Representation type and stable equivalence of Morita type for finite-dimensional algebras
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.
scientific article; zbMATH DE number 1336822