JournalscmhVol. 72 , No. 2DOI 10.1007/s000140050016

Stable equivalence preserves representation type

  • Henning Krause

    Universität Bielefeld, Germany
Stable equivalence preserves representation type cover

Abstract

Given two finite dimensional algebras Λ\Lambda and Γ\Gamma, it is shown that Λ\Lambda is of wild representation type if and only if Γ\Gamma is of wild representation type provided that the stable categories of finite dimensional modules over Λ\Lambda and Γ\Gamma are equivalent. The proof uses generic modules. In fact, a stable equivalence induces a bijection between the isomorphism classes of generic modules over Λ\Lambda and Γ\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.