Multiplicities of graded families of ideals on Noetherian local rings

TL;DR

This paper generalizes the classical multiplicity of ideals to graded families of ideals in Noetherian local rings, extending key theorems and providing new proofs that do not rely on volume or Okounkov body theory.

Contribution

It introduces a new notion of multiplicity for graded families of ideals, extending classical results and simplifying proofs without volume or Okounkov body methods.

Findings

Generalized multiplicity for graded families of ideals.

Extended classical theorems like Rees and Minkowski inequalities.

Provided new proofs independent of volume and Okounkov bodies.

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 $R$.