Compatibility of rigid and algebraic Poincar´ e duality

M

M^{(M )•an}−→ “lim←−”

M

J^{(M )••}=:J^••

ofM^•anin ProC (P^an). There is a canonical morphism of complexes of objects of ProC (P^an),I^••an−→ J^••

(which is an isomorphism of D(ProC (P^an))). On the other hand, if I^•(Ω_P^•an) is an injective resolu-tion of Ω^•_Pan as a CP^•an-Module, then we have a canonical morphism of complexes of objects of C (P^an), E^•(Ω^•_P)^an−→ I^•(Ω_P^•an) (again an isomorphism of D(C (P^an))). Therefore we have the following diagram of (pairings of) double complexes

Hom^••Ω^•_{P an}(I^••an, E^•(Ω^•_P)^an)⊗ I^••an−−−→ (Γ_XE^•(Ω_P^•))^an

 y

 y Hom^••Ω^•_{P an}(I^••an, I^•(Ω^•_Pan)) ⊗ I^••an−−−→ Γ_X^anI^•(Ω_P^•an)

x



 y

Hom^••Ω_{P an}^• (J^••, I^•(Ω_P^•an)) ⊗ J^•• −−−→ Γ_X^anI^•(Ω_P^•an) . We notice that the canonical morphisms of bicomplexes

Hom^••Ω^•_{P an}(I^••an, I^•(Ω_P^•an))−−−→ Hom^•Ω^•_{P an}(M^•an, I^•(Ω_P^•an))←−−− Hom^••Ω^•_{P an}(J^••, I^•(Ω_P^•an)) induce isomorphisms of the associated spectral sequences (see for example [HL, 4.2]). Therefore, taking the associated diagram of pairings of spectral sequences and composing with the canonical GAGA morphism of pairings of spectral sequences induced by

Hom^••Ω_P^•(I^••, E^•(Ω_P^•)) ⊗ I^•• −−−→ Γ_XE^•(Ω_P^•)

 y

 y

 y

ε_∗Hom^••Ω^•_{P an}(I^••an, E^•(Ω^•_P)^an)⊗ ε_∗I^••an−−−→ ε_∗Γ_XE^•(Ω_P^•)^an,

we complete the proof. 

5.4.3. Theorem. For any i, the canonical comparison isomorphisms αi of (5.0.4) and αⁱ_c of (5.2.1) are compatible with Poincar´e duality, i.e. they fit in a commutative diagram

H_i^DR(X) ⊗ HDR,cⁱ (X) −−−→ C

α_iok ok^αic k

H_i^DR(X^an)⊗ HDR,cⁱ (X^an)−−−→ C . Proof. We apply the previous proposition, withM^•= “lim

←−”

Mj^{(M )}_! Ω^•

X_W^{(M )}. We obtain a commutative diagram

H_i^DR(X) ⊗ HDR,cⁱ (X) −−−→ H²ⁿ

X(P, Ω_P^•) = H₀^DR(X)

αiok ok^αic ok^α0

H_i^DR(X^an)⊗ HDR,cⁱ (X^an)−−−→ H²ⁿ_X^an(P^an, Ω^•_Pan) = H₀^DR(X^an).

We consider the algebraic trace map Tr : H²ⁿ

X(P, Ω_P^•)−→ C and the map Tr^an: H²ⁿ

X^an(P^an, Ω_P^•an)−→ C uniquely defined by fitting in the commutative diagram

H²ⁿ

X(P, Ω_P^•) −−−→ C^Tr





 y

∼=

H²ⁿ

X^an(P^an, Ω^•_Pan)−−−→

Tr^an

C

Composing with Tr and Tr^an, we obtain the diagram of the statement. 

6. Compatibility of rigid and algebraic Poincar´ e duality.

6.1. Setting. Let V be a discrete valuation ring of mixed characteristics (0, p). As usual, let K and k be, respectively, the fraction field and the residue field of V . All V -schemes will be meant to be

separated and of finite type, and all morphisms between them will be assumed to be V -morphisms. Let X be aV -scheme, and let j : X ,→ X be an open immersion in a proper V -scheme. Let i : X ,→ P be a closed immersion of X in aV -scheme P . We assume that there is an open smooth V -subscheme W of P containing X as a closed subscheme. Let h : C ,→ X be a closed V -subscheme of X, whose support is exactly |X| r |X|.

6.2. Notation. For a V -scheme T , we will use the standard notation Tk (resp. TK) for the special (resp. the generic) fiber. The formal V -scheme completion of T along a closed V -subscheme S, will be denoted by T_/S. In the special case of S = T_k, we will write bT for T_/Tk. We freely use the notions and notation of [B.96]. In particular, the (Raynaud) generic fiber of a formal schemeT over Spf V (for the p-adic topology on V ) is denoted by TK. We point out that we take this notion in the extended sense of [B.96, 0.2.6]. If T is aV -scheme, the tube of a locally closed subset U of Tk in ( bT )K=: bTK will be denoted by ]U [

Tb. We use as much as possible a functorial notation for these constructions. For example, for C,→ X^h ,→ P asⁱ before, we get

(6.2.1) ]hk[

Pb : ]Ck[

Pb,−−−→ ]Xk[

Pb , h_{P /}: P_/Ck,−−−→ P_/Xk, (6.2.2) (h_{P /})K: (P_/Ck)K,−−−→(P_/Xk)K , and (i_/X

k)K: bXK,−−−→(P_/Xk)K . Notice that, by [B.96, 0.2.7], ]hk[

Pb identifies with (hP /)K.

For a K-scheme of finite type T , we denote by T^anthe associated rigid analytic space and let

(6.2.3) ε = ε_T : T^an−−−→ T

be the natural morphism of locally ringed G-spaces. For a coherent sheafE on T we set E^an= ε^∗_T(E ). When T is a properV -scheme,TbK ∼= (TK)^an. For example, the map (i_/X

k)K in (6.2.2) identifies with the natural closed immersion of rigid K-analytic spaces X^an_K in ]X_k[

Pb.

6.3. Cohomology with compact supports. The data in 6.1 permit to simultaneously calculate the algebraic De Rham cohomology with compact supports of XK/K, H_DR,c^• (XK/K), and the rigid cohomology with compact supports of Xk/K, H_rig,c^• (Xk/K).

The algebraic De Rham cohomology with proper supports of XK/K may be calculated from the complex of abelian sheaves on XK

(6.3.1) Rlim

M >N←−

IC^N_K^−•Ω_P^•

K

I_X^M−•

K

∼= (Ω_P^•

K)_/X

K→(ΩP^•_K)_/CK

tot

so that

(6.3.2) H_DR,c^• (XK/K) = H^•(XK,

(Ω_P^•_K)_/X

K→(ΩP^•_K)/C_K



tot) . 6.4. The complex of coherentOP_K^an-Modules

((Ω^•_P

K/I_X^N−•

K

)^an→(ΩP^•_K/IC^NK^−•)^an)_tot∼= ((Ω_P^•an K/I_X^Nan−•

K

)→(Ω^•P_K^an/IC^N_K^an−•))_tot, where I_X^an

K (resp. IC_K^an) is theOP_K^an-Ideal corresponding to the closed analytic subspace X^an_K (resp. C_K^an) of P_K^anmay also be regarded as a complex of abelian sheaves on X^an_K. By GAGA over K and (1.3.1) on X_K, we then have

(6.4.1) H_DR,c^• (XK/K) = lim

←−N

H^•(X^an_K, ((Ω_P^•an K/I_X^Nan−•

K

)−−→(ΩP^•_K^an/IC^N_K^an−•))tot) .

6.5. We have canonical flat morphisms ofV -formal schemes (6.5.1) P_/Ck−−−→ P^{hP /} _/Xk−−−→ P^{iP /} /P_k=: bP and open immersions of rigid K-analytic spaces

(6.5.2) (P_/Ck)K

(h_{P /})K

,−−−→(P_/Xk)K (i_{P /})K

,−−−→(P/Pk)K = bPK . 32

The last sequence identifies via [B.96, 0.2.7] with the natural sequence of open immersions of rigid K-analytic spaces

(6.5.3) ]Ck[

Pb,−−−→]Xk[

Pb,−−−→ bPK . Let us consider the diagram

P/C_k analytification in the sense of [B.96, 0.2.6], and using [B.96, 0.2.7], we obtain a complex of coherent sheaves on (P_/Ck)_K =]C_k[

/Xk)K is taken in bidegree (0, 0). By definition [B.86, 3.3], (6.5.5) H_rig,c^• (X_k/K) := H^•_]X 6.6. Theorem (Cospecialization morphism). Under the previous assumptions, there is a canonical functorial K-linear map

cosp^•: H_c,rig^• (Xk/K)−−−→ HDR,c^• (XK/K) . This map will be called cospecialization.

Proof. The proof is based on the following diagram of locally ringed G-spaces C_K ,−−−−−−−−−−−−−−−−−→ XK ,−−−−−−−−−−−−−→ P_K We notice that for any r and N there are isomorphisms

(6.6.1) (]hk[ We deduce the existence of maps

(]h_k[

Pb)_∗-acyclic, the previous maps are in fact surjective.) These we may view as a map

of complexes on X^an_K. Combining the two, we get morphisms

and, passing to the limit, maps of complexes of abelian sheaves on X^an_K (6.6.2) ((i_/X

We deduce morphisms of hypercohomology groups H_c,rig^• (Xk/K) := H^•(]Xk[

which induce the cospecialization morphism. 

6.7. The specialization morphism (from [BB]). For the convenience of the reader, we recall the construction given in [BB] of the specialization morphism between algebraic and rigid-analytic homologies (or cohomologies supported in a closed subset). In the notation of 6.1, let U be the open complement of X in W and u be the open immersion U ,→ W . If we take an injective resolution I^• of Ω_W^•

K, and an injective resolutionJ^• of Ω^•_Wan

K, then we have a canonical morphism

(6.7.1) ε⁻¹_W We sum up the previous remarks into the following proposition.

6.7.4. Proposition. For any injective resolutionI^• of Ω_W^•

K andJ^• of Ω_W^• an

Via the functoriality map induced by ε_WK, we deduce a morphism of complexes of K-vector spaces (6.7.5) Γ(WK, I^•→ uK∗u⁻¹_K I^•

This is the specialization morphism

(6.7.6) sp^•: H_DR,X^• _K(WK/K)−−−→ Hrig,X^• k(Wk/K) .

6.8. Another construction of the cospecialization morphism. In order to compare Poincar´e dualities in the algebraic and rigid contexts, it is convenient to exhibit a construction of the cospecialization morphism, closer to the previous definition of the specialization map.

6.8.1. We place ourselves in the previous setting, and let W be a proper V -scheme containing W as an open subscheme. We give here some structure of V -scheme to W r W (resp. to W r U) such that W r W ,→ W r U ,→ W be closed immersions; we denote by l : W r W ,→ W the composite immersion. Then where RjW_K! indicates the composite R lim

←− ◦jW_K!. Then

On the other hand, we may define H_DR,c^• (X_K^an/K) := H^•(P_K^an, ((Ω_P^•an The canonical map ε_W

K : W^an_K→ WK induces a canonical morphism taking cohomology over W^an_K and composing with the canonical functorial map induced by ε_W

K we have the morphism

(6.8.3) H_DR,c^• (XK/K)−−−→ HDR,c^• (X_K^an) which is an isomorphism, as already said, by a GAGA argument over K.

6.8.4. From the commutative diagram

W^an_K rW_K^{an l}

as in (6.6.2), we have a canonical morphism Applying this for W and U , we obtain a morphism

(6.8.7) Taking hypercohomology, we find a canonical morphism

(6.8.8) H_rig,c^• (X_k/K)−−−→ HDR,c^• (X_K^an) ;

finally, composing with the isomorphism H_DR,c^• (X_K^an) ∼= H_DR,c^• (X_K/K) we obtain the cospecialization mor-phism of the theorem.

6.9. Theorem (Compatibility of algebraic and rigid Poincar´e dualities). For i = 0, . . . , 2n, the canonical morphisms of specialization

sp²ⁿ⁻ⁱ: H_DR,X²ⁿ⁻ⁱ

K(W_K/K)−→ Hrig,X²ⁿ⁻ⁱ_k(W_k/K) of [BB] and of cospecialization

cospⁱ: H_rig,cⁱ (X_k/K)−→ HDR,cⁱ (X_K/K) are compatible with Poincar´e pairings

h , i : H_DR,X²ⁿ⁻ⁱ_K(WK/K)⊗ HDR,cⁱ (XK/K)−−−→ K and

h , i : Hrig,X²ⁿ⁻ⁱ_k(Wk/K)⊗ Hrig,cⁱ (Xk/K)−−−→ K

in the sense that, for α∈ HDR,X²ⁿ⁻ⁱ_K(WK/K) and β∈ Hrig,cⁱ (Xk/K), one hashsp²ⁿ⁻ⁱα, βi = hα, cospⁱβi . Proof. In the notation of 6.1, we may and will assume P proper and P_K smooth. So, in the second construction of the cospecialization morphism, we take W = P . In this special case, by its very definition, the trace map Tr^rig_W

k : H_rig,c²ⁿ (W_k/K)−→ K of [B.97, 2] is the composite H_rig,c²ⁿ (Wk/K)−−−→ H^∼= rig²ⁿ(Pk/K)

Tr^rig

−−−→ KPk

and sits in a commutative diagram

H_DR,c²ⁿ (WK/K)−−−→ H^∼= DR²ⁿ (PK/K)−−−→ K^TrPK

The pairing in the rigid context is constructed by Berthelot in [B.97, 2.2] using the adjunction between the functors j†_W and Γ_]Wk[

bP

([B.97, 2.1]). Taking an injective resolutionJ^•of Ω_P^•an

K as a complex of sheaves of K-vector spaces on P_K^an, and a pairing J^• ⊗ J^•→ J^• extending the wedge product, one obtains a pairing

j†_WJ^•⊗ Γ]W_k[

bPJ^•−−−→ J^• . This, applied to the smooth schemes Wk and Uk, leads to the pairing



where j†_WJ⁰(resp. Γ_]Wk[

bPJ⁰) is placed in bidegree (0, 0). Taking cohomology, we get the pairings (6.9.1) H_p^rig(Xk/K)⊗ Hrig,c^p (Xk/K)−−−→ Hrig²ⁿ(Pk/K) .

In the present algebraic context, the pairing (3.5.1) becomes, via (2.6.1), jW_K∗E(Ω_W^•

K)⊗ “lim←−”

N

I_P^N_K⁻_rW^• _KΩ_P^•

K−→ E(ΩP^•_K) . This pairing is also induced by an extension E(Ω_P^•

K)⊗ ΩP^•_K→ E(ΩP^•_K) of the wedge product. So, we get the canonical pairing

jW_K∗E(Ω_W^• _K)⊗ RjW_K!Ω_W^•_K−−−→ E(ΩP^•_K) . This, applied to the smooth schemes WK and UK, leads to the pairing

jW_K∗E(Ω^•_WK)→ jU_K∗E(Ω_U^•_K)

tot⊗ RjU_K!Ω_U^•_K→ RjW_K!Ω^•_WK

tot−−−→ E(Ω^•PK) . Taking cohomology, we obtain the pairings

(6.9.2) H_p^DR(X_K/K)⊗ HDR,c^p (X_K/K)−−−→ HDR²ⁿ(P_K/K) .

Now the compatibility of (6.9.2) and (6.9.1) is obvious via proposition 6.7.4, (6.8.2) and (6.8.7). 

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Nel documento Poincar´ e Duality for Algebraic De Rham Cohomology. (pagine 31-38)