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The Weak Lefschetz Property for Tensor Products of Artinian Monomial Algebras and Its Applications to Lollipop Graphs

Tran Quang Hoa, Nguyen Duy Phuoc, Tran Nguyen Thanh Son

TL;DR

This work analyzes when the weak Lefschetz property (WLP) holds for Artinian algebras arising from graphs, focusing on tensor products with complete quadratic monomial algebras and on lollipop graphs $L_{m,n}$. It proves a general criterion: for $B= rac{oldsymbol{k}[x_1, olinebreak\dots,x_n]}{(x_1, olinebreak\dots,x_n)^2} ensor_oldsymbol{k} A$ with $oldsymbol{ extell}'=x_1+ olinebreak\nolinebreak olinebreak x_n+oldsymbol{ extell}$, the maps $oldsymbol{ extell}'$ are injective/surjective precisely when the corresponding maps $oldsymbol{ extell}$ and $oldsymbol{ extell}^2$ on $A$ have the same property; this yields a way to determine WLP for $A(L_{m,n})$ via exact sequences and inductive arguments. The independence polynomial of $L_{m,n}$ is shown to be unimodal with mode in $iglrace rakl_n, rakl_n+1igrrace$, enabling a precise parameter-based WLP classification: in characteristic zero, $A(L_{m,n})$ has the WLP exactly for specified small values of $m$ and $n$ (three explicit families). Overall, the paper links graph-theoretic invariants (independence polynomials) to Lefschetz properties, delivering a complete WLP characterization for lollipop-graph algebras and contributing a general framework for tensor-product Lefschetz analysis in monomial settings.

Abstract

In this paper, we investigate the weak Lefschetz property for tensor products of Artinian monomial algebras and complete quadratic monomial algebras. As an application, we classify the weak Lefschetz property of the Artinian algebras $A(L_{m,n})$, which are defined by the edge ideals of the lollipop graphs $L_{m,n}$ together with the squares of the variables.

The Weak Lefschetz Property for Tensor Products of Artinian Monomial Algebras and Its Applications to Lollipop Graphs

TL;DR

This work analyzes when the weak Lefschetz property (WLP) holds for Artinian algebras arising from graphs, focusing on tensor products with complete quadratic monomial algebras and on lollipop graphs . It proves a general criterion: for with , the maps are injective/surjective precisely when the corresponding maps and on have the same property; this yields a way to determine WLP for via exact sequences and inductive arguments. The independence polynomial of is shown to be unimodal with mode in , enabling a precise parameter-based WLP classification: in characteristic zero, has the WLP exactly for specified small values of and (three explicit families). Overall, the paper links graph-theoretic invariants (independence polynomials) to Lefschetz properties, delivering a complete WLP characterization for lollipop-graph algebras and contributing a general framework for tensor-product Lefschetz analysis in monomial settings.

Abstract

In this paper, we investigate the weak Lefschetz property for tensor products of Artinian monomial algebras and complete quadratic monomial algebras. As an application, we classify the weak Lefschetz property of the Artinian algebras , which are defined by the edge ideals of the lollipop graphs together with the squares of the variables.
Paper Structure (8 sections, 18 theorems, 63 equations, 1 figure, 1 table)

This paper contains 8 sections, 18 theorems, 63 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The weak Lefschetz property
  4. Graph theory
  5. Artinian algebras associated to graphs
  6. WLP of tensor products of Artinian monomial algebras and complete quadratic monomial algebras
  7. The independence polynomial of lollipop graphs
  8. WLP of Artinian algebras associated to lollipop graphs

Key Result

Theorem 1.2

Suppose that $\operatorname{char}(\Bbbk)=0$. Let where $A=\dfrac{\Bbbk[y_1,y_2,\ldots,y_m]}{J}$ is an Artinian monomial algebra over $\Bbbk$ of socle degree $D>0$. Set $\ell=y_1+y_2+\cdots+y_m$ and $\ell'= x_1+x_2+\cdots + x_n+\ell$. Then, for all $i\in \{1,2,\ldots,D-1\}$, the multiplication map is injective (respectively, surjective) if and only if the multiplication maps are both injective (

Figures (1)

  • Figure 1: Lollipop graph $L_{m,n}$

Theorems & Definitions (34)

  • Theorem 1.2: Theorem \ref{['generalization of lollipop']}
  • Theorem 1.3: Theorem \ref{['WLP for lollipop']}
  • Definition 2.1
  • Proposition 2.2: MMN2011
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: HMMNWW2013
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 24 more