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Integral crystalline cohomology
over very ramified valuation rings

Author: Gerd Faltings
Journal: J. Amer. Math. Soc. 12 (1999), 117-144
MSC (1991): Primary 14F30, 14L05
MathSciNet review: 1618483
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Abstract: We explain how to set up an integral version ($\mathbb{Z}_{p}$ as opposed to $\mathbb{Q}_{p}$) of Fontaine's comparison between crystalline and étale cohomology, over $p$-adic fields with arbitrary ramification index. The main results then are that Fontaine's map respects integrality of Tate-cycles, and a construction of versal deformations of $p$-divisible groups with Tate-cycles. An appendix deals with finite generation of crystalline cohomology.

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  • [B] P. Berthelot, Cohomologie Cristalline des Schémas de Caractéristique $p>0$, Springer Lecture Notes 407, 1974. MR 52:5676
  • [BBM] P. Berthelot, L. Breen, W. Messing, Théorie de Dieudonné Cristalline II, Springer Lecture Notes 930, 1982. MR 85k:14023
  • [BO] P. Berthelot, A. Ogus, $F$-Isocrystals and de Rham Cohomology. I, Invent. math. 72, 1983, pp. 159-199. MR 85e:14025
  • [D] V.G. Drinfeld, Coverings of $p$-adic symmetric regions, Functional Analysis and its Applications 10, 1976, pp. 107-115. MR 54:10281
  • [Fa1] G. Faltings, $p$-adic Hodge-theory, Journal of the AMS 1, 1988, pp. 255-299. MR 89g:14008
  • [Fa2] 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
  • [Fa3] G. Faltings, $F$-isocrystals on open varieties, Grothendieck's 60'th birthday Festschrift, Birkhäuser, Boston, 1990. MR 92f:14015
  • [Fa4] G. Faltings, Crystalline cohomology of semistable curves, Journal of algebraic geometry 1, 1991, pp. 61-82, Correction in Journal of algebraic geometry 1, 1991, p. 427. MR 93e:14025; MR 93k:14028
  • [Fa5] G. Faltings, Crystalline cohomology of semistable curves - the $\mathbb Q _{p}$-theory, Journal of algebraic geometry 6, 1997, pp. 1-18. CMP 98:05
  • [Fa6] G. Faltings, Hodge-Tate Structures and Modular Forms, Math. Ann., vol. 278, 1987, pp. 133-149. MR 89e:11033
  • [Fo1] J.M. Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, Astérisque 65, 1979, pp. 3-80. MR 82k:14046
  • [Fo2] J.M. Fontaine, Sur certaines types de représentations $p$-adiques du groupe de Galois d'un corps local, construction d'un anneau de Barsotti-Tate, Ann. of Math. 115, 1982, pp. 529-577. MR 84d:14010
  • [Fo3] J.M. Fontaine, Le corps des périodes $p$-adiques, Prépublications Paris-Orsay, 1993, pp. 93-39.
  • [FL] J.M. Fontaine, G. Laffaille, Construction de représentations $p$-adiques, Ann. scient. ENS 15, 1982, pp. 547-608. MR 85c:14028
  • [H1] O. Hyodo, A note on $p$-adic étale cohomology in the semi-stable reduction case, Invent. math. 91, 1988, pp. 543-557. MR 89h:14017
  • [H2] O. Hyodo, On the de Rham-Witt complex attached to a semi-stable family, Comp. Math. 78, 1991, pp. 241-260. MR 93c:14022
  • [HK] O. Hyodo, K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, Asterisque No. 223, 1994, pp. 221-268 MR 95k:14034
  • [I] L. Illusie, Déformations des groupes de Barsotti-Tate, Astérisque 127, 1985, pp. 151-198. CMP 17:17
  • [K] K. Kato, Logarithmic structures of Fontaine-Illusie, in Algebraic Analysis, Geometry, and Number Theory (J.I. Igusa, ed.), Johns Hopkins Univ. Press, Baltimore 1989. CMP 97:16
  • [MM] B. Mazur, W. Messing, Universal extensions and one dimensional crystalline cohomology, Springer Lecture Notes 370, 1974. MR 51:10350
  • [Me] W. Messing, The crystals associated to Barsotti-Tate groups, Springer Lecture Notes 264, 1972. MR 50:337
  • [Mo] A. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. 72, 1993, pp. 301-337. MR 95a:14022
  • [T] T. Tsuji, Syntomic complexes and $p$-adic vanishing cycles, J. Reine Angew. Math. 472, 1996, pp. 69-138. MR 98a:14032
  • [V] H. Voskuil, Ultrametric uniformization and symmetric spaces, Thesis, Groningen, 1990.

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Additional Information

Gerd Faltings
Affiliation: Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germany

Keywords: $p$-adic \'etale cohomology, crystalline cohomology, $p$-divisible groups
Received by editor(s): February 22, 1994
Received by editor(s) in revised form: April 3, 1998
Article copyright: © Copyright 1999 American Mathematical Society

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