Asymptotic behaviour of integral closures, quintasymptotic primes and ideal topologies

TL;DR

This paper investigates the asymptotic behavior of integral closures and quintasymptotic primes in Noetherian rings, establishing conditions for the equivalence of certain ideal topologies and linking these to local cohomology vanishing.

Contribution

It provides a characterization of when ideal topologies are equivalent based on quintasymptotic primes and relates this to the vanishing of local cohomology modules in local rings.

Findings

Topologies defined by integral closures are equivalent iff the multiplicative set avoids quintasymptotic primes.

Local cohomology vanishes iff a certain multiplicative set intersects the maximal ideal and topologies are equivalent.

Main result generalizes Marti-Farre's theorem on ideal topologies and cohomology.

Abstract

Let $R$ be a commutative Noetherian ring, $N$ a finitely generated $R$-module and $I$ an ideal of $R$. The set $\bar{Q^*}(I, N)$, the quintasymptotic primes of $I$ with respect to $N$, was originally introduced by McAdam \cite{Mc2}. Also, the ideal $I_a^{(N)}$, the integral closure of $I$ with respect to $N$, was introduced by R.Y. Sharp et al. in \cite{STY}. The purpose of this paper is to show that, whenever $S$ is a multiplicatively closed subset of $R$ then the topologies defined by $\{(I^n)_a^{(N)}\}_{n\geq1}$ and $\{S((I^n)_a^{(N)})\}_{n\geq1}$ are equivalent if and only if $S$ is disjoint from the quintasymptotic primes of $I$ with respect to $N$. In addition, using this result, we also show that, if $(R, \mathfrak{m})$ is local and $N$ is quasi-unmixed, then the local cohomology module $H^{\dim N}_I(N)$ vanishes if and only if there exists a multiplicatively closed subset $S$ of $R$ such that $\mathfrak{m} \cap S \neq \emptyset$ and the topologies induced by $\{(I^n)_a^{(N)}\}_{n\geq1}$ and $\{S((I^n)_a^{(N)})\}_{n\geq1}$ are equivalent. As a special of this characterization we obtain the main result of Marti-Farre \cite{MF}.