Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Auslander-Reiten components with bounded short cycles   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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The categories of this paper are: astro-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph