## On nonexistence of oriented cycles in Auslander-Reiten quivers.(English)Zbl 0559.16017

Let us recall that, given an Artin algebra A, $$\beta$$ (A) is the maximum number of indecomposable components in the middle term of an Auslander- Reiten sequence in mod A which are neither projective nor injective. In this paper, the author continues his research on representation finite algebras with $$\beta$$ (A)$$\leq 2$$ [see his papers with Z. Pogarzały; Proc. Lond. Math. Soc., III. Ser. 47, 463-479 (1983; reviewed above) and J. Waschbüsch; J. Reine Angew. Math. 345, 172-181 (1983; Zbl 0511.16021)].
The main question is to determine when, for a representation finite algebra A, the Auslander-Reiten quiver $$\Gamma_ A$$ has no oriented cycles. Using the main result in the first of the above mentioned papers, the question is solved for the case $$\beta$$ (A)$$\leq 2$$ in the following way.
Theorem. If A is a factor of a hereditary algebra, the following are equivalent. (1) A is representation finite, $$\beta$$ (A)$$\leq 2$$ and $$\Gamma_ A$$ has no oriented cycles. (2) A is biserial and there is no special family of local modules in ind A. - Condition (2) is a sort of combinatorial condition on two series of modules, one consisting of uniserial modules and the other of local non uniserial modules, expressed in terms of socs and tops.
Reviewer: H.A.Merklen

### MSC:

 16Gxx Representation theory of associative rings and algebras 16P20 Artinian rings and modules (associative rings and algebras) 16P10 Finite rings and finite-dimensional associative algebras 16Exx Homological methods in associative algebras 16D80 Other classes of modules and ideals in associative algebras

### Citations:

Zbl 0559.16016; Zbl 0511.16021
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