Path ideals of rooted trees and their graded Betti numbers

Let Γ be a rooted (and directed) tree, and let t be a positive integer. The path ideal It(Γ) is generated by monomials that correspond to directed paths of length (t−1) in Γ. In this paper, we study algebraic properties and invariants of It(Γ). We give a recursive formula to compute the graded Betti numbers of It(Γ) in terms of path ideals of subtrees. We also give a general bound for the regularity, explicitly compute the linear strand, and investigate when It(Γ) has a linear resolution.