On the degree in categories of complexes of fixed size

TL;DR

This paper investigates the degrees of irreducible morphisms between complexes in categories derived from artin algebras, providing criteria for finiteness and characterizations related to algebra types.

Contribution

It introduces methods to compute degrees of morphisms in complexes of fixed size and characterizes when these categories are of finite type for hereditary algebras.

Findings

Criteria for finite left and right degrees of morphisms.

Conditions under which kernels and cokernels belong to the same category.

Characterization of finite type categories based on morphism degrees.

Abstract

We consider $\Lambda$ an artin algebra and $n \geq 2$. We study how to compute the left and right degrees of irreducible morphisms between complexes in a generalized standard Auslander-Reiten component of ${\mathbf{C_n}({\rm proj}\, \Lambda)}$ with length. We give conditions under which the kernel and the cokernel of irreducible morphisms between complexes in $\mathbf{C_n}({\rm proj}\, \Lambda)$ 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 $\mathbf{C_n}({\rm proj} \,H)$ is of finite type.