direction via the composition

Y f|U Y G (U) → Gx −−→ Ix, x∈U x∈U and one can check that these are inverse to each other. Now, given an inclusion

0 → F → G , we apply HomSh(X)(·, I ) and use the bijection above to obtain a diagram

HomSh(X)(G , I ) / HomSh(X)(F , I )

Q Q x∈X HomAb(Gx,Ix) / x∈X HomAb(Fx,Ix) whose vertical maps are bijections; we want to show that the top row is surjective, but this is equivalent to the bottom row being surjective. But since we have inclusions 0 → Fx → Gx for each x ∈ X,

HomAb(Gx,Ix) → HomAb(Fx,Ix) is surjective since each Ix is an injective abelian group.

Remark 2.32. One should be careful to note that in the above proof, it is not true that Ix = Ix; the stalk Ix depends very much on the topology of X. Deﬁnition 2.33. A resolution F → I • of a sheaf F is called injective if the I i are injective sheaves for all i ≥ 0. Corollary 2.34. Every sheaf of abelian groups has an injective resolution.

8 Proof. Let F ∈ Ob Sh(X). By Lemma 2.30, there is an injective sheaf I 0 and an exact sequence 0 → F → I 0. We may consider the sheaf I 0/F and ﬁnd an injective object I 1 with 0 → I 0/F → I 1 exact; then

0 → F → I 0 → I 1 is exact. The resolution is obtained by continuing along in this manner.

Lemma 2.35. Let ϕ : F → G be a morphism of sheaves and G → J • an injective resolution. (a) If F → S • is any resolution, then there is a morphism of complexes ϕ• : S • → J • such that

ϕ F / G

S • / J • ϕ•

commutes. (b) If ψ• : S • → J • is any other morphism of complexes making the above diagram commute, then ϕ• and ψ• are homotopic. Proof. Exercise. Lemma 2.36. Let

0 → F → G → H → 0 be an exact sequence of sheaves on X. Then there exist injective resolutions I •, J •, K • of F , G , H , respectively and a commutative diagram

0 0 0

0 / F / G / H / 0

0 / I • / J • / K • / 0, where the bottom row is an exact sequence of complexes. Proof. Exercise.

2.7 Sheaf Cohomology Given a sheaf F , we may take its group of global sections Γ(X, F ), and this deﬁnes the global section functor Γ(X, ·): Sh(X) → Ab. Lemma 2.37. The functor Γ(X, ·) is left exact in the sense that if

0 → F → G → H is an exact sequence of sheaves, then

0 → Γ(X, F ) → Γ(X, G ) → Γ(X, H ) is exact as a sequence of abelian groups.

9 Proof. The exact sequence says that F is the kernel of the map G → H . Exactness at Γ(X, F ) simply says that the global sections of F are a subgroup of those of G ; exactness at Γ(X, G ) says that Γ(X, F ) is the kernel of Γ(X, G ) → Γ(X, H ), but the kernel of this is the group of global sections of the kernel presheaf, which is precisely F , by Lemma 2.14. Deﬁnition 2.38. Let X be a topological space and let F ∈ Ob Sh(X) be a sheaf of abelian groups on X. Let 0 → F → I • be an injective resolution of F . The ith (sheaf) cohomology group of F is deﬁned as

Hi(X, F ) := Hi(Γ(X, I •)).

Lemma 2.39. The deﬁnition above is independent (up to isomorphism) of the choice of injective resolution of F . Proof. Exercise. Lemma 2.40. Let

0 → F → G → H → 0 be an exact sequence of sheaves with F ﬂasque. Then

0 → Γ(X, F ) → Γ(X, G ) → Γ(X, H ) → 0 is exact. Proof. Exercise. Theorem 2.41. The sheaf cohomology groups satisfy the following properties. (a) We have H0(X, F ) =∼ Γ(X, F ). (b) If I is an injective sheaf, then Hi(X, I ) = 0 if i > 0. (c) If ϕ : F → G is a sheaf morphism, then for i ≥ 0, there is a group homomorphism

Hi(ϕ): Hi(X, F ) → Hi(X, G )

such that

0 (i) ϕ = ϕX : F (X) → G (X); i 1 1 (ii) H ( F ) = Hi(X,F) for all i ≥ 0; (iii) if ψ : G → H is a sheaf morphism, then Hi(ψ ◦ ϕ) = Hi(ψ) ◦ Hi(ϕ).

(d) If

0 → F → G → H → 0

is a short exact sequence of sheaves, then there is a long exact sequence

0 → H0(X, F ) → H0(X, G ) → H0(X, H ) → H1(X, F ) → H1(X, G ) → H1(X, H ) → · · ·

such that if

0 / F / G / H / 0

0 / P / Q / R / 0

10 is a commutative diagram with exact rows, then

0 / Γ(X, F ) / Γ(X, G ) / Γ(X, H ) / H1(X, F ) / H1(X, G ) /

0 / Γ(X, P) / Γ(X, Q) / Γ(X, R) / H1(X, P) / H1(X, Q) / commutes.

Proof. If F → I • is an injective resolution, then

0 → F → I 0 → I 1 is exact, so by left exactness of Γ(X, ·) (Lemma 2.37), we have

d0 0 → Γ(X, F ) → Γ(X, I 0) −→ Γ(X, I 1), but H0(X, F ) is by deﬁnition ker d0 (since there is no −1-term), but the diagram says that this is precisely Γ(X, F ). This proves (a). If I is an injective sheaf, then one can take an injective resolution with I 0 = I and I i = 0 for i > 0; then it is obvious that Hi(X, I ) = 0 for i > 0. This proves (b). Given injective resolutions F → I •, G → J •, Lemma 2.35 gives a morphism of complexes ϕ• : I • → J • making

ϕ F / G

I • / J • ϕ• commute, unique up to homotopy. We then take global sections and get a map of complexes of • • • abelian groups ϕX : Γ(X, I ) → Γ(X, J ), and we deﬁne

i i • i i i i H (ϕ) := H (ϕ ): H (X, F ) = H Γ(X, I ) → H (X, G ) = H Γ(X, J ) .

0 0 0 One sees that property (i) holds by since ϕX will take ker dI to kerJ and that (ii) and (iii) hold by uniqueness (i.e., the fact that the relevant maps of complexes are homotopic). This proves (c). Suppose we have an exact sequence as given. Then we may use Lemma 2.40 to obtain an exact sequence of complexes

0 → I • → J • → K • → 0, where I •, J •, K • are injective resolutions of F , G , H , respectively. If we apply Γ(X, ·), then the rows remain exact, since each I i is injective and hence ﬂasque, by Lemma 2.28. Therefore we obtain a diagram with exact rows

q0 p0 0 / Γ(X, I 0) / Γ(X, J 0) / Γ(X, K 0) / 0

0 0 0 dF dG dH q1 p1 0 / Γ(X, I 1) / Γ(X, J 1) / Γ(X, K 1) / 0

1 1 1 dF dG dH 0 / Γ(X, I 2) / Γ(X, J 2) / Γ(X, K 2) / 0, q2 p2

11 so we can apply the snake lemma, which gives an exact sequence

Γ(X, I 1) Γ(X, J 1) Γ(X, K 1) 0 → Γ(X, F ) → Γ(X, G ) → Γ(X, H ) → 0 → 0 → 0 . im dF im dG im dH 1 1 0 1 0 Note that H (X, F ) = ker dF /im dF ⊆ Γ(X, I )/im dF , so we need to show that the image of the connecting homomorphism lies in H1(X, F ). Recall that if z ∈ Γ(X, H ) then to obtain its 0 0 0 0 1 image in coker dF , we choose y ∈ Γ(X, J ) with p (y) = z and then observe that dG (y) = q (x) 1 0 for some x ∈ Γ(X, I ), and x¯ ∈ cokerdF is the image of z. If x is obtained like this, then we have 2 1 1 1 1 0 q (dF x) = dG (q x) = dG (dG y) = 0, and since q2 is injective, we see that indeed x¯ ∈ H1(X, F ). Then it is not hard to see that 1 1 1 1 1 1 q , p map ker dF , ker dG to ker dG , ker dH , respectively, so we may replace the last three terms by H1(X, F ) → H1(X, G ) → H1(X, H ). One must also check that this preserves exactness: for 1 1 1 1 example, if y ∈ ker dG represents a class y¯ ∈ H (X, G ) in the kernel of H (X, G ) → H (X, H ), 1 1 0 0 then the snake lemma tells us that there is some x ∈ Γ(X, I ) with q x = y + dG y for some y0 ∈ Γ(X, J 0). But then

1 0 0 1 1 2 1 0 = dG (y + dG y ) = dG q x = q dF x

2 1 1 and since q is injective, dF x = 0, and so x¯ ∈ H (X, F ), which proves that exactness is main- tained. One continues in this manner to obtain the existence of the long exact sequence. The last statement is left as an exercise.

2.8 Acyclic Sheaves Deﬁnition 2.42. A sheaf S is called acyclic if

Hi(X, S ) = 0 for i > 0. If F is any sheaf, then a resolution 0 → F → S • is called an acyclic resolution if each of the S i is acyclic. Proposition 2.43. If F → S • is any resolution of F , then there are canonical maps

i • i H Γ(X, S ) → H (X, F ) for i ≥ 0, which are isomorphisms if S • is acyclic. Proof. Lemma 2.35 gives us the existence of the a morphism of complexes ϕ• : S • → I • such that

F / S •

ϕ• F / I • commutes and also states that any other such morphism is homotopic to ϕ•, so we get a well- deﬁned map as claimed. i i i−1 i i i−1 0 0 Let K := ker dS = im dS and L := ker dI = im dI and note that K = L = F . For i ≥ 0, we have commutative diagrams

0 / K i / S i / K i+1 / 0

0 / L i / I i / L i+1 / 0

12 with exact rows. Let us take i = 0 and take global sections to get:

0 / F (X) / S 0(X) / K 1(X) / H1(X, F ) / H1(X, S 0) / H1(X, K 1) / H2(X, F )

0 / F (X) / I 0(X) / L 1(X) / H1(X, F ) / H1(X, I 0) / H1(X, L 1) / H2(X, F ).

1 1 0 Noting that Γ(X, K ) = ker dS ,X , assuming that S is acyclic, we get an isomorphism

1 • 1 0 ∼ 1 H (Γ(X, S )) = ker dS ,X /im dS ,X −→ H (X, F ),

∼ compatible with the map to I 0. Furthermore, for i ≥ 1, we have isomorphisms Hi(X, K 1) −→ Hi(X, L 1) compatible with isomorphisms to Hi+1(X, F ). Taking i = 1 and global sections, we get

0 / K 1(X) / S 1(X) / K 2(X) / H1(X, K 1) / H1(X, S 1) / H1(X, K 2) / H2(X, K 1)

0 / L 1(X) / I 1(X) / L 2(X) / H1(X, L 1) / H1(X, I 1) / H1(X, L 2) / H2(X, K 1). and repeating the same arguments proves the statement.

2.9 Soft and Fine Sheaves From now on, we will assume that X is a paracompact Hausdorff space. Note that topological (and hence smooth and complex) manifolds are such.

2.9.1 Soft Sheaves Deﬁnition 2.44. A sheaf F is soft if for any closed set S ⊆ X, the restriction map

F (X) → F (S) is surjective, i.e., any section over S can be extended to a global section. Lemma 2.45. Flasque sheaves are soft.

Proof. Let F be a ﬂasque sheaf, S ⊆ X a closed set and let s ∈ F (S) = colimU⊇SF (U). Then s is represented by some s¯ ∈ F (U) for some U ⊇ S. Since F is ﬂasque, there is s˜ ∈ F (X) that restricts to s¯ and hence to s. Theorem 2.46. If

0 → F → G → H → 0 is an exact sequence of sheaves and F is soft, then the induced sequence of global sections

0 → F (X) → G (X) → H (X) → 0 is exact. Proof. Exercise.

Corollary 2.47. If F and G are soft and

0 → F → G → H → 0 is exact, then H is soft.

13 Proof. Let S ⊆ X be closed and s ∈ H (s). Consider the above as a sequence of sheaves on S: then F |S remains soft (since any closed set in S is also closed in X). Then applying the Theorem for the sequence over S, there is a section of t ∈ G (S) which maps to s. Since G is soft, there is a section t˜∈ G (X) restricting to t. Then the image of t˜∈ H (X) will map to s. Corollary 2.48. If

d0 d1 0 → S0 −→ S1 −→· · · is an exact sequence of soft sheaves, then

0 → Γ(X, S0) → Γ(X, S1) → · · · is also exact. Proof. Let K i := ker di = im di+1. We ﬁrst show by induction that K i is soft for i ≥ 1. We have K 1 = S 0 and this is soft by assumption. Now, we have exact sequences

0 → K i → S i → K i+1 → 0, and by Corollary 2.47, K i+1 is soft if K i is. Therefore, Theorem 2.46 gives exact sequences

0 → Γ(X, K i) → Γ(X, S i) → Γ(X, K i+1) → 0, and it follows that the exact sequence of global sections is exact. Corollary 2.49. Soft sheaves are acyclic. Proof. Let S be a soft sheaf, S → I • an injective resolution. Then since S = ker(I 0 → I 1), we have an exact sequence

0 → I 0/S → I 1 → I2 → · · · and since injective sheaves are ﬂasque (Lemma 2.28) and ﬂasque sheaves are soft (Lemma 2.45), applying Corollary 2.48, we obtain the conclusion.

2.9.2 Fine Sheaves

Definition 2.50. Let U = {Ui}i∈I be a locally finite open cover of X (i.e., every point of X has a neighbourhood which intersects only finitely many Ui). Recall that a partition of unity subordinate to U is a collection of continuous functions {fi : X → [0, 1]}i∈I such that for each i ∈ I,

supp fi := {x ∈ X : fi(x) 6= 0} ⊆ Ui and for any x ∈ X, X fi(x) = 1. i∈I

[Note that since the cover is locally finite, this is a finite sum for any x ∈ X.] It is a fact that a Hausdorff space is compact if and only if every open cover admits a locally finite refinement with a subordinate partition of unity; in the case that X is a smooth manifold, then the partition of unity may be taken to be smooth. A sheaf F is fine if for any locally finite open cover {Ui}i∈I of X, there exists a family of sheaf morphisms {ηi : F → F }i∈I such that P 1 (a) i∈I ηi = F ;

(b) supp ηi ⊆ Ui, where supp ηi = {x ∈ X : ηi,x 6= 0}.

14 The family {ηi}i∈I is called a partition of unity of F subordinate to the covering {Ui}i∈I . The following is then clear. 0 0 0 Lemma 2.51. Let C = CX be the sheaf of continuous R-valued functions on X (so that C (U) is a ring for every open U ⊆ X). If F is a sheaf such that F (U) is a C 0(U)-modules for all open U in such a way scalar multiplication is compatible with the restriction maps (in which case we call F a sheaf of C 0-modules), then F is ﬁne. Remark 2.52. It is clear that in the above, if X is a smooth manifold, then we may replace C 0 with AX,R or AX , the sheaf of rings of R- or C-valued functions. However, even if X is a com- 0 plex manifold, we cannot replace C by OX , since partitions of unity will never be holomorphic functions. Example 2.53. The following sheaves are ﬁne

Vk ∗ (a) T X, or more generally, any locally free sheaf of AX -modules, when X is a smooth manifold. p,q (b) AX when X is a complex manifold. Proposition 2.54. Fine sheaves are soft.

Proof. Let F be a ﬁne sheaf, S ⊆ X closed and s ∈ F (S). Let {Ui} be an open covering of S and let si ∈ F (Ui) be such that

si|S∩Ui = s|S∩Ui . ` Let U0 = X \ S and s0 := 0. Then {Ui} {U0} is an open covering of X; since X is paracompact, we may assume that this covering is locally ﬁnite, and that there is a partition of unity {ηi} subordinate to it. Now, for each i, ηi(si) ∈ F (Ui) and this vanishes on an open neighbourhood of X \ Ui, so it can be extended to a section on all of X, which we will also denote by ηi(si). If we set X s˜ := ηi(si) ∈ F (X), i then s˜ maps to s. Corollary 2.55. (a) If X be a smooth manifold, then

i i H (X,RX ) = HdR(X,R),

where the left side is the sheaf cohomology groups of the constant sheaf RX (for R = R or C) and the right are the de Rham cohomology groups. (b) If X is a complex manifold and E is a holomorphic vector bundle over X, then

i i 0,• H (X,E) =∼ H A (X,E) ,

and in particular, one has Dolbeault’s theorem: q p ∼ p,q H (X, ΩX ) = H (X), p where the left side is the sheaf cohomology of the holomorphic vector bundle ΩX and the right side is the Dolbeault cohomology group. i Corollary 2.56. (a) If X is a smooth manifold, then H (X,RX ) = 0 for i > dimR X. (b) If X is a complex manifold and E a holomorphic vector bundle on X, then Hi(X,E) = 0 for

i > dimC X.