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

TL;DR

This paper explores the connection between log-crystalline cohomology of semi-stable affine schemes over a valuation ring and Galois cohomology, utilizing Faltings' almost étale extensions and Fontaine rings.

Contribution

It establishes a new relationship between log-crystalline cohomology and Galois cohomology via Fontaine rings in the semi-stable reduction context.

Findings

Relates log-crystalline cohomology to Galois cohomology of the geometric generic fibre.

Uses Faltings' theory of almost étale extensions to bridge cohomological theories.

Provides a framework for understanding semi-stable reduction through Fontaine rings.

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 $X=\text{Spec}(R)$ with semi-stable reduction to the Galois cohomology of the fundamental group of the geometric generic fibre $\pi_1(X_{\bar{K}})$ with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings' theory of almost \'etale extensions.