Cohen-Macaulay and Gorenstein Path Ideals of Trees

Abstract

Let R=k[x1,…,xn], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Γ be a directed rooted tree. We characterize all such trees whose path ideals are unmixed and Cohen-Macaulay. Moreover, we show that R/It(Γ) is Gorenstein if and only if the Stanley-Reisner simplicial complex of It(Γ) is a matroid.