and the De Rham-Weil theorem

We present here the most basic facts concerning , and restrict our- selves to paracompact spaces, i.e. Hausdorff topological spaces admitting locally finite partitions of unity for any open covering. The theory of sheaves is due to (between 1940 and 1945; he was then a war prisoner in Austria). Analytic sheaves were then developped by Oka, Cartan and Serre. For the applications to algebraic geom- etry involving Zariski (which is non Hausdorff), a more general approach of cohomology due to Grothendieck is needed, but the paracompact theory usually suffices for analytic geometry. Here, all sheaves are implicitly supposed to be at least sheaves of abelian groups, and correspondingly, sheaf morphisms are morphisms of sheaves of abelian groups.

§ 1. Cechˇ Cohomology

§ 1.A. Definitions Í Let X be a , A a sheaf of abelian groups on X, and = (Uα)α∈I an open covering of X. For the sake of simplicity, we set

Uα0α1...αq = Uα0 ∩ Uα1 ∩ ... ∩ Uαq .

q ˇ A The group C (Í, ) of Cech q-cochains is the set of families

c =(cα0α1...αq ) ∈ A(Uα0α1...αq ). q+1 (α0,...,αYq)∈I

q A The group structure on C (Í, ) is the obvious one deduced from the addition law on

ˇ q q q+1

Í A Í A sections of A. The Cech differential δ : C ( , ) −→ C ( , ) is defined by the formula

q j (1.1) (δ c)α0...αq+1 = (−1) c , α0...αj ...αq+1 ↾Uα0...αq+1 06Xj6q+1 b

q q A and we set C (Í, )=0, δ = 0 for q < 0. In degrees 0 and 1, we get for example

0 (1.2) q =0, c =(cα), (δ c)αβ = cβ − cα ↾Uαβ , ′ 1 (1.2 ) q =1, c =(cαβ), (δ c)αβγ = cβγ − cαγ + cαβ ↾Uαβγ .

Easy verifications left to the reader show that δq+1 ◦ δq = 0. We get therefore a cochain

• ˇ

A Í complex C (Í, ), δ , called the complex of Cech cochains relative to the covering . 

ˇ Í (1.3) Definition. The Cech cohomology group of A relative to is

q q •

A Í A H (Í, ) = H C ( , ) .  Formula (1.2) shows that the set of Cechˇ 0-cocycles is the set of families (cα) ∈

A(Uα) such that cβ = cα on Uα ∩ Uβ. Such a family defines in a unique way a global

Qsection f ∈ A(X) with f↾Uα = cα. Hence

0

A A

(1.4) H (Í, ) = (X). Í Now, let Î =(Vβ)β∈J be another open covering of X that is finer than ; this means that there exists a map ρ : J → I such that Vβ ⊂ Uρ(β) for every β ∈ J. Then we can

• • •

A Î A define a morphism ρ : C (Í, ) −→ C ( , ) by

(1.5) (ρqc) = c ; β0...βq ρ(β0)...ρ(βq ) ↾Vβ0...βq

the commutation property δρ• = ρ•δ is immediate. If ρ′ : J → I is another refinement • ′• map such that Vβ ⊂ Uρ′(β) for all β, the morphisms ρ , ρ are homotopic. To see this,

q q q−1

A Î A we define a map h : C (Í, ) −→ C ( , ) by

q j ′ ′ (h c) − = (−1) c − . β0...βq 1 ρ(β0)...ρ(βj )ρ (βj )...ρ (βq 1) ↾Vβ0...βq−1 06Xj6q−1

The identity δq−1 ◦ hq + hq+1 ◦ δq = ρ′q − ρq is easy to verify. Hence ρ• and

′• Î ρ induce a map depending only on Í, :

q • q ′• q q

A Î A (1.6) H (ρ ) = H (ρ ): H (Í, ) −→ H ( , ).

q q

Í A Now, we want to define a H (X, A) of the groups H ( , ) by means of the refinement mappings (1.6). In order to avoid set theoretic difficulties, the coverings

used in this definition will be considered as subsets of the power set È(X), so that the collection of all coverings becomes actually a set.

ˇ q (1.7) Definition. The Cech cohomology group H (X, A) is the direct limit

q q

Í A H (X, A)= lim H ( , )

−→ Í

when Í runs over the collection of all open coverings of X. Explicitly, this means that q the elements of H (X, A) are the equivalence classes in the disjoint union of the groups

ˇ q q q A Í A Î A H (Í, ), with an element in H ( , ) and another in H ( , ) identified if their

q

A Ï Í Î images in H (Ï, ) coincide for some refinement of the coverings and .

• B If ϕ : A → is a sheaf morphism, we have an obvious induced morphism ϕ :

• •

A Í B C (Í, ) −→ C ( , ), and therefore we find a morphism

q • q q

A Í B

H (ϕ ): H (Í, ) −→ H ( , ).

B C Let 0 → A → → → 0 be an of sheaves. We have an exact sequence of groups

q q q

A Í B Í C (1.8) 0 −→ C (Í, ) −→ C ( , ) −→ C ( , ), but in general the last map is not surjective, because every in C(Uα0,...,αq ) need

• •

Í B Í C not have a lifting in B(Uα0,...,αq ). The image of C ( , ) in C ( , ) will be denoted

•

C C B

C (Í, ) and called the complex of liftable cochains of in . By construction, the B sequence

q q q

A Í B Í C

(1.9) 0 −→ C (Í, ) −→ C ( , ) −→ C ( , ) −→ 0 B

is exact, thus we get a corresponding long exact sequence of cohomology

q q q q+1

A Í B Í C Í A

(1.10) H (Í, ) −→ H ( , ) −→ H ( , ) −→ H ( , ) −→··· .

B Ê

(1.11) Proposition. Let A be a over a sheaf of rings on X. Assume Ê that Ê is a soft sheaf (i.e., by definition, that admits locally finite partitions of unity

q A for every open covering of X). Then H (Í, )=0 for every q > 1 and every open

covering Í =(Uα)α∈I of X. Í Proof. Let (ψα)α∈I be a in Ê subordinate to , i.e. Supp(ψα) ⊂ Uα

q q q−1

Í A Í A and α ψα =1Ê , the sum being locally finite. We define h : C ( , ) −→ C ( , ) by P

q (1.12) (h c)α0...αq−1 = ψν cνα0...αq−1 Xν∈I

where ψν cνα0...αq−1 is extended by 0 on Uα0...αq−1 ∩ ∁Uν. It is clear that

q−1 q q (δ h c)α0...αq = ψν cα0...αq − (δ c)να0...αq , Xν∈I  i.e. δq−1hq + hq+1δq = Id. Hence δqc = 0 implies δq−1hqc = c if q > 1. 

§ 1.B. Cechˇ Cohomology on Paracompact Spaces We prove here that Cechˇ cohomology theory behaves well on paracompact spaces, namely, we get exact sequences of cohomology for any short exact sequence of sheaves.

(1.13) Proposition. Assume that X is paracompact. If

B C 0 −→ A −→ −→ −→ 0

is a short exact sequence of sheaves, there is a “long” exact sequence

q q q ˇ q+1

B C A H (X, A) −→ H (X, ) −→ H (X, ) −→ H (X, ) −→···

which is the direct limit of the exact sequences (1.10) over all coverings Í. Proof. We have to show that the natural map

q q

C Í C lim H (Í, ) −→ lim H ( , ) −→ B −→

is an isomorphism. This follows easily from the following lemma, which says essentially B that every cochain in C becomes liftable in after a refinement of the covering.

q

Í C (1.14) Lifting lemma. Let Í =(Uα)α∈I be an open covering of X and c ∈ C ( , ).

If X is paracompact, there exists a finer covering Î = (Vβ)β∈J and a refinement map

q q C

ρ : J → I such that ρ c ∈ C (Î, ).

B Í Proof. Since Í admits a locally finite refinement, we may assume that itself is locally

finite. There exists an open covering Ï =(Wα)α∈I of X such that W α ⊂ Uα. For every point x ∈ X, we can select an open neighborhood Vx of x with the following properties: a) if x ∈ Wα, then Vx ⊂ Wα ; b) if x ∈ Uα or if Vx ∩ Wα 6= ∅, then Vx ⊂ Uα ;

q B c) if x ∈ Uα0...αq , then cα0...αq ∈ C (Uα0...αq , C) admits a lifting in (Vx).

Indeed, a) (resp. c)) can be achieved because x belongs to only finitely many sets Wα B (resp. Uα), and so only finitely many sections of C have to be lifted in . b) can be ′ achieved because x has a neighborhood Vx that meets only finitely many sets Uα ; then we take ′ ′ r Vx ⊂ Vx ∩ Uα ∩ (Vx W α). U\α∋x U\α6∋x

Choose ρ : X → I such that x ∈ Wρ(x) for every x. Then a) implies Vx ⊂ Wρ(x), so Í Î =(Vx)x∈X is finer than , and ρ defines a refinement map. If Vx0...xq 6= ∅, we have

6 6 Vx0 ∩ Wρ(xj ) ⊃ Vx0 ∩ Vxj 6= ∅ for 0 j q,

thus Vx0 ⊂ Uρ(x0)...ρ(xq ) by b). Now, c) implies that the section cρ(x0)...ρ(xq ) admits a

q

B B

lifting in B(Vx0 ), and in particular in (Vx0...xq ). Therefore ρ c is liftable in . 

B C (1.15) Corollary. Let 0 −→ A −→ −→ −→ 0 be an exact sequence of sheaves on 1

X, and let U be a paracompact open subset. If H (U, A)=0, there is an exact sequence

B C 0 −→ Γ(U, A) −→ Γ(U, ) −→ Γ(U, ) −→ 0.

§ 1.C. Leray’s Theorem for acyclic coverings We assume here that we are here in a topological space X such that every open subset U is paracompact (one can show that every metrizable space has this property). By

ˇ

B C definition of Cech cohomology, for every exact sequence of sheaves 0 → A → → → 0 there is a commutative diagram

q q q q+1 q+1

A Í B Í C Í A Í B

H (Í, )−→ H ( , )−→ H ( , )−→ H ( , )−→ H ( , ) B (1.16)

q q q q+1 q+1

B C A B H (X, A)−→ H (X, )−→ H (X, ) −→ H (X, )−→ H (X, ). y y y y y in which the vertical maps are the canonical arrows to the inductive limit.

s (1.17) Theorem (Leray). Assume that H (Uα0...αt , A)=0 for all indices α0,...,αt

q q

A A and all s > 1. Then H (Í, ) ≃ H (X, ) for every q > 0. A We say that the covering Í is acyclic (with respect to ) if the hypothesis of Th. 1.17

is satisfied. Leray’s theorem asserts that the cohomology groups of A on X can be computed by means of an arbitrary acyclic covering (if such a covering exists), without using the direct limit procedure.

Proof. By induction on q, the result being obvious for q = 0. Consider the exact sequence

B C B

0 → A → → → 0 where is the sheaf of non necessarily continuous sections of

C B A B A A and = / . As is acyclic, the hypothesis on and the long exact sequence of

s • •

Í C Í C cohomology imply H (U , C)=0for s > 1, t > 0. Moreover C ( , ) = C ( , ) e α0...αt B thanks to Cor. 1.15 applied on each open set Uα0...αq . The induction hypothesis in degree q and diagram (1.16) give

q q q+1

B Í C Í A H (Í, )−→ H ( , )−→ H ( , )−→ 0 ≃ ≃

q q q+1

C A H (X, B)−→ H (X, )−→ H (X, )−→ 0, y y y

q+1 q+1

A A hence H (Í, ) −→ H (X, ) is also an isomorphism. 

1 1

A A (1.18) Remark. The morphism H (Í, ) −→ H (X, ) is always injective. Indeed, (1.10) yields

0 0 ≃ 1

C Í B Í A

H (Í, )/ Im H ( , ) −→ H ( , ) B

0  0 ≃ 1 

B A H (X, C)/ Imy H (X, ) −→ H (X,y )

0 0 0 0

B B B Í C C

and H (Í, ) = H (X, ) = Γ(X, ), while H ( , ) −→ H (X, ) is an injection. B

1 1

A Î A As a consequence, the refinement mappings H (Í, ) → H ( , ) are also injective. 

§ 2. The De Rham-Weil isomorphism theorem

• A Let (Ä ,d) be a of a sheaf , that is a complex of sheaves such that we have an exact sequence

0 1

0 d 1 d 2

Ä Ä Ä 0 −→ A −→ −→ −→ −→··· .

q s q Ä We assume in addition that all Ä are acyclic on X, i.e. H (X, ) =0 for all q > 0 and

q q 0

Z A s > 1. Set Z = ker d . Then = and for every q > 1 we get a short exact sequence

q−1

q−1 q−1 d q

Ä Z 0 −→ Z −→ −→ −→ 0.

Theorem 1.5 yields an exact sequence

q−1 s,q

s q−1 d s q ∂ s+1 q−1 s+1 q−1

Z Z Ä (2.1) H (X, Ä ) −→ H (X, ) −→ H (X, ) → H (X, )=0.

If s > 1, the first group is also zero and we get an isomorphism

s,q s q ≃ s+1 q−1 Z ∂ : H (X, Z ) −→ H (X, ).

0 q−1 q−1 0 q q

Ä Z Z For s = 0 we have H (X, Ä ) = (X) and H (X, ) = (X) is the q-cocycle • 0,q group of Ä (X), so the connecting map ∂ gives an isomorphism

0,q

q • q q−1 q−1 ∂ 1 q−1

Z Ä Z H Ä (X) = (X)/d (x) −→ H (X, ).  e The composite map ∂q−1,1 ◦···◦ ∂1,q−1 ◦ ∂0,q therefore defines an isomorphism e 0,q 1,q−1 q−1,1

q • ∂ 1 q−1 ∂ ∂ q 0 q

Z Z A (2.2) H Ä (X) −→H (X, ) −→ ··· −→ H (X, )=H (X, ).  e

This isomorphism behaves functorially with respect to morphisms of resolutions. Our B assertion means that for every sheaf morphism ϕ : A → and every morphism of

• • • Å resolutions ϕ : Ä −→ , there is a commutative diagram

s • s A H Ä (X) −→ H (X, )  (2.3) Hs(ϕ•) Hs(ϕ)

s  • s  B H yÅ (X) −→ H (yX, ). 

q q q q+1

Å Å If Ï = ker(d : → ), the functoriality comes from the fact that we have commutative diagrams

s,q

q−1 q−1 q s q ∂ s+1 q−1

Ä Z Z Z 0 →Z → → → 0 , H ( X, ) −→ H (X, ) ϕq−1 ϕq−1 ϕq Hs(ϕq) Hs+1(ϕq−1)

    s,q 

y q−1 yq−1 y q s y q ∂ s+1 y q−1

Å Ï Ï Ï 0 →Ï → → → 0 , H (X, ) −→ H (X, ).

• A (2.4) De Rham-Weil isomorphism theorem. If (Ä ,d) is a resolution of by q sheaves Ä which are acyclic on X, there is a functorial isomorphism

q • q A H Ä (X) −→ H (X, ).  

(2.5) Example: . Let X be a n-dimensional paracompact differential manifold. Consider the resolution

0 d 1 q d q+1 n

E E E E 0 → R → E → →···→ → →···→ → 0

∞ given by the d acting on germs of C differential q-forms (cf. Exam- ple 2.2). The De Rham cohomology groups of X are precisely

q q •

R E (2.6) HDR(X, ) = H (X) . 

q q

E E All sheaves E are X -modules, so is acyclic by Cor. 4.19. Therefore, we get an isomorphism

q R ≃ q R (2.7) HDR(X, ) −→ H (X, ) from the De Rham cohomology onto the cohomology with values in the R. ∞ Instead of using C differential forms, one can consider the resolution of R given by the exterior derivative d acting on currents:

′ d ′ ′ d ′ ′

D D D D R D 0 → → n → n−1 →···→ n−q → n−q−1 →···→ 0 → 0.

′ E The sheaves Dq are also X -modules, hence acyclic. Thanks to (2.3), the inclusion

q ′ D E ⊂ n−q induces an isomorphism

q • q ′ D (2.8) H E (X) ≃ H n−•(X) ,   both groups being isomorphic to Hq(X, R). The isomorphism between cohomology of differential forms and singular cohomology (another topological invariant) was first es- tablished by [De Rham 1931]. The above proof follows essentially the method given by [Weil 1952], in a more abstract setting. As we will see, the isomorphism (2.7) can be put under a very explicit form in terms of Cechˇ cohomology. We need a simple lemma.

(2.9) Lemma. Let X be a paracompact differentiable manifold. There are arbitrarily

fine open coverings Í = (Uα) such that all intersections Uα0...αq are diffeomorphic to convex sets.

′ Proof. Select locally finite coverings Ωj ⊂⊂ Ωj of X by open sets diffeomorphic to n concentric euclidean balls in R . Let us denote by τjk the transition diffeomorphism from ′ 2 the coordinates in Ωk to those in Ωj. For any point a ∈ Ωj, the x 7→ |x − a| 2 computed in terms of the coordinates of Ωj becomes |τjk(x) − τjk(a)| on any patch Ωk ∋ a. It is clear that these functions are strictly convex at a, thus there is a euclidean ′ ′ ball B(a, ε) ⊂ Ωj such that all functions are strictly convex on B(a, ε) ∩ Ωk ⊂ Ωk (only

a finite number of indices k is involved). Now, choose Í to be a (locally finite) covering ′ of X by such balls Uα = B(aα, εα) with Uα ⊂ Ωρ(α). Then the intersection Uα0...αq is defined in Ωk, k = ρ(α0), by the equations

2 2 |τjk(x) − τjk(aαm )| < εαm

where j = ρ(αm), 0 6 m 6 q. Hence the intersection is convex in the open coordinate

chart Ωρ(α0). 

Let Ω be an open subset of Rn which is starshaped with respect to the origin. Then • the De Rham complex R −→ E (Ω) is acyclic: indeed, the Poincar´elemma yields a

q q q−1 E homotopy operator k : E (Ω) −→ (Ω) such that

1 q q−1 n k fx(ξ1,...,ξq−1) = t ftx(x,ξ1,...,ξq−1) dt, x ∈ Ω, ξj ∈ R , Z0 0 0

k f = f(0) ∈ R for f ∈ E (Ω).

q •

R E R Hence HDR(Ω, )=0 for q > 1. Now, consider the resolution of the constant sheaf on X, and apply the proof of the De Rham-Weil isomorphism theorem to Cechˇ cohomol-

ogy groups over a covering Í chosen as in Lemma 2.9. Since the intersections Uα0...αs

ˇ s q q−1 q

Z E are convex, all Cech cochains in C (Í, ) are liftable in by means of k . Hence

s,q−s s q−s s+1 q−s−1

Z Í Z for all s = 1,...,q we have ∂ : H (Í, ) −→ H ( , ) for s > 1 and we get a resulting isomorphism

q−1,1 1,q−1 0,q q ≃ q

R Í R ∂ ◦···◦ ∂ ◦ ∂ : HDR(X, ) −→ H ( , ) e We are going to compute the connecting homomorphisms ∂s,q−s and their inverses ex- plicitly.

s q−s s Z Let c in C (Í, ) such that δ c = 0. As cα0...αs is d-closed, we can write c =

q−s q−s s q−s−1 E d(k c) where the cochain k c ∈ C (Í, ) is defined as the family of sections q−s q−s−1 s q−s s q−s s k cα0...αs ∈ E (Uα0...αs ). Then d(δ k c) = δ (dk c) = δ c = 0 and

s,q−s s q−s ˇ s+1 q−s−1 Z ∂ {c} = {δ k c} ∈ H (Í, ).

q ≃ q

R Í R The isomorphism HDR(X, ) −→ H ( , ) is thus defined as follows: to the cohomology q 0 class {f} of a closed q-form f ∈ E (X), we associate the cocycle (cα)=(f↾Uα ) ∈

0 q Z C (Í, ), then the cocycle

1 q 0 q 0 1 q−1 Z cαβ = k cβ − k cα ∈ C (Í, ),

s s q−s Z and by induction cocycles (cα0...αs ) ∈ C (Í, ) given by

s+1 j q−s s (2.10) cα ...α = (−1) k c on Uα0...αs+1 . 0 s+1 α0...αj ...αs+1 06Xj6s+1 b

q q q Í Í R R The image of {f} in H ( , ) is the class of the q-cocycle (cα0...αq ) in C ( , ).

∞ ˇ Í Conversely, let (ψα)bea C partition of unity subordinate to . Any Cech cocycle

s+1 q−s−1 s s q−s−1

Z Í E c ∈ C (Í, ) can be written c = δ γ with γ ∈ C ( , ) given by

γα0...αs = ψν cνα0...αs , Xν∈I

(cf. Prop. 1.11)), thus {c′} =(∂s,q−s)−1{c} can be represented by the cochain c′ = dγ ∈

s q−s Z C (Í, ) such that

′ q−s−1 cα0...αs = dψν ∧ cνα0...αs =(−1) cνα0...αs ∧ dψν. Xν∈I Xν∈I

For a reason that will become apparent later, we shall in fact modify the sign of our s,q−s q−s−1 q isomorphism ∂ by the factor (−1) . Starting from a class {c} ∈ H (Í, R), we

0 q Z obtain inductively {b} ∈ H (Í, ) such that

(2.11) bα0 = cν0...νq−1α0 dψν0 ∧ ... ∧ dψνq−1 on Uα0 ,

ν0,...,νXq−1

q R corresponding to {f} ∈ HDR(X, ) given by the explicit formula

(2.12) f = ψνq bνq = cν0...νq ψνq dψν0 ∧ ... ∧ dψνq−1 .

Xνq ν0X,...,νq The choice of sign corresponds to (2.2) multiplied by (−1)q(q−1)/2.

(2.13) Example: groups. Let X be a C-analytic manifold

p,q ∞ C of n, and let E be the sheaf of germs of differential forms of type (p, q) with complex values. For every p =0, 1,...,n, the Dolbeault-Grothendieck lemma shows p,• ′′ p that (E ,d ) is a resolution of the sheaf ΩX of germs of holomorphic forms of degree p on X. The Dolbeault cohomology groups of X are defined by

p,q q p,•

(2.14) H (X, C) = H E (X) .  p,q Since the sheaves E are acyclic, we get the Dolbeault isomorphism theorem, originally proved by Dolbeault in 1953, which relates d′′-cohomology and sheaf cohomology:

p,q C ≃ q p (2.15) H (X, ) −→ H (X, ΩX ).

The case p = 0 is especially interesting:

0,q q (2.16) H (X, C) ≃ H (X, ÇX).

p,q ′ D As in the case of De Rham cohomology, there is an inclusion E ⊂ n−p,n−q and the ′ ′′ p complex of currents (Dn−p,n−•,d ) defines also a resolution of ΩX . Hence there is an isomorphism:

p,q q p,• q ′ D C E (2.17) H (X, ) = H (X) ≃ H n−p,n−•(X) .