Tilted special biserial algebras
type: Item
instance of: scholarly article
title: Tilted special biserial algebras
zbMATH Open document ID: 0949.16011
author: Francois Huard
author: Shiping Liu
published in: Journal of Algebra
publication date: 2000-11-28T00:00:00Z
review text: Let \(k\) be an algebraically closed field, and \(A\) be a finite dimensional basic \(k\)-algebra. In this paper, the authors assume that \(A\) is special biserial (in the sense of \textit{A. Skowroński} and \textit{J. Waschbüsch} [J. Reine Angew. Math. 345, 172-181 (1983; Zbl 0511.16021)]), and give combinatorial criteria allowing to verify whether \(A\) is tilted or not. They had already considered [in ``Tilted string algebras'' (preprint)] the case where \(A\) is a string algebra, they may thus assume that it is not. Then, \(A\) is tilted if and only if it is quasi-tilted. The proof rests on using the (well-known) description of the indecomposable \(A\)-modules in order to obtain combinatorial conditions for an indecomposable module to have projective, or injective, dimension at most one. The authors then obtain necessary and sufficient conditions for the global dimension of \(A\) to be at most two. They deduce their main results from the homological characterisation of quasi-tilted algebras.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 1340567
zbMATH Keywords: special biserial algebras
zbMATH Keywords: finite dimensional basic algebras
zbMATH Keywords: indecomposable modules
zbMATH Keywords: string algebras
zbMATH Keywords: global dimensions
zbMATH Keywords: quasi-tilted algebras
reviewed by: Ibrahim Assem
MaRDI profile type: Publication
full work available at URL: jabr.1998.7828
OpenAlex ID: W1973618817
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DOI: 10.1006/JABR.1998.7828
description: scientific article; zbMATH DE number 1340567