Illinois Journal of Mathematics

Monomial sequences of linear type

Hamid Kulosman

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Abstract

Let $R$ be a Noetherian commutative ring, $\langle a_1,\dots, a_n\rangle$ a sequence of elements of $R$, $I=(a_1,\dots,a_n)$ the ideal generated by the elements $a_i$ and $I_i=(a_1,\dots,a_i)$, $i=0,1,\dots, n$, the ideal generated by the first $i$ elements of the sequence. A c-sequence is a sequence $\langle a_1,\dots, a_n\rangle$ which satisfies the condition \[ [I_{i-1}I^k : a_i] \cap I^k=I_{i-1}I^{k-1} \] for every $i \in\{1, \dots, n\}$ and every $k\geq1$. It generates an ideal of linear type. We characterize c-sequences in terms of the corresponding sequences in the Rees algebra of the ideal generated by the elements of the sequence. We then characterize monomial c-sequences of three terms.

Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1213-1221.

Dates
First available in Project Euclid: 18 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258554358

Digital Object Identifier
doi:10.1215/ijm/1258554358

Mathematical Reviews number (MathSciNet)
MR2595763

Zentralblatt MATH identifier
1182.13005

Subjects
Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13B25: Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 13A15: Ideals; multiplicative ideal theory 13C13: Other special types

Citation

Kulosman, Hamid. Monomial sequences of linear type. Illinois J. Math. 52 (2008), no. 4, 1213--1221. doi:10.1215/ijm/1258554358. https://projecteuclid.org/euclid.ijm/1258554358


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References

  • D. Costa, Sequences of linear type, J. Algebra 94 (1985), 256–263.
  • M. Fiorentini, On relative regular sequences, J. Algebra 18 (1971), 384–389.
  • W. Heinzer, A. Mirbagheri, L. J. Ratliff Jr. and K. Shah, Parametric decomposition of monomial ideals, II, J. Algebra 187 (1997), 120–149.
  • J. Herzog, A. Simis and W. Vasconcelos, Approximation complexes of blowing-up rings, J. Algebra 74 (1982), 466–493.
  • J. Herzog, A. Simis and W. Vasconcelos, Koszul homology and blowing-up rings, Commutative algebra (Trento, 1981), Lecture Notes in Pure and Applied Mathematics, vol. 84, Dekker, New York, 1983, pp. 79–169.
  • C. Huneke, On the symmetric and Rees algebra of an ideal generated by a d-sequence, J. Algebra 62 (1980), 268–275.
  • C. Huneke, Symbolic powers of prime ideals and special graded algebras, Comm. Algebra 9 (1981), 339–366.
  • C. Huneke, The theory of d-sequences and powers of ideals, Adv. Math. 46 (1982), 249–279.
  • M. Kühl, On the symmmetric algebra of an ideal, Manuscripta Math. 37 (1982), 49–60.
  • H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1989.
  • Z. Tang, On certain monomial sequences, J. Algebra 282 (2004), 831–842.
  • G. Valla, On the symmetric and Rees algebras of an ideal, Manuscripta Math. 30 (1980), 239–255.