Cohen-Macaulay and Gorenstein path ideals of trees

TL;DR

This paper characterizes when path ideals of directed rooted trees are Cohen-Macaulay, unmixed, or Gorenstein, linking algebraic properties to combinatorial structures like matroids.

Contribution

It provides a complete characterization of trees with Cohen-Macaulay and Gorenstein path ideals, connecting algebraic properties to combinatorial features.

Findings

Characterization of trees with Cohen-Macaulay path ideals

Conditions for path ideals to be Gorenstein involving matroids

Identification of when path ideals are unmixed

Abstract

Let $R=k[x_{1},\ldots,x_{n}]$, where $k$ is a field. The path ideal (of length $t\geq 2$) 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.