Uniqueness of uniform decompositions in exact categories
type: Item
instance of: scholarly article
title: Uniqueness of uniform decompositions in exact categories
zbMATH Open document ID: 1461.18005
author: Mustafa Kemal Berktaş
author: Septimiu Crivei
author: Fatma Kaynarca
author: D. Keskin-Tütüncü
published in: Journal of Pure and Applied Algebra
publication date: 2021-03-03T00:00:00Z
review text: The authors extend uniqueness theorems on uniform decompositions of \textit{H. Krause} [Expo. Math. 33, No. 4, 535--549 (2015; Zbl 1353.18011)], \textit{L. Diracca} and \textit{A. Facchini} [J. Pure Appl. Algebra 172, No. 2--3, 183--191 (2002; Zbl 1006.18010)] (Krull-Remak Schmidt Theorem in abelian categories) to certain Quillen exact categories. Two objects \(A\) and \(B\) of an exact category \((\mathcal{A}, \mathcal{E})\) are (i) \(\mathcal{E}\)-essentially related if there exists \(\mathcal{E}\)-essential inflations \(A' \rightarrow A\) and \(B' \rightarrow B\) such that \(A' \cong B'\) and (ii) in the same \(\mathcal{E}\)-monogeny class if there exist inflations \(A \rightarrow B\) and \(B \rightarrow A.\) For an exact category \((\mathcal{A}, \mathcal{E}),\) let \(\mathcal{I}\) be the class of objects \(A\) such that \(A\) has an \(\mathcal{E}\)-uniform \(\mathcal{E}\)-injective envelope \(\mathcal{E}(A)\) with local endomorphism ring, and there exists an inflation \(A \rightarrow \mathcal{E}(A).\) Let \((\mathcal{A},\mathcal{E})\) be a weakly idempotent complete exact category, and let \(A = \bigoplus_{i=1}^m A_i\) and \(B = \bigoplus_{j=1}^n B_j\) be objects of \(\mathcal{A}\) such that \(A_i\)'s and \(B_j\)'s \(\in \mathcal{I}.\) If \((\mathcal{A}, \mathcal{E})\) had finite intersections and \(A,B\) are \(\mathcal {E}\)-essentially related, then the authors prove that \(m= n\) and there is a permutation \(\sigma\) of \(\{1,2,\dots, n\}\) such that \(A_i\) and \(B_{\sigma(i)}\) are \(\mathcal{E}\)-essentially related for every \(i \in \{1,2, \dots, n \}.\) If \(A\) and \(B\) are in the same \(\mathcal{E}\)-monogeny class, then they show that \(m = n\) and there is a permutation \(\sigma\) of \(\{1,2,\dots, n\}\) such that \(A_i\) and \(B_{\sigma(i)}\) are in the same \(\mathcal{E}\)-monogeny class for every \(i \in \{1,2, \dots, n \}.\) Moreover they generalize Krause's theorem to quasi abelian categories and give further applications to abelian categories.
reviewed by: Intan Muchtadi-Alamsyah
Mathematics Subject Classification ID: 18E05
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 18C35
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 7317124
zbMATH Keywords: Quillen exact category
zbMATH Keywords: uniform decomposition
zbMATH Keywords: (quasi-)abelian category
MaRDI profile type: Publication
full work available at URL: j.jpaa.2020.106621
OpenAlex ID: W3100404656
cites work: Q4293501
cites work: A uniqueness theorem in a finitely accessible additive category
cites work: On pure Goldie dimensions
cites work: Reduction of exact structures
cites work: Intersections, sums, and the Jordan-Hölder property for exact categories
cites work: Exact categories
cites work: Locally finitely presented additive categories
cites work: Gruson-Jensen Duality for Idempotent Rings
cites work: Covers in finitely accessible categories
cites work: Maximal exact structures on additive categories revisited
cites work: Uniqueness of monogeny classes for uniform objects in abelian categories
cites work: Additive categories of locally finite representation type
cites work: The Jordan-Hölder property and Grothendieck monoids of exact categories
cites work: Krull-Schmidt fails for serial modules
cites work: Q5599870
cites work: Properties of direct summands of modules
cites work: Locally finitely presented categories and functor rings
cites work: LEFT COTORSION RINGS
cites work: PURE-INJECTIVE ENVELOPES
cites work: Chain complexes and stable categories
cites work: Exactly definable categories
cites work: Functors on locally finitely presented additive categories
cites work: Uniqueness of uniform decompositions in abelian categories
cites work: Krull-Schmidt categories and projective covers
cites work: Q5508993
cites work: Q3632824
cites work: Definable additive categories: purity and model theory
cites work: Higher algebraic K-theory: I
cites work: Almost abelian categories
cites work: On the maximal exact structure of an additive category
cites work: Q4245741
cites work: Flat covers of modules
DOI: 10.1016/J.JPAA.2020.106621
description: scientific article; zbMATH DE number 7317124