On the degree of irreducible morphisms

Abstract

We study the degree of irreducible morphisms in generalized standard convex components of the Auslander–Reiten quiver of an artin algebra with the property that paths with the same origin and end vertices have equal length. We call the components with this last property components with length. In particular, we give two criteria to determine wether the degree of such an irreducible morphism f is finite or infinite. One of them is given in terms of the compositions of f with non-zero maps between modules in the component. The other states that the left degree of an irreducible map f is finite if and only if Ker f belongs to the component. We apply our results to irreducible morphisms over artin algebras of finite representation type and over tame hereditary algebras.