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**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.

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 |