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Some Effective Methods in the Openness of Loci for Cohen-Macaulay and Gorenstein Properties

  • Fabio Rossi
  • Walter Spangher
Part of the Progress in Mathematics book series (PM, volume 94)

Abstract

In this paper we present some effective methods in order to compute the P-locus of a quotient A of a polynomial ring S = k[X], where P is any of the following properties: “Cohen-Macaulay (C. M.)”, “Gorenstein”, “C. M. type ≤ r”, “Complete intersection (C. I.)”. It is well known that the previous loci are open being A an excellent ring (see, for instance, [GM]).

Keywords

Prime Ideal Local Ring Complete Intersection Polynomial Ring Regular Ring 
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References

  1. [A]
    L. L. Avramov, Flat morphisms of complete intersections, Soviet Math. Dokl. 16 (1975), 1413–1417.MATHGoogle Scholar
  2. [Ba]
    D. A. Bayer, The division algorithm and the Hilbert scheme, Ph. D. Thesis, Harvard (1982).Google Scholar
  3. [BSS]
    D. Bayer, Ma. Stillmann, Mi. Stillmann, “Macaulay user manual,” 1989.Google Scholar
  4. [Bo]
    N. Bourbaki, “Algébre,” Ch. IV-V, Hermann, Paris, 1959.MATHGoogle Scholar
  5. [GM]
    S. Greco, M. G. Marinari, Nagata’s criterion and openness of loci for Gorenstein and complete intersection, Math. Z. 160 (1978), 207–216.MathSciNetMATHCrossRefGoogle Scholar
  6. [GNN]
    R. Gilmer, B. Nashier, W. Nichols, On the heights of prime ideals under integral extensions, Arch. Math. 52 (1989), 47–52.MathSciNetMATHCrossRefGoogle Scholar
  7. [HK]
    J. Herzog, E. Kunz, “Der kanonische Modul eines Cohen-Macaulay Ringes,” Lecure Notes in Math. 238, Springer, 1971.Google Scholar
  8. [K]
    I. Kaplansky, “Commutative rings,” Allyn and Bacon, 1970.Google Scholar
  9. [M1]
    H. Matsumura, “Commutative algebra,” (2 edn.), Benjamin, 1980.Google Scholar
  10. [M2]
    H. Matsumura, “Commutative ring theory,” Cambridge U. Press, 1986.Google Scholar
  11. [MM]
    E. Mayr, A. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. Math. 46 (1982), 305–329.MathSciNetMATHCrossRefGoogle Scholar
  12. [Mu]
    D. Mumford, “Introduction to algebraic geometry,” preliminary version of first 3 chapters, Harvard Univ. mimeogr.Google Scholar
  13. [No]
    D.G. Northcott, “Finite free resolutions,” Cambridge U. Press, 1976.Google Scholar
  14. [Sa]
    J. D. Sally, “Numbers of generators of ideals in local rings,” Lect. Notes Pure Appl. Math. 35, Marcel Dekker, 1978.Google Scholar
  15. [S]
    P. Samuel, “Méthodes d’algèbre abstraite en géométrie algébrique,” Springer, 1955.Google Scholar
  16. [Se]
    A. Seidenberg, Contractions in algebra, Trans. Amer. Math. Soc 197 (1974), 273–313.MathSciNetMATHCrossRefGoogle Scholar
  17. [Sh]
    I. R. Shafarevich, “Basic algebraic geometry,” Springer, 1974.Google Scholar
  18. [Sp]
    D. Spear, A constructive approach to commutative ring theory, in “Proc. 1977 Macsyma User’s conference,” 1977, pp. 369–376.Google Scholar
  19. [Va]
    W. V. Vasconcelos, “Divisor theory in module categories,” North Holland, Amsterdam, 1974.MATHGoogle Scholar
  20. [ZS]
    O. Zariski, P. Samuel, “Commutative algebra,” Vol. II, Van Nostrand, New York — London, 1968.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Fabio Rossi
    • 1
  • Walter Spangher
    • 1
  1. 1.Dipartimento di Scienze MatematicheUniversità degli Studi di TriesteTriesteItalia

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