Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T01:34:52.579Z Has data issue: false hasContentIssue false
  • English
  • Français

The co-localization of an Artinian module

Published online by Cambridge University Press:  20 January 2009

Leif Melkersson  and
Peter Schenzel
Leif Melkersson
Affiliation:
University of LundDepartment of MathematicsBox 118, S-221 00 Lund, Sweden
Peter Schenzel
Affiliation:
Max-Planck-GesellschaftAG Algebraische Geometrie und ZahlentheorieJägerstr. 10-11D-10 117 Berlin, Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a multiplicative set S of a commutative ring R we define the co-localization functor HomR(Rs,⋅). It is a functor on the category of R-modules to the category of Rs-modules. It is shown to be exact on the category of Artinian R-modules. While the co-localization of an Artinian module is almost never an Artinian Rs-module it inherits many good properties of A, e.g. it has a secondary representation. The construction is applied to the dual of a result of Bourbaki, a description of asymptotic prime divisors and the co-support of an Artinian module.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 1 , February 1995 , pp. 121 - 131
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Bourbaki, N., Théorie des ensembles. Chap 3. sec. ed. (Hermann, Paris, 1963).Google Scholar
2.Bourbaki, N., Algébre commutative. Chap 3. et 4. (Hermann, Paris, 1967).Google Scholar
3.Kirby, D., Coprimary decomposition of Artinian modules, J. London Math. Soc. (2) 6 (1973), 571–576.CrossRefGoogle Scholar
4.Lafon, J. P., Algèbre commutative: Langages géométrique et algébrique (Hermann, Paris, 1977).Google Scholar
5.Macdonald, I. G., Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973), 23–43.Google Scholar
6.Matsumura, H., Commutative ring theory (Cambridge University Press, Cambridge, 1986).Google Scholar
7.Melkersson, L., On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Cambridge Philos. Soc. 107 (1990), 267–271.CrossRefGoogle Scholar
8.Northcott, D. G., Generalized Koszul complexes and Artinian modules, Quart. J. Math. Oxford (2) 23 (1972), 289–297.CrossRefGoogle Scholar
9.Rotman, J., An introduction to homological algebra (Academic Press, New York, San Francisco, London, 1979).Google Scholar
10.Sharp, R. Y., Asymptotic behaviour of certain sets of attached prime ideals, J. London Math. Soc. (2) 34 (1986), 212–218.CrossRefGoogle Scholar
11.Sharp, R. Y., A method for the study of Artinian modules, with an application to asymptotic behavior, in: Commutative Algebra (Math. Sciences Research Inst. Publ., No. 15, Springer-Verlag, 1989), 443–465.CrossRefGoogle Scholar
12.Taherizadeh, A.-J., On asymptotic values of certain sets of attached prime ideals, Glasgow Math. J. 30 (1988), 293–300.CrossRefGoogle Scholar
13.Zöschinger, H., Über koassoziierte Primideale, Math. Scand. 63 (1988), 196–211.CrossRefGoogle Scholar