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The Grothendieck duality theorem
via Bousfield's techniques
and Brown representability

Author: Amnon Neeman
Journal: J. Amer. Math. Soc. 9 (1996), 205-236
MSC (1991): Primary 14F05, 55P42
MathSciNet review: 1308405
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Abstract: Grothendieck proved that if $f:X\longrightarrow Y$ is a proper morphism of nice schemes, then $Rf_*$ has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by patching local data. Deligne proved the existence of the adjoint by a global argument, and Verdier showed that this global adjoint may be computed locally. In this article we show that the existence of the adjoint is an immediate consequence of Brown's representability theorem. 1It follows almost as immediately, by ``smashing'' arguments, that the adjoint is given by tensor product with a dualising complex. Verdier's base change theorem is an easy consequence.

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  • 1. P. Berthelot, A. Grothendieck and L. Illusie, SGA 6: Théorie des intersections et théorème de Riemann--Roch, Lecture Notes in Math., vol. 225, Springer--Verlag, Heidelberg, 1971. MR 50:7133
  • 2. M. Bökstedt and A. Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), 209--234. MR 94f:18008
  • 3. A.K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133--150. MR 52:1676
  • 4. ------, The localization of spectra with respect to homology, Topology 18 (1979), 257--281. MR 80m:55006
  • 5. ------, The Boolean algebra of spectra, Comm. Math. Helv. 54 (1979), 368--377. MR 81a:55015
  • 6. E.H. Brown, Abstract homotopy theory, Trans. Amer. Math. Soc. 119 (1965), 79--85. MR 32:452
  • 7. P. Deligne, Cohomology à support propre en construction du foncteur $f^!$, Appendix to: Residues and Duality, Lecture Notes in Math., vol. 20, Springer-Verlag, Heidelberg, 1966, pp. 404--421. MR 36:5145
  • 8. R. Hartshorne, Residues and duality, Lecture Notes in Math., vol. 20, Springer--Verlag, Heidelberg, 1966. MR 36:5145
  • 9. R. Kiehl, Ein ``Descente''--Lemma und Grothendiecks Projektionssatz für nicht--noethersche Schemata, Math Ann. 198 (1972), 287--316. MR 52:3165
  • 10. J. Lipman, Notes on derived categories and derived functors, preprint.
  • 11. ------, Non--noetherian Grothendieck duality, preprint.
  • 12. A. Neeman, Stable homotopy as a triangulated functor, Invent. Math. 109 (1992), 17--40. MR 93j:55022
  • 13. ------, The connection between the K--theory localisation theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. 25 (1992), 547--566. MR 93k:18015
  • 14. A. Neeman and V.Voevodsky, Triangulated categories, incomplete preprint.
  • 15. N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), 121--154. MR 89m:18013
  • 16. R. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift ( a collection of papers to honor Grothendieck's 60'th birthday), Vol. 3, Birkhäuser, 1990, pp. 247--435. MR 92f:19001
  • 17. J.--L. Verdier, Base change for twisted inverse images of coherent sheaves, Collection: Algebraic Geometry (Internat. Colloq.), Tata Inst. Fund. Res., Bombay, 1968, pp. 393--408. MR 43:227

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Additional Information

Amnon Neeman
Affiliation: Department of Mathematics University of Virginia Charlottesville, Virginia 22903

Received by editor(s): January 24, 1994
Received by editor(s) in revised form: December 2, 1994
Additional Notes: The author’s research was partly supported by NSF grant DMS–9204940
Article copyright: © Copyright 1996 American Mathematical Society