Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1875-1942.

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=Spec(R) with good or semi-stable reduction to the Galois cohomology of the fundamental group π 1 (X 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.

Soit K un corps de caractéristique nulle, complet pour une valuation discrète, à corps résiduel parfait de caractéristique p>0, et soit K + son anneau d’entiers. Nous montrons que la cohomologie log-cristalline de la fibre spéciale de certains K + -schémas affines X=Spec(R) à réduction bonne ou semi-stable se calcule comme la cohomologie galoisienne du groupe fondamental π 1 (X K ¯ ) de la fibre générique géométrique de X à valeurs dans un anneau de Fontaine construit à partir de R. Ce calcul est basé sur la théorie des revêtements presque étales de Faltings.

DOI: 10.5802/aif.2661
Classification: 14F30
Keywords: p-adic Hodge theory, almost étale extensions, crystalline cohomology, log-structures
Lodh, Rémi Shankar 1

1 The University of Utah Department of Mathematics 155 S 1400 E Salt Lake City UT 84112 (USA)
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Lodh, Rémi Shankar. Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1875-1942. doi : 10.5802/aif.2661. http://archive.numdam.org/articles/10.5802/aif.2661/

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