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

Skowroński, Andrzej

Acta Univ. Carol., Math. Phys. 25, No. 1, 45-52 (1984).

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

Keywords:

Artin algebra; number of indecomposable components; middle term; Auslander-Reiten sequence; representation finite algebras; hereditary algebra; oriented cycles; uniserial modules

Citations:

Zbl 0559.16016; Zbl 0511.16021