A note on stable equivalences of Morita type

We investigate when an exact functor F congruent to - circle times M-Lambda(Gamma) : mod-Lambda -> mod-Gamma which induces a stable equivalence is pan of a stable equivalence of Morita type. If Lambda and Gamma 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 Lambda and Gamma, it is possible to replace Lambda by a Morita equivalent k-algebra Delta such that Gamma is a subring of Delta and the induction and restriction functors; induce inverse stable equivalences. (c) 2006 Elsevier B.V. All rights reserved.