On the weak Lefschetz Property of graded modules over $K[x,y]$

TL;DR

This paper investigates conditions under which graded modules over the polynomial ring in two variables exhibit the Weak Lefschetz Property, providing an algorithm for testing and characterizing indecomposable modules with specific Hilbert functions.

Contribution

It introduces an algorithm to test the WLP for graded modules with fixed Hilbert functions and characterizes indecomposable modules with Hilbert function (h0,h1) as having the WLP.

Findings

Indecomposable modules with Hilbert function (h0,h1) possess the WLP.

An algorithm is provided to test the WLP for graded modules with given Hilbert functions.

Not all non-cyclic modules over $K[x,y]$ have the WLP, but specific conditions ensure it.

Abstract

It is known that graded cyclic modules over $S=K[x,y]$ have the Weak Lefschetz Property (WLP). This is not true for non-cyclic modules over $S$. The purpose of this note is to study which conditions on $S$-modules ensure the WLP. We give an algorithm to test the WLP for graded modules with fixed Hilbert function. In particular, we prove that indecomposable graded modules over $S$ with the Hilbert function $(h_0,h_1)$ have the WLP.