Exact Krull-Schmidt categories with finitely many indecomposables
type: Item
instance of: scholarly article
title: Exact Krull-Schmidt categories with finitely many indecomposables
zbMATH Open document ID: 1495.18012
author: Wolfgang Rump
published in: Mathematische Zeitschrift
publication date: 2022-01-25T00:00:00Z
review text: An Ext-category is an exact category with enough projectives and enough injectives such that every split epimorphism has a kernel. Notice that many naturally arising exact categories are, in fact, Ext-categories. Let \(\mathcal{A}\) be an Ext-category with the Krull-Schmidt property. It is of interest to understand when the set \(\mathrm{ind}(\mathcal{A})\) of (isomorphism classes of) indecomposable objects in \(\mathcal{A}\) is finite. A main result in the paper under review gives, under some technical assumptions on \(\mathcal{A}\), several interesting conditions which are all equivalent to finiteness of the set \(\mathrm{ind}(\mathcal{A})\). One of these equivalent conditions is that the homotopy category \(\mathrm{Ext}(\mathcal{A})\) of conflations in \(\mathcal{A}\) (which is proved to be abelian) is a length category, meaning that every object in this category has finite length. A second equivalent condition is that the category \(\mathrm{mod}(\mathcal{A})\) of finitely presented \(\mathcal{A}\)-modules is abelian and noetherian. A third equivalent condition is that \(\mathcal{A}\) has almost split sequences, and the homotopy category \(\mathsf{M}(\mathcal{A})\) of morphisms in \(\mathcal{A}\) is \(L\)-finite. This condition, of which the last part is somewhat technical (however, it is explained well in Section 4), is really the key condition in the author's proof. One application of this result yields several different characterizations of when a Cohen-Macaulay order \(\Lambda\) over a complete regular local ring \(R\) is representation finite. The last section of the paper contains an interesting characterization of Ext-categories \(\mathcal{A}\) of finite type, which means that one has \(\mathcal{A} = \mathrm{add}(A)\) for some object \(A \in \mathcal{A}\) (here the Krull-Schmidt property is not assumed).
reviewed by: Henrik Holm
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 18G25
zbMATH DE Number: 7463807
zbMATH Keywords: Cohen-Macaulay order
zbMATH Keywords: exact category
zbMATH Keywords: \(L\)-functor
zbMATH Keywords: triadic category
zbMATH Keywords: almost split sequence
zbMATH Keywords: isolated singularity
zbMATH Keywords: representation-finite
zbMATH Keywords: non-commutative resolution
zbMATH Keywords: Gorenstein ring
zbMATH Keywords: Gorenstein projective
MaRDI profile type: Publication
full work available at URL: s00209-021-02710-0
OpenAlex ID: W3184937425
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Exact Krull-Schmidt categories with finitely many indecomposables (DOI)
DOI: 10.1007/S00209-021-02710-0

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