On the global dimension of the category of diagrams of modules
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.
diagrams of objects of a small category over a category
scientific article; zbMATH DE number 1432170
scientific article