Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicities of graded families of ideals on Noetherian local rings

Steven Dale Cutkosky

Abstract

Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$. In this article, we give a generalization of the multiplicity $e(I)$ of an $m_R$-primary ideal $I$ of $R$ to a multiplicity $e(\mathcal I)$ of a graded family of $m_R$-primary ideals $\mathcal I$ in $R$. This multiplicity gives the classical multiplicity $e(I)$ if $\mathcal I=\{I^n\}$ is the $I$-adic filtration, and agrees with the volume, $\displaystyle \lim_{n\rightarrow \infty}d!\frac{\ell(R/I_n) }{n^d}$ for $R$ such that the volume always exists as a limit. We will show in this paper that many of the classical theorems for the multiplicity of an ideal generalize to this multiplicity, including mixed multiplicities, the Rees theorem and the Minkowski inequality and equality. We give simple proofs which are independent of the theory of volumes and Okounkov bodies for all of our results, with the one exception being the proof of the Minkowski equality. We do this by interpreting the multiplicity of graded families of $m_R$-primary ideals as a limit of intersection products on the family of $R$-schemes which are obtained by blowing up $m_R$-primary ideals in $R$.

Multiplicities of graded families of ideals on Noetherian local rings

Abstract

Let be a -dimensional Noetherian local ring with maximal ideal . In this article, we give a generalization of the multiplicity of an -primary ideal of to a multiplicity of a graded family of -primary ideals in . This multiplicity gives the classical multiplicity if is the -adic filtration, and agrees with the volume, for such that the volume always exists as a limit. We will show in this paper that many of the classical theorems for the multiplicity of an ideal generalize to this multiplicity, including mixed multiplicities, the Rees theorem and the Minkowski inequality and equality. We give simple proofs which are independent of the theory of volumes and Okounkov bodies for all of our results, with the one exception being the proof of the Minkowski equality. We do this by interpreting the multiplicity of graded families of -primary ideals as a limit of intersection products on the family of -schemes which are obtained by blowing up -primary ideals in .
Paper Structure (13 sections, 38 theorems, 192 equations)

This paper contains 13 sections, 38 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Mixed multiplicities of graded $m_R$-families
  3. Valuations, integral closure and saturation
  4. Rees's Theorem
  5. Minkowski inequalities and equality
  6. One dimensional Noetherian local rings
  7. Intersection products of graded $m_R$-families
  8. Multiplicities of graded $m_R$-families
  9. A comparison of volume and multiplicity on Noetherian local rings
  10. valuations, integral closure and valuation saturation
  11. A saturated filtration which is not real divisorial
  12. Rees's Theorem
  13. Minkowski inequality and equality

Key Result

Theorem 1.1

(Theorem 1.1 C2) Suppose that $N(\hat{R})$ is the nilradical of the $m_R$-adic completion $\hat{R}$ of $R$. Then the limit exists for any graded $m_R$-family $\mathcal{I}=\{I_n\}$ on $R$ if and only if $\dim N(\hat{R}) < d$.

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 59 more