Monogenic bialgebras over finite fields and rings of Witt vectors
type: Item
instance of: scholarly article
title: Monogenic bialgebras over finite fields and rings of Witt vectors
zbMATH Open document ID: 0988.16026
DOI: 10.1016/S0022-4049(00)00163-8
author: Alan Koch
published in: Journal of Pure and Applied Algebra
publication date: 2002-04-07T00:00:00Z
reviewed by: Manfred Kolster
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 13K05
zbMATH DE Number: 1660389
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Witt vectors
zbMATH Keywords: Dieudonné modules
zbMATH Keywords: local rings
zbMATH Keywords: cocommutative bialgebras
zbMATH Keywords: numbers of lifts
zbMATH Keywords: Honda systems
MaRDI profile type: Publication
cites work: Congruence-torsion subgroups of dimension one formal groups
cites work: Q4097403
cites work: Q4097401
cites work: Q5328221
cites work: Lifting Witt subgroups to characteristic zero
cites work: Witt subgroups and cyclic Dieudonné modules killed by \(p\).
cites work: Deformations and liftings of finite, commutative group schemes
cites work: Group schemes of prime order
cites work: Q3884236
full work available at URL: s0022-4049(00)00163-8
OpenAlex ID: W2046275129
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review text: Let \(k=\mathbb{F}_{p^l}\) be a finite field. The author studies Hopf algebras over \(k\), which are monogenic, i.e. generated as algebras by a single element, and local-local, i.e. both \(H\) and its linear dual \(H^*\) are local rings. The first result gives a complete classification of these Hopf algebras: There is a one-to-one correspondence between monogenic local-local Hopf algebras over \(k\) and the set NEWLINE\[NEWLINE\mathbb{Z}^+\cup\{(n,r,z)\mid n\geq 1,\;r\leq n-1,\;0\leq z\leq p^d-2,\;d=\gcd(l,r+1)\}.NEWLINE\]NEWLINE The proof uses the one-to-one correspondence between finite local-local cocommutative bialgebras and Dieudonné modules. The author then studies lifts of these algebras to characteristic \(0\). He shows that \(H\) lifts to the ring \(W(k)\) of Witt vectors with coefficients in \(k\) if and only if \(H\) corresponds under the above classification to a triple \((n,r,z)\) with \(r+1\) dividing \(n\). Furthermore, if a lift is possible, the author computes the precise number of lifts. The proof uses the correspondence between lifts and finite Honda systems attached to Dieudonné modules and finite submodules of \(W(k)\) [cf. \textit{J.-M. Fontaine}, C. R. Acad. Sci., Paris, Sér. A 280, 1423-1425 (1975; Zbl 0331.14023)].
description: scientific article; zbMATH DE number 1660389