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

TL;DR

The paper proves the existence of Roller Coaster algebras: artinian Gorenstein algebras with unconstrained first halves of their Hilbert series, realized via Nagata idealization of monomial algebras from whiskered graphs. It develops a framework around Simplicial Perazzo forms and Macaulay duality to produce Gorenstein algebras that fail the weak Lefschetz property, including a broad supply of Koszul examples. Key techniques connect graph theory (independence, whiskering) with algebraic properties (G-quadratic, Koszul, WLP/SLP) through edge ideals, independence complexes, and Perazzo constructions. The results show that the space of possible first-half Hilbert sequences for such algebras is effectively unconstrained, with implications for the structure of Gorenstein and Koszul algebras in high socle degree. Overall, the work unifies combinatorial and algebraic methods to generate large families of counterexamples and to advance understanding of Lefschetz properties in monomial and quadratic settings.

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.

Theorems & Definitions (56)

• Theorem 1.1: Roller Coaster theorem for artinian Gorenstein algebras
• Theorem 1.2
• Conjecture 1.3
• Conjecture 1.4
• Theorem 2.1: DE2009
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.3