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Normality of monomial ideals in three variables

Maki Ataka, Naoyuki Matsuoka

Abstract

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three and higher, this is no longer true in general, including monomial ideals in polynomial rings. In this paper, we study the normality of integrally closed monomial ideals in the polynomial ring $k[x,y,z]$ over a field $k$. We prove that every such ideal with at most seven minimal monomial generators is normal, thereby giving a sharp bound for normality in this setting. The proof is based on a detailed case-by-case analysis, combined with valuation-theoretic and combinatorial methods via Newton polyhedra.

Normality of monomial ideals in three variables

Abstract

An ideal in a Noetherian ring is called \textit{normal} if is integrally closed for all . Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three and higher, this is no longer true in general, including monomial ideals in polynomial rings. In this paper, we study the normality of integrally closed monomial ideals in the polynomial ring over a field . We prove that every such ideal with at most seven minimal monomial generators is normal, thereby giving a sharp bound for normality in this setting. The proof is based on a detailed case-by-case analysis, combined with valuation-theoretic and combinatorial methods via Newton polyhedra.
Paper Structure (17 sections, 17 theorems, 50 equations, 1 figure, 4 tables)

This paper contains 17 sections, 17 theorems, 50 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. The Case $\mu_A(I) \le 5$
  5. Strategy for proving the main result in the cases $\mu_A(I)=6$ and $7$
  6. The Case $\mu_A(I) =6$
  7. Classification of integrally closed monomial ideals
  8. Determining $(I^2:\mathfrak{m})\setminus I^2$
  9. Finding $\lambda =(p,q,r)$ with $\operatorname{ord}_\lambda(g) < \operatorname{ord}_\lambda(I^2)$
  10. The Case $\mu_A(I) =7$
  11. Classification of integrally closed monomial ideals
  12. Determining $(I^2:\mathfrak{m})\setminus I^2$
  13. Finding $\lambda =(p,q,r)$ with $\operatorname{ord}_\lambda(g) < \operatorname{ord}_\lambda(I^2)$
  14. Further consequences and structures
  15. Extension to higher dimensions
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $A = k[x_1, x_2, \ldots, x_d]$ be a polynomial ring over a field $k$ of characteristic zero. Let $I$ be a zero-dimensional ideal of $A$ generated by $d+3$ homogeneous polynomials. If $I$ is integrally closed, then $I$ is normal.

Figures (1)

  • Figure 1: Typical Newton polyhedra in two variables with $\mu_A(I) = 4$

Theorems & Definitions (35)

  • Theorem 1.2: EndoHongUlrich
  • Corollary
  • Remark 1.3
  • Lemma 2.1: cf. SH
  • Theorem 2.2: Z, ZS
  • Theorem 2.3: RRV, SH
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 25 more