Characteristic \(p\) analogue of modules with finite crystalline height
type: Item
instance of: scholarly article
title: Characteristic \(p\) analogue of modules with finite crystalline height
zbMATH Open document ID: 1167.11043
published in: Pure and Applied Mathematics Quarterly
publication date: 2009-04-14T00:00:00Z
review text: Let \(K\) be a complete valuation field with perfect residue field \(k\) of characteristic \(p>0\) and let \(K_{\text{sep}}\) be a separable closure of \(K\). Set \(\Gamma_ K={\text{Gal}}(K_{\text{sep}}/K)\) and let \(\Gamma_ K^ {(v)}\) be the ramification group with upper ramification number \(v\). Let \(R\) denote the valuation ring of \(K\). When \(K\) is of characteristic \(0\), let \({\text{MG}}_ K^ h\) be the category of finite \({\mathbb Z}_ p[\Gamma _ K]\)-modules with crystalline height \(h\) with \(h\in{\mathbb N}\). \textit{J.-M. Fontaine} [Invent. Math. 81, 515--538 (1985; Zbl 0612.14043)] proved that if \(G\in {\text{MG}} _ K^ 1\), \(p^ N G= 0\) and \(v>e(N+1/(p-1)) -1\), then \(\Gamma _ K^ {(v)}\) acts trivially on \(G\). Here \(e\) denotes the ramification index of \(K\) over \({\mathbb Q}_ p\). When \(K\) is of characteristic \(p\), the author [Compos. Math. 142, No. 4, 867--888 (2006; Zbl 1102.14032)] proved that if \(G={\mathcal G}(K_{\text{sep}})\) where \({\mathcal G}\) is an \(O\)-strict module over \(R\) (concept introduced by \textit{G. Faltings} [Mosc. Math. J. 2, No. 2, 249--279 (2002; Zbl 1013.11079)]) and \(\pi^ N G=0\), then for \(v>e(N+1/(q-1))-1\), \(\Gamma _ K^{(v)}\) acts trivially on \(G\), where \(k\supseteq {\mathbb F}_ q\), \(O={\mathbb F}_ q\pi\subseteq R\) and \(\pi\in R\) is not invertible. This result is the analogue to Fontaine's estimate in the unequal characteristic case. In the equal characteristic case, to see the interpretation of higher étale cohomology, it seems natural to consider Galois modules arising from Anderson motives. Let \({\text{MG}}(O)_ K\) be the category of \(O\)-modules of finite type with continuous \(O\)-linear action of \(\Gamma _ K\). Then \({\text{MG}}^ h(O)_ K\) is the corresponding category of \(O[\Gamma _ K]\)-modules of ``finite crystalline height'' \(h\), with \(h\in {\mathbb N}\). The first main result in this paper is that if \(G\in {\text{MG}}^ h(O)_ K\) then there is \(\pi\)-divisible g…
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 14F99
zbMATH DE Number: 5543156
zbMATH Keywords: local fields
zbMATH Keywords: upper ramification numbers
zbMATH Keywords: ramification filtration
zbMATH Keywords: ramification groups
zbMATH Keywords: crystalline representations
zbMATH Keywords: Anderson motives
zbMATH Keywords: valuations
author: Victor Abrashkin
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
DOI: 10.4310/PAMQ.2009.V5.N1.A14
description: scientific article; zbMATH DE number 5543156