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The regularity of monomial ideals and their integral closures

Yijun Cui, Cheng Gong, Guangjun Zhu

Abstract

Let $I$ be a monomial ideal in a polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$ with $n=2$ or $3$, and let $\overline{I}$ be its integral closure. We will show that $\text{reg} (\overline{I}) \le \text{reg} (I)$. Furthermore, if $I$ is generated by elements of degree $d$, then $\text{reg} (I)=d$ if and only if $I$ has linear quotients.

The regularity of monomial ideals and their integral closures

Abstract

Let be a monomial ideal in a polynomial ring over a field with or , and let be its integral closure. We will show that . Furthermore, if is generated by elements of degree , then if and only if has linear quotients.

Paper Structure

This paper contains 3 sections, 21 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regularity of monomial ideals and their integral closures

Key Result

Lemma 2.1

(V) Let $I\subset S$ be a monomial ideal with $\mathcal{G}(I)=\{{\mathbf x}^{{{\mathbf b}}_1},\ldots, {\mathbf x}^{{{\mathbf b}}_m}\}$. Then $\overline{I}$ is generated by the monomials ${\mathbf x}^{{\mathbf a}}$, where ${\mathbf a}=(\lceil a_1 \rceil,\dots,\lceil a_n \rceil)$ with $(a_1,\ldots,a_n

Theorems & Definitions (43)

  • Conjecture 1.1
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 33 more