On stable equivalences induced by exact functors

Liu, Yuming

doi:10.1090/S0002-9939-05-08157-8

On stable equivalences induced by exact functors
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Proc. Amer. Math. Soc. 134 (2006), 1605-1613 Request permission

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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