Roller Coaster Gorenstein algebras and Koszul algebras failing 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.