Betti numbers of powers of path ideals of cycles

TL;DR

This paper investigates the algebraic properties of powers of path ideals of cycles, proving they have linear resolutions and providing explicit Betti number formulas for specific cases.

Contribution

It establishes that powers of cycle path ideals have linear resolutions and derives explicit Betti number formulas for cases where m equals n-1 or n-2.

Findings

$J_{n,m}^t$ has a linear free resolution.

Explicit Betti number formulas are provided for $m = n-1, n-2$ cases.

The results deepen understanding of algebraic invariants of cycle path ideals.

Abstract

Let $J_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n}x_1\cdots x_{m-1})$ be the $m$-path ideal of a cycle of length $n \ge 5$ over a polynomial ring $S = k[x_1,\ldots,x_n]$. Let $t\geq 1$ be an integer. We show that $J_{n,m}^t$ has a linear free resolution and give a precise formula for all of its Betti numbers when $m = n-1, n-2$.