Dieudonné module structures for ungraded and periodically graded Hopf rings.
type: Item
instance of: scholarly article
title: Dieudonné module structures for ungraded and periodically graded Hopf rings.
zbMATH Open document ID: 1210.16031
author: Rui Miguel Saramago
published in: Algebras and Representation Theory
publication date: 2010-11-23T00:00:00Z
review text: There are several different theories of Dieudonné modules, used to classify different classes of group schemes over perfect fields of positive characteristic. Hence, there are different types of Dieudonné modules for different classes of Hopf algebras. In the work under review, the author discusses many of the different categorical equivalences between Dieudonné modules and Hopf algebras. Additionally, the study is extended to Hopf rings and their equivalence to Dieudonné rings (which, as one might imagine, are Dieudonné modules with additional structure). Let \(H\) be a graded, connected, bicommutative (a.k.a. commutative and cocommutative, or Abelian) Hopf algebra over the finite field \(\mathbb{F} _p\). Then its corresponding Dieudonné module is \(D_*H=\{\Hom(H(n),H)\} _{n\geq 1}\), where \(H(n)=\mathbb{F}_p[x_0,\dots,x_k]\) where \(n\) strongly divides \(p^k\), with comultiplication defined in such a way as to make all Witt polynomials primitive. The Frobenius and Verschiebung on \(H(n)\) induce maps \(F\) and \(V\) on \(D_*H\) with the properties that \(FV=VF=p\) and for each \(x\in D_*H\) there is an \(n\geq 0\) such that \(V^nx=0\). Thus \(D_*H\) is a graded module over \(R:=\mathbb{F}_p[F,V]/(FV-p)\), and one can view \(D_*\) as a covariant functor between these Hopf algebras and graded \(R\)-modules which induces an equivalence of categories. Let \(U_*\) be the inverse correspondence. By generalizing to the category of Hopf rings \(\{H_n\}_{n\in\mathbb{Z}}\) where each \(H_n\) is a Hopf algebra of the type above the equivalence generalizes to Dieudonné rings by putting additional structure on the \(R\)-modules. Indeed, the necessary multiplications \(\circ_{ij}\colon H_i\otimes H_j\to H_{i+j}\) induce products \(D_*(H_i)\otimes D_*(H_j)\to D_*(H_{i+j})\) in Hopf algebras, and it is shown that \(U_*D_*H=\{U_*D_*H_n\}_{n\in\mathbb{Z}}\) is a Hopf ring equivalent to \(H\). Similarly, given a Dieudonné ring \(\{M_n\}_{n\in\mathbb{Z}}\) the maps \(\circ_{ij}'\co…
reviewed by: Alan Koch
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16W70
Mathematics Subject Classification ID: 57T05
Mathematics Subject Classification ID: 18E10
zbMATH DE Number: 5818577
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Dieudonné modules
zbMATH Keywords: Hopf rings
zbMATH Keywords: Dieudonné rings
zbMATH Keywords: group schemes over perfect fields
zbMATH Keywords: equivalences of categories
MaRDI profile type: Publication
full work available at URL: s10468-009-9135-8
OpenAlex ID: W2052373582
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DOI: 10.1007/S10468-009-9135-8
description: scientific article; zbMATH DE number 5818577