Krull-Remak-Schmidt decompositions in Hom-finite additive categories
arXiv:2209.00337 · doi:10.1016/j.exmath.2022.12.003
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 for which there is a commutative unital ring , such that each Hom-set in is a finite length -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.
v2: 17 pages, added reference Chen--Ye--Zhang, typos corrected and some minor changes. v1: 17 pages, comments welcome!