Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T07:21:02.494Z Has data issue: false hasContentIssue false
  • English
  • Français

Derived functors of the torsion functor and local cohomology of noncommutative rings

Part of: Abelian categories Commutative algebra: Homological methods Homological algebra

Published online by Cambridge University Press:  09 April 2009

Jonathan S. Golan  and
Jacques Raynaud
Jonathan S. Golan
Affiliation:
Department of MathematicsUniversity of Haifa31999 Haifa, Israel
Jacques Raynaud
Affiliation:
Départment de MathématiquesUniversité Claude-Bernard(Lyon I) 69622 Villeurbanne Cedex, France
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.

Let R be an associative ring which is not necessarily commutative. For any torsion theory τ on the category of left R-modules and for any nonnegative integer n we define and study the notion of the nth local cohomology functor with respect to τ. For suitably nice rings a bound for the nonvanishing of these functors is given in terms of the τ-dimension of the modules.

MSC classification

Secondary: 18E25: Derived functors and satellites 18G10: Resolutions; derived functors 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 18E40: Torsion theories, radicals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 2 , October 1983 , pp. 162 - 177
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Albu, T. and Nastasescu, C., ‘Local cohomology and torsion theory. I’, Rev. Roumaine Math. Pures Appl. 26 (1981), 3–14.Google Scholar
[2]Albu, T. and Nastasescu, C., ‘Some aspects of non-noetherian local cohomology’, Comm. Algebra 8 (1980), 1539–1560.CrossRefGoogle Scholar
[3]Barou, G., ‘Cohomologie locale des algèbres enveloppants d'algèbres de Lie nilpotentes’, Séminaire d' Algèbre Paul Dubreil, edited by Malliavin, M. (Lecture Notes in Mathematics 641, Springer-Verlag, Berlin, 1978).Google Scholar
[4]Beachy, J., ‘On the torsion theoretic support of a module’, Hokkaido Math. J. 6 (1977), 16–27.Google Scholar
[5]Bijan-Zadeh, M. H., ‘Torsion theories and local cohomology over commutative noetherian rings’, J. London Math. Soc. (2) 19 (1979), 402–410.CrossRefGoogle Scholar
[6]Cahen, P.-J., ‘Commutative torsion theory’, Trans. Amer. Math. Soc. 184 (1973), 73–85.Google Scholar
[7]Dickson, S., ‘Direct decompositions of radicals’, Proceedings of the conference on categorical algebra, La Jolla, 1965 (Springer-Verlag, Berlin, 1966).Google Scholar
[8]Golan, J. S., Localization of noncommutative rings (Marcel Dekker, New York, 1975).Google Scholar
[9]Golan, J. S., ‘A Krull-like dimension for noncommutative rings’, Israel J. Math. 19 (1974), 297–304.Google Scholar
[10]Golan, J. S., ‘A characterization of left semiartinian rings’, Bull. Austral. Math. Soc. 11 (1974), 425–428.CrossRefGoogle Scholar
[11]Golan, J. S., Decomposition and dimension in module categories (Marcel Dekker, New York, 1977).Google Scholar
[12]Golan, J. S., Structure sheaves over a noncommutative ring (Marcel Dekker, New York, 1980).Google Scholar
[13]Golan, J. S. and Raynaud, J., ‘Dimension de Gabriel et TTK-dimension de modules’, C. R. Acad. Sci. Paris 278 (1974), A1603–A1606.Google Scholar
[14]Kato, T., ‘Rings of U-dominant dimension ≥ 1’, Tôhoku Math. J. 21 (1969), 321–327.Google Scholar
[15]Maury, G. and Raynaud, J., Ordres maximaux au sens de K. Asano (Lecture Notes in Mathematics 808, Springer-Verlag, Berlin, 1980).Google Scholar
[16]Miller, R. W., ‘Finitely generated projective modules and TTF classes’, Pacific J. Math. 64 (1976), 505–515.Google Scholar
[17]Raynaud, J., Localisations et spectres d'anneaux (Docteur d'Etat thesis, Université Claude-Bernard, Lyon, 1976).Google Scholar
[18]Sharp, R. Y., ‘Local cohomology theory in commutative algebra’, Quart. J. Math. Oxford Ser. (2) 21 (1970), 425–434.Google Scholar
[19]Stenström, B., Rings of quotients (Springer-Verlag, Berlin, 1975).CrossRefGoogle Scholar
[20]Storrer, H., ‘Torsion theories and dominant dimension’, appended to Lambek, J., Torsion theories, additive semantics, and rings of quotients (Lecture Notes in Mathematics 177, Springer-Verlag, Berlin, 1971).Google Scholar
[21]Suominen, K., ‘Localization of sheaves and Cousin complexes’, Acta Mathematica 131 (1973), 27–41.CrossRefGoogle Scholar
[22]Van Oystaeyen, F. ‘Note on the torsion theory at a prime ideal of a left noetherian ring’, J. Pure Appl. Algebra 6 (1975), 297–304.Google Scholar