\(H\)-module endomorphism rings
type: Item
instance of: scholarly article
title: \(H\)-module endomorphism rings
zbMATH Open document ID: 0838.16034
DOI: 10.1016/0022-4049(94)00076-U
author: Yinhuo Zhang
author: Fred Van Oystaeyen
published in: Journal of Pure and Applied Algebra
publication date: 1995-12-04T00:00:00Z
review text: Let \(H\) be a finite-dimensional Hopf algebra and \(A/B\) an \(H\)-Galois extension. Then for any right \(A\)-module \(M\) there is a ring isomorphism \(\text{End}_A(M\otimes_BA)\simeq\text{End}_B(M)\# H\). \(\text{End}_B(M)/\text{End}_A(M)\) is an \(H^*\)-extension. If moreover \(H\) is unimodular, \(\text{End}_B(M)/\text{End}_A(M)\) is Galois if and only if \(M\otimes_BA\) weakly divides \(M\), and \(\text{End}_B(M)\) has a trace 1 element if and only if \(M\) weakly divides \(M\otimes_BA\) (the unimodularity is not actually needed for the last assertion). In the case where \(M\) and \(M\otimes_BA\) weakly divide each other, the rings \(\text{End}_A(M)\) and \(\text{End}_A(M\otimes_BA)\) are Morita equivalent. A Higman criterion for existence of trace 1 elements in the \(H\)-module algebra \(\text{End}_B(M)\) is established. Some results of Dade (for the graded case) and Schneider are obtained as applications. Finally the stable Clifford theory for graded rings is extended to the case of Hopf extensions.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 810234
zbMATH Keywords: finite-dimensional Hopf algebras
zbMATH Keywords: \(H\)-Galois extensions
zbMATH Keywords: unimodularity
zbMATH Keywords: existence of trace 1 elements
zbMATH Keywords: \(H\)-module algebras
zbMATH Keywords: stable Clifford theory
zbMATH Keywords: graded rings
zbMATH Keywords: Hopf extensions
reviewed by: Serban Raianu
MaRDI profile type: Publication
cites work: Hopf Galois extensions, smash products, and Morita equivalence
cites work: Group-graded rings and modules
cites work: The equivalence of various generalizations of group rings and modules
cites work: Hopf extensions of algebras and Maschke type theorems
cites work: Cleft cqmodule algebras and hopf modules
cites work: Q3897166
cites work: Q3957087
cites work: The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite
cites work: Representation theory of Hopf Galois extensions
cites work: Q5590089
cites work: Inductions and coinductions for Hopf extensions
cites work: Hopf frobenius extensions of algebras
full work available at URL: 0022-4049(94)00076-u
OpenAlex ID: W1998022243
description: scientific article; zbMATH DE number 810234