Multiplicity sequence and integral dependence

TL;DR

This paper establishes a fundamental equivalence between integral closure and multiplicity sequences of ideals in certain rings, and introduces a principle for specialization of integral dependence based on multiplicity sequence constancy.

Contribution

It proves that ideals have the same integral closure if and only if their multiplicity sequences are identical, and develops a specialization principle for integral dependence.

Findings

Equivalence of integral closure and multiplicity sequence in specific rings

A new principle for specialization of integral dependence

Conditions for integral dependence via multiplicity sequence constancy

Abstract

We prove that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of Specialization of Integral Dependence, which gives a condition for integral dependence in terms of the constancy of the multiplicity sequence in families.