Advertisement

Mathematische Annalen

, Volume 331, Issue 3, pp 487–503 | Cite as

A formula for the core of an ideal

  • Claudia Polini
  • Bernd UlrichEmail author
Article

Abstract.

The core of an ideal is the intersection of all its reductions. For large classes of ideals I we explicitly describe the core as a colon ideal of a power of a single reduction and a power of I.

Keywords

Large Classis Single Reduction Colon Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aberbach, I.M., Huneke, C.: A theorem of Briançon-Skoda type for regular local rings containing a field. Proc. Amer. Math. Soc. 124, 707–713 (1996)CrossRefGoogle Scholar
  2. 2.
    Corso, A., Polini, C., Ulrich, B.: The structure of the core of ideals. Math. Ann. 321, 89–105 (2001)Google Scholar
  3. 3.
    Corso, A., Polini, C., Ulrich, B.: Core and residual intersections of ideals. Trans. Amer. Math. Soc. 354, 2579–2594 (2002)CrossRefGoogle Scholar
  4. 4.
    Corso, A., Polini, C., Ulrich, B.: The core of projective dimension one modules. Manuscripta Math. 111, 427–433 (2003)CrossRefGoogle Scholar
  5. 5.
    Heinzer, W., Johnston, B., Lantz, D.: First coefficient domains and ideals of reduction number one. Comm. Algebra 21, 3797–3827 (1993)Google Scholar
  6. 6.
    Herzog, J., Simis, A., Vasconcelos, W.V.: Koszul homology and blowing-up rings. In: Commutative Algebra, Proceedings: Trento 1981, Greco/Valla (eds.), Lecture Notes in Pure and Applied Mathematics 84, Marcel Dekker, New York, 1983, pp. 79–169Google Scholar
  7. 7.
    Herzog, J., Vasconcelos, W.V., Villarreal, R.: Ideals with sliding depth. Nagoya Math. J. 99, 159–172 (1985)Google Scholar
  8. 8.
    Hochster, M., Huneke, C.: Indecomposable canonical modules and connectedness. Contemp. Math. 159, 197–208 (1994)Google Scholar
  9. 9.
    Huckaba, S.: Reduction numbers for ideals of higher analytic spread. Math. Proc. Camb. Phil. Soc. 102, 49–57 (1987)Google Scholar
  10. 10.
    Huneke, C.: Linkage and Koszul homology of ideals. Amer. J. Math. 104, 1043–1062 (1982)Google Scholar
  11. 11.
    Huneke, C., Swanson, I.: Cores of ideals in 2-dimensional regular local rings. Michigan Math. J. 42, 193–208 (1995)CrossRefGoogle Scholar
  12. 12.
    Huneke, C., Trung, N.V.: On the core of ideals. PreprintGoogle Scholar
  13. 13.
    Hyry, E.: Coefficient ideals and the Cohen-Macaulay property of Rees algebras. Proc. Amer. Math. Soc. 129, 1299–1308 (2001)CrossRefGoogle Scholar
  14. 14.
    Hyry, E., Smith, K.: On a non-vanishing conjecture of Kawamata and the core of an ideal. Amer. J. Math. 125, 1349–1410 (2003)Google Scholar
  15. 15.
    Lipman, J.: Stable ideals and Arf rings. Amer. J. Math. 93, 649–685 (1971)Google Scholar
  16. 16.
    Lipman, J.: Adjoints of ideals in regular local rings. Math. Res. Lett. 1, 739–755 (1994)Google Scholar
  17. 17.
    Polini, C., Ulrich, B.: Linkage and reduction numbers. Math. Ann. 310, 631–651 (1998)CrossRefGoogle Scholar
  18. 18.
    D. Rees, Sally, J.D.: General elements and joint reductions. Michigan Math. J. 35, 241–254 (1988)CrossRefGoogle Scholar
  19. 19.
    Trung, N.V.: Reduction exponent and degree bound for the defining equations of graded rings. Proc. Amer. Math. Soc. 101, 229–236 (1987)Google Scholar
  20. 20.
    Ulrich, B.: Artin-Nagata properties and reductions of ideals. Contemp. Math. 159, 373–400 (1994)Google Scholar
  21. 21.
    Ulrich, B.: Ideals having the expected reduction number. Amer. J. Math. 118, 17–38 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations