Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras.
type: Item
instance of: scholarly article
title: Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras.
zbMATH Open document ID: 1346.16026
author: Alexandru Chirvasitu
published in: Algebra \& Number Theory
publication date: 2014-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.6701
reviewed by: Stefaan Caenepeel
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16T15
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 6348603
zbMATH Keywords: cosemisimple Hopf algebras
zbMATH Keywords: faithfully flat Hopf algebras
zbMATH Keywords: cosemisimple Hopf subalgebras
zbMATH Keywords: module coalgebras
MaRDI profile type: Publication
OpenAlex ID: W2008565461
arXiv ID: 1110.6701
DOI: 10.2140/ANT.2014.8.1179
review text: The authors study the following question, asked by \textit{S. Montgomery} [Hopf algebras and their actions on rings. Providence: AMS (1993; Zbl 0793.16029)]: let \(A\) be a Hopf subalgebra of a Hopf algebra \(H\) over a field; is \(H\) faithfully flat over \(A\)? The question goes back to one of Kaplansky's conjectures, and has a rich history, documented well in the introduction of the paper.NEWLINENEWLINE The main result is the following: if \(H\) is cosemisimple, then \(H\) is faithfully flat over every Hopf subalgebra \(A\). Moreover, the left \(H\)-module coalgebra \(C=H/HA^+\) is cosemisimple. A second result is the following: if the coradical of \(H\) is a Hopf algebra, then \(H\) is faithfully flat over its cosemisimple Hopf subalgebras. In the final Section 3, the left inverse of the inclusion \(A\to H\) is studied, in the case where \(A\) and \(H\) are CQG algebras.
description: scientific article; zbMATH DE number 6348603