Mixed Multiplicities of Multi-Graded Algebras Over Noetherian Local Rings

TL;DR

This paper explores the relationship between mixed multiplicities of multi-graded algebras over Noetherian local rings and Hilbert-Samuel multiplicities, providing formulas that unify and extend previous results in the field.

Contribution

It expresses mixed multiplicities in terms of Hilbert-Samuel multiplicities, offering new formulas that generalize earlier work by Trung and Verma.

Findings

Mixed multiplicities are expressed via Hilbert-Samuel multiplicities.

Provides formulas for mixed multiplicities of ideals.

Extends the main results of Trung-Verma 2007.

Abstract

Let S be a finitely generated standard multi-graded algebra over a Noetherian local ring A. This paper first expresses mixed multiplicities of S in term of Hilbert-Samuel multiplicity that explained the mixed multiplicities S as the Hilbert-Samuel multiplicities for quotient modules of S. As an application, we get formulas for the mixed multiplicities of ideals that covers the main result of Trung-Verma 2007 [TV].