Epsilon multiplicity for Noetherian graded algebras

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 Cutkosky’s result concerning epsilon multiplicity is obtained as a corollary of our main theorem. We also produce a multigraded generalization of a result due to Dao and Monataño about monomial ideals.