A note on stable equivalences of Morita type

Abstract

We investigate when an exact functor F ∼ = − ⊗ Λ M Γ : mod-Λ → mod-Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard's for self-injective algebras. As a corollary, we see that the two functors given by tensoring with the bimodules in a stable equivalence of Morita type are right and left adjoints of one another, provided that these bimodules are indecomposable. This fact has many interesting consequences for stable equivalences of Morita type. In particular, we show that a stable equivalence of Morita type induces another stable equivalence of Morita type between certain self-injective algebras associated to the original algebras. We further show that when there exists a stable equivalence of Morita type between Λ and Γ , it is possible to replace Λ by a Morita equivalent k-algebra ∆ such that Γ is a subring of ∆ and the induction and restriction functors induce inverse stable equivalences.

Derived equivalences and stable equivalences of Morita type, II

“…(i) For each algebra A, there is an associated self-injective algebra (see Subsection 2.3), which we denote by ∆ A . The result [11,Theorem 4.2] shows that if A/rad(A) and B/rad(B) are separable then every stable equivalence of Morita type between A and B restricts to a stable equivalence of Morita type between the associated self-injective algebras ∆ A and ∆ B . As an immediate consequence of Theorem 1.3, we have the following corollary.…”

On singular equivalences of Morita type

“…In order to prove Theorem 3.1, we shall use the following technical notion, motivated by Dugas and Martinez-Villa [12]. (v) A direct summand of a strongly right nonsingular bimodule is also strongly right nonsingular.…”

“…First, we shall prove in Theorem 3.1 that under mild conditions a singular equivalence of Morita type gives rise to a bi-adjoint pair. This section is inspired by an analogous approach by Alex Dugas and Roberto Martinez-Villa [12]. Then we shall investigate Hochschild homology and show in Theorem 4.1 that Hochschild homology of a finite dimensional algebra over a field and in strictly positive degrees is invariant under a singular equivalence of Morita type.…”

Simple-Minded Systems in Stable Module Categories

“…where A P A and B Q B are projective bimodules. Proof Note that under the assumption of this lemma, by [7] we can assume that both (N ⊗ A −, M ⊗ B −) and (M ⊗ B −, N ⊗ A −) are adjoint pairs. In particular, N ⊗ A − and M ⊗ B − maps projective (injective, respectively) modules to projective (injective, respectively) modules, and P and Q are projective-injective bimodules.…”