A generalization of Sen's theory.
type: Item
instance of: scholarly article
title: A generalization of Sen's theory.
zbMATH Open document ID: 1072.11089
published in: Mathematische Annalen
publication date: 2004-07-07T00:00:00Z
review text: Let \(K\) be a local field of characteristic \(0\) with residue field \(k\) of characteristic \(p>0\). Let \({\mathbb C}\) be the completion of an algebraic closure \(\overline{K}\) of \(K\) and let \(G\) be the Galois group \({\text{Gal}}( \overline{K}/K)\). Then \({\mathbb C}\) is algebraically closed and the action of \(G\) extends continuously to \({\mathbb C}\). The main goal of the paper under review is to generalize the results of \textit{S. Sen} [Invent. Math. 62, 89--116 (1980; Zbl 0463.12005)] obtained in case \(k\) is perfect. In this work the author does not assume \(k\) to be perfect and assumes that \([k: k^p]= p^h < \infty\), \(h\geq 0\). Let \({\mathbf {Rep}}_E (J)\) denote the \(E^J\)-linear additive category of finite dimensional \(E\)-representations, where \(E\) is a topological field on which a topological group \(J\) acts. A description of the category of \({\mathbb C}\)-representations is given, generalizing Sen's theory which corresponds to the case \(h=0\). Let \(t_1,\dots, t_h\) be elements of the ring of integers of \(K\) such that \(k=k^p[\overline{t}_1, \dots, \overline{t}_h]\). Let \(K_\infty\) be the field generated by the \(p^n\)-th roots of unity and those of \(t_i\) for all \(i=1,\dots, h\) and \(n\) any positive integer and let \(L\) be the closure of \(K_\infty\) in \({\mathbb C}\). Let \(H\) and \(\Gamma\) denote the Galois groups \({\text{Gal}}(\overline {K}/K_\infty)\) and \({\text{Gal}} (K_\infty/K)\) respectively. For any \(L\)-representation \(X\) of \(\Gamma\), let \(X^{\text{f}}\) denote the union of all finite-dimensional \(K_ \infty\)-subspaces of \(X\) which are stable under the action of \(\Gamma\). The main results of the paper are described by the following two theorems. (1) We have \({\mathbb C}^H=L\) and for every \({\mathbb C}\)-representation \(W\) of \(H\), the \({\mathbb C}\)-linear map \({\mathbb C}\otimes _L W^H\to W\) induced by the inclusion of \(W^H\) into \(W\) is an isomorphism. The functor \(W\to W^H\) …
Mathematics Subject Classification ID: 11S25
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 2078206
zbMATH Keywords: \(p\)-adic cohomology
zbMATH Keywords: \(p\)-adic Galois representations
zbMATH Keywords: \(p\)-adic Lie groups
zbMATH Keywords: \(p\)-adic representations
author: Olivier Brinon
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: s00208-003-0472-3
OpenAlex ID: W2126102672
DOI: 10.1007/S00208-003-0472-3
description: scientific article; zbMATH DE number 2078206