Filter-regular sequences and mixed multiplicities

TL;DR

This paper investigates conditions under which mixed multiplicities of multigraded modules over Artinian local rings are positive, providing characterizations and applications to ideals, and connecting to classical results in algebraic geometry.

Contribution

It characterizes positive mixed multiplicities using filter-regular sequences and length computations, extending previous results and unifying various classical theorems.

Findings

Provides criteria for positivity of mixed multiplicities.

Characterizes mixed multiplicities via lengths of modules.

Recovers classical results of Risler, Teissier, Trung, and Verma.

Abstract

Let $S$ be a finitely generated standard multigraded algebra over an Artinian local ring $A$; $M$ a finitely generated multigraded $S$-module. This paper answers to the question when mixed multiplicities of $M$ are positive and characterizes them in terms of lengths of $A$-modules. As an application, we get results on mixed multiplicities of ideals (Proposition 4.4 and Theorem 4.5), and recover the results of Risler and Teissier [Te](see Remark 4.8), Trung and Verma [TV](see Remark 4.6).