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

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 A for which there is a commutative unit...

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Bibliographic Details
Published in:Expositiones mathematicae 2023-03, Vol.41 (1), p.220-237
Main Author: Shah, Amit
Format: Article
Language:English
Subjects:
Additive category
Hom-finite category
Idempotent
Krull-Remak-Schmidt decomposition
Krull-Schmidt category
Split idempotents
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Summary: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 A for which there is a commutative unital ring k, such that each Hom-set in 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.
ISSN:0723-0869
DOI:10.1016/j.exmath.2022.12.003