Skip to main content
SpringerLink
Log in

Cohen-Macaulay and Gorenstein property of Rees algebras of non-singular equimultiple prime ideals

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We give criteria for the Cohen-Macaulay and Gorenstein property of Rees algebras of height 2 non-singular equimultiple prime ideals in terms of explicite representations of the associated graded rings. As consequences, we show that in general, the Cohen-Macaulay resp. Gorenstein property of such Rees algebras does not imply the Cohen-Macaulay resp. Gorenstein property of the base ring and that these properties depend upon the characteristic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cavalieri, M., and Niesi, G.: On Serre's conditions in the form ring of an ideal. J. Math. Kyoto Univ.21, 537–546 (1981)

    MathSciNet  Google Scholar 

  2. Gräbe, H.-J.: The Gorenstein property depends on characteristic. Beitr. Algebra Geometrie17, 169–174 (1984)

    MATH  Google Scholar 

  3. Herrmann, M., and Ikeda, S.: On the Gorenstein property of Rees algebras. Manuscripta Math.59, 471–490 (1983)

    Article  MathSciNet  Google Scholar 

  4. Herrmann, M., Ikeda, S. and Orbanz, U.: Multiplicity and blowing-ups. Springer 1988

  5. Herrmann, M., Ribbe, J. and Trung, N.V.: Rees algebras of non-singular equimultiple prime ideals. Nagoya Math. J., to appear

  6. Herzog, J., Simis, A., and Vasconcelos, W.: On the canonical module of the Rees algebra and the associated graded ring of an ideal. J. Algebra105, 285–302 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hoa, L.T.: The Gorenstein property depends upon characteristic for affine semi-group rings. Arch. Math.56, 228–235 (1991)

    Article  MATH  Google Scholar 

  8. Hochster, M.: Criteria for the equality of ordinary and symbolic powers of primes. Math. Z.133, 53–65 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hochster, M., and Ratliff, L.J.: Five theorems on Macaulay rings. Pacific Math. J.44, 147–172 (1973)

    MATH  MathSciNet  Google Scholar 

  10. Huneke, C.: The theory of d-sequences and powers of ideals. Adv. Math.46, 249–279 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  11. Huneke, C.: On the associated graded ring of an ideal. Illinois J. Math.26, 121–137 (1982)

    MATH  MathSciNet  Google Scholar 

  12. Ikeda, S.: On the Gorensteiness of Rees algebras over local rings. Nagoya Math. J.102, 135–154. (1986)

    MATH  MathSciNet  Google Scholar 

  13. Ikeda, S., and Herrmann, M.: Remarks of lifting of Cohen-Macaulay property. Nagoya Math. J92, 121–132. (1983)

    MATH  MathSciNet  Google Scholar 

  14. Ikeda, S., and Trung, N.V.: When is the Rees algebra Cohen-Macaulay?. Comm. Algebra17 (12), 2893–2922 (1989)

    MATH  MathSciNet  Google Scholar 

  15. Northcott, D.G., and Rees, D.: Reductions of ideals in local rings. Math. Proc. Cambridge Phil. Soc.50, 145–158 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  16. Reisner, A.: Cohen-Macaulay quotient rings of polynomial rings. Adv. Math.21, 30–49 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  17. Trung, N.V.: Über die Übertragung der Ringeigenschaften zwischen R undR[u]/(F). Math. Nachr.92, 215–229 (1979)

    MATH  MathSciNet  Google Scholar 

  18. Trung, N.V.: From associated graded modules to blowing-ups of Cohen-Macaulay modules. J. Math. Kyoto Univ.24, 611–622 (1984)

    MATH  MathSciNet  Google Scholar 

  19. Trung, N.V.: Toward a theory of generalized Cohen-Macaulay modules. Nagoya Math. J.102, 229–234 (1987)

    Google Scholar 

  20. Trung, N.V.: Reduction exponent and degree bounds for the defining equations of graded rings. Proc. Amer. Math. Soc.101, 229–234 (1987)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Bò Hô, Box 631, Hanoi, Vietman

    Ngô Viêt Trung & Duong Quôc Viêt

Authors
  1. Ngô Viêt Trung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Duong Quôc Viêt
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to the memrory of Professor Lê Van Thiêm

Professor Lê Van Thiêm was the first directorof Hanoi Institute of Mathematics

Rights and permissions

Reprints and permissions

About this article

Cite this article

Trung, N.V., Viêt, D.Q. Cohen-Macaulay and Gorenstein property of Rees algebras of non-singular equimultiple prime ideals. Manuscripta Math 76, 147–167 (1992). https://doi.org/10.1007/BF02567752

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02567752

Keywords

Search

Navigation