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Cofiniteness and vanishing of local cohomology modules

Published online by Cambridge University Press:  24 October 2008

Craig Huneke  and
Jee Koh
Craig Huneke
Affiliation:
Department of Mathematics, Purdue University, W. Lafayette, IN 47907, U.S.A.
Jee Koh
Affiliation:
Department of Mathematics, University of Indiana, Bloomington, IN 47405, U.S.A.
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Extract

Let R be a noetherian local ring with maximal ideal m and residue field k. If M is a finitely generated R-module then the local cohomology modules are known to be Artinian. Grothendieck [3], exposé 13, 1·2 made the following conjecture:

If I is an ideal of R and M is a finitely generated R-module, then HomR (R/I, ) is finitely generated.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 3 , November 1991 , pp. 421 - 429
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Brodmann, M.. Einige Ergebnisse aus der Lokalen Kohomologietheorie und Ihre Anwendung. Osnabrück. Schrift. Math. no. 5 (University of Osnabrück, 1983).Google Scholar
[2]Grothendieck, A.. Local Cohomology, notes by R. Hart shorne. Lecture Notes in Math. vol. 862 (Springer-Verlag, 1966).Google Scholar
[3]Grothendieck, A.. Cohomologie Locale des Faisceaux et Theoremes de Lefshetz Locaux et Globaux (North Holland, 1969).Google Scholar
[4]Hartshorne, R.. Affine duality and cofiniteness. Invent. Math. 9 (1970), 145–164.CrossRefGoogle Scholar
[5]Hartshorne, R.. Cohomological dimension of algebraic varieties. Ann. of Math. 88 (1968), 403–450.CrossRefGoogle Scholar
[6]Hartshorne, R. and Speiser, R.. Local cohomological dimension in characteristic p. Ann. of Math. 105 (1977), 45–79.CrossRefGoogle Scholar
[7]Huneke, C. and Lyubeznik, G.. On the vanishing of local cohomology. Invent. Math. 102 (1990), 73–93.CrossRefGoogle Scholar
[8]Matlis, E.. Injective modules over noetherian rings. Pacific J. Math. 8 (1958), 511–528.CrossRefGoogle Scholar
[9]Ogus, A.. Local cohomological dimension of algebraic varieties. Ann. of Math. 98 (1973), 327–365.CrossRefGoogle Scholar
[10]Hartshorne, C. and Szpiro, L.. Dimension projective finie et cohomologie locale. Inst. Hautes Études Sci. Publ. Math. 42 (1973), 323–395.Google Scholar
[11]Rotman, J.. An Introduction to Homological Algebra (Academic Press, 1979).Google Scholar
[12]Serre, J. P.. Algebra Locale; Multiplicities. Lecture Notes in Math. vol. 11 (Springer-Verlag, 1965).Google Scholar