Abstract
Let (A,m) be an analytically unramified formally equidimensional Noetherian local ring with depthA≥2. Let I be an m-primary ideal and set I⁎ to be the integral closure of I. Set G⁎(I)=⨁n≥0(In)⁎/(In+1)⁎ be the associated graded ring of the integral closure filtration of I. We prove that depthG⁎(In)≥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 s0 such that for all s≥s0 and J is an integrally closed ideal strictly containing (ms)⁎ then we have a strict inequality μ(J)<μ((ms)⁎) (here μ(J) is the number of minimal generators of J).
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