×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique \?>0, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin-New York, 1974 (French). · Zbl 0298.14012
[2] Pierre Berthelot, Lawrence Breen, and William Messing, Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, vol. 930, Springer-Verlag, Berlin, 1982 (French). · Zbl 0753.14041
[3] P. Berthelot and A. Ogus, \?-isocrystals and de Rham cohomology. I, Invent. Math. 72 (1983), no. 2, 159 – 199. · Zbl 0516.14017 · doi:10.1007/BF01389319 · doi.org
[4] V. G. Drinfel\(^{\prime}\)d, Coverings of \?-adic symmetric domains, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 29 – 40 (Russian).
[5] Gerd Faltings, \?-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255 – 299. · Zbl 0764.14012
[6] G. Faltings, Crystalline cohomology and \(p\)-adic Galois-representations, Proc. 1st JAMI-conference (ed. J.I. Igusa), pp. 25-81, John Hopkins (1989). CMP 97:16 · Zbl 0805.14008
[7] Gerd Faltings, \?-isocrystals on open varieties: results and conjectures, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 219 – 248. · Zbl 0736.14004
[8] Gerd Faltings, Crystalline cohomology of semistable curves, and \?-adic Galois-representations, J. Algebraic Geom. 1 (1992), no. 1, 61 – 81. Gerd Faltings, Correction to: ”Crystalline cohomology of semistable curves, and \?-adic Galois-representations” [J. Algebraic Geom. 1 (1992), no. 1, 61 – 81; MR1129839 (93e:14025)], J. Algebraic Geom. 1 (1992), no. 3, 427.
[9] G. Faltings, Crystalline cohomology of semistable curves - the \(\mathbb Q _{p}\)-theory, Journal of algebraic geometry 6, 1997, pp. 1-18. CMP 98:05
[10] Gerd Faltings, Hodge-Tate structures and modular forms, Math. Ann. 278 (1987), no. 1-4, 133 – 149. · Zbl 0646.14026 · doi:10.1007/BF01458064 · doi.org
[11] Jean-Marc Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, Journées de Géométrie Algébrique de Rennes. (Rennes, 1978) Astérisque, vol. 65, Soc. Math. France, Paris, 1979, pp. 3 – 80 (French). · Zbl 0429.14016
[12] Jean-Marc Fontaine, Sur certains types de représentations \?-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), no. 3, 529 – 577 (French). · Zbl 0544.14016 · doi:10.2307/2007012 · doi.org
[13] J.M. Fontaine, Le corps des périodes \(p\)-adiques, Prépublications Paris-Orsay, 1993, pp. 93-39.
[14] Jean-Marc Fontaine and Guy Laffaille, Construction de représentations \?-adiques, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 4, 547 – 608 (1983) (French). · Zbl 0579.14037
[15] Osamu Hyodo, A note on \?-adic étale cohomology in the semistable reduction case, Invent. Math. 91 (1988), no. 3, 543 – 557. · Zbl 0619.14013 · doi:10.1007/BF01388786 · doi.org
[16] Osamu Hyodo, On the de Rham-Witt complex attached to a semi-stable family, Compositio Math. 78 (1991), no. 3, 241 – 260. · Zbl 0742.14015
[17] Osamu Hyodo and Kazuya Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, Astérisque 223 (1994), 221 – 268. Périodes \?-adiques (Bures-sur-Yvette, 1988). · Zbl 0852.14004
[18] L. Illusie, Déformations des groupes de Barsotti-Tate, Astérisque 127, 1985, pp. 151-198. CMP 17:17
[19] K. Kato, Logarithmic structures of Fontaine-Illusie, in Algebraic Analysis, Geometry, and Number Theory , Johns Hopkins Univ. Press, Baltimore 1989. CMP 97:16 · Zbl 0776.14004
[20] B. Mazur and William Messing, Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics, Vol. 370, Springer-Verlag, Berlin-New York, 1974. · Zbl 0301.14016
[21] William Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972. · Zbl 0243.14013
[22] A. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. 72 (1993), no. 2, 301 – 337 (French). · Zbl 0834.14010 · doi:10.1215/S0012-7094-93-07211-0 · doi.org
[23] Takeshi Tsuji, Syntomic complexes and \?-adic vanishing cycles, J. Reine Angew. Math. 472 (1996), 69 – 138. · Zbl 0838.14015 · doi:10.1515/crll.1996.472.69 · doi.org
[24] H. Voskuil, Ultrametric uniformization and symmetric spaces, Thesis, Groningen, 1990. · Zbl 0718.22006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.