Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-09T06:56:44.908Z Has data issue: false hasContentIssue false

On Grothendieck's local duality theorem

Published online by Cambridge University Press:  24 October 2008

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

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
Copyright
Copyright © Cambridge Philosophical Society 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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), 197204.Google Scholar
(4)Matlis, E.Injective modules over Noetherian rings. Pacific J. Math. 8, (1958), 511528.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), 323395.CrossRefGoogle Scholar
(7)Roberts, P.Two applications of dualizing complexes over local rings. Ann. Scient. Éc.Norm. Sup. 9, (1976), 103106.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), 425434.CrossRefGoogle Scholar
(10)Sharp, R. Y.Some results on the vanishing of local cohomology modules. Proc. London Math. Soc. 30, (1975), 177195.Google Scholar
(11)Sharp, R. Y.Dualizing complexes for commutative Noetherian rings. Math. Proc. Cambridge Philos. Soc. 78, (1975), 369386.Google Scholar
(12)Sharp, R. Y.A commutative Noetherian ring which possesses a dualizing complex is acceptable. Math. Proc. Cambridge Philos. Soc. 82, (1977), 197213.Google Scholar
(13)Sharpe, D. W. and Vámos, P.Injective modules (Cambridge University Press, 1972).Google Scholar