Monomial Almost Complete Intersections and the Strong Lefschetz Property

TL;DR

This paper investigates when monomial almost complete intersections possess the strong Lefschetz property, providing classifications and proofs for specific cases using Hilbert series analysis.

Contribution

It offers a complete classification for certain monomial almost complete intersections regarding the SLP and proves that symmetric Hilbert series cases always have the SLP.

Findings

Complete classification for monomial almost complete intersections with support in two variables.

Proof that symmetric Hilbert series monomial almost complete intersections always have the SLP.

Utilization of Hilbert series as a key tool in analyzing the SLP.

Abstract

Motivated by the foundational result that a monomial complete intersection has the strong Lefschetz property (SLP) in characteristic zero, it is natural to ask when monomial almost complete intersections have the SLP. In this paper, using the Hilbert series as a central tool, we investigate the strong Lefschetz property for certain monomial almost complete intersections, those with the non-pure-power generator having support in two variables, and those with symmetric Hilbert series. In the former case, we give a complete classification for when the SLP holds, and in the latter case, we prove that such algebras always have the SLP.