Some Formulas for Epsilon Multiplicity in Local Rings

TL;DR

This paper establishes the existence of epsilon multiplicity in certain Noetherian local rings and extends the volume-multiplicity formula to this context, broadening the understanding of multiplicity invariants.

Contribution

It proves the existence of epsilon multiplicity under new conditions and generalizes the volume equals multiplicity formula for this invariant.

Findings

Epsilon multiplicity exists when the nilradical of the completion has nonmaximal dimension.

The volume equals multiplicity formula is extended to this setting.

Provides new tools for studying multiplicities in local algebra.

Abstract

We prove that the epsilon multiplicity exists in a Noetherian local ring whenever the nildradical of the completion of R has nonmaximal dimension. We also extend the volume equals multiplicity formula for the epsilon multiplicity to this setting.