Relative stable equivalences of Morita type
The stable category of a finite-dimensional algebra \(\Lambda\) is obtained from the module category \(\bmod \Lambda\) by taking the quotient modulo all morphisms factoring through projective modules. Stable categories occur in representation theory and stable equivalences have been studied and used in representations of finite groups and representation theory of finite-dimensional algebras. A theory of stable equivalences is still little known to us. Stable equivalences of Morita type, were first introduced by \textit{M. Broué} [NATO ASI Ser., Ser. C, Math. Phys. Sci. 424, 1--26 (1994; Zbl 0827.20007)], is a special class of stable equivalences. \textit{J. Rickard} [J. Lond. Math. Soc., II. Ser. 43, No. 1, 37--48 (1991; Zbl 0683.16030)] showed that a derived equivalence between two self-injective algebras induces a stable equivalence of Morita type. In the paper under review, the author introduces a notion of relative stable equivalence of Morita type. He proves that if two finite-dimensional algebras \(A\) and \(B\) are stably equivalent of Morita type (i.e., there are two bimodules \(_AN_B\) and \(_BM_A\) which are projective as left modules and as right modules such that there are bimodule isomorphisms \(_AN\otimes_B M_A\cong\ _AA_A\oplus\ _AP_A\) and \(_BM\otimes_A N_B\cong\ _BB_B\oplus\ _BQ_B\), where \(_AP_A\) and \(_BQ_B\) are projective bimodules), then the trivial extensions \(A\ltimes P\) and \(B\ltimes Q\) are \((A, B)\)-relatively stably equivalent of Morita type. This shows that relative stable equivalence of Morita type appears naturally in the context of stable equivalence of Morita type. The author also produces a new relative stable equivalence of Morita type from given ones. This gives a generalization of a result proved by \textit{Y. Liu} and \textit{C. Xi} [Trans. Am. Math. Soc. 358, No. 6, 2537--2560 (2006; Zbl 1100.16012)] for stable equivalences of Morita type.
relative stable equivalence of Morita type
scientific article; zbMATH DE number 6851638
scientific article