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On the global dimension of the category of commutative diagrams in an abelian category. (English. Russian original) Zbl 0874.18008

Sib. Math. J. 37, No. 5, 1041-1051 (1996); translation from Sib. Mat. Zh. 37, No. 5, 1181-1194 (1996).
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. H.-J. Baues and G. Wirsching, J. Pure Appl. Algebra 38, 187-211 (1985; Zbl 0587.18006)]. 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 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.

MSC:

18G20 Homological dimension (category-theoretic aspects)
18A25 Functor categories, comma categories
18E10 Abelian categories, Grothendieck categories
05E25 Group actions on posets, etc. (MSC2000)
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References:

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