Gluing of idempotents, radical embeddings and two classes of stable equivalences.
Let \(A\) be an Artin algebra, and \(B\) be a subalgebra of \(A\), having the same radical. The authors study a class of stable equivalences between \(A\) and \(B\), which are not of Morita type. More precisely, these stable equivalences are given by bimodules, which are projective on one side, but not on the other side. This recovers, reinterprets and extends the results of \textit{R. Martínez-Villa} [from Representation theory II, Proc. 2nd int. Conf., Ottawa 1979, Lect. Notes Math. 832, 396-431 (1980; Zbl 0443.16019)]. The interesting point is that the stable equivalences the authors construct are not induced by a pair of adjoint functors. The authors also study an alternative stable category defined as a quotient modulo semi-simple modules. The authors show that in the situation they study a stable equivalence modulo semi-simple modules exists precisely when there is a stable equivalence modulo projectives. However, all stable equivalences modulo semi-simple modules are given by induction and restriction (and thus by a pair of adjoint functors). In either situation the authors give a precise criterion when the corresponding stable equivalence exists.
scientific article; zbMATH DE number 5304613
scientific article