Relative stable equivalences of Morita type
type: Item
instance of: scholarly article
title: Relative stable equivalences of Morita type
zbMATH Open document ID: 1448.16021
author: Zygmunt Pogorzały
published in: Mathematische Zeitschrift
publication date: 2018-03-21T00:00:00Z
review text: The stable category of a finite-dimensional algebra \(\Lambda\) is obtained from the module category \(\bmod \Lambda\) by taking the quotient modulo all morphisms factoring through projective modules. Stable categories occur in representation theory and stable equivalences have been studied and used in representations of finite groups and representation theory of finite-dimensional algebras. A theory of stable equivalences is still little known to us. Stable equivalences of Morita type, were first introduced by \textit{M. Broué} [NATO ASI Ser., Ser. C, Math. Phys. Sci. 424, 1--26 (1994; Zbl 0827.20007)], is a special class of stable equivalences. \textit{J. Rickard} [J. Lond. Math. Soc., II. Ser. 43, No. 1, 37--48 (1991; Zbl 0683.16030)] showed that a derived equivalence between two self-injective algebras induces a stable equivalence of Morita type. In the paper under review, the author introduces a notion of relative stable equivalence of Morita type. He proves that if two finite-dimensional algebras \(A\) and \(B\) are stably equivalent of Morita type (i.e., there are two bimodules \(_AN_B\) and \(_BM_A\) which are projective as left modules and as right modules such that there are bimodule isomorphisms \(_AN\otimes_B M_A\cong\ _AA_A\oplus\ _AP_A\) and \(_BM\otimes_A N_B\cong\ _BB_B\oplus\ _BQ_B\), where \(_AP_A\) and \(_BQ_B\) are projective bimodules), then the trivial extensions \(A\ltimes P\) and \(B\ltimes Q\) are \((A, B)\)-relatively stably equivalent of Morita type. This shows that relative stable equivalence of Morita type appears naturally in the context of stable equivalence of Morita type. The author also produces a new relative stable equivalence of Morita type from given ones. This gives a generalization of a result proved by \textit{Y. Liu} and \textit{C. Xi} [Trans. Am. Math. Soc. 358, No. 6, 2537--2560 (2006; Zbl 1100.16012)] for stable equivalences of Morita type.
reviewed by: Alireza Nasr-Isfahani
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16E30
zbMATH DE Number: 6851638
zbMATH Keywords: stable equivalence of Morita type
zbMATH Keywords: relative stable equivalence of Morita type
zbMATH Keywords: trivial extension algebra
MaRDI profile type: Publication
full work available at URL: s00209-017-1899-7
OpenAlex ID: W2614090542
cites work: Q4318498
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cites work: Representation theory. A homological algebra point of view.
DOI: 10.1007/S00209-017-1899-7
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description: scientific article; zbMATH DE number 6851638