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 \(p=2\).


14F30 \(p\)-adic cohomology, crystalline cohomology
14L05 Formal groups, \(p\)-divisible groups


Zbl 0805.14008
Full Text: DOI


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