A generalization of Sen–Brinon’s theory

Abstract

Let K be a complete discrete valuation field of mixed characteristic and k be its residue field of prime characteristic p > 0. We assume that [k : k ^p] = p ^h < ∞. Let G _K be the absolute Galois group of K and ${\mathcal{R}}$ be a Banach algebra over ${\mathbb{C}_p:=\widehat{\overline{K}}}$ with a continuous action of G _K . When k is perfect (i.e. h = 0), Sen studied the Galois cohomology ${{\rm H}^1(G_K, \mathcal{R}^\ast)}$ and Sen’s operator associated to each class (Sen Ann Math 127:647–661, 1988). In this paper we generalize Sen’s theory to the case h ≥ 0 by using Brinon’s theory (Brinon Math Ann 327:793–813, 2003). We also give another formulation of Brinon’s theorem (à la Colmez).