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 $\ depth \ A \geq 2$. Let $I$ be an $\mathfrak{m}$-primary ideal and set $I^*$ to be the integral closure of $I$. Set $G^*(I) = \bigoplus_{n\geq 0} (I^n)^*/(I^{n+1})^*$ be the associated graded ring of the integral closure filtration of $I$. We prove that $\ depth \ G^*(I^n) \geq 2$ 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 $s \geq s_0$ and $J$ is an integrally closed ideal \emph{strictly} containing $(\mathfrak{m}^s)^*$ then we have a strict inequality $\mu(J) < \mu((\mathfrak{m}^s)^*)$ (here $\mu(J)$ is the number of minimal generators of $J$).