On the Lefschetz Property for quotients by monomial ideals containing squares of variables

Abstract

Let $\Delta$ be an (abstract) simplicial complex on $n$ vertices. One can define the Artinian monomial algebra $A(\Delta )=\mathbb{k}[x_1,\dots ,x_n]/⟨x_1^2,\dots ,x_n^2,I_{\Delta }⟩$, where $\mathbb{k}$ is a field of characteristic $0$ and $I_{\Delta }$ is the Stanley-Reisner ideal associated to $\Delta$. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of $A(\Delta )$ in terms of the simplicial complex $\Delta$. We are able to completely analyze when WLP holds in degree $1$, complementing work by Migliore, Nagel and Schenck in [MNS2020]. We give a complete characterization of all $2$-dimensional pseudomanifolds $\Delta$ such that $A(\Delta )$ satisfies WLP. We also construct Artinian Gorenstein algebras that fail WLP by combining our results and the standard technique of Nagata idealization.

13 pages, 3 figures

References in corpus (2)

• Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles
• Grünbaum coloring and its generalization to arbitrary dimension

Cited by in corpus (1)

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