Degrees in Auslander-Reiten components with almost split sequences of at most two middle terms

Abstract

We consider A to be an artin algebra. We study the degrees of irreducible morphisms between modules in Auslander-Reiten components Gamma having only almost split sequences with at most two indecomposable middle terms, that is, alpha(Gamma) leq 2. We prove that if f:X rightarrow Y is an irreducible epimorphism of finite left degree with X or Y indecomposable, then there exists a module Z in Gamma and a morphism varphi in Re^{n}(Z,X) and not in Re^{n+1}(Z,X) for some positive integer n such that f varphi=0. In particular, for such components if A is a finite dimensional algebra over an algebraically closed field and f=(f_1,f_2)^t:X rightarrow Y_1 oplus Y_2 is an irreducible morphism then we show that d_l(f)=d_l(f_1)+ d_l(f_2). We also characterize the artin algebras of finite representation type with alpha(Gamma_A) leq 2 in terms of a finite number of irreducible morphisms with finite degree.