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

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.