Krull-Remak-Schmidt decompositions in Hom-finite additive categories

TL;DR

This paper proves that in Hom-finite additive categories, having split idempotents and local endomorphism rings for indecomposables are equivalent to the category being Krull-Schmidt, ensuring unique decompositions into indecomposables.

Contribution

It establishes the equivalence between Krull-Schmidt property, split idempotents, and local endomorphism rings in Hom-finite categories.

Findings

Hom-finite categories are Krull-Schmidt iff they have split idempotents.

Indecomposable objects in such categories have local endomorphism rings.

The paper provides a proof of these equivalences.

Abstract

An additive category in which each object has a Krull-Remak-Schmidt decomposition -- that is, a finite direct sum decomposition consisting of objects with local endomorphism rings -- is known as a Krull-Schmidt category. A Hom-finite category is an additive category $\mathcal{A}$ for which there is a commutative unital ring $k$, such that each Hom-set in $\mathcal{A}$ is a finite length $k$-module. The aim of this note is to provide a proof that a Hom-finite category is Krull-Schmidt, if and only if it has split idempotents, if and only if each indecomposable object has a local endomorphism ring.