# Colength, multiplicity, and ideal closure operations

## TL;DR

This paper establishes that in a certain class of local rings, the equality of colength and multiplicity for integrally closed ideals characterizes regularity, using the interplay between multiplicity and ideal closure operations.

## Contribution

It provides a new criterion for regularity in local rings based on colength and multiplicity of integrally closed ideals, linking these invariants through closure operations.

## Key findings

- Equality of colength and multiplicity implies regularity in certain rings.
- Relationship between multiplicity and closure operations is key to the proof.
- Characterization of regularity via ideal invariants.

## Abstract

In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then $R$ is regular. We deduce this using the relationship between multiplicity and various ideal closure operations.