Rocky Mountain Journal of Mathematics

Asymptotic behavior of integral closures, quintasymptotic primes and ideal topologies

Reza Naghipour and Peter Schenzel

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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

Permanent link to this document
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

Keywords
Integral closure ideal topologies local cohomology quintasymptotic prime

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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References

  • S.H. Ahn, Asymptotic primes and asymptotic grade on modules, J. Algebra \bf174 (1995), 980–998.
  • M. Brodmann and R.Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998.
  • F.W. Call, On local cohomology modules, J. Pure Appl. Alg. \bf43 (1986), 111–117.
  • F.W. Call and R.Y. Sharp, A short proof of the local Lichtenbaum-Hartshorne theorem on the vanishing of local cohomology modules, Bull. Lond. Math. Soc. \bf18 (1986), 261–264.
  • K. Divaani-Aazar and P. Schenzel, Ideal topologies, local cohomology and connectedness, Math. Proc. Cambr. Philos. Soc. \bf131 (2001), 211–226.
  • T. Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscr. Math. \bf104 (2001), 519–525.
  • M. Marti-Ferre, Symbolic powers and local cohomology, Mathematika \bf42 (1995), 182–187.
  • H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986.
  • S. McAdam, Asymptotic prime divisors, Lect. Notes Math. \bf1023 (1983).
  • ––––, Quintasymptotic primes and four results of Schenzel, J. Pure Appl. Alg. \bf47 (1987), 283–298.
  • S. McAdam and L.J. Ratliff, Essential sequences, J. Algebra \bf95 (1985), 217–235.
  • A.A. Mehrvarz, R. Naghipour and M. Sedghi, Quintasymptotic primes, local cohomology and ideal topologies, Colloq. Math. 106 (2006), 25–37.
  • M. Nagata, Local rings, Interscience Tracts, New York, 1961.
  • R. Naghipour and M. Sedghi, A characterization of Cohen-Macaulay modules and local cohomology, Archives Math. \bf87 (2006), 303–308.
  • D.G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambr. Philos. Soc. \bf50 (1954), 145–158.
  • L.J. Ratliff, Jr., On asymptotic prime divisors, Pacific J. Math. \bf111 (1984), 395–413.
  • P. Schenzel, Symbolic powers of prime ideals and their topology, Proc. Amer. Math. Soc. 93 (1985), 15–20.
  • ––––, Finiteness of relative Rees rings and asymptotic prime divisors, Math. Nachr. \bf129 (1986), 123–148.
  • ––––, Explicit computations around the Lichtenbaum-Hartshorne vanishing theorem, Manuscr. Math. \bf78 (1993), 57–68.
  • ––––, On the use of local cohomology in algebra and geometry, in Six lectures on commutative algebra, Progr. Math. 166 (1998).
  • R.Y. Sharp, Linear growth of primary decompositions of integral closures, J. Algebra 207 (1998), 276–284.
  • R.Y. Sharp, Y. Tiras and M. Yassi, Integral closures of ideals relative to local cohomology modules over quasi-unmixed local rings, J. Lond. Math. Soc. \bf42 (1990), 385–392.