On asymptotic depth of integral closure filtration and an application

TL;DR

This paper investigates the asymptotic depth properties of the integral closure filtration in certain local rings and applies these results to compare minimal generator counts of integrally closed ideals.

Contribution

It establishes that the depth of the associated graded ring of the integral closure filtration stabilizes at least 2 for large indices and applies this to derive inequalities for minimal generators of integrally closed ideals.

Findings

Depth of G^*(I^n) stabilizes at ≥ 2 for large n

Existence of a threshold s_0 for minimal generator inequalities

Comparison of minimal generators for certain integrally closed ideals

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 $J$).