The Generators of a Colon Ideal with an Application to the Weak Lefschetz Property for Monomial Almost Complete Intersections in Three Variables

TL;DR

This paper determines explicit generators for a colon ideal related to monomial almost complete intersections and uses this to analyze the weak Lefschetz property, confirming a conjecture in new cases.

Contribution

It provides explicit formulas for generators of a specific colon ideal and links these to the weak Lefschetz property for certain monomial ideals.

Findings

Explicit formulas for colon ideal generators.

Failure of WLP characterized by polynomial determinant.

Confirmed conjecture in new cases for level algebras.

Abstract

Much progress has been made in classifying when the weak Lefschetz property holds for $A=\mathbb{F}[x,y,z]/I$ where $\text{char}(\mathbb{F})=0$ and $I=(x_{1}^{d_{1}},y^{d_{2}},z^{d_{3}},x^{a_{1}}y^{a_{2}}z^{a_{3}})$ is a monomial almost complete intersection. We connect this problem to the setting of two variables through a certain relation. In so doing, we are led to determine explicit formulas for the generators of the colon ideal $(x^{d_{1}},y^{d_{2}}):(x+y)^{a_{3}}$. With these generators in hand, we construct a matrix and show that failure of WLP for $A$ is dictated by the vanishing of a certain polynomial (namely the determinant of our matrix) when $A$ is level. We further show in the level case that a conjecture first posed by Migliore, Mir\'o-Roig, and Nagel is true in a few new cases.