Commentarii Mathematici Helvetici

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Volume 72, Issue 2, 1997, pp. 266–284
DOI: 10.1007/s000140050016

Published online: 1997-06-30

Stable equivalence preserves representation type

Henning Krause[1]

(1) Universit├Ąt 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