The strong Lefschetz property of certain modules over Clements-Lindstr\"om rings

TL;DR

This paper develops a new method using combinatorial lemmas to analyze the Lefschetz properties of modules over Clements-Lindstr"om rings, proving strong Lefschetz properties in specific cases.

Contribution

It introduces a novel approach based on the Lindstr"om-Gessel-Viennot Lemma to study Lefschetz properties of modules over Clements-Lindstr"om rings, including new results for characteristic zero.

Findings

Modules over certain Clements-Lindstr"om rings have the strong Lefschetz property in characteristic zero.

Homogeneous ideals in two-dimensional Clements-Lindstr"om rings possess the strong Lefschetz property.

Type two monomial ideals of codimension three exhibit the strong Lefschetz property.

Abstract

We introduce a method for studying the Lefschetz properties for $k[x,y]$-modules based on the Lindstr\"om-Gessel-Viennot Lemma. In particular, we prove that certain modules over Artinian Clements-Lindstr\"om rings in characteristic zero have the strong Lefschetz property. In particular, we show that every homogeneous idea in a Clements-Lindstr\"om ring of embedding dimension two has the strong Lefschetz property. As an application, we study the strong Lefschetz property of type two monomial ideals of codimension three.