## Rocky Mountain Journal of Mathematics

### Asymptotic behavior of integral closures, quintasymptotic primes and ideal topologies

#### Abstract

Let $R$ be a Noetherian ring, $N$ a finitely generated $R$-module and $I$ an ideal of $R$. It is shown that the sequences $Ass _R R/(I^n)_a^{(N)}$, $Ass _R (I^n)_a^{(N)}/ (I^{n+1})^{(N)}_a$ and $Ass _R (I^n)_a^{(N)}/ (I^n)_a$, $n= 1,2, \ldots$, of associated prime ideals, are increasing and ultimately constant for large $n$. Moreover, it is shown that, if $S$ is a multiplicatively closed subset of $R$, then the topologies defined by $(I^n)_a^{(N)}$ and $S((I^n)_a^{(N)})$, $n\geq 1$, are equivalent if and only if $S$ is disjoint from the quintasymptotic primes of $I$. By using this, 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 $(I^n)_a^{(N)}$ and $S((I^n)_a^{(N)})$, $n\geq 1$, are equivalent.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 2 (2018), 551-572.

Dates
First available in Project Euclid: 4 June 2018

https://projecteuclid.org/euclid.rmjm/1528077633

Digital Object Identifier
doi:10.1216/RMJ-2018-48-2-551

Mathematical Reviews number (MathSciNet)
MR3810209

Zentralblatt MATH identifier
06883481

Subjects
Primary: 13B20 13E05: Noetherian rings and modules

#### Citation

Naghipour, Reza; Schenzel, Peter. Asymptotic behavior of integral closures, quintasymptotic primes and ideal topologies. Rocky Mountain J. Math. 48 (2018), no. 2, 551--572. doi:10.1216/RMJ-2018-48-2-551. https://projecteuclid.org/euclid.rmjm/1528077633

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