Epsilon multiplicity for graded algebras

TL;DR

This paper proves that the relative epsilon multiplicity for reduced standard graded algebras over excellent local rings exists as a limit, clarifying its behavior and extending previous results.

Contribution

It establishes the existence of the epsilon multiplicity as a limit in a broad algebraic setting, improving understanding of its properties.

Findings

Epsilon multiplicity exists as a limit for reduced standard graded algebras over excellent local rings.

The paper generalizes and extends Cutkosky's results on epsilon multiplicity.

Provides new insights into the behavior of epsilon multiplicity in algebraic geometry.

Abstract

The notion of $\varepsilon$-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 standard graded algebras over an excellent local ring exists as a limit. We also obtain some important special cases of Cutkosky's results concerning $\varepsilon$-multiplicity, as corollaries of our main theorem.