Abstract : The aim of this thesis is to give a complete proof of the tame inertia Serre's conjecture which gives constraints (in relation to e and r) on the Galois group of cohomology H^r_et(X_Kbar, Z/pZ) where X is a proper smooth variety with semi-stable reduction on a p-adic field K with absolute ramification index e.
In order to do that, we establish, in the case er < p-1, a period isomophism linking the former étale cohomology group and a group of log-crystalline cohomology of the special fiber of X. Then, we show that this group defines an object of the category M^r introduced by Breuil. Finally, the conclusion follows from a careful study of the objects of M^r.
The last chapter of this thesis (which is independent) is devoted to the construction of a duality on the category M^r.