Left-right projective bimodules and stable equivalences of Morita type

Abstract

Abstract. We study a connection between left-right projective bimodules and stable equivalences of Morita type for finite-dimensional associative algebras over a field. Some properties of the category of all finite-dimensional left-right projective bimodules for selfinjective algebras are also given.Introduction. Let K be a fixed field. In the representation theory of finite-dimensional associative K-algebras with identity, stable equivalences of Morita type seem to be of particular relevance. They play a substantial role in the representation theory of finite groups (see [7,12,15]). It was observed by Rickard that if two self-injective K-algebras are derived equivalent then they are stably equivalent of Morita type [14].Rickard [14] showed that for two derived equivalent K-algebras their Hochschild cohomology algebras are isomorphic. Happel also proved it but did not publish. This result was generalized in [13] to self-injective Kalgebras A and B which are stably equivalent of Morita type. The main trick used in the proof of this generalization rests heavily on an easy observation that there are two very small subcategories (they consist only of one Ω-orbit) which are stably equivalent; one of them is contained in the category of finite-dimensional A-bimodules, the other in the category of finite-dimensional B-bimodules. An important property of these subcategories is that they consist only of bimodules which are projective as left modules and as right modules. Following Butler and King [8] we call such bimodules left-right projective. They were first studied by Auslander and Reiten [5]. It is natural to ask whether a stable equivalence of Morita type between two finite-dimensional K-algebras A and B induces a stable equivalence between their categories of left-right projective bimodules.One of the main aims of this paper is to give an affirmative answer to the last question. We show even more: existence of a stable equivalence between two finite-dimensional K-algebras is equivalent to existence of a special stable equivalence between the categories of left-right projective bimodules.

On bimodules determining stable equivalences

“…Assume that A N B ∈ lrp(A ⊗ K B op ) determines a stable equivalence of A and B and is indecomposable. Then we infer by [7] that the functor − ⊗ A N induces an equivalence lrp(A e ) −→ lrp(A ⊗ K B op ). Suppose that A N B is not minimal in the class lrp(A ⊗ K B op ).…”

Stable auto-equivalences for local symmetric algebras

“…This functor is exact if and only if A M is projective, and it takes projective A-modules to projective B-modules if and only if M B is projective. Following [23], we call the bimodule A M B left-right projective if A M and M B are both projective. For such bimodules the functor − ⊗ A M B induces an exact functor between triangulated categories mod-A → mod-B.…”

A note on stable equivalences of Morita type

“…Hence, α is induced by an exact functor between module categories if and only if it is induced by a functor of the form − ⊗ Λ M Γ for a bimodule M which is projective as a right Γ -module and as a left Λ-module. Pogorzaly has already studied bimodules with this property in connection with stable equivalence in [15], and we shall continue to call such bimodules left-right projective. Clearly, every projective bimodule is left-right projective, but not conversely.…”