On the global dimension of the category of commutative diagrams in an abelian category
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.
scientific article; zbMATH DE number 1023918
scientific article