# Stable equivalence preserves representation type

Research paper by H. Krause

Indexed on: 01 Sep '97Published on: 01 Sep '97Published in: Commentarii Mathematici Helvetici

#### 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.