Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicities and degree functions in local rings via intersection products

Steven Dale Cutkosky, Jonathan Montaño

TL;DR

This work develops a self-contained intersection-theoretic framework over Noetherian local rings that interprets Hilbert–Samuel multiplicities, degree functions, and mixed multiplicities via birational geometry. The authors construct an intersection product $(-)_R$ and prove a Ramanujam-type formula $e(I)=-((I\mathcal{O}_Y)^d)_R$, extending it to arbitrary Noetherian local rings, and show that mixed multiplicities arise as intersection numbers, yielding multilinearity. They connect these geometric constructions to Rees’ degree-function theorem and Rees valuations, providing geometric proofs and corollaries, including 2-dimensional results related to Teissier–Rees–Sharp theory and negative-definite intersection matrices. The framework unifies multiplicity theory with intersection theory on birational models, offering self-contained proofs and new insights, including an explicit correspondence between integrally closed ideals on 2D local rings and resolution data through the semigroup $S(R)$ and group $G(R)$.

Abstract

We prove a theorem on the intersection theory over a Noetherian local ring $R$, which gives a new proof of a classical theorem of Rees about degree functions. To obtain this, we define an intersection product on schemes that are proper and birational over such rings $R$, using the theory of rational equivalence developed by Thorup, and the Snapper-Mumford-Kleiman intersection theory for proper schemes over an Artinian local ring. Our development of this product is essentially self-contained. As a central component of the proof of our main theorem, we extend to arbitrary Noetherian local rings a formula by Ramanujam that computes Hilbert-Samuel multiplicities. In the final section, we express mixed multiplicities in terms of intersection theory and conclude from this that they satisfy a certain multilinearity condition. Then we interpret some theorems of Rees and Sharp and of Teissier about mixed multiplicities over $2$-dimensional excellent local rings in terms of our intersection product.

Multiplicities and degree functions in local rings via intersection products

TL;DR

This work develops a self-contained intersection-theoretic framework over Noetherian local rings that interprets Hilbert–Samuel multiplicities, degree functions, and mixed multiplicities via birational geometry. The authors construct an intersection product and prove a Ramanujam-type formula , extending it to arbitrary Noetherian local rings, and show that mixed multiplicities arise as intersection numbers, yielding multilinearity. They connect these geometric constructions to Rees’ degree-function theorem and Rees valuations, providing geometric proofs and corollaries, including 2-dimensional results related to Teissier–Rees–Sharp theory and negative-definite intersection matrices. The framework unifies multiplicity theory with intersection theory on birational models, offering self-contained proofs and new insights, including an explicit correspondence between integrally closed ideals on 2D local rings and resolution data through the semigroup and group .

Abstract

We prove a theorem on the intersection theory over a Noetherian local ring , which gives a new proof of a classical theorem of Rees about degree functions. To obtain this, we define an intersection product on schemes that are proper and birational over such rings , using the theory of rational equivalence developed by Thorup, and the Snapper-Mumford-Kleiman intersection theory for proper schemes over an Artinian local ring. Our development of this product is essentially self-contained. As a central component of the proof of our main theorem, we extend to arbitrary Noetherian local rings a formula by Ramanujam that computes Hilbert-Samuel multiplicities. In the final section, we express mixed multiplicities in terms of intersection theory and conclude from this that they satisfy a certain multilinearity condition. Then we interpret some theorems of Rees and Sharp and of Teissier about mixed multiplicities over -dimensional excellent local rings in terms of our intersection product.

Paper Structure

This paper contains 13 sections, 42 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Intersection theory over local rings
  3. Cycles
  4. Intersection Theory over ${\operatorname{Spec}}(R)$
  5. Intersection products over a local ring
  6. Intersection theory on proper schemes over an Artinian ring
  7. Multiplicities of ideals as intersection products
  8. Degree functions
  9. Theorems of Rees on degree functions
  10. Analytically unramified local domains
  11. Local rings
  12. Mixed multiplicities of ideals as intersection products
  13. The semigroup and group of integrally closed ideals on a $2$-dimensional local ring

Key Result

Theorem A

Let $(R,\mathfrak{m}_R,\mathbb{k})$ be a $d$-dimensional local ring and $I\subset R$ be an $\mathfrak{m}_R$-primary ideal. Let $\pi:Y\rightarrow {\operatorname{Spec}}(R)$ be a birational projective morphism such that $I\mathcal{O}_Y$ is invertible. Then

Theorems & Definitions (79)

  • Theorem A: \ref{['TheoremMultInt']}
  • Theorem 1.1: Rees, Re, Re2
  • Theorem B: \ref{['Theorem2']}
  • Corollary 1: \ref{['CorThm2']}
  • Theorem C: \ref{['ThmExc']}
  • Theorem D: \ref{['thm:MixedMult']}
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 69 more