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Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case

Rémi Shankar Lodh — 2011

Annales de l’institut Fourier

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.

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