Galois cohomology of Fontaine rings

Abstract

Let $V$ be a complete discrete valuation ring of mixed characteristic. We express the crystalline cohomology of the special fibre of certain smooth affine $V$-schemes $X=Spec(R)$ tensored with an appropriate ring of $p$-adic periods as the Galois cohomology of the fundamental group of the geometric generic fibre $\pi_1(X_{\bar{V}[1/p]})$ with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings' approach to $p$-adic Hodge theory (the theory of almost étale extensions). Using this we deduce maps from $p$-adic étale cohomology to crystalline cohomology of smooth $V$-schemes. The results are more general, as the semi-stable case is also considered. In the end we derive an alternative proof of the theorem of Tsuji (the semi-stable conjecture of Fontaine-Jannsen).