Revista Matemática Iberoamericana

Asymptotic behaviour of monomial ideals on regular sequences

Monireh Sedghi

Full-text: Open access

Abstract

Let $R$ be a commutative Noetherian ring, and let $\mathbf{x}= x_1, \ldots, x_d$ be a regular $R$-sequence contained in the Jacobson radical of $R$. An ideal $I$ of $R$ is said to be a monomial ideal with respect to $\mathbf{x}$ if it is generated by a set of monomials $x_1^{e_1}\ldots x_d^{e_d}$. The monomial closure of $I$, denoted by $\widetilde{I}$, is defined to be the ideal generated by the set of all monomials $m$ such that $m^n\in I^n$ for some $n\in \mathbb{N}$. It is shown that the sequences $\mathrm{Ass}_RR/\widetilde{I^n}$ and $\mathrm{Ass}_R\widetilde{I^n}/I^n$, $n=1,2, \ldots,$ of associated prime ideals are increasing and ultimately constant for large $n$. In addition, some results about the monomial ideals and their integral closures are included.

Article information

Source
Rev. Mat. Iberoamericana, Volume 22, Number 3 (2006), 955-962.

Dates
First available in Project Euclid: 22 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1169480036

Mathematical Reviews number (MathSciNet)
MR2320407

Zentralblatt MATH identifier
1115.13016

Subjects
Primary: 13B20 13B21: Integral dependence; going up, going down

Keywords
monomial ideals integral closures monomial closures

Citation

Sedghi, Monireh. Asymptotic behaviour of monomial ideals on regular sequences. Rev. Mat. Iberoamericana 22 (2006), no. 3, 955--962. https://projecteuclid.org/euclid.rmi/1169480036


Export citation

References

  • Brodmann, M.: Asymptotic stability of $\Ass(M/I^nM)$. Proc. Amer. Math. Soc. 74 (1979), 16-18.
  • Bruns, W. and Herzog, J.: Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge, 1993.
  • Heinzer, W., Mirbagheri, A., Ratliff, L. J., Jr. and Shah, K.: Parametric decomposition of monomial ideals II. J. Algebra 187 (1997), 120-149.
  • Kaplansky, I.: Commutative rings. Revised edition. The University of Chicago Press, Chicago, Ill.-London, 1974.
  • Kiyek, K. and Stückrad, J.: Integral closure of monomial ideals on regular sequences. (Proceedings of the International Conference on Algebraic Geometry and Singularities, Sevilla, 2001). Rev. Mat. Iberoamericana 19 (2003), 483-508.
  • Matsumura, H.: Commutative ring theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge, 1986.
  • McAdam, S.: Asymptotic prime divisors. Lecture Notes in Mathematics 1023. Springer-Verlag, Berlin, 1983.
  • Northcott, D. G. and Rees, D.: Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50 (1954), 145-158.
  • Ratliff, L. J., Jr.: On asymptotic prime divisors. Pacific J. Math. 111 (1984), no. 2, 395-413.
  • Sharp, R. Y.: Linear growth of primary decompositions of integral closures. J. Algebra 207 (1998), 276-284.
  • Vasconcelos, W. V.: Computational methods in commutative algebra and algebraic geometry. Algorithms and Computation in Mathematics 2. Springer-Verlag, Berlin, 1998.