On the degree in categories of complexes of fixed size

Abstract

We consider $\Lambda$ an artin algebra and $n\ge 2$. We study how to compute the left and right degrees of irreducible morphisms between complexes in a generalized standard Auslander-Reiten component of ${\mathbf{C}}_{\mathbf{n}}(\mathrm{proj}\Lambda )$ with length. We give conditions under which the kernel and the cokernel of irreducible morphisms between complexes in ${\mathbf{C}}_{\mathbf{n}}(\mathrm{proj}\Lambda )$ belong to such a category. For a finite dimensional hereditary algebra $H$ over an algebraically closed field, we determine when an irreducible morphism has finite left (or right) degree and we give a characterization, depending on the degrees of certain irreducible morphisms, under which ${\mathbf{C}}_{\mathbf{n}}(\mathrm{proj}H)$ is of finite type.