On a theorem of Kisin
type: Item
instance of: scholarly article
title: On a theorem of Kisin
zbMATH Open document ID: 1296.11159
author: Floric Tavares Ribeiro
author: Alexander A. Beilinson
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2013-10-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1205.2913
full work available at URL: S1631073X1300068X
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 6216316
zbMATH Keywords: crystalline representations
zbMATH Keywords: discrete valuation field
MaRDI profile type: Publication
OpenAlex ID: W1974033988
arXiv ID: 1205.2913
cites work: Q4254096
cites work: Q4325442
cites work: Q4325443
cites work: Crystalline representations and F-crystals
DOI: 10.1016/J.CRMA.2013.04.002
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review text: Let \(K\) be a complete discrete valuation field of mixed characteristic with perfect residue field. We fix a uniformizer \(\pi\) of \(K\) and consider \(K_\xi = K(\pi^{1/p^\infty})\). The main result of this paper is to give a short and new proof of the following theorem of \textit{M. Kisin} [Prog. Math. 253, 457--496 (2006; Zbl 1184.11052)]: the restriction functor from the category of crystalline representations of \(G_K\) to the category of representations of \(G_{K_\xi}\) is fully faithful. A simple use of Shapiro's lemma reduces the theorem to showing that, if \(R\) is the representation given by inducing the trivial representation of \(G_{K_\xi}\) to \(G_K\), the only non-trivial crystalline \(G_K\)-submodule of \(R\) is just \(\mathbb Q_p\). As \(R\) can be identified with the space of continuous functions on \(G_K/G_{K_\xi}\), the authors prove that the union of all finite-dimensional \(G_K\)-submodules of \(R\) are locally polynomial functions; the union of all semistable \(G_K\)-submodules are polynomial functions; and finally the only nontrivial crystalline \(G_K\)-submodule is the set of constant functions.NEWLINENEWLINEThe paper is very-well written and it only has one page and a half!
description: scientific article; zbMATH DE number 6216316