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

Let $K$ be a field of characteristic zero complete for a discrete valuation,\nwith perfect residue field of characteristic $p>0$, and let $K^+$ be the\nvaluation ring of $K$. We relate the log-crystalline cohomology of the special\nfibre of certain affine $K^+$-schemes $X=\\text{Spec}(R)$ with semi-stable\nreduction to the Galois cohomology of the fundamental group of the geometric\ngeneric fibre $\\pi_1(X_{\\bar{K}})$ with coefficients in a Fontaine ring\nconstructed from $R$. This is based on Faltings' theory of almost \\'etale\nextensions.\n