Construction of semi-stable \(p\)-adic representations
type: Item
instance of: scholarly article
title: Construction of semi-stable \(p\)-adic representations
zbMATH Open document ID: 1010.14004
author: Pierre Colmez
author: Jean-Marc Fontaine
published in: Inventiones Mathematicae
publication date: 2001-01-04T00:00:00Z
review text: Let \(K\) be a field of characteristic zero complete with respect to some discrete valuation. If \(K_0\) denotes the field of fractions of the associated ring of Witt vectors, then \(K/K_0\) is a finite totally ramified field extension. A \(p\)-adic representation of the Galois group \(G_K=\text{Gal} (\overline K/K)\) is given by a \(\mathbb{Q}_p\)-vector space of finite dimension, together with a continuous linear action of \(G_K\) on it. Among those representations are the so-called semi-stable representations, which are related to étale cohomology theory in algebraic geometry, and for any semi-stable \(p\)-adic representation, there is an associated algebraic object, namely its so-called \((\varphi, N)\)-module \(D\). The correspondence that associates to a representation \(V\) its filtred \((\varphi,N)\)-module \(D\) defines a tensor equivalence between the category of semi-stable \(p\)-adic representations of \(G_K\) and a full subcategory of the category of \((\varphi,N)\)-modules, which is the so-called category of admissible filtred \((\varphi,N)\)-modules. The second author of the present paper, \textit{J.-M. Fontaine}, has studied this functor in some of his earlier works between 1979 and 1994. However, up to now, the entire theory did not appear to be complete, as no explicit description of the category of admissible filtred \((\varphi,N)\)-modules had been achieved so far. In the paper under review, the authors prove a fact that had been conjectured in the past. Namely, one can define the category of weakly admissible filtred \((\varphi, N)\)-modules, which seems to be easier to come to grips with, and then the conjecture states that weak admissibility is indeed the same as ordinary admissibility. The fact that this equivalence holds is a consequence from theorem A proved in the present paper. This principal theorem, which transpires a wealth of important corollaries, with a view to semi-stable \(p\)-adic representations of the Galois group \(G_K\), is p…
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 1443591
zbMATH Keywords: Tannaka categories
zbMATH Keywords: crystalline cohomology
zbMATH Keywords: \(p\)-adic étale cohomology
zbMATH Keywords: \(p\)-adic representation of the Galois group
zbMATH Keywords: category of admissible filtred modules
zbMATH Keywords: Witt vectors
reviewed by: Werner Kleinert
MaRDI profile type: Publication
full work available at URL: s002220000042
OpenAlex ID: W2042283093
Wikidata QID: Q105575916
DOI: 10.1007/S002220000042
description: scientific article; zbMATH DE number 1443591