The composite of irreducible morphisms in standard components

Abstract

In this work, we consider standard components of the Auslander– Reiten quiver with trivial valuation. We give a characterization of when there are n irreducible morphisms between modules in such a component with non-zero composite belonging to the n + 1-th power of the radical. We prove that a necessary condition for their existence is that it has to be a non-zero cycle or a non-zero bypass in the component. For directed algebras, we prove that the composite of n irreducible morphisms between indecomposable modules belongs to a greater power of the radical, greater than n, if and only if it is zero.