On asymptotic depth of integral closure filtration and an application

Let (A, m) be an analytically unramified formally equidimensional Noetherian local ring with depth A >= 2. Let I be an m-primary ideal and set I* to be the integral closure of I. Set G*(I) = circle plus(n >= 0)(I-n)*/(In+1)* be the associated graded ring of the integral closure filtration of I. We prove that depth G* (I-n) >= 2 for all n >> 0. As an application we prove that if A is also an excellent normal domain containing an algebraically closed field isomorphic to A/m then there exists so such that for all s >= s(0) and J is an integrally closed ideal strictly containing (m(s))* then we have a strict inequality mu(J) < mu((m(s))*) (here mu(J) is the number of minimal generators of J). (C) 2020 Elsevier Inc. All rights reserved.