Conjecture de l'inertie mod\'{e}r\'{e}e de Serre

Abstract

Let K be a local field of mixte characteristics. We assume that the residue field is perfect. Let X\_K be a proper smooth scheme over K admitting an integer model X which is proper and semi-stable. In this article, we prove a period isomorphism linking the \'{e}tale cohomology of X\_Kbar with coefficients in Z/p^n and the log-crystalline cohomology of the special fiber of X. Nevertheless, we have a restriction on the absolute ramification of K and the degree of the cohomologies. We apply the theory to deduce a complete proof of the Serre conjecture on the tame inertia.