Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-30T20:24:23.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Some homological properties of complete modules

Published online by Cambridge University Press:  24 October 2008

Anne-Marie Simon
Anne-Marie Simon
Affiliation:
Service d'algèbre C.P. 211, Université Libre de Bruxelles, Campus Plaine, Boulevard du Triomphe, B-1050 Bruxelles, Belgium
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper A is a commutative noetherian ring, a an ideal of A and the A- modules are given the a-adic topology.

It is a general feeling that completeness is a kind of finiteness condition. We make precise that feeling and, after a result concerning the homology of a complex of complete modules which can be used in place of Nakayama's Lemma, we establish analogies between complete modules and finitely generated ones, with respect to flat dimension, injective dimension, Bass numbers and the Koszul complex. This is particularly clear in the local case, where we have also some partial information on the support of a complete module. With respect to dimension however, the analogy fails, as shown by an example.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 2 , September 1990 , pp. 231 - 246
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Atiyah, M. F. and MacDonald, I. G.. Introduction to Commutative Algebra (Addison-Wesley, 1969).Google Scholar
[2]Bartijn, J.. Flatness, completions, regular sequences, un ménage à trois. Thesis, Utrecht (1985).Google Scholar
[3]Bartijn, J. and Strooker, J. R.. Modifications monomiales. In Séminaire d'Algèbra P. Dubreil et M. P. Malliavin. Lecture Notes in Math. vol. 1029 (Springer-Verlag, 1983), pp. 192–217.CrossRefGoogle Scholar
[4]Bass, H.. On the ubiguity of Gorenstein rings. Math. Z. 82 (1963), 8–28.Google Scholar
[5]Bourbaki, N.. Algèbre Commutative (Hermann, 1961).Google Scholar
[6]Bourbaki, N.. Algèbre, chapitre 10, Algèbre Homologique (Masson, 1981).Google Scholar
[7]Cartan, H. and Eilenberg, S.. Homological Algebra. Princeton Math. Ser. no. 19 (Princeton University Press, 1956).Google Scholar
[8]Copson, E. T.. Metric Spaces. Cambridge Tracts in Math. and Math. Physics no. 57 (Cambridge University Press, 1968).Google Scholar
[9]Foxby, H. B.. Bounded complexes of flat modules. J. Pure Appl. Algebra 14 (1979), 149–172.Google Scholar
[10]Hartshorne, R.. Residues and Duality. Lecture Notes in Math. vol. 20 (Springer-Verlag, 1966).CrossRefGoogle Scholar
[11]Hochster, M.. Topics in the Homological Theory of Modules over Commutative Rings. CBMS Regional Conf. Ser. in Math. no. 24 (American Mathematical Society, 1975).Google Scholar
[12]Matlis, E.. The Koszul complex and duality. Comm. Algebra 1 (1974), 87–144.Google Scholar
[13]Matlis, E.. The higher properties of R-sequences. J. Algebra 50 (1978), 77–112.CrossRefGoogle Scholar
[14]Matsumura, H.. Commutative Ring Theory. Cambridge Studies in Advanced Math. (Cambridge University Press, 1986).Google Scholar
[15]Peskine, C. et al. Szipiro, L.. Dimension projective finite et cohomologie locale. Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119.Google Scholar
[16]Raynaud, M. et al. Gruson, L.. Critères de platitude et de projectivité. Invent. Math. 13 (1971), 1–89.Google Scholar
[17]Roberts, P.. Homological Invariants of Modules over Commutative Rings. (Presses de l'Université de Montréal, 1980).Google Scholar
[18]Roberts, P.. Le théorème d'intersection. CR. Acad. Sci. Paris Sér. I 304 (1987), 177–180.Google Scholar
[19]Sharp, R. Y. and Vámos, P.. Baire's category theorem and prime avoidance in complete local rings. Arch. Math. (Basel) 44 (1985), 243–248.Google Scholar
[20]Sharp, R. Y.. Cohen–Macaulay properties for balanced big Cohen–Macaulay modules. Math. Proc. Cambridge Philos. Soc. 90 (1981), 229–238.CrossRefGoogle Scholar
[21]Zarzuela, S.. Systems of parameters for non finitely generated modules and Big Cohen–Macaulay modules. Mathematika 35 (1988), 207–215.Google Scholar