Une application de Corps des Normes

Abstract

Let k be a perfect field of characteristic p > 0, K$_0$ = Frac(W(k)), π a uniformizer in K$_0$ and π$_n$ ∈ K $_0$ (n∈ N) such that π$_0$ = π and π$^p$$_n$ $_ + 1$ = π$_n$. We write K$_∞$ = ∪$_n$ $_∈N$ K$_0$ (π$_n$), H$_∞$ = Gal (K$_0$/ K$_∞$ and G = Gal(K$_0$/ K$_0$). The main result of this paper is that the functor ’restriction of the Galois action‘ from the category of crystalline representations of G with Hodge–Tate weights in an interval of length [les ] p − 2 to the category of p-adic representations of H$_∞$ is fully faithful and its essential image is stable by sub-object and quotient. The proof uses the comparison between two ways of building mod. p representations of H$_∞$: one thanks to the norm field of K$_∞$, the other thanks to some categories of ’filtered‘ modules with divided powers previously introduced by the author.