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
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28.
D. Buchsbaum and D. Eisenbud, What makes a complex exact?, to appear in J. of Alg.
_____, Remarks on ideals and resolutions, preprint.
W. L. Chow, On unmixedness theorem, Amer. J. Math. 86 (1964), 799–822.
D. Ferrand and M. Raynaud, Fibres formelles d'un anneau local Noethérien, Annales Sci. de l'École Normale Supérieure 3 (1970), 295–312.
A. Grothendieck (with J. Dieudonné), Éléments de géométrie algébrique, IV. (Seconde partie), Publications mathématiques de l'I. H. E. S. no 24, Paris, 1965.
A. Grothendieck (notes by R. Hartshorne), Local cohomology, Springer-Verlag Lecture Notes in Mathematics No. 41, 1967.
M. Hochster, Grade-sensitive modules and perfect modules, preprint.
_____, Contracted ideals from integral extensions, preprint.
I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, 1971.
_____, Topics in commutative ring theory, I. and III., duplicated notes.
G. Levin and W. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math. 25 (1968), 315–323.
M. Nagata, Local rings, Interscience Tracts 13, New York, 1962.
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.
D. Rees, The grade of an ideal or module, Proc. of the Cambridge Philosophical Society 53 (1957), 28–42.
J. P. Serre, Algèbre locale. Multiplicités. Springer-Verlag Lecture Notes in Mathematics No. 11, 1965.
R. Y. Sharp, Application of dualizing complexes to finitely generated modules of finite injective dimension, preprint.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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
