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

  • Jonathan S. Golan (a1) and Jacques Raynaud (a2)

Abstract

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.

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References

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[1]Albu, T. and Nastasescu, C., ‘Local cohomology and torsion theory. I’, Rev. Roumaine Math. Pures Appl. 26 (1981), 314.
[2]Albu, T. and Nastasescu, C., ‘Some aspects of non-noetherian local cohomology’, Comm. Algebra 8 (1980), 15391560.
[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).
[4]Beachy, J., ‘On the torsion theoretic support of a module’, Hokkaido Math. J. 6 (1977), 1627.
[5]Bijan-Zadeh, M. H., ‘Torsion theories and local cohomology over commutative noetherian rings’, J. London Math. Soc. (2) 19 (1979), 402410.
[6]Cahen, P.-J., ‘Commutative torsion theory’, Trans. Amer. Math. Soc. 184 (1973), 7385.
[7]Dickson, S., ‘Direct decompositions of radicals’, Proceedings of the conference on categorical algebra, La Jolla, 1965 (Springer-Verlag, Berlin, 1966).
[8]Golan, J. S., Localization of noncommutative rings (Marcel Dekker, New York, 1975).
[9]Golan, J. S., ‘A Krull-like dimension for noncommutative rings’, Israel J. Math. 19 (1974), 297304.
[10]Golan, J. S., ‘A characterization of left semiartinian rings’, Bull. Austral. Math. Soc. 11 (1974), 425428.
[11]Golan, J. S., Decomposition and dimension in module categories (Marcel Dekker, New York, 1977).
[12]Golan, J. S., Structure sheaves over a noncommutative ring (Marcel Dekker, New York, 1980).
[13]Golan, J. S. and Raynaud, J., ‘Dimension de Gabriel et TTK-dimension de modules’, C. R. Acad. Sci. Paris 278 (1974), A1603–A1606.
[14]Kato, T., ‘Rings of U-dominant dimension ≥ 1’, Tôhoku Math. J. 21 (1969), 321327.
[15]Maury, G. and Raynaud, J., Ordres maximaux au sens de K. Asano (Lecture Notes in Mathematics 808, Springer-Verlag, Berlin, 1980).
[16]Miller, R. W., ‘Finitely generated projective modules and TTF classes’, Pacific J. Math. 64 (1976), 505515.
[17]Raynaud, J., Localisations et spectres d'anneaux (Docteur d'Etat thesis, Université Claude-Bernard, Lyon, 1976).
[18]Sharp, R. Y., ‘Local cohomology theory in commutative algebra’, Quart. J. Math. Oxford Ser. (2) 21 (1970), 425434.
[19]Stenström, B., Rings of quotients (Springer-Verlag, Berlin, 1975).
[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).
[21]Suominen, K., ‘Localization of sheaves and Cousin complexes’, Acta Mathematica 131 (1973), 2741.
[22]Van Oystaeyen, F.Note on the torsion theory at a prime ideal of a left noetherian ring’, J. Pure Appl. Algebra 6 (1975), 297304.
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