Degrees of irreducible morphisms and finite-representation type

Abstract

We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree in terms of the existence of certain annihilator maps. This is used to prove our main theorem: An algebra is of finite representation type if and only if for every indecomposable projective the inclusion of the radical in the projective has finite right degree, which is equivalent to requiring that for every indecomposable injective the epimorphism from the injective to its quotient by its socle has finite left degree. As an application of the techniques we develop, we study the behavior of the composite of paths of irreducible morphisms between indecomposable modules.