On locally finite partially ordered sets of dimension 3 (Q1288306)

The least upper bound \(n\) of the set of naturals for which each of the functors \(\text{Ext}^n_\mathcal A (A,-)\: \mathcal A\to\text{Ab}\) differs from the zero functor is said to be the projective dimension \(\text{pd }A\) of an object \(A\) of the abelian category \(\mathcal A\). For a small category \({\mathcal C}\), the functor \(L{\mathcal C}\: {\mathcal C}^{op}\times{\mathcal C}\to\text{Ab}\) is defined which assigns, to each pair of objects \(x\) and \(y\) of the category \({\mathcal C}\), a free abelian group, whose base is the set of all morphisms \({\mathcal C}(x,y)\) of the object \(x\) into the object \(y\) of the category \({\mathcal C}\). The projective dimension of the object \(L{\mathcal C}\) in the category of functors \({\mathcal C}^{op}\times{\mathcal C}\to\text{Ab}\) is called the Hochschild--Mitchell dimension of the category \({\mathcal C}\) and is denoted by \(\text{dim }{\mathcal C}\). For each object \(A\) of an abelian category \(\mathcal A\), the functor \(A\otimes (-)\:\text{Ab}\to\mathcal A\) is defined. Let \(A{\mathcal C}\: {\mathcal C}^{op}\times{\mathcal C} \to\mathcal A\) be the composition of the functors \(L{\mathcal C}\) and \(A\otimes (-)\). In the article under review, the inequality \(\text{pd }A{\mathcal C}\leq\text{dim }{\mathcal C} +\text{pd } A\) is proved for every locally finite partially ordered set \({\mathcal C}\). In addition, it is proved that \(\text{pd } A{\mathcal C}=3+\text{pd }A\) if \(\text{dim }{\mathcal C}=3\). The last statement gives an answer to a question raised by \textit{B. Mitchell} [Adv. Math. 8, 1-161 (1972; Zbl 0232.18009)].