Monomial ideals, almost complete intersections and the Weak Lefschetz Property

TL;DR

This paper investigates the Weak Lefschetz property in monomial ideals, revealing its dependence on field characteristic and arithmetic properties of exponents, and highlights the complexity of establishing this property.

Contribution

It provides new insights into how the Weak Lefschetz property varies with field characteristic and monomial exponent vectors in monomial ideals.

Findings

Weak Lefschetz property depends on the characteristic of the ground field.

The property is influenced by arithmetic properties of monomial exponents.

Examples demonstrate subtle variations in the property for different ideals.

Abstract

Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field, and on arithmetic properties of the exponent vectors of the monomials.