On stable equivalences of Morita type for finite dimensional algebras

Abstract:

In this paper, we assume that algebras are finite dimensional algebras with 1 over a fixed field $k$ and modules over an algebra are finitely generated left unitary modules. Let $A$ and $B$ be two algebras (where $k$ is a splitting field for $A$ and $B$) with no semisimple summands. If two bimodules $_{A}M_{B}$ and $_{B}N_{A}$ induce a stable equivalence of Morita type between $A$ and $B$, and if $N\otimes _{A}-$ maps any simple $A$-module to a simple $B$-module, then $N\otimes _{A}-$ is a Morita equivalence. This conclusion generalizes Linckelmann’s result for selfinjective algebras. Our proof here is based on the construction of almost split sequences.