Skip to main content

Cohen-macaulay modules

Part of the Lecture Notes in Mathematics book series (LNM,volume 311)

Abstract

The object of this paper is to discuss the conjecture, which will be abbreviated (E), that every complete local ring of dimension n possesses a finitely generated module of depth n. It is noted that several conjectures which have been open for some time follow from (E), and the connection of (E) with Serre's conjecture on multiplicities over regular local rings is discussed. In fact Serre's conjecture is proved for dimension ≤ 4 using the ideas under consideration.

A number of proofs of (E) for the two-dimensional case are given, and some possible methods for handling the general case are discussed. One of these is proposed as particularly worthy of study and is applied to an interesting class of examples in dimension 3 to obtain modules of depth 3. These examples do not yield easily to other techniques.

Keywords

  • Exact Sequence
  • Local Ring
  • Betti Number
  • Projective Dimension
  • Ring Homomorphism

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.

This research was supported in part by National Science Foundation Grant GP-29224X.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. D. Buchsbaum and D. Eisenbud, What makes a complex exact?, to appear in J. of Alg.

    Google Scholar 

  3. _____, Remarks on ideals and resolutions, preprint.

    Google Scholar 

  4. W. L. Chow, On unmixedness theorem, Amer. J. Math. 86 (1964), 799–822.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. D. Ferrand and M. Raynaud, Fibres formelles d'un anneau local Noethérien, Annales Sci. de l'École Normale Supérieure 3 (1970), 295–312.

    MathSciNet  MATH  Google Scholar 

  6. A. Grothendieck (with J. Dieudonné), Éléments de géométrie algébrique, IV. (Seconde partie), Publications mathématiques de l'I. H. E. S. no 24, Paris, 1965.

    Google Scholar 

  7. A. Grothendieck (notes by R. Hartshorne), Local cohomology, Springer-Verlag Lecture Notes in Mathematics No. 41, 1967.

    Google Scholar 

  8. M. Hochster, Grade-sensitive modules and perfect modules, preprint.

    Google Scholar 

  9. _____, Contracted ideals from integral extensions, preprint.

    Google Scholar 

  10. I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, 1971.

    MATH  Google Scholar 

  11. _____, Topics in commutative ring theory, I. and III., duplicated notes.

    Google Scholar 

  12. G. Levin and W. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math. 25 (1968), 315–323.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. M. Nagata, Local rings, Interscience Tracts 13, New York, 1962.

    Google Scholar 

  14. C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Thesis (Orsay, Serie A, No d'Ordre 781), to appear in Publ. I. H. E. S.

    Google Scholar 

  15. D. Rees, The grade of an ideal or module, Proc. of the Cambridge Philosophical Society 53 (1957), 28–42.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. J. P. Serre, Algèbre locale. Multiplicités. Springer-Verlag Lecture Notes in Mathematics No. 11, 1965.

    Google Scholar 

  17. R. Y. Sharp, Application of dualizing complexes to finitely generated modules of finite injective dimension, preprint.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1973 Springer-Verlag

About this paper

Cite this paper

Hochster, M. (1973). Cohen-macaulay modules. In: Brewer, J.W., Rutter, E.A. (eds) Conference on Commutative Algebra. Lecture Notes in Mathematics, vol 311. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0068925

Download citation

  • DOI: https://doi.org/10.1007/BFb0068925

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06140-3

  • Online ISBN: 978-3-540-38340-6

  • eBook Packages: Springer Book Archive