Graded Morita theory for group coring and graded Morita-Takeuchi theory.
type: Item
instance of: scholarly article
title: Graded Morita theory for group coring and graded Morita-Takeuchi theory.
zbMATH Open document ID: 1259.16049
published in: Taiwanese Journal of Mathematics
publication date: 2012-08-01T00:00:00Z
full work available at URL: http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/2006
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16T15
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 6062764
zbMATH Keywords: graded Morita contexts
zbMATH Keywords: group corings
zbMATH Keywords: Hopf group coalgebras
zbMATH Keywords: graded Morita-Takeuchi contexts
author: Guohua Liu
author: Shuanhong Wang
reviewed by: Blas Torrecillas
MaRDI profile type: Publication
OpenAlex ID: W4230035690
DOI: 10.11650/TWJM/1500406678
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review text: Let \(G\) be a group and \(A\) an associative unital algebra over a field \(k\). A \(G\)-group \(A\)-coring \(\mathbf C\) is a family \((C_\alpha)_{\alpha\in G}\) of \(A\)-bimodules together with a family of bimodule maps \(\Delta_{\alpha,\beta}\colon C_{\alpha\beta}\to C_\alpha\otimes C_\beta\) and \(\varepsilon\colon C_e\to A\) (\(e\) is the unit of \(G\)), such that \((\Delta_{\alpha,\beta}\otimes_A\text{id}_{C_\gamma})\circ\Delta_{\alpha\beta,\gamma}=(\text{id}_{C_\alpha}\otimes_A\Delta_{\beta,\gamma})\circ\Delta_{\alpha,\beta\gamma}\) and \((\text{id}_{C_\alpha}\otimes_A\varepsilon)\circ\Delta_{\alpha,e}=\text{id}_{C_\alpha}=(\varepsilon\otimes_A\text{id}_{A_\alpha})\circ\Delta_{e,\alpha}\). This notion was introduced by \textit{S. Caenepeel, K. Janssen} and \textit{S. H. Wang} [Appl. Categ. Struct. 16, No. 1-2, 65-96 (2008; Zbl 1145.16017)] as a generalization of group coalgebras and Hopf group coalgebras.NEWLINENEWLINE In this paper, the authors construct a graded Morita context asociated to a comodule over a group coring. This Morita context connects the dual graded ring of a group coring and the graded endomorphism ring of any group coring comodule. In the last section, a graded version of Morita-Takeuchi's equivalence theorem is obtained.
description: scientific article; zbMATH DE number 6062764