Lefschetz properties of monomial ideals with almost linear resolution

TL;DR

This paper investigates the Lefschetz properties of artinian monomial ideals with almost linear resolutions, providing new insights into their algebraic structure and confirming a conjecture for such ideals.

Contribution

It proves a conjecture regarding Lefschetz properties for monomial ideals with minimal free resolutions that are linear for at least n-2 steps.

Findings

Confirmed the conjecture of Eisenbud, Huneke, and Ulrich for almost linear resolutions.

Established conditions under which the Weak and Strong Lefschetz Properties hold.

Connected the minimal free resolution structure to Lefschetz properties in monomial ideals.

Abstract

We study the WLP and SLP of artinian monomial ideals in $S=\mathbb{K}[x_1,\dots ,x_n]$ via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of $S/I$ is linear for at least $n-2$ steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.