Stable equivalence preserves representation type

Abstract

Given two finite dimensional algebras $\Lambda$ and $\Gamma$, it is shown that $\Lambda$ is of wild representation type if and only if $\Gamma$ is of wild representation type provided that the stable categories of finite dimensional modules over $\Lambda$ and $\Gamma$ are equivalent. The proof uses generic modules. In fact, a stable equivalence induces a bijection between the isomorphism classes of generic modules over $\Lambda$ and $\Gamma$ , and the result follows from certain additional properties of this bijection. In the second part of this paper the Auslander-Reiten translation is extended to an operation on the category of all modules. It is shown that various finiteness conditions are preserved by this operation. Moreover, the Auslander-Reiten translation induces a homeomorphism between the set of non-projective and the set of non-injective points in the Ziegler spectrum. As a consequence one obtains that for an algebra of tame representation type every generic module remains fixed under the Auslander-Reiten translation.