A note on stable equivalences of Morita type.
Let \(\Lambda\) be a finite dimensional algebra over a field \(k\), \(\text{mod\,}\Lambda\) the category of finitely generated left \(\Lambda\)-modules, \(\underline{\text{mod}}\,\Lambda\) the quotient category of \(\text{mod\,}\Lambda\) modulo the projectives, and \(\pi_\Lambda\colon\text{mod\,}\Lambda\to\underline{\text{mod}}\,\Lambda\) the projection functor. Let \(\Gamma\) be another finite dimensional \(k\)-algebra and suppose that there exist an exact functor \(F\colon\text{mod\,}\Lambda\to\text{mod\,}\Gamma\) and a stable equivalence \(\alpha\colon\underline{\text{mod}}\,\Lambda\to\underline{\text{mod}}\,\Gamma\) satisfying \(\pi_\Gamma F=\alpha\pi_\Lambda\). Then it is well known that \(F\cong-\otimes_\Lambda M\) where \(M\) is a \(\Lambda\)-\(\Gamma\)-bimodule that is projective as left \(\Lambda\)-module and as right \(\Gamma\)-module, and the question is whether there exists a \(\Gamma\)-\(\Lambda\)-bimodule \(N\), projective as left \(\Gamma\)-module and as right \(\Lambda\)-module, that gives rise to isomorphisms \(M\otimes_\Gamma N\cong\Lambda\oplus P\) and \(N\otimes_\Lambda M\cong\Gamma\oplus Q\) of bimodules where \(P\) is a projective \(\Lambda\)-bimodule, and \(Q\) is a projective \(\Gamma\)-bimodule. If the indicated bimodule \(N\) exists, \(\alpha\) is said to be a stable equivalence of `Morita type' [\textit{M. Broué}, NATO ASI Ser., Ser. C, Math. Phys. Sci. 424, 1-26 (1994; Zbl 0827.20007)]. Suppose that the semisimple quotients \(\Lambda/\text{rad\,}\Lambda\) and \(\Gamma/\text{rad\,}\Gamma\) are separable. If \(\Lambda\) and \(\Gamma\) are selfinjective algebras then, for all \(M\), the stable equivalence \(\alpha\) is of Morita type [\textit{J. Rickard}, CMS Conf. Proc. 23, 157-178 (1998; Zbl 0914.20010)]. The authors of the article under review give up the assumption that \(\Lambda\) and \(\Gamma\) are selfinjenctive, and obtain necessary and sufficient conditions on the bimodule \(M\) in order for \(\alpha\) to be a stable equivalence of Morita type. As consequences, they show that if \(\alpha\) is a stable equivalence of Morita type, then (1) it induces a stable equivalence of Morita type between certain selfinjective algebras associated with \(\Lambda\) and \(\Gamma\) and (2) there exists a \(k\)-algebra \(\Delta\) that is Morita equivalent to \(\Lambda\) and contains \(\Gamma\) as a \(k\)-subalgebra such that the induction and restriction functors for the inclusion \(\Gamma\subset\Delta\) induce inverse stable equivalences between \(\text{mod\,}\Gamma\) and \(\text{mod\,}\Delta\).
scientific article; zbMATH DE number 5121383
scientific article