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Rémi Shankar Lodh
Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case
(Revêtements presque étales d’anneaux de Fontaine et cohomologie log-cristalline dans le cas semi-stable)
Annales de l'institut Fourier, 61 no. 5 (2011), p. 1875-1942, doi: 10.5802/aif.2661
Article PDF | Reviews MR 2961843 | Zbl pre06032129
Class. Math.: 14F30
Keywords: $p$-adic Hodge theory, almost étale extensions, crystalline cohomology, log-structures
[1] F. Andreatta & O. Brinon, “Acyclicité géométrique d’un $B_{\operatorname{cris}}$ relatif”, Preprint, 2007
[2] P. Berthelot, Cohomologie cristalline des schémas de caractéristique $p>0$, Lecture notes in mathematics 407, Springer, 1974 MR 384804 | Zbl 0298.14012
[3] P. Berthelot & A. Ogus, Notes on crystalline cohomology, Princeton University Press, 1978 MR 491705 | Zbl 0383.14010
[4] N. Bourbaki, Algèbre commutative, Masson, 1985
[5] C. Breuil, “Topologie log-syntomique, cohomologie log-cristalline, et cohomologie de Čech”, Bull. S.M.F. 124 (1996) no. 4, p. 587-647 Numdam | MR 1432059 | Zbl 0865.19004
[6] G. Faltings, Crystalline cohomology and $p$-adic Galois representations, in J.-I. Igusa, ed., Algebraic Analysis, Geometry and Number Theory, Johns Hopkins University Press, 1989, p. 25-80 MR 1463696 | Zbl 0805.14008
[7] G. Faltings, “Almost étale extensions”, Astérisque 279 (2002), p. 185-270 MR 1922831 | Zbl 1027.14011
[8] J.-M. Fontaine, “Le corps des périodes $p$-adiques”, Astérisque 223 (1994), p. 59-111 MR 1293971 | Zbl 0940.14012
[9] O. Gabber & L. Ramero, Almost ring theory, Lecture notes in mathematics 1800, Springer, 2003 MR 2004652 | Zbl 1045.13002
[10] A. Grothendieck & J. Dieudonné, “Éléments de géométrie algébrique”, Publ. math. I.H.E.S. 4,8,11,17,20,24,28,32 (1960-1967) Numdam | Zbl 0153.22301
[11] L. Illusie, “Complexe de de Rham-Witt et cohomologie cristalline”, Ann. Sci. E.N.S. (4ème série) 12 (1979), p. 501-661 Numdam | MR 565469 | Zbl 0436.14007
[12] K. Kato, Logarithmic structures of Fontaine-Illusie, in J.-I. Igusa, ed., Algebraic Analysis, Geometry and Number Theory, Johns Hopkins University Press, 1989, p. 191-224 MR 1463703 | Zbl 0776.14004
[13] K. Kato, “Semi-stable reduction and $p$-adic étale cohomology”, Astérisque 223 (1994), p. 269-293 MR 1293975 | Zbl 0847.14009
[14] J.-P. Serre, Corps locaux, Hermann, 1968 MR 354618
[15] J.-P. Serre, Local Algebra, Springer Monographs in Mathematics, Springer, 2000 MR 1771925 | Zbl 0959.13010
Rémi Shankar Lodh
Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case
(Revêtements presque étales d’anneaux de Fontaine et cohomologie log-cristalline dans le cas semi-stable)
Annales de l'institut Fourier, 61 no. 5 (2011), p. 1875-1942, doi: 10.5802/aif.2661
Article PDF | Reviews MR 2961843 | Zbl pre06032129
Class. Math.: 14F30
Keywords: $p$-adic Hodge theory, almost étale extensions, crystalline cohomology, log-structures
Résumé - Abstract
Let $K$ be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic $p>0$, and let $K^+$ be the valuation ring of $K$. We relate the log-crystalline cohomology of the special fibre of certain affine $K^+$-schemes $X=\operatorname{Spec}(R)$ with good or semi-stable reduction to the Galois cohomology of the fundamental group $\pi _1(X_{\bar{K}})$ of the geometric generic fibre with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings’ theory of almost étale extensions.
Bibliography
[2] P. Berthelot, Cohomologie cristalline des schémas de caractéristique $p>0$, Lecture notes in mathematics 407, Springer, 1974 MR 384804 | Zbl 0298.14012
[3] P. Berthelot & A. Ogus, Notes on crystalline cohomology, Princeton University Press, 1978 MR 491705 | Zbl 0383.14010
[4] N. Bourbaki, Algèbre commutative, Masson, 1985
[5] C. Breuil, “Topologie log-syntomique, cohomologie log-cristalline, et cohomologie de Čech”, Bull. S.M.F. 124 (1996) no. 4, p. 587-647 Numdam | MR 1432059 | Zbl 0865.19004
[6] G. Faltings, Crystalline cohomology and $p$-adic Galois representations, in J.-I. Igusa, ed., Algebraic Analysis, Geometry and Number Theory, Johns Hopkins University Press, 1989, p. 25-80 MR 1463696 | Zbl 0805.14008
[7] G. Faltings, “Almost étale extensions”, Astérisque 279 (2002), p. 185-270 MR 1922831 | Zbl 1027.14011
[8] J.-M. Fontaine, “Le corps des périodes $p$-adiques”, Astérisque 223 (1994), p. 59-111 MR 1293971 | Zbl 0940.14012
[9] O. Gabber & L. Ramero, Almost ring theory, Lecture notes in mathematics 1800, Springer, 2003 MR 2004652 | Zbl 1045.13002
[10] A. Grothendieck & J. Dieudonné, “Éléments de géométrie algébrique”, Publ. math. I.H.E.S. 4,8,11,17,20,24,28,32 (1960-1967) Numdam | Zbl 0153.22301
[11] L. Illusie, “Complexe de de Rham-Witt et cohomologie cristalline”, Ann. Sci. E.N.S. (4ème série) 12 (1979), p. 501-661 Numdam | MR 565469 | Zbl 0436.14007
[12] K. Kato, Logarithmic structures of Fontaine-Illusie, in J.-I. Igusa, ed., Algebraic Analysis, Geometry and Number Theory, Johns Hopkins University Press, 1989, p. 191-224 MR 1463703 | Zbl 0776.14004
[13] K. Kato, “Semi-stable reduction and $p$-adic étale cohomology”, Astérisque 223 (1994), p. 269-293 MR 1293975 | Zbl 0847.14009
[14] J.-P. Serre, Corps locaux, Hermann, 1968 MR 354618
[15] J.-P. Serre, Local Algebra, Springer Monographs in Mathematics, Springer, 2000 MR 1771925 | Zbl 0959.13010
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310