Epsilon multiplicity for graded algebras

Abstract

The notion of ε -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 ε -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 ε -multiplicity, as corollaries of our main theorem.