Path ideals of rooted trees and their graded Betti numbers

TL;DR

This paper investigates algebraic properties of path ideals in rooted trees, providing recursive formulas for Betti numbers, bounds for regularity, and criteria for linear resolutions.

Contribution

It introduces a recursive method to compute graded Betti numbers and characterizes when the path ideal has a linear resolution.

Findings

Recursive formula for graded Betti numbers

General bound for the regularity

Characterization of linear resolutions

Abstract

Let $\Gamma$ be a rooted tree and let $t$ be a positive integer. We study algebraic invariants and properties of the path ideal generated by monomial corresponding to paths of length $(t-1)$ in $\Gamma$. In particular, we give a recursive formula to compute the graded Betti numbers, a general bound for the regularity, an explicit computation of the linear strand, and we characterize when this path ideal has a linear resolution.