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 I t ( 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 I t ( Γ ) . We give a recursive formula to compute the graded Betti numbers of I t ( 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 I t ( Γ ) has a linear resolution.