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On Grothendieck's local duality theorem

Published online by Cambridge University Press:  24 October 2008

M. H. Bijan-Zadeh  and
R. Y. Sharp
M. H. Bijan-Zadeh
Affiliation:
University of Sheffield
R. Y. Sharp
Affiliation:
University of Sheffield
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Extract

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 3 , May 1979 , pp. 431 - 437
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Hall, J. E.Fundamental dualizing complexes for commutative Noetherian rings. Quart. J. Math. Oxford 30 (1979) (To appear.)Google Scholar
(2)Hartshorne, R.Residues and duality (Springer, Lecture Notes in Mathematics, no. 20, 1966).CrossRefGoogle Scholar
(3)Macdonald, I. G. and Sharp, R. Y.An elementary proof of the non-vanishing of certain local cohomology modules. Quart. J. Math. Oxford 23, (1972), 197–204.Google Scholar
(4)Matlis, E.Injective modules over Noetherian rings. Pacific J. Math. 8, (1958), 511–528.Google Scholar
(5)Northcott, D. G.An introduction to homological algebra (Cambridge University Press, 1960).Google Scholar
(6)Peskine, C. and Szpiro, L.Dimension projective finie et cohomologie locale. Inst. Hautes Études Sci. Publ. Math. 42, (1973), 323–395.CrossRefGoogle Scholar
(7)Roberts, P.Two applications of dualizing complexes over local rings. Ann. Scient. Éc.Norm. Sup. 9, (1976), 103–106.CrossRefGoogle Scholar
(8)Schenzel, P.Dualizing complexes and systems of parameters. J. Algebra (To appear.)Google Scholar
(9)Sharp, R. Y.Local cohomology theory in commutative algebra. Quart. J. Math. Oxford 21, (1970), 425–434.CrossRefGoogle Scholar
(10)Sharp, R. Y.Some results on the vanishing of local cohomology modules. Proc. London Math. Soc. 30, (1975), 177–195.Google Scholar
(11)Sharp, R. Y.Dualizing complexes for commutative Noetherian rings. Math. Proc. Cambridge Philos. Soc. 78, (1975), 369–386.Google Scholar
(12)Sharp, R. Y.A commutative Noetherian ring which possesses a dualizing complex is acceptable. Math. Proc. Cambridge Philos. Soc. 82, (1977), 197–213.Google Scholar
(13)Sharpe, D. W. and Vámos, P.Injective modules (Cambridge University Press, 1972).Google Scholar