Actions of Picard groups on graded rings
type: Item
instance of: scholarly article
title: Actions of Picard groups on graded rings
zbMATH Open document ID: 0941.16027
author: Jeremy Haefner
author: Ángel del Río
published in: Journal of Algebra
publication date: 1999-11-18T00:00:00Z
review text: Let \(R\) be a \(G\)-strongly graded ring, \(G\) any group. Let \(A\) denote \(R_e\). In this paper, the authors describe when strongly graded rings \(R\) and \(R'\) are graded isomorphic, graded equivalent (i.e., the categories of graded modules are equivalent by a functor that commutes with suspensions), or graded \(A\)-isomorphic (i.e., the isomorphism is the identity on \(A=R_e=R_e'\)). They show that \(R\) and \(R'\) are graded equivalent iff \(R'\) is graded isomorphic to \(R^C\), \(R\) acted on by a graded Morita context \(C\). Also they show that the difference between the classes of rings graded equivalent to \(R\) and graded isomorphic to \(R\) (graded equivalent to \(R\) and graded \(A\)-isomorphic to \(R\), graded isomorphic to \(R\) and graded \(A\)-isomorphic to \(R\), respectively) can be described by quotients of \(\text{Pic}(A)\) or \(\Aut(A)\). There exist graded equivalent strongly \(G\)-graded rings \(R\) and \(R'\) with \(A=R_e=R_e'\) and \(R_g\cong R_g'\) as \(A\)-bimodules for all \(g\) but \(R\) not graded isomorphic to \(R'\). Now the authors look for \(R'\) graded equivalent to \(R\) but ``simpler'' in the sense that \(R_e'\) is better understood or \(R'\) is a crossed product. In particular, if \(R_e\) is semiperfect, \(R'\) can be chosen so that \(R_e'\) is the basic ring of \(R_e\). If \(R\) and thus \(R_e\) is semisimple, \(R'\) is a crossed product over a finite product of division rings and can be described explicitly.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1348173
zbMATH Keywords: graded equivalences
zbMATH Keywords: strongly graded rings
zbMATH Keywords: categories of graded modules
zbMATH Keywords: graded Morita contexts
zbMATH Keywords: crossed products
reviewed by: Margaret Beattie
MaRDI profile type: Publication
OpenAlex ID: W2076508293
cites work: Q4847007
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cites work: Global dimensions of crossed products
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DOI: 10.1006/JABR.1999.7862
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description: scientific article; zbMATH DE number 1348173