On the Weak-Lefschetz property for Artinian Gorenstein algebras

TL;DR

This paper investigates the conditions under which Artinian Gorenstein algebras of codimension 3 exhibit the Weak Lefschetz property, identifying sequences and Betti sequences that guarantee this property.

Contribution

It establishes that many Gorenstein sequences and Betti sequences in codimension 3 ensure the WLP, and shows that general algebras with these sequences typically possess the property.

Findings

Many Gorenstein sequences force the WLP in codimension 3.

Certain Gorenstein Betti sequences are compatible with sequences that force the WLP.

General algebras with these Betti sequences have the WLP.

Abstract

We deal with the Weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension $3.$ We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence $H$ of codimension 3 we found several Gorenstein Betti sequences compatible with $H$ which again force the WLP. Finally we show that for every Gorenstein Betti sequence the general Artinian standard graded Gorenstein algebra with such Betti sequence has the WLP.