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Integral crystalline cohomology over very ramified valuation rings. (English) Zbl 0914.14009
Let $$V$$ be a complete discrete valuation ring with $$\pi$$ one of its uniformisers. Denote by $$k=V/\pi\cdot V$$ the residue field and assume $$k$$ is perfect of characteristic $$p>0$$; actually $$p$$ is assumed to be odd in most of the paper. Write $$K$$ for the fraction field of $$V$$ and assume $$K$$ has characteristic zero. $$V_0=W(k)\subset V$$ will denote the ring of Witt vectors, $$\varphi$$ its Frobenius, $$K_0$$ its field of fractions, and $$e=[K:K_0]$$ the ramification degree. It follows from Fontaine’s comparison theory for $$p$$-adic étale cohomology and crystalline cohomology that in the unramified situation, for a smooth and proper $$V$$-scheme $$X$$, $$V$$ a $$p$$-adic discrete valuation ring, there exists a ring $$B(V)$$ and an isomorphism $H^*_{\text{ét}}(X\otimes_V\overline{K},{\mathbb Q}_p)\otimes_{{\mathbb Q}_p}B(V)\cong H^*_{\text{crys}} (X/V_0)\otimes_{V_0}B(V).$ In the underlying paper one is interested in a generalization of Fontaine’s situation to both ramified valuation rings and more general coefficients. So let $$X$$ be a proper and smooth $$V$$-scheme, $$V$$ totally ramified over $$V_0$$, with uniformiser $$\pi$$ satisfying the minimal equation $$f(\pi)=0$$. Let $$R_V$$ be the PD-hull of $$V$$, i.e. the PD-completion of the ring obtained by adjoining divided powers of $$f$$ to $$R=V_0[[T]]$$ , where $$f(T)=T^e+\text{l.o.t.}$$ is an Eisenstein polynomial. One can define the relative crystalline cohomology of $$X/R_V$$. Let $$\mathcal E$$ be a crystal of vector bundles on $$X/R_V$$, e.g. if $$X$$ lifts to a smooth formal scheme $$\mathcal X$$ over $$R_V$$, $$\mathcal E$$ corresponds to a vector bundle $$\mathcal E_{\mathcal X}$$ with an integrable connection $$\nabla$$. One may calculate $$H^*(X/R_V,\mathcal E)$$ by de Rham complexes. More generally, using filtered crystals $$\mathcal E$$ (i.e. such that $$\mathcal E_{\mathcal X}$$ is filtered with $$\nabla$$ satisfying Griffiths tranversality), the associated graded of the complex representing $$H^*(X/R_V,\mathcal E)$$ is a module over $$\text{gr}_F(R_V)$$. Then, forming the derived tensor product with $$gr^0_F(R_V)=V$$, one obtains the hypercohomology of $$X$$ with values in the associated graded of the de Rham complex $$\mathcal E_X\otimes\Omega^*_{X/V}$$. This is called the Hodge cohomology of $$\mathcal E_X$$, written $$H^*(X,\mathcal E_X\otimes\Omega^*_{X/V})$$. The following assumption is basic:
$$(*)$$ $$H^*(X,\mathcal E_X\otimes\Omega^*_{X/V})$$ is torsion-free over $$V$$ and its spectral sequence degenerates.
The crystalline cohomology $$H^*(X/R_V,\mathcal E)$$ can be represented by a finite complex $$M^*(X/R_V,\mathcal E)$$ of filtered free $$R_V$$-modules. $$R_V$$ has a Frobenius action $$\phi$$ given by $$\phi(T)=T^p$$. This extends to $$X$$ and $$\mathcal E$$. One defines a Frobenius crystal (F-crystal for short) as a crystal $$\mathcal E$$ with an isomorphism $$\phi_X^*(\mathcal E)[1/p]\cong\mathcal E[1/p]$$. Crystalline cohomology satisfies Poincaré duality and the dual of an $$F$$-crystal is an $$F$$-crystal again. One can state a first result:
Let $$X/V$$ be proper and smooth, and let $$\mathcal E$$ be a filtered F-crystal on $$X$$ such that $$(*)$$ holds. Then the crystalline cohomology $$H^*(X/R_V,\mathcal E)$$ is represented by a complex $$M^*(X/R_V,\mathcal E)$$ of filtered free $$R_V$$-modules, with trivial differentials. Furthermore, inverting $$p$$, all $$H^i(X/R_V,\mathcal E)[1/p]$$ are induced from $$K_0$$-vector spaces $$H^i_0$$ with Frobenius automorphism $$\Phi_0$$.
One may introduce a logarithmic structure on the base giving rise to semistable $$X$$. A corresponding result holds such that there is a nilpotent $$N\in\text{ End}(H^i_0)$$ such that the connection becomes $$\nabla= d+N\cdot dt/T$$ and it satisfies $$p\cdot\Phi_0\cdot N=N\cdot\Phi_0$$. Another variant is the one ‘with a divisor at infinity’. One may carry over the theory of the category $$\mathcal{MF}_{[0,p-2]}$$ defined by G. Faltings [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf. Baltimore 1988, 25-80 (1989; Zbl 0805.14008)], provided one restricts to objects without torsion. One starts with the category $$\mathcal{MF}_{[0,a]}(V)$$, $$a\leq p-2$$, of filtered $$R_V$$-modules equipped with nilpotent (mod $$p$$) connection $$\nabla$$ and Frobenius $$\Phi$$ satisfying suitable properties, and then for a smooth $$V$$-scheme $$X$$ one defines the category $$\mathcal{MF}_{[0,a]}(X/R_V)$$ whose objects are filtered crystals $$\mathcal E$$ with Frobenius maps $$\Phi_i:\Phi^*(F^i(\mathcal E))\rightarrow\mathcal{E}$$, ($$\ldots$$). It follows that for such $$\mathcal E$$ satisfying $$(*)$$ the cohomology $$H^b(X/R_V,\mathcal E)$$, $$a+b\leq p-2$$, lies in $$\mathcal{MF}_{[0,p-2]}(V)$$.
Next comes the definition and properties of Fontaine’s famous ring $$B^+(V)$$ with its filtration $$F^{\bullet}$$. One defines a map $$\alpha: {\mathbb Q}_p(1)\rightarrow B^+(V)^*$$ with logarithm $$\beta:{\mathbb Z}_p(1)\rightarrow F^1(B^+(V))$$, such that $$\Phi\cdot\beta=p\cdot \Phi$$. $$\beta_0$$ denotes the image of a generator of $${\mathbb Z}_p(1)$$. With an object $$M\in\mathcal{MF}_{[0,p-2]}(V)$$ one associates a $${\mathbb Z}_p$$-module with continuous $$\text{Gal}(\overline{K}/K)$$-action by ${\mathbb D}(M)=\operatorname{Hom}_{R_V,F,\Phi_*}(M,D^+(V))=\operatorname{Hom}_{B^+(V),F,\Phi_*}(M\otimes_{R_V}B^+(V),B^+(V)),$ with dual $${\mathbb D}(M)^*$$. One has the result:
(i) $${\mathbb D}(M)$$ is free over $${\mathbb Z}_p$$, of rank $$\text{rk}_{R_V}(M)$$;
(ii) For $$M\in\mathcal{MF}_{[0,a]}(V)$$ one has $\beta_0^a\cdot{\mathbb D}(M)^*\otimes_{{\mathbb Z}_p}B^+(V)\subset M\otimes_{R_V}B^+(V)\subset{\mathbb D}(M)^*\otimes_{{\mathbb Z}_p}B^+(V),$ where the inclusions are strict for the filtrations;
(iii) The functor $${\mathbb D}$$ from $$\mathcal{MF}_{[0,p-2]}(V)$$ to $${\mathbb Z}_p \text{-Gal}(\overline{K}/K)$$-modules is fully faithful.
The result also holds for ‘divisors at infinity’.
To come to a comparison of crystalline and étale cohomology for a proper and smooth $$V$$-scheme $$X$$, of relative dimension $$b$$, let $$\mathcal E\in\mathcal{MF}_{[0,a]}(X/R_V)$$, $$a+b\leq p-2$$. Assume $$(*)$$ holds. Define the smooth étale $${\mathbb Z}_p$$-sheaf $${\mathbb L}$$ on $$X\otimes_VK$$ by $${\mathbb L}={\mathbb D}(\mathcal E)^*$$. Then: Assume $$(*)$$ holds. For each $$i$$, the étale cohomology $$H^i(X\otimes_V\overline{K},{\mathbb L})$$ is isomorphic to $${\mathbb D}(H^i(X/R_V,\mathcal E))^*$$.
The same holds in the logarithmic case of ‘divisors at infinity’, both for usual cohomology and cohomology with compact support. In particular, the étale cohomology has no $$p$$-torsion as well.
The whole theory may be applied to $$p$$-divisible groups $$H$$ over $$V$$. It leads to a filtered F-crystal $$M=M(H)$$ over $$R_V$$, and a canonical map $$\rho:M(H)\otimes B^+(V)\rightarrow H^1_{\text{ét}}(H)\otimes B^+(V)$$ which is shown to be a functorial injection, respecting Frobenius, filtrations and Galois actions. Furthermore, $$\text{Coker}(\rho)$$ is annihilated by $$\beta_0$$. For $$p>2$$, the Tate module $$T_p(H)$$ is shown to be equal to $${\mathbb D}(M(H))$$. Defining an étale Tate $$r$$-cycle as a Galois-invariant class $$\psi_{\text{ét}}\in(H^1_{\text{ét}}(H))^{\otimes 2r}(r)$$ and a crystalline Tate $$r$$-cycle as a class $$\psi_{\text{crys}}\in F^r(M(H)^{\otimes 2r})$$ which is $$\nabla$$-parallel and fixed by $$\Phi_r=\Phi/p^r$$, one has Fontaine’s result that $$\rho(\psi_{\text{crys}}\otimes 1)=\psi_{\text{ét}}\otimes\beta^r$$. It follows that for $$r\leq p-2$$, the comparison between crystalline and étale cohomology respects integrality, i.e. $$\psi_{\text{crys}}$$ is integral iff $$\psi_{\text{ét}}$$ is integral.
Let $$H_0$$ be a $$p$$-divisible group over $$V_0$$ with dual $$H^*_0$$. If $$\dim(H_0)=d$$ and $$\dim(H^*_0)=d^*$$, one defines the height $$h=h(H_0):=d+d^*$$. Let $$M_0$$ be the Dieudonné module associated to $$H_0$$. Let $$n=dd^*$$. Then there is a versal deformation $$H$$ of $$H_0\otimes_{V_0}k$$ over $$A=V_0[[t_1,\ldots,t_n]]$$. Let $$M=M(H)=H^1_{DR}(H/A)\in\mathcal{MF}_{[0,1]}(A)$$ be the associated Dieudonné module. If $$\phi:A\rightarrow A$$ denotes the lift of Frobenius to $$A$$ with $$\phi(t_i)=t_i^p$$, there exists a canonical $$\phi$$-linear endomorphism $$\Phi:M\rightarrow M$$ inducing an isomorphism $$\Phi:(M+p^{-1}\cdot F)\otimes_{A\phi}A\cong M$$, where the restriction $$\Phi| F=p\cdot\Phi_1$$. Let $$R$$ be a complete local $$V_0$$-algebra with residue field $$k$$, and a PD-ideal $$I\subset R$$ such that all divided powers of $$I$$ are $$p$$-adically closed. Assume $$R$$ is of the form $$R=V_0[[x_1,\ldots,x_r]]$$, $$I=(0)$$, $$\phi_R(x_i)=x_i^p$$, $$\phi_R| V_0=\phi$$, and suppose on $$R$$ a free module $$M_R$$ of rank $$d+d^*$$ , a direct summand $$F_R\subset M_R$$ of rank $$d$$ and an isomorphism $$\Phi_R:\widetilde{M}\otimes_{R\phi}R\cong M$$, $$\widetilde{M}=M+p^{-1}\cdot F$$, are given. Also, assume the canonical pushforwards to $$V_0$$ are isomorphic. Then: There exists a lifting $$\alpha:A\rightarrow R$$ such that $$(M_R,F_R,\Phi_R)$$ is the pushforward of $$(M,F,\Phi)$$. In particular, $$(M_R,F_R,\Phi_R)$$ is induced by a deformation of $$H_0$$, and $$M_R$$ admits a unique connection $$\nabla_R$$ such that $$\Phi_R$$ is $$\nabla_R$$-horizontal, and $$\nabla_R$$ is induced from $$\nabla$$ on $$M$$.
The paper closes with an appendix on basic results on finiteness, and an explanation on how to treat the prime $$p=2$$.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14L05 Formal groups, $$p$$-divisible groups
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