Symmetric algebras stably equivalent to the trivial extensions of tubular algebras
type: Item
instance of: scholarly article
title: Symmetric algebras stably equivalent to the trivial extensions of tubular algebras
zbMATH Open document ID: 0867.16005
author: Zygmunt Pogorzały
author: Andrzej Skowroński
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: Let \(K\) be an algebraically closed field, \(A\) a finite dimensional \(K\)-algebra and \(T(A)\) the trivial extension of \(A\) by its minimal cogenerator bimodule \(DA=\text{Hom}_K(A,K)\). Denote by mod-\(A\) the category of finitely generated right \(A\)-modules, and by \(\underline{\text{mod}}\)-\(A\) the stable category of mod-\(A\). The main result of the paper is the following theorem. Assume that \(A\) is a tubular algebra and \(\Lambda\) a symmetric algebra. Then \(\Lambda\) is stably equivalent to \(T(A)\) if and only if \(\Lambda\) is stably equivalent to \(T(E)\), where \(E\) is a tubular algebra which is tilting-cotilting equivalent to \(A\). Two corollaries are also derived. The proof relies on the characterization of tubular algebras as a cycle finite algebra with many families of sincere tubes, given by the second author [J. Pure Appl. Algebra 103, No. 1, 105-116 (1995; Zbl 0841.16020)], and on the investigation, made in the first part of the paper, of the structure of \(\underline{\text{mod}}\)-\(T(C)\), where \(C\) is a canonical tubular algebra.
reviewed by: Andrei Marcus
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 883751
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: minimal cogenerators
zbMATH Keywords: categories of finitely generated right modules
zbMATH Keywords: stable categories
zbMATH Keywords: tubular algebras
zbMATH Keywords: symmetric algebras
zbMATH Keywords: stable equivalences
zbMATH Keywords: tilting-cotilting equivalent algebras
zbMATH Keywords: sincere tubes
MaRDI profile type: Publication
full work available at URL: jabr.1996.0111
OpenAlex ID: W2014098403
DOI: 10.1006/JABR.1996.0111
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description: scientific article; zbMATH DE number 883751