Epsilon multiplicity and analytic spread of filtrations

Abstract

We extend the epsilon multiplicity of ideals defined by Ulrich and Validashti to epsilon multiplicity of filtrations and show that under mild assumptions, this multiplicity exists as a limit. We show that in rather general rings, the epsilon multiplicity of a $\mathbb{Q}$-divisorial filtration is positive if and only if the analytic spread of the filtration is maximal (equal to the dimension of the ring). The condition that filtrations $\mathcal{J}\subset \mathcal{I}$ have the same epsilon multiplicity is considered, and we find conditions ensuring that the filtrations have the same integral closure.