Representation type and stable equivalence of Morita type for finite-dimensional algebras
type: Item
instance of: scholarly article
title: Representation type and stable equivalence of Morita type for finite-dimensional algebras
zbMATH Open document ID: 0937.16021
DOI: 10.1007/PL00004671
author: Henning Krause
published in: Mathematische Zeitschrift
publication date: 2000-03-09T00:00:00Z
review text: The author considers two finite dimensional algebras \(\Lambda\) and \(\Gamma\) over an algebraically closed field \(k\), which are stably equivalent of Morita type in the sense that there are bimodules \(B\) in \(\text{mod }\Lambda^{\text{op}}\otimes_k\Gamma\) and \(C\) in \(\text{mod }\Gamma^{\text{op}}\otimes_k\Lambda\) such that the functors \(-\otimes_\Lambda B\colon\text{mod }\Lambda\to\text{mod }\Gamma\) and \(-\otimes_\Gamma C\colon\text{mod }\Gamma\to\text{mod }\Lambda\) induce mutually inverse equivalences between the stable categories \(\underline{\text{mod}} \Lambda\) and \(\underline{\text{mod}} \Gamma\). An example of this situation was previously considered by \textit{I. Assem} and \textit{J.~A. de la Peña} [Fundam. Math. 149, No. 2, 171-181 (1996; Zbl 0851.16013)], who showed that if \(\Lambda\) and \(\Gamma\) are two finite dimensional algebras tilted from each other, then the corresponding trivial extensions \(T\Lambda\) and \(T\Gamma\) are stably equivalent of Morita type and have the same representation type. Motivated by these results but using, instead of the geometric methods of Assem-de la Peña, the `generical' definition of the representation type of an algebra due to \textit{W. Crawley-Boevey} [Proc. Lond. Math. Soc., III. Ser. 63, No. 2, 241-265 (1991; Zbl 0741.16005)], the author proves that if \(\Lambda\) and \(\Gamma\) are stably equivalent of Morita type, then they have the same representation type.
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16G10
zbMATH DE Number: 1336822
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: stable equivalences
zbMATH Keywords: representation types
zbMATH Keywords: generic modules
zbMATH Keywords: endofinite modules
zbMATH Keywords: pure-injective modules
zbMATH Keywords: Ziegler spectra
zbMATH Keywords: stable categories
zbMATH Keywords: trivial extensions
reviewed by: José L. Gómez-Pardo
MaRDI profile type: Publication
Recommended article: Stable Equivalence and Generic Modules
Recommended article: Construction of stable equivalences of Morita type for finite-dimensional algebras I
Recommended article: On constructing stably equivalent functors
Recommended article: A note on stable equivalences of Morita type.
Recommended article: Algebras stably equivalent to trivial extensions of hereditary algebras of type $Ã_n$
Recommended article: Relative stable equivalences of Morita type
Recommended article: Tilting functors and stable equivalences for selfinjective algebras
Recommended article: Tensor functors and finite representation type
Recommended article: Tilting theory and functor categories. I: Classical tilting.
Recommended article: Invariants under stable equivalences of Morita type.
description: scientific article; zbMATH DE number 1336822