Asymptotic behaviour of integral closures, quintasymptotic primes and ideal topologies

Abstract

Let $R$ be a commutative Noetherian ring, $N$ a finitely generated $R$-module and $I$ an ideal of $R$. The set $\overline{Q^{\ast }}(I,N)$, 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 $\{(I^n{)}_a^{(N)}{\}}_{n\ge 1}$ and $\{S((I^n{)}_a^{(N)}){\}}_{n\ge 1}$ 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 $H_I^{\dim N}(N)$ vanishes if and only if there exists a multiplicatively closed subset $S$ of $R$ such that $\mathfrak{m}\cap S\ne \varnothing$ and the topologies induced by $\{(I^n{)}_a^{(N)}{\}}_{n\ge 1}$ and $\{S((I^n{)}_a^{(N)}){\}}_{n\ge 1}$ are equivalent. As a special of this characterization we obtain the main result of Marti-Farre \cite{MF}.

15 pages, to appear in: Rocky Mountain Journal of Mathematics

Cited by in corpus (1)

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