A note on joint reductions and mixed multiplicities

TL;DR

This paper interprets mixed multiplicities of ideals in a noetherian local ring as multiplicities of joint reductions and superficial sequences, generalizing Rees's theorem and providing new insights into their structure.

Contribution

It introduces a new interpretation of mixed multiplicities as multiplicities of joint reductions and superficial sequences, extending Rees's theorem.

Findings

Mixed multiplicities can be viewed as multiplicities of joint reductions.

Established a connection between mixed multiplicities and Rees's superficial sequences.

Generalized Rees's theorem on mixed multiplicity.

Abstract

Let $(A, \frak m)$ be a noetherian local ring with maximal ideal $\frak{m}$ and infinite residue field $k = A/\frak{m}.$ Let $J$ be an $\frak m$-primary ideal, $I_1,...,I_s$ ideals of $A$, and $M$ a finitely generated $A$-module. In this paper, we interpret mixed multiplicities of $(I_1,..., I_s,J)$ with respect to $M$ as multiplicities of joint reductions of them. This generalizes the Rees's theorem on mixed multiplicity (Theorem 2.4). As an application we show that mixed multiplicities are also multiplicities of Rees's superficial sequences.