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 the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds $\Delta$ such that $A(\Delta )$ satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization.

KEYWORDS:

• Artinian algebra
• bipartite graph
• face 2-colorable
• Gorenstein ring
• Grünbaum coloring
• Lefschetz properties
• monomial ideal
• pseudomanifolds
• simplicial complex
• Stanley-Reisner ring
• triangulation

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 05C25
• 05E40
• 13E10
• 13F55
• 13H10

Acknowledgments

We thank Aaron Dall, Alessio D’Alí, Serge Lawrencenko, Hal Schenck, Alexandra Seceleanu, and an anonymous referee for some helpful correspondence on the topics of this work. We also thank Daniel Erman and an anonymous referee for their valuable comments.