A note on stable equivalences of Morita type

We investigate when an exact functor F≅−⊗ΛMΓ:mod-Λ→mod-ΓΓ which induces a stable equivalence is part of a stable equivalence of Morita type. If ΛΛ and ΓΓ are finite dimensional algebras over a field kk 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 ΛΛ and ΓΓ, it is possible to replace ΛΛ by a Morita equivalent kk-algebra ΔΔ such that ΓΓ is a subring of ΔΔ and the induction and restriction functors induce inverse stable equivalences.