Methods of Calculating Cohomological and Hochschild–mitchell Dimensions of Finite Partially Ordered Sets

Mitchell characterized all finite partially ordered sets with incidence ring of Hochschild dimension 0, 1, and 2. Cheng characterized all finite partially ordered sets of cohomological dimension one. There are no conjectures in other dimensions. This article contains the algorithms for calculating the dimensions of finite partially ordered sets by elementary operations over rows and columns of matrices with integer entries. 1. Cohomological dimension Denote by N the set of nonnegative integers, Z the additive group of integers, Ab the category of abelian groups and homomorphisms. For any subset M ⊆ N the sup will be considered in {−1} ∪N ∪ {∞}. Thus we let sup ∅ = −1. Let (X,⩽) be a partially ordered set (shortly poset). We consider X as a small category with objects ObX = X, in which for every x, y ∈ X the set X(x, y) of morphisms x → y consists of one element if x ⩽ y in X, and X(x, y) = ∅, otherwise. If x ⩽ y and x ̸= y, then we write x < y. If neither x ⩽ y nor y ⩽ x, then we say that x and y are incompatible. If every pair of distinct elements in X are incompatible, then we say that X is discrete. If for every x, y ∈ X there exists a sequence x0 ⩽ x1 ⩾ x2 ⩽ · · · ⩾ x2n of elements in X such that x0 = x and x2n = y, then X is connected. Every subset Y ⊆ X will be considered as the poset in which y1 ⩽ y2 for y1, y2 ∈ Y if and only if y1 ⩽ y2 in X. The maximal connected subsets of X are called the connected components. Denote by Ab the category of functors X → Ab. For every F ∈ Ab and n ∈ N we have the abelian groups