# Degrees of Irreducible Morphisms over Perfect Fields

## TL;DR

This paper introduces a new concept of the degree of morphisms in module categories over perfect fields, linking it to natural transformations and the radical filtration, with implications for understanding irreducible morphisms and their compositions.

## Contribution

It extends the notion of degree of irreducible morphisms to a broader context using natural transformations, providing new criteria for finiteness and generalizing existing results.

## Key findings

- Degree of morphisms is finite iff associated natural transformation has a representable kernel.
- Generalizations of known results on degrees of irreducible morphisms over perfect fields.
- Application to compositions of paths of irreducible morphisms and radical powers.

## Abstract

The module category of any artin algebra is filtered by the powers of its radical, thus defining an associated graded category. As an extension of the degree of irreducible morphisms, this text introduces the degree of morphisms in the module category in terms of the induced natural transformations between representable functors on this graded category. When the ground ring is a perfect field, and the given morphism behaves nicely with respect to covering theory (as do irreducible morphisms with indecomposable domain or indecomposable codomain), it is shown that the degree of the morphism is finite if and only if its associated natural transformation has a representable kernel. As a corollary, generalisations of known results on the degrees of irreducible morphisms over perfect fields are given. Finally, this study is applied to the composition of paths of irreducible morphisms in relationship to the powers of the radical.