Roller Coaster Gorenstein algebras and Koszul algebras failing the weak Lefschetz property

TL;DR

This paper constructs new examples of artinian Gorenstein algebras, called Roller Coaster algebras, demonstrating that many quadratic monomial and G-quadratic Gorenstein algebras do not satisfy the weak Lefschetz property, challenging existing assumptions.

Contribution

It introduces Roller Coaster algebras via Nagata idealization and shows that quadratic monomial and G-quadratic Gorenstein algebras often fail the weak Lefschetz property.

Findings

Existence of Roller Coaster Gorenstein algebras with unconstrained Hilbert series

Quadratic monomial algebras rarely have the weak Lefschetz property

Large family of G-quadratic Gorenstein algebras fail the weak Lefschetz property

Abstract

Inspired by the Roller Coaster Theorem from graph theory, we prove the existence of artinian Gorenstein algebras with unconstrained Hilbert series, which we call Roller Coaster algebras. Our construction relies on Nagata idealization of quadratic monomial algebras defined by whiskered graphs. The monomial algebras are interesting in their own right, as our results suggest that artinian level algebras defined by quadratic monomial ideals rarely have the weak Lefschetz property. In addition, we discover a large family of G-quadratic Gorenstein algebras failing the weak Lefschetz property.