Epsilon multiplicity for Noetherian graded algebras

TL;DR

This paper proves that the relative epsilon multiplicity for reduced Noetherian graded algebras over an excellent local ring exists as a limit, and introduces mixed epsilon multiplicity for monomial ideals.

Contribution

It establishes the existence of epsilon multiplicity as a limit in a broad algebraic setting and extends the concept to mixed epsilon multiplicity for monomial ideals.

Findings

Relative epsilon multiplicity exists as a limit for reduced Noetherian graded algebras.

A corollary of the main theorem recovers a result of Cutkosky on epsilon multiplicity.

Introduces the notion of mixed epsilon multiplicity for monomial ideals.

Abstract

The notion of epsilon multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that the relative epsilon multiplicity of reduced Noetherian graded algebras over an excellent local ring exists as a limit. An important special case of a result of Cutkosky concerning epsilon multiplicity, is obtained as a corollary of our main theorem. We also develop the notion of mixed epsilon multiplicity for monomial ideals.