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On stable equivalences induced by exact functors

Author: Yuming Liu
Journal: Proc. Amer. Math. Soc. 134 (2006), 1605-1613
MSC (2000): Primary 16G10; Secondary 16G70
Published electronically: December 5, 2005
MathSciNet review: 2204270
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Abstract: Let $ A$ and $ B$ be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence $ \alpha$ between $ A$ and $ B$ such that $ \alpha$ is induced by exact functors. We present a nice correspondence between indecomposable modules over $ A$ and $ B$. As a consequence, we have the following: (1) If $ A$ is a self-injective algebra, then so is $ B$; (2) If $ A$ and $ B$ are finite dimensional algebras over an algebraically closed field $ k$, and if $ A$ is of finite representation type such that the Auslander-Reiten quiver of $ A$ has no oriented cycles, then $ A$ and $ B$ are Morita equivalent.

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Yuming Liu
Affiliation: School of Mathematical Sciences, Beijing Normal University, 100875 Beijing, People’s Republic of China

Keywords: Stable equivalence induced by exact functors, simple module, indecomposable module
Received by editor(s): September 28, 2004
Received by editor(s) in revised form: January 11, 2005
Published electronically: December 5, 2005
Communicated by: Martin Lorenz
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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