Commentarii Mathematici Helvetici
Full-Text PDF (314 KB) | Metadata | Table of Contents | CMH summary
Published online: 1997-06-30
Stable equivalence preserves representation typeHenning Krause (1) Universität Bielefeld, Germany
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.
Keywords: Stable equivalence, generic module, Ziegler spectrum, Auslander-Reiten translation
Krause Henning: Stable equivalence preserves representation type. Comment. Math. Helv. 72 (1997), 266-284. doi: 10.1007/s000140050016