A linear formula for the generalized multiplicity sequence

TL;DR

This paper establishes a linear formula for generalized multiplicity sequences of ideals in local rings, providing a numerical characterization of integral closure and extending Rees' theorem.

Contribution

It introduces a linear formula for generalized multiplicity sequences and characterizes integral closure for ideals in local rings.

Findings

Derived a linear formula expressing multiplicity sequences as local multiplicities.

Proved that equality of multiplicity sequences implies ideal reduction.

Extended Rees' theorem to a broader class of ideals and modules.

Abstract

For an arbitrary ideal I in a local ring R and a finitely generated R-module M, we prove a formula expressing each generalized multiplicity sequence c_k(I,M) as a linear combination of certain local multiplicities. As a consequence, when M is formally equidimensional, we prove that if I is contained in J and c_k(I,M)=c_k(J,M) for all k, then I is a reduction of (J,M). The converse of this statement is also known to be true by a result of Ciuperca. This theorem gives a complete numerical characterization of the integral closure, generalizing a well known theorem of Rees.