Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Forcing the weak Lefschetz property for equigenerated monomial ideals

Nasrin Altafi, Samuel Lundqvist

Abstract

We classify the minimal number of generators of artinian equigenerated monomial ideals $I$ such that $\Bbbk[x_1,\ldots,x_n]/I$ is forced to have the weak Lefschetz property.

Forcing the weak Lefschetz property for equigenerated monomial ideals

Abstract

We classify the minimal number of generators of artinian equigenerated monomial ideals such that is forced to have the weak Lefschetz property.
Paper Structure (15 sections, 31 theorems, 96 equations)

This paper contains 15 sections, 31 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and an overview of the paper
  3. Constructing algebras failing the WLP
  4. Ideals with $\mu \geq n+2$ minimal generators and the failure of the WLP
  5. The case $\mu \in \left[2n-1,\beta(n,d) \right]$
  6. The case $\mu \in \left[n+2, 2n-2 \right]$
  7. Ideals with $n+1$ minimal generators and the failure of the WLP
  8. The case $n=3$
  9. The case $n\geq 4$, $n$ even
  10. The case $n\geq 5$, $n$ odd
  11. Presence of the WLP
  12. The case $n=3$
  13. The case $n\geq 4$
  14. Proof of the main result
  15. A final remark

Key Result

Theorem 1.1

Let $n \geq 3$ and $d \geq 2$ be integers and let $S=\Bbbk[x_1,\ldots,x_n]$, where $\Bbbk$ is a field of characteristic zero. Let and let Then for every there exists an artinian ideal $I$ in $S$ minimally generated by $\mu$ elements of degree $d$ such that $S/I$ fails the WLP. Moreover, the bounds are sharp, which means that if $\mu \notin \Sigma_{n,d}$, then for every artinian monomial ideal $

Theorems & Definitions (61)

  • Theorem 1.1
  • proof
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • proof : Second proof of Theorem \ref{['thm:tensor']}
  • Remark 3.1
  • Corollary 3.1
  • proof
  • Remark 3.2
  • ...and 51 more