Mixed Multiplicities of Filtrations

TL;DR

This paper extends the theory of mixed multiplicities from $m_R$-primary ideals to more general filtrations in Noetherian local rings, constructing a polynomial that encodes these multiplicities and proving classical inequalities hold in this broader context.

Contribution

It introduces a generalized framework for mixed multiplicities of filtrations, including the construction of a real polynomial and the extension of key classical theorems.

Findings

Existence of a polynomial encoding mixed multiplicities under certain conditions.

Classical Minkowski inequalities hold for filtrations.

Generalization of mixed multiplicity theory to non-Noetherian filtrations.

Abstract

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $R$ is less than the dimension of $R$, which holds for instance if $R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $m_R$-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and Rees and Sharp.