On the global dimension of the category of commutative diagrams in an abelian category
type: Item
instance of: scholarly article
title: On the global dimension of the category of commutative diagrams in an abelian category
zbMATH Open document ID: 0874.18008
DOI: 10.1007/BF02110735
published in: Siberian Mathematical Journal
publication date: 1997-07-22T00:00:00Z
review text: For a small category \({\mathcal C}\) and an abelian category \({\mathcal A}\) let \({\mathcal C} {\mathcal A}\) denote the abelian category of functors \({\mathcal C} \to {\mathcal A}\) and of natural transformations of them. Let, further, \(\dim {\mathcal C}\) be the Hochschild-Mitchell dimension of \({\mathcal C}\) [cf. \textit{H.-J. Baues} and \textit{G. Wirsching}, J. Pure Appl. Algebra 38, 187-211 (1985; Zbl 0587.18006)]. \textit{B. Mitchell} [Adv. Math. 8, 1-161 (1972; Zbl 0232.18009)] proved the inequality \(\dim {\mathcal C} \leq\text{gl.dim } {\mathcal C} {\mathcal A}-\)gl.dim \({\mathcal A}\) valid for any abelian category \({\mathcal A}\) with sums. The example constructed by Spiers in his Ph.D. thesis shows that the inequality can be strict and that its right side can depend on \({\mathcal A}\). In the present paper, the author gives the characterization (in terms of the homology groups \(H_n (]x,y[)\), \(x,y\in {\mathcal C})\) of finite posets \({\mathcal C}\) considered as small categories such that \(\dim {\mathcal C}= \text{gl.dim} {\mathcal C} {\mathcal A}-\)gl.dim \({\mathcal A}\) for any abelian category \({\mathcal A}\). Finite posets are described for which the above inequality is strict. For any two small categories \({\mathcal C}\) and \({\mathcal D}\) the above paper by \textit{B. Mitchell} gives the estimation \(\dim {\mathcal C} \times {\mathcal D} \leq\dim {\mathcal C} +\dim {\mathcal D}\). For locally finite posets the author finds the conditions under which this estimation is fulfilled as an equality and as a strict inequality.
Mathematics Subject Classification ID: 18G20
Mathematics Subject Classification ID: 18A25
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 05E25
zbMATH DE Number: 1023918
zbMATH Keywords: projective dimension
zbMATH Keywords: global dimension
zbMATH Keywords: nerve
zbMATH Keywords: interval
zbMATH Keywords: small category
zbMATH Keywords: abelian category
zbMATH Keywords: Hochschild-Mitchell dimension
zbMATH Keywords: homology groups
author: Ahmet A. Khusainov
reviewed by: Feliks R. Bobovich
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1023918