On Stable Equivalences of Morita Type between Representation-finite Selfinjective Algebras

Throughout this paper k denotes an algebraically closed field, and all algebras considered here are assumed to be basic, connected, finite-dimensional k-algebras with identity. For an algebra Λ, we denote by modΛ and by modΛ the category of finitedimensional (right) Λ-modules and its stable category, respectively. Recall that for two algebras Λ and Π, an equivalence modΛ → modΠ is called a stable equivalence from Λ to Π. The algebras Λ and Π are said to be stably equivalent if there exists a stable equivalence between them. Following Broué [4] (see also Linckelmann [5]) a stable equivalence φ : modΛ → modΠ is called of Morita type if there exist bimodules ΛMΠ, ΠNΛ such that the following three conditions are satisfied: (1) ΛM,MΠ, ΠN and NΛ are projective modules; (2) (a) M ⊗Π N ∼= Λ⊕ P as Λ-bimodules for some projective Λ-bimodule P , and (b) N ⊗Λ M ∼= Π⊕Q as Π-bimodules for some projective Π-bimodule Q; and (3) φ is lifted to the functor -⊗ΛM , i.e., the diagram modΛ -⊗ΛM −−−→ modΠ