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

Hailong Dao, Ritika Nair

Abstract

Let $Δ$ be an (abstract) simplicial complex on $n$ vertices. One can define the Artinian monomial algebra $A(Δ) = \Bbbk[x_1, \ldots, x_n]/ \langle x_1^2, \ldots, x_n^2, I_Δ \rangle$, where $\Bbbk$ is a field of characteristic $0$ and $I_Δ$ is the Stanley-Reisner ideal associated to $Δ$. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of $A(Δ)$ in terms of the simplicial complex $Δ$. 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 $Δ$ such that $A(Δ)$ satisfies WLP. We also construct Artinian Gorenstein algebras that fail WLP by combining our results and the standard technique of Nagata idealization.

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

Abstract

Let be an (abstract) simplicial complex on vertices. One can define the Artinian monomial algebra , where is a field of characteristic and is the Stanley-Reisner ideal associated to . In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of in terms of the simplicial complex . We are able to completely analyze when WLP holds in degree , complementing work by Migliore, Nagel and Schenck in [MNS2020]. We give a complete characterization of all -dimensional pseudomanifolds such that satisfies WLP. We also construct Artinian Gorenstein algebras that fail WLP by combining our results and the standard technique of Nagata idealization.
Paper Structure (7 sections, 18 theorems, 8 equations)

This paper contains 7 sections, 18 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminary definitions
  3. Weak Lefschetz Property in degree $1$
  4. Weak Lefschetz Property in degree $d$ for Pseudomanifolds
  5. Pseudomanifold without boundary
  6. Pseudomanifold with boundary
  7. Examples of Artinian Gorenstein algebras that fail WLP

Key Result

Theorem 1.1

(Theorem deg1) Let ${\Delta}$ be a simplicial complex, $G({\Delta})$ the $1$-skeleton of ${\Delta}$, and $A=A({\Delta})$ the Artinian ring defined by the Stanley-Reisner ideal of ${\Delta}$ plus the squares of all variables.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Example 2.5
  • Proposition 2.6
  • ...and 37 more