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arXiv · 1701.02794

Auslander-Reiten components with bounded short cycles

Shiping Liu·Jinde Xu
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Abstract

We study Auslander-Reiten components of an artin algebra with bounded short cycles, namely, there exists a bound for the depths of maps appearing on short cycles of non-zero non-invertible maps between modules in the given component. First, we give a number of combinatorial characteri\-zations of almost acyclic Auslander-Reiten components. Then, we show that an Auslander-Reiten component with bounded short cycles is closely related to the connec\-ting component of a tilted quotient algebra. In particular, the number of such components is finite and each of them is almost acyclic with only finitely many DTr-orbits. As an application, we show that an artin algebra is representation-finite if and only if its module category has bounded short cycles. This includes a well known result of Ringel's, saying that a representation-directed algebra is representation-finite.

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BibTeXRIS

Shiping Liu, Jinde Xu. 2017-01-10. Auslander-Reiten components with bounded short cycles. https://arxiv.org/abs/1701.02794

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