On asymptotic depth of integral closure filtration and an application

Abstract

Let $(A,\mathfrak{m})$ be an analytically unramified formally equidimensional Noetherian local ring with $depthA\ge 2$. Let $I$ be an $\mathfrak{m}$-primary ideal and set $I^{\ast }$ to be the integral closure of $I$. Set $G^{\ast }(I)={⨁}_{n\ge 0}(I^n{)}^{\ast }/(I^{n+1}{)}^{\ast }$ be the associated graded ring of the integral closure filtration of $I$. We prove that $depth{G}^{\ast }(I^n)\ge 2$ for all $n\gg 0$. 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 $s_0$ such that for all $s\ge s_0$ and $J$ is an integrally closed ideal \emph{strictly} containing $({\mathfrak{m}}^s{)}^{\ast }$ then we have a strict inequality $\mu (J)<\mu (({\mathfrak{m}}^s{)}^{\ast })$ (here $\mu (J)$ is the number of minimal generators of $J$).