On modules with cyclic vertices in the Auslander-Reiten quiver

Abstract

In studying indecomposable modules of a finite-dimensional K-algebra where K is a field, the Auslander-Reiten quiver r has proved to be a powerful tool. If the algebra is a group algebra KG (or a block B), then properties of the group are related to the graph structure of the Auslander -Reiten quiver. For example, the graph f(KG) (or T(B)) is finite if and only if a Sylow-p-subgroup of G (or a defect group of B) is cyclic, where p is the characteristic of K. It is now well known that an indecomposable KG-module has a vertex, which is a minimal subgroup Q of G such that A4 is Q-projective; and Q is unique up to G-conjugation [7]. We ask whether there are constraints on the position of modules with a given vertex in the Auslander-Reiten quiver. Here we are concerned with this problem for modules with cyclic vertices which are not projective. There is not much to say in the case when the defect group of the block is cyclic; in this case, all its modules have cyclic vertices. We assume therefore that the block containing these modules has a non-cyclic defect group. Assume that M is indecomposable with a cyclic vertex, and that A4 is not projective. It is well known that M is n-periodic, hence r-periodic. (For group algebras, the Auslander-Reiten translate r is the same as 02, where 52 is the usual Heller operator.) By a more general theorem [2, p. 1631, the component--O say-of the stable Auslander-Reiten quiver containing M, is a “tube” [ 151. Given such a tube 0, then the vertices in each row form a single r-orbit. The “end” of the tube is the unique row where each vertex has only one predecessor. That is, if M is a module corresponding to a vertex [it41 in 0 and O+rM-+E+M-+O (*I 289 002 l-8693/86 $3.00