Lifting Witt subgroups to characteristic zero
type: Item
instance of: scholarly article
title: Lifting Witt subgroups to characteristic zero
zbMATH Open document ID: 0919.14030
author: Alan Koch
published in: The New York Journal of Mathematics
publication date: 1998-10-28T00:00:00Z
full work available at URL: https://eudml.org/doc/119878
full work available at URL: Vol4.html
review text: Let \(k\) be a perfect field of characteristic \(p>0\) with \(p\) odd, let \(W(k)\) denote the ring of Witt vectors over \(k\), and let \(W_n\) denote the group scheme of Witt vectors of finite length \(n\). The author studies the question, when a connected subgroup scheme of \(W_n\) lifts to \(W(k)\) in terms of Dieudonné modules. The Dieudonné ring \(E\) associated to \(k\) is the non-commutative ring \(E= W(k)[F,V]\) with relations \(FV=VF =p\), \(Fw = w^{\sigma}F\), \(wV = Vw^{\sigma}\) for \(w \in W(k)\), where \(\sigma\) raises each component of \(w\) to its \(p\)-th power. Under the anti-equivalence between affine commutative unipotent \(k\)-group schemes and certain modules over \(E\) [see \textit{M. Demazure} and \textit{P. Gabriel}, ``Groupes algébriques'', Tome I (1970; Zbl 0203.23401)] the connected subgroup schemes \(G\) of \(W_n\) correspond to cyclic Dieudonné modules, i.e., modules of the form \(E/I\) for some ideal \(I \subset E\). The author now characterizes the connected subgroup schemes \(G\) of \(W_n\), which lift to \(W(k)\), in terms of the structure of the associated cyclic Dieudonné modules. The main tool is the use of finite Honda systems, and is based on work of \textit{J.-M. Fontaine} [C. R. Acad. Sci., Paris, Sér. A 280, 1273-1276 (1975; Zbl 0331.14022)].
reviewed by: Manfred Kolster
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 13K05
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 1215962
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: Dieudonné modules
zbMATH Keywords: group scheme
zbMATH Keywords: Dieudonné ring
zbMATH Keywords: finite Honda systems
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1215962