Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Matthew Davidson Booth, Adela Vraciu

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, Miró-Roig, and Nagel is true in a few new cases.

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

Abstract

Much progress has been made in classifying when the weak Lefschetz property holds for where and 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 . With these generators in hand, we construct a matrix and show that failure of WLP for is dictated by the vanishing of a certain polynomial (namely the determinant of our matrix) when is level. We further show in the level case that a conjecture first posed by Migliore, Miró-Roig, and Nagel is true in a few new cases.
Paper Structure (3 sections, 11 theorems, 141 equations)

This paper contains 3 sections, 11 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Generators of the Colon Ideal
  3. Application to WLP in 3-Variable Monomial ACIs

Key Result

Theorem 2.2

Consider the ideal $(x^{d_{1}},y^{d_{2}}):(x+y)^{a}$ where $2\leq d_{1}\leq d_{2}$ and $1\leq a\leq d_{1}+d_{2}-2$. Setting $k=d_{2}-d_{1}$, the aforementioned colon ideal has the following generators in $P$:

Theorems & Definitions (26)

  • Conjecture 1.1
  • Theorem 2.2
  • proof
  • Lemma 2.4
  • proof
  • proof
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • proof
  • ...and 16 more