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On stable equivalences of Morita type for finite dimensional algebras

Author: Yuming Liu
Journal: Proc. Amer. Math. Soc. 131 (2003), 2657-2662
MSC (2000): Primary 16D20; Secondary 16G20
Published electronically: February 6, 2003
MathSciNet review: 1974320
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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.

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Additional Information

Yuming Liu
Affiliation: Department of Mathematics, Beijing Normal University, 100875 Beijing, People’s Republic of China

Received by editor(s): January 11, 2002
Received by editor(s) in revised form: April 3, 2002
Published electronically: February 6, 2003
Communicated by: Martin Lorenz
Article copyright: © Copyright 2003 American Mathematical Society