The strong Lefschetz property in codimension two

TL;DR

This paper investigates the strong Lefschetz property for artinian quotients of polynomial rings in two variables, establishing new bounds, classifications, and conditions under which the property holds regardless of field characteristic.

Contribution

It improves bounds for when artinian quotients have the strong Lefschetz property, classifies sharp cases, and characterizes monomial ideals that always possess the property in two variables.

Findings

Monomial ideals in two variables always have the strong Lefschetz property regardless of characteristic.

The bounds for characteristic in which the property holds are improved for monomial ideals.

Lexsegment ideals are precisely those monomial ideals in two variables that always have the strong Lefschetz property.

Abstract

Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of monomial ideals. Using this we classify when both bounds are sharp. Moreover, we prove that the artinian quotient of a monomial ideal in $K[x,y]$ always has the strong Lefschetz property, regardless of the characteristic of the field, exactly when the ideal is lexsegment. As a consequence we describe a family of non-monomial complete intersections that always have the strong Lefschetz property.