On Stable Equivalence of Tensor Products of Algebras

Abstract

The paper investigates the following problem. Let bimodules N, M yield a stable equivalence of Morita type between self-injective K-algebras A and E. Further, let bimodules S, T yield a stable equivalence of Morita type between self-injective K-algebras B and F. Then we want to know whether the functor M ⊗ A − ⊗ B S: mod(A ⊗ K B op ) → mod(E ⊗ K F op ) induces a stable equivalence between A ⊗ K B op and E ⊗ K F op . There is given a reduction of this problem to some smaller subcategories for self-injective algebras. Moreover, new invariants of stable equivalences of Morita type are constructed in a general case of arbitrary finite-dimensional algebras over a field.