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Normality criteria for monomial ideals

Luis A. Dupont, Humberto Muñoz-George, Rafael H. Villarreal

Abstract

In this paper we study the normality of monomial ideals using linear programming and graph theory. We give normality criteria for monomial ideals, for ideals generated by monomials of degree two, and for edge ideals of graphs and clutters and their ideals of covers.

Normality criteria for monomial ideals

Abstract

In this paper we study the normality of monomial ideals using linear programming and graph theory. We give normality criteria for monomial ideals, for ideals generated by monomials of degree two, and for edge ideals of graphs and clutters and their ideals of covers.
Paper Structure (7 sections, 30 theorems, 97 equations, 2 figures)

This paper contains 7 sections, 30 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normality criteria for monomial ideals
  4. Ideals generated by monomials of degree $2$
  5. Normality of ideals of covers of graphs
  6. Examples
  7. Procedures

Key Result

Theorem 2.1

(Ful1, monalg-rev) If $\Gamma\subset\mathbb{N}^s$ is a finite set of points and $\mathcal{S}={\mathbb Z}\Gamma\cap {\mathbb R}_+\Gamma$, then the following hold:

Figures (2)

  • Figure 1: Graph $G$ is an odd antihole with $7$ vertices.
  • Figure 2: Graph $\overline{G}$ consists of two antiholes joined by a vertex.

Theorems & Definitions (65)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 55 more