A note on stable equivalences of Morita type.
type: Item
instance of: scholarly article
title: A note on stable equivalences of Morita type.
zbMATH Open document ID: 1130.16005
author: Alex S. Dugas
author: Roberto Martinez-Villa
published in: Journal of Pure and Applied Algebra
publication date: 2007-02-02T00:00:00Z
review text: Let \(\Lambda\) be a finite dimensional algebra over a field \(k\), \(\text{mod\,}\Lambda\) the category of finitely generated left \(\Lambda\)-modules, \(\underline{\text{mod}}\,\Lambda\) the quotient category of \(\text{mod\,}\Lambda\) modulo the projectives, and \(\pi_\Lambda\colon\text{mod\,}\Lambda\to\underline{\text{mod}}\,\Lambda\) the projection functor. Let \(\Gamma\) be another finite dimensional \(k\)-algebra and suppose that there exist an exact functor \(F\colon\text{mod\,}\Lambda\to\text{mod\,}\Gamma\) and a stable equivalence \(\alpha\colon\underline{\text{mod}}\,\Lambda\to\underline{\text{mod}}\,\Gamma\) satisfying \(\pi_\Gamma F=\alpha\pi_\Lambda\). Then it is well known that \(F\cong-\otimes_\Lambda M\) where \(M\) is a \(\Lambda\)-\(\Gamma\)-bimodule that is projective as left \(\Lambda\)-module and as right \(\Gamma\)-module, and the question is whether there exists a \(\Gamma\)-\(\Lambda\)-bimodule \(N\), projective as left \(\Gamma\)-module and as right \(\Lambda\)-module, that gives rise to isomorphisms \(M\otimes_\Gamma N\cong\Lambda\oplus P\) and \(N\otimes_\Lambda M\cong\Gamma\oplus Q\) of bimodules where \(P\) is a projective \(\Lambda\)-bimodule, and \(Q\) is a projective \(\Gamma\)-bimodule. If the indicated bimodule \(N\) exists, \(\alpha\) is said to be a stable equivalence of `Morita type' [\textit{M. Broué}, NATO ASI Ser., Ser. C, Math. Phys. Sci. 424, 1-26 (1994; Zbl 0827.20007)]. Suppose that the semisimple quotients \(\Lambda/\text{rad\,}\Lambda\) and \(\Gamma/\text{rad\,}\Gamma\) are separable. If \(\Lambda\) and \(\Gamma\) are selfinjective algebras then, for all \(M\), the stable equivalence \(\alpha\) is of Morita type [\textit{J. Rickard}, CMS Conf. Proc. 23, 157-178 (1998; Zbl 0914.20010)]. The authors of the article under review give up the assumption that \(\Lambda\) and \(\Gamma\) are selfinjenctive, and obtain necessary and sufficient conditions on the bimodule \(M\) in order for \(\alpha\) to be a stable equivalence of …
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D20
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 5121383
zbMATH Keywords: stable equivalences of Morita type
zbMATH Keywords: categories of modules
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: selfinjective algebras
reviewed by: Mark Kleiner
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.01.007
OpenAlex ID: W2032087765
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DOI: 10.1016/J.JPAA.2006.01.007
description: scientific article; zbMATH DE number 5121383