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

TL;DR

This paper characterizes when the Artinian monomial algebra associated with a simplicial complex satisfies the Weak Lefschetz Property, focusing on degree 1 and 2-dimensional pseudomanifolds, and constructs examples that fail WLP.

Contribution

It provides a complete characterization of WLP for $A( riangle)$ in degree 1 and for 2-dimensional pseudomanifolds, and introduces methods to construct Gorenstein algebras that do not satisfy WLP.

Findings

WLP holds in degree 1 under specific conditions.

Complete characterization of WLP for 2-dimensional pseudomanifolds.

Construction of Gorenstein algebras failing WLP.

Abstract

Let $\Delta$ be an (abstract) simplicial complex on $n$ vertices. One can define the Artinian monomial algebra $A(\Delta) = \Bbbk[x_1, \ldots, x_n]/ \langle x_1^2, \ldots, x_n^2, I_{\Delta} \rangle$, where $\Bbbk$ 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.