(1) In particular : (2) Hi(U, DXF•[−n]) =Ri−nHom(H•c(U,F•),Z

0.1. Third (and last) characterization of Deligne’ s extension. LetF^• ∈D⁺(ZX).

Then for U ∈Op_X and i∈Z:

RHom(ZU, D_XF^•) = RHom(ZU, RHom(F, ω_X))

=RHom(ZU⊗^LF^•, p^!_XZ)

=RHom(RpX!F_U^•,Z) . (1)

In particular :

(2) Hⁱ(U, DXF^•[−n]) =Rⁱ⁻ⁿHom(H^•c(U,F^•),Z) .

Suppose now that F^• ∈D⁺(QX) (hereQ could be replaced by any field). Then : Hⁱ(U, D_XF^•[−n]) = Homⁱ⁻ⁿ(H^•c(U,F^•),Q)

=Hⁿ⁻ⁱc (U,F^•)^∗ , (3)

where we used thatD(Q−V ec) is the sum indexed byZof copies ofQ−V ec, and where

∗ denotes the dual in Q−V ec.

This yields :

∀x∈X,∀i∈Z, Hⁱ(D_XF^•[−n])_x 'Hⁿ⁻ⁱ(i^!_xF^•)^∗ . The following corollary follows immediately :

Corollary 0.1.1. Let E be a local system on some open dense submanifold of X whose complement has dimension at least 2. The set of axioms (AX2)E,p is equivalent to the following set (AX3)E,p for F^• ∈DGSh(X) :

(i) HⁱF^• = 0 for i <0 ; F^• is X-clc for some PL-pseudomanifold stratification X of X ; there exists an open dense subset submanifold U of X of codimension at least 2 on which E is defined such that F_|U^• ' E_U.

(ii) dim suppH^jF^• ≤n−p⁻¹(j) for all j >0.

(iii) dim suppH^jD_XF^•[−n]≤n−q⁻¹(j) for all j >0.

Theorem 0.1.2. (1) The set of conditions(AX3)E,pdeterminesF^•inD^b(ZX)uniquely and is satisfied by P_p(E).

(2) If moreover we work inD(QX)thenF^• satisfies(AX3)E,p if and only ifD_XF^•[−n]

satisfies(AX3)E^∗⊗Or,q whereE^∗ denotes the local systemHom(E,QU2)and qis the perversity dual to p.

Proof. The first part follows from the analogous result for (AX2)E,p.

For the second statement , letG^• =D_XF^•[−n]. Then D_XG^•[−n]'D_XD_XF^• 'F^• by biduality. Thus (AX3)E,p(ii) forF^• is equivalent to (AX3)E^∗⊗Or,q(iii) forG^• and similarly exchanging (ii) and (iii). Hence it remains to show thatG^•satisfies (AX3)E^∗⊗Or,q(i). But :

HⁱG_x^• =Hⁱ(D_XF^•[−n])_x =Hⁿ⁻ⁱ(i^!_xF^•)^∗ = 0 fori <0 . Moreover G^• isX-clc if F^• isX-clc by theorem??. Finally :

G_|U^•₂ =D_XF^•[−n]|U₂ =D_U2F_|U^•₂E[−n]' E^∗ ⊗ Or_U2 .

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Corollary 0.1.3(Poincar´e duality, cf. theorem??). LetE be aQ-local system on a dense open submanifold of X with complement of codimension at least 2. Then

I^qHⁱ(X,E^∗⊗ Or) =I^pH_cⁿ⁻ⁱ(X,E)^∗ . Proof.

I^qHⁱ(X,E^∗⊗ Or) =Hⁱ(X,P_q^•((E^∗⊗ Or)⊗ Or)) =Hⁱ(X,P_q^•((E ⊗ Or)^∗⊗ Or))

=Hⁱ(X, D_XP_p^•(E ⊗ Or)[−n])

=Hⁿ⁻ⁱc (X,P_p^•(E ⊗ Or))^∗ by equation (3)

=I^pH_cⁿ⁻ⁱ(X,E)^∗ .

Examples 0.1.4. In this section we assume that X is normal.

TakingE =Or, p=t and q= 0 we obtain :

Hⁱ(X,QX) =I⁰Hⁱ(X,QX) =I^qHⁱ(X,Or^∗⊗ Or)

=I^pH_cⁿ⁻ⁱ(X,Or)^∗ =I^tH_cⁿ⁻ⁱ(X,Or)^∗ =H_i^c(X,QX)^∗ . TakingE =QX, p= 0 andq=t we get :

Hn−i(X,QX) = I^tHⁱ(X,Or) =I^qHⁱ(X,Q^∗X ⊗ Or)

=I^pH_cⁿ⁻ⁱ(X,QX)^∗ =I⁰H_cⁿ⁻ⁱ(X,QX)^∗ =H_cⁿ⁻ⁱ(X,QX) . 0.2. Pairings. Let us first consider the functoriality of Deligne’s extension.

Proposition 0.2.1. Let f₂ : E −→ F be a morphism of local sytems on U₂. Let p, q be perversities satisfyingp≤q. Then f₂ extends in a unique way to a morphism in D(ZX) :

f :P_p^•(E)−→ P_q^•(F) .

Proof. Let L^• := P_p^•(E) and M^• := P_q^•(F). By induction it is enough to show that f_k:L^•_k −→M_k^• ∈D(ZUk) extends uniquely to f_k+1 :L^•_k+1 −→M_k+1^• ∈D(ZUk+1) (k ≥2).

By definition ofM_k+1^• one has

Hom_D(ZUk+1)(L^•_k+1, M_k+1^• ) = Hom_D(ZUk+1)(L^•_k+1, τ≤q(k)Rj_k∗M_k^•) . On the other hand one has a natural sequence of homomorphisms :

Hom_D(ZUk+1)(L^•_k+1, τ≤q(k)Rj_k∗M_k^•)−→^φ Hom_D(ZUk+1)(L^•_k+1, Rj_k∗M_k^•)'Hom_D(ZUk)(L^•_k, M_k^•) , where the last isomorphism is given by adjunction.

Note that τ≤q(k)L^•_k+1 =L^•_k+1 as q(k) ≥ p(k) and L^•_k+1 :=τ≤p(k)Rjk∗L^•_k. The following easy lemma (exercice) applied to C = D(Z^Uk+1), A= L^•_k+1, B =Rjk∗M_k^• and m = q(k), implies thatφ is an isomorphism. The result follows.

Lemma 0.2.2. Let C be a triangulated category, A ∈ C and m ∈ Z. Suppose that the natural morphism τ≤mA−→A is an isomorphism in C. Then for any B ∈ C the natural homomorphism

HomC(A, τ≤mB)−→HomC(A, B) is an isomorphism.

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Proposition 0.2.3. Let µ2 : E ⊗ F −→ G be a pairing of local systems on U2 and let p, q, r be perversities such thatp+q ≤r. Then there exists a unique morphism inD^b(Z^X)

µ:P_p^•(E)⊗^LP_q^•(F)−→ P_r^•(G) which co¨ıncide with µ2 on U2.

Proof. First notice the following lemma, left as an exercice :

Lemma 0.2.4. Let A be an Abelian category and D(A) its derived category. Let A, B ∈ D(A) satisfying Hⁱ(A) = 0 for i > 0 and Hⁱ(B) = 0 for i < 0. Then the natural homomorphism

HomD(A)(A, B)−→HomA(H⁰(A),H⁰(B)) is an isomorphism.

We apply this lemma to A =DGSh(X), noticing that Hⁱ(E ⊗^LF) = 0 for i >0 and H⁰(E ⊗^LF) ' E ⊗ F. Thus µ₂ can be seen as µ₂ : E ⊗^LF −→ G in D^b(U₂) and the statement of the proposition makes sense.

Once more we proceed by induction. Let L^• :=P_p^•(E),M^• :=P_q^•(F), N^• :=L^•⊗^LM^• and Q^• :=P_r^•(G) . We prove by induction the extension ofµ_k:N_k^• −→R^•_k ∈D^b(ZUk) to µ_k+1 ∈D^b(ZUk+1).

We claim that τ≤r(k)N_k+1^• −→ N_k+1^• is a quasi-isomorphism. Applying lemma 0.2.2 it then follows that

Hom_D(ZUk+1)(N_k+1^• , Q^•_k+1 =τ≤r(k)Rj_k∗Q^•_k)−→Hom_D(ZUk)(N_k^•, Q^•_k) is an isomorphism. Thus µ_k uniquely extends toµ_k+1.

To show the claim, notice thatN_k+1^• is quasi-isomorphic toL^•_k+1⊗^LM_k+1^• . Asτ_≤p(k)L^•_k+1 −→

L^•_k+1 and τ_≤q(k)M_k+1^• −→ M_k+1^• are quasi-isomorphism we can choose for computing L^•_k+1 ⊗^L M_k+1^• flat resolutions of L^•_k+1 and M_k+1^• vanishing in degree higher than p(k) andq(k) respectively. The total complex formed from these resolutions vanishes in degree larger than r(k)≥p(k) +q(k).

From proposition 0.2.3 we obtain various pairings in hypercohomology. In particular we get a map

Hⁱc(X,P_p^•(E))⊗H^j(X,P_q^•(F))−→H^i+jc (X,P_r^•(G)) .

TakingF =E^∗⊗ Or, G=Or and µ₂ :E ⊗(E^∗⊗ Or)−→ Or the canonical pairing we obtain the pairing

I^pH_cⁱ(X,E ⊗ Or)⊗I^qH^j(X,E^∗)−→I^rH_c^i+j(X,QX) described in theorem ??.