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.