The Verdier Duality Theoremfor di�erentiable manifolds

Matteo Staccone

Contents

Contents 1

1 Introduction 2

2 Preliminaries 4

3 Proof of the Duality Theorem 6

3.1 The main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 An application 13

1

1 Introduction

For every sheaf F and for every open set U , we de�ne for every compact K ⊂ U the sheaf ΓK(U,F )as being the kernel of the restriction map

Γ(U,F )→ Γ(U \K,F ).

Then we de�ne the compactly supported sheaf associated to F as

Γc(U,F ) := lim−→K⊂U

ΓK(U,F ).

Now we consider the complex of sheaves of the di�erential forms over the di�erentiable manifold X

of real dimension n, which is denoted by A •(X), and we construct the following complex:

Γc(X,An−n ⊗1X F )

dF−−→ · · · dF−−→ Γc(X,An−1 ⊗1X F )

dF−−→ Γc(X,An−0 ⊗1X F )

Here the di�erential operator dF is de�ned as

dF := d⊗D

where d is the standard exterior di�erentiation and D sends F to 0. Furthermore, 1X is the locallyconstant sheaf on X, and F a sheaf of 1-modules.

We denote the cohomology of this complex by: H−qc (X,F ).

De�nition 1.0.1 Given a long exact sequence of sheaves 0 → G → I0 → I1 → . . . such that Iq isinjective for all q ≥ 0, we denote the cohomology of

RHomD(X)(F ,G ) := HomX(F , I•) : HomX(F , I0)→ HomX(F , I1)→ · · ·

as ExtqX(F ,G ).

Remark 1.0.2 We remark that De�nition 1.0.1 does not depend on the choice of injective resolutionG → I•, due to well known mapping properties of injective resolutions, which are unique up tohomotopy. Such homotopy induces an isomorphism on the level of cohomology in the derived categoryof sheaves of 1X -modules on X: D≥0(X). We denote by

RHomD(X)(F ,G )

the complex HomX(F , I•) in D+(X), and by ExtiX(F ,G ) its cohomology groups:

HiRHomD(X)(F ,G ) = RiHomD(X)(F ,G ) = HiHomX(F , I•).

The notation we are adopting is used to denote the right-derived functor of a left-exact functor on anabelian category with enough injective objects. In our case the functor in question is HomX(F ,−) onthe abelian category Shv(M) of sheaves of 1M -modules on M . Such derived functors is an universalδ-functors.

2

Let M be a di�erentiable manifold of dimension n, and let 1M be the locally constant sheaf, and Fa sheaf of 1M−modules. The Verdier duality theorem asserts that there is a pairing:

H−qc (M,F )× ExtqM (F , oM )→ H−0c (M, oM )∼−→ R

where oM is the orientation sheaf, the last map H−0c (M, oM )∼−→ R consisting in integration, which

yields a duality isomorphism:

ExtqX(F , oM )∼→ HomR(H−qc (M,F ),R).

3

2 Preliminaries

Lemma 2.0.1 Let f : X → Y be a continuous map of topological spaces, F a sheaf of sets on X.

We de�ne the pushforwardpresheaf of F along f by assigning to any open subset V ⊂ Y ,

(f∗F )(V ) := F (f−1(V )).

We have that (f∗F ) is a sheaf of sets on Y .

Proof. Let {Vi}i∈I be an open covering of V .

We take two sections s, t ∈ (f∗F )(V ) such that s|Vi = t|Vi for all i ∈ I and we claim that s = t. Since

f−1(V ) = f−1(⋃

Vi

)=⋃f−1(Vi)

we obtain(f∗F )(V ) = F

(⋃f−1(Vi)

)⊂∏

F (f−1(Vi)),

so s, t can be taught as vectors, and each entry of s and t are equal. It must be the case that s = t,therefore.

We now take si ∈ (f∗F )(Vi) for all i ∈ I such that

si|Vi∪Vj = sj |Vi∪Vj , ∀i, j ∈ I

and we want to �nd a global section s which restricts to si, for all i ∈ I. We have:

(f∗F )(Vi) = F (f−1(Vi))

so si|Vi∪Vj = sj |Vi∪Vj is equivalent to si|f−1(Vi)∪f−1(Vj) = sj |f−1(Vi)∪f−1(Vj) since ∪f−1(Vi) is an open

cover of f−1(V ), and we already know that F is a sheaf. Therefore, there exists a section s in(f∗F )(V ) = F (f−1(V )) which does the desired job.

Lemma 2.0.2 Let i : Z ↪→ X be a closed immersion, let U = X −Z, then let j : U ↪→ X be the open

complement. For F a sheaf of sets on U we de�ne for V ⊂ X open

(j!F )(V ) = {s ∈ (j∗F )(V ) = F (U ∩ V ) | supp(s) is closed in V }.

Then (j!F ) is a sheaf.

Proof. First we observe that we have an injective map of presheaves of sets on U :

j!F → j∗F

by design. It follows that if we let {Vi}i∈I be an open covering of V , then two sections in (j!F )(V )which are locally equal, are equal, because so are in F (U ∩ V ).

We now take si ∈ (j!F )(Vi) for all i ∈ I such that

si|Vi∪Vj = sj |Vi∪Vj ∀i, j ∈ I

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and we want to �nd a section s ∈ (j!F )(V ) restricting to the si's. si ∈ (j!F )(Vi) is equivalent tosi ∈ F (U ∪ V ) such that supp(si) is closed in Vi, so the condition si|Vi∪Vj = sj |Vi∪Vj is equivalent tosi|U∩(Vi∪Vj) = sj |U∩(Vi∪Vj) and since F is a sheaf on U , there exists a global section s which coincideswith si on U ∪ Vi and each si has its support closed in Vi, so s has its support closed in V =

⋃i∈IVi,

since being closed is a local property.

Lemma 2.0.3 For any sheaf F of 1X-modules on X, there is a family of open embeddings

{jα : Uα ↪→ X|α ∈ I},

where U = {Uα} is a cover of X, such that there is a surjection∐α∈I

(jα)!(1Uα) � F .

Proof. Let I be the set of pairs

(s, x) where s ∈ Fx, then

for each α ∈ I there is

an open neighbourhood jα : Uα ↪→ X

of x such that s ∈ F (Uα), so

∐α∈I

(jα)!117→s−−−→ F

is well de�ned, and surjective.

5

3 Proof of the Duality Theorem

We denote with Dq(U) := HomR((A n−qc ⊗1 oM (U),R) for every open set U , and more generally we

denote with (DF )q(U) := HomR((A n−q ⊗1 F )c(U),R) for every open set U .

Proposition 3.0.1 (DF )q is a sheaf of 1M -modules on M .

Proof. Let {Uα → X} be an open cover of M .

We consider the sequence of di�erentiable maps:

M ←f

∐α0

Uα0

δ0⇔δ1

∐α0<α1

Uα0α1 · · ·

where δi is the inclusion which ignores the ith open set. This induces a sequence on the di�erentialforms with compact support sheaves:

(∗) 0← (A n−q ⊗1 F )c(M)←f∗

(A n−q ⊗1 F )c(∐α0

Uα0)δ0∗−δ1∗← (A n−q ⊗1 F )c(

∐α0<α1

Uα0α1)

and passing to its dual we obtain:

(∗∗) 0→ (DF )q(M)f∗−→ (DF )q(

∐α0

Uα0)δ∗0−δ

∗1−−−−→ (DF )q(

∐α0<α1

Uα0α1)

So (DF )q is a sheaf if and only if (∗∗) is exact, and it is exact if (∗) is exact. Thus, it is su�ces toshow that (∗) is an exact sequence of sheaves of 1M (M)-modules. In order to do this, we need toshow that im(δ0∗ − δ1∗) = ker(f∗). In order to show this, in turn, we have:

f∗ ◦ δ0∗ = f∗ ◦ δ1∗

we show thatf ◦ δ0 = f ◦ δ1.

This follows easily by the fact that the following is a �ber square:

qα0<α1

Uα0α1= U ×X U Uα0

Uα1 X

δ0

δ1f

f

We are left to show that:ker(f∗) ⊂ im(δ0∗ − δ1∗).

Let(wα0)α0 ∈ (A n−q ⊗1 F )c(

∐α0

Uα0)

Then (wα0)α0

by f∗ is sent to ∑α0

wα0 ∈ (A n−q ⊗1 F )c(M)

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and by hypothesis we know that∑α0

wα0= 0. We consider the partition of unity ρα1

subordinate to

the open cover, so we have ∀α1

(wα0ρα1

)α0α1∈ (A n−q ⊗F )c(

∐α0<α1

Uα0α1).

Then(δ0∗)(wα0

⊗ ρα1)α0

=∑α1 6=α0

ρα1wα0

and(δ1∗)(wα1

⊗ ρα0)α1

=∑α1 6=α0

ρα0wα1

.

Therefore, we have

[(δ0∗ − δ1∗)(wρ)]α0=

∑α1 6=α0

ρα1wα0−∑α1 6=α0

ρα0wα1

= wα0(1− ρα0)− ρα0(−wα0)

= wα0 − ρα0wα0 + ρα0wα0

= wα0

where we have exploited the linearity of tensor product.

Theorem 3.0.2 There is a natural isomorphism of sheaves of 1M -modules:

Γ(M, (DF )q)'−→ HomM (F ,D1q).

Proof. Let us consider an element

LM ∈ (DF )q(M) = HomR((A n−q ⊗1 F )c(M),R)

so LM is a map LM : (An−q ⊗1 F )c(M)→ R.

Now we need to de�ne a map between F (U) and D1q(U): for every open set j : U ↪→ M letω ∈ A n−q

c (U) and s ∈ F (U), then ω ⊗ s ∈ (A n−q ⊗F )c(U) and for ! the push forward of compactsupport j!(ω ⊗ s) ∈ (A n−q ⊗F )c(M) so we de�ne λU (s)(ω) := LM (j!(ω ⊗ s)). In order λ to yield amap of sheaves, we need to show that the following diagram commutes for every open set i : V ↪→ U :

F (U) D1q(U)

F (V ) D1q(V )

λU

res res

λV

which for ω ∈ (A n−q ⊗F )c(V ) and s ∈ F (U) is equivalent to

i∗(λU (s))(ω) = (λU (s))(i!ω) = LM (j!(s⊗i!ω)) = LM (j!i!(i∗s⊗ω)) = LM ((ji)!(i

∗s⊗ω)) = λV (i∗s)(ω)

7

Now we de�ne the sheaf map TF as TF : (DF )q → HomM (F ,D1q⊗oM ) and TF is de�ned for everyopen set U as TF ,U : (DF )q(U) → HomM (F (U),D1q(U)) which sends LU to λU . Now we provethat TF is a natural transormation:

Fact For every couple of sheaves of 1M -modules A and B such that there exists a map Af−→ B the

following diagram commutes:

Hom((A n−q ⊗1 B)c(M),R) Hom((A n−q ⊗1 B)c(M),R)

HomM (B,D1q) HomM (A,D1q)

Hom(id⊗f)

TB(U) TA(U)

Hom(f)

Proof. Let's consider µ : A n−qc → R and σB : B → R, then µ⊗σB is an element of HomM ((A n−q⊗1

B)c,R) and we have that (µ⊗ σB)(ω ⊗ sB) is sent to

λBU (sB)(ω) = µ(ω)⊗ σB(sB),

whereω ⊗ sB ∈ (A n−q ⊗B)c

and λBU : B(U)→ D1q(U) sendssB 7→ µ(−)⊗ σB(sB).

Moreover, we have λAU (sA)(ω) = µ(ω)⊗ (σB ◦ f)(sA), where f ◦ sA = sB

TA ◦ Hom(id ⊗ f,R)(µ ⊗ σB) = TA(µ ⊗ (σB ◦ f)) = µ(−) ⊗ (σB ◦ f). On the other hand, we have(Hom(f,D1q) ◦ TB)(µ⊗ σB) = µ(−)⊗ (σB ◦ f), thus the diagram commutes.

Now, by Lemma 2.1.3 , we know that there exists a surjection G :=∐α∈I

(jα)!(1Uα)φ� F . ker(φ) is

still a sheaf so there exists a surjection

R :=∐β∈J

(jβ)!(1Uβ ) � ker(φ).

Then we obtain

Rg� ker(φ)

h↪→ G

and de�ning ψ := h ◦ g we have the following exact sequence:

Rψ→ G

φ→ F → 0.

Applying the two controvariant functors Hom(−,D1q) and HomM ((A n−q ⊗1 −)c,R) to the exactsequence we obtain the following diagram:

0 Hom((A n−q ⊗1 F )c(M),R) Hom((A n−q ⊗1 G)c(M),R) Hom((A n−q ⊗1 R)c(M),R)

0 HomM (F ,D1q) HomM (G,D1q) HomM (R,D1q)

r

TF

s

TG TR

r′ s′

8

Since T is a natural trasformation the diagram commutes. Furthermore, we observe that TR and TGare two isomorphisms:∏

α

HomM ((A n−q ⊗ (j)!(1Uα),R)TG→∏α

HomM ((j)!(1Uα),D1q)

and

for all open sets Uα we have the isomorphism by adjuction:

HomM ((j)!(1Uα),D1q)∼−→ HomUα(1Uα , (jα)∗(D1q)) = Γ(Uα,D1q)

The same way it can be shown that TR is an isomorphism. Now it remains to show that TF is injectiveand surjective.

We choose f ∈ HomM ((A n−q⊗1F )c,R) such that f ∈ ker(TF ). Then we have TF (f) = 0. Applyingr′ we have r′(TF (f)) = 0 and because of the commutativity of the relevant diagram, we have

r′(TF (f)) = TG(r(f)) = 0.

Now, TG is injective and so r′(f) = 0. Since r′ is also injective, we have f = 0, as desired. Injectivityis proved.

We now choose h ∈ HomM (F ,D1q), and then r′(h) ∈ HomM (G,D1q) and TG is surjective, whichimplies that there exists g ∈ HomM ((A n−q ⊗1 G)c,R) such that r′(h) = TG(g). Applying s′ and bythe exactness of the sequence we obtain

0 = s′(r′(h)) = s′(TG(g)) = TR(s(g)),

so s(g) = 0 and this implies that there exists h′ ∈ HomM ((A n−q ⊗1 F )c,R) such that r(h′) = g. Itfollows TF (h′) = h, which achieves surjectivity, as desired.

Corollary 3.0.3 There is an isomorphism of sheaves of 1M -modules:

(D1)q∼−→ Dq ⊗ oM .

Proof. From the theorem

Dq ∼−→ HomM (oM ,D1q)

and o⊗2M∼−→ 1M .

Proposition 3.0.4 D1q∼−→ Dq ⊗ oM is an injective sheaf of 1M -modules on M .

Proof. Given an exact sequence of sheaves of 1M -modules:

0→ F ′ → F → F ′′ → 0

we need to show that the following sequence of R-vector spaces is exact:

9

0→ HomM (F ′′,D1q)→ HomM (F ,D1q)→ HomM (F ′,D1q)→ 0.

By the theorem we know that this is equivalent to showing that

0→ HomM ((A n−q ⊗1 F ′′)c,R)→ HomM ((A n−q ⊗1 F )c,R)→ HomM ((A n−q ⊗1 F ′)c,R)→ 0

is exact. This last sequence is exact if

0→ (A n−q ⊗1 F ′)c → (A n−q ⊗1 F )c → (A n−q ⊗1 F ′′)c → 0

is exact, where we already know that it is exact at the last step. For a given open set U , using thede�nition of compact support forms, we have for every compact K ⊂ U the exact sequence

0→ FK(U)→ F (U)→ F (U \K).

This way we have the following diagram:

0 (A n−q ⊗1 F ′)(U \K) (A n−q ⊗1 F )(U \K) (A n−q ⊗1 F ′′)(U \K) 0

0 (A n−q ⊗1 F ′)(U) (A n−q ⊗1 F )(U) (A n−q ⊗1 F ′′)(U) 0

0 (A n−q ⊗1 F ′)K(U) (A n−q ⊗1 F )K(U) (A n−q ⊗1 F ′′)K(U) 0

. 0 0 0

α′′ β′′

α′

δ

β′δ′ δ′′

α

γ

β

γ′ γ′′

Here the �rst two rows and the three columns are exact, and we want to show that the last row isexact, already knowing that it is exact at the last step.

We �rst check that we have im(α) ⊂ ker(β).

We take x ∈ (A n−q ⊗1 F ′)K(U) and we want to show that β(α(x)) = 0. By commutativity of theabove diagram, we have: γ′α(x) = α′γ(x). Applying β′ we obtain

0 = β′α′γ(x) = β′γ′α(x) = γ′′βα(x),

but γ′′ is injective so βα(x) = 0.

We next check that we have: ker(β) ⊂ im(α).

We take x ∈ (A n−q ⊗1 F )K(U) such that β(x) = 0. Applying γ′′ we have 0 = γ′′β(x) = β′γ′(x).Since the middle row is exact, there exists y ∈ (A n−q ⊗1 F ′)(U) such that α′(y) = γ′(x). Now wehave α′′δ(y) = δ′α′(y) = δ′γ′(x) = 0 since the second column is exact. But α′′ is injective so δ(y) = 0and the column is exact, which implies that there exists y′ ∈ (A n−q⊗1 F ′)K(U) such that γ(y′) = y.Therefore

γ′α(y′) = α′γ(y′) = α′(y) = γ′(x)

and since γ′ is injective we have α(y′) = x, as desired.

We now claim α is injective.

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We take x ∈ (An−q ⊗1 F ′)K(U) such that α(x) = 0. Then, 0 = γ′α(x) = α′γ(x), but we know thatα′ is injective then γ(x) = 0 and since γ is injective x = 0, as desired.

We have shown that for all compact K ⊂ U , the sequence

0→ (A n−q ⊗1 F ′)K(U)→ (An−q ⊗1 F )K(U)→ (An−q ⊗1 F ′′)K(U)→ 0

is exact. Since this sequence is exact for all K, we conclude that it is exact also taking the directlimit, and we conclude.

3.1 The main Theorem

Theorem 3.1.1 (Verdier Duality) For any real di�erentiable manifold M , sheaf of 1M -modules

F , q ∈ Z, we have a natural and canonical isomorphism:

ExtqX(F , oM )∼−→ HomR(H−qc (M,F ),R).

Proof. We observe that:

HomR(H−qc (M,F ),R)∼−→ Hq(Hom((A n−• ⊗1 F )c(M),R) = Hq((DF )•)

since the dual of the cohomology is the homology of the dual. Now we take an injective resolution

0→ oM → I•

thenExtqM (F , oM ) = Hq(HomM (F , I•)).

We observe that:ExtqM (F , oM ) = Hq(F , oM )

for any injective resolution0→ oM → I•

as the de�nition of Ext's does not depend on the choice of I•. We want to �nd maps

αp : (DF )q → Iq

for all q ≥ 0 such that they induce an isomorphism on cohomology groups. In particualar, if αp isitself an isomorphism then we automatically have an isomorphism on the level of cohomology groups(that is, in the derived category D(M)). Since we know that Ext's de�nition does not depend on thechoice of I•, we choose

Iq = D1q ⊗1 oM

and we already know that

(DF )q(M)αq−→ HomM (F ,D1q ⊗1 oM )

is an isomorphism for all q ≥ 0. In order to �nish the proof we need to show that

0→ oM → D0 ⊗1 oMg−→ D1 ⊗1 oM → · · ·

is an injective resolution. We already know that

D• ⊗1 oM

11

is a complex and we know from the Poincaré duality that if U is di�eomorphic to Rn then theco-homology of

Γ(U,Dq)

is Γ(U,1) in degree 0 and zero otherwise, i.e. Dq is an injective resolution of 1 since such opens forma basis for the topology of a manifold. Thus Dq ⊗ oM is an injective resolution of oM .

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4 An application

We discuss now an application of the Verdier Duality Theorem.

For any locally constant sheaf E of 1-modules on M , we consider the complex

0→ A 0 ⊗1 EdE−−→ A 1 ⊗1 E

dE−−→ · · ·

where dE = d ⊗ D and d is the standard exterior di�erentiation and D sends E to 0. We de�neHq(M,E ) to be the cohomology of the above complex.

Corollary 4.0.1 If i : N ↪→ M is a closed submanifold of codimension d, there is a natural iso-

morphism

Hq(N,E ∨ ⊗1 oN )'−→ Extq+dM (i∗E , oM )

for any locally constant sheaf E on N .

Proof. Since we know that the dual of Extq+dM (i∗E , oM ) is

Hm−(q+d)c (M, i∗E ) = Hn+d−(q+d)

c (M, i∗E ) = Hn−qc (N,E )

and that the dual of Hq(N,E ∨ ⊗ oN ) is Hn−qc (N,E ), then the isomorphism between Hn−q

c (M, i∗E )and Hn−q

c (N,E ) induces the desired natural isomorphism.

13