Cohen-Macaulay and Gorenstein path ideals of trees

Abstract

Let $R=k[x_1,\dots ,x_n]$, where $k$ is a field. The path ideal (of length $t\ge 2$) of a directed graph $G$ is the monomial ideal, denoted by $I_t(G)$, whose generators correspond to the directed paths of length $t$ in $G$. Let $\Gamma$ be a directed rooted tree. We characterize all such trees whose path ideals are unmixed and Cohen-Macaulay. Moreover, we show that $R/I_t(\Gamma )$ is Gorenstein if and only if the Stanley-Reisner simplicial complex of $I_t(\Gamma )$ is a matroid.

10 pages, 4 figures, to appear in Algebra Colloquium. arXiv admin note: text overlap with arXiv:0708.3111 by other authors