On the global dimension of the category of diagrams of modules
type: Item
instance of: scholarly article
title: On the global dimension of the category of diagrams of modules
zbMATH Open document ID: 0938.18001
DOI: 10.1007/BF02674306
published in: Siberian Mathematical Journal
publication date: 2000-04-16T00:00:00Z
review text: A subset \(X\subseteq I\) of a partially ordered set \(I\) is called coinitial if, for every \(i\in I\), there exists an \(x\in X\) such that \(x\leq i\). The coinitiality \(\text{ci}(I)\) of a partially ordered set \(I\) is the least cardinality of coinitial subsets of \(I\). Recall that a category \(\mathcal C\) is called a small category if the class of its objects \(\text{Ob }{\mathcal C}\) is a set. Let \(\mathcal A\) be an arbitrary category and let \(\mathcal C\) be a small category. The diagrams of objects of \(\mathcal A\) over \(\mathcal C\) are the functors \({\mathcal C}\rightarrow {\mathcal A}\). Morphisms of diagrams are natural transformations of the category; they make a proper square of morphisms commutative. The category of diagrams is denoted by \({\mathcal A}^{\mathcal C}\). The author considers a partially ordered set \(I\) as a category whose objects are elements of \(I\) and morphisms \(i\rightarrow j\) are pairs of elements \(i\leq j\). For an abelian category \(\mathcal A\), the author uses the notion of global dimension \(\text{gl.dim }{\mathcal A}\) introduced by \textit{B. Mitchell} [Adv. Math. 8, 1-161 (1972; Zbl 0232.18009)]. Let \(K\) be a commutative Noetherian ring with unit \(1\), let \(I\) be a linearly ordered set, and let \(\text{Mod}^I_K\) be the category of diagrams of \(K\)-modules over \(I\). It was proven [\textit{H. Brune}, Manuscr. Math. 26, 141-154 (1978; Zbl 0412.18003)] that if the least upper bound of the coinitialities of the subsets of \(I\) bounded from below is equal to a cardinal number \(\aleph_n\) then the inequality \(\text{gl.dim } \text{Mod}^I_K\leq n+2+ \text{gl.dim } K\) holds. The author proves that if the ring \(K\) is a Dedekind domain or a Noetherian local ring then this inequality transforms into equality.
Mathematics Subject Classification ID: 18A25
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 18A10
Mathematics Subject Classification ID: 18B35
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 18G20
zbMATH DE Number: 1432170
zbMATH Keywords: coinitiality of a partially ordered set
zbMATH Keywords: diagrams of objects of a small category over a category
zbMATH Keywords: global dimension
author: Ahmet A. Khusainov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02674306
OpenAlex ID: W2080360125
description: scientific article; zbMATH DE number 1432170