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

DOI:
https://doi.org/10.1090/S0894-0347-96-00174-9

MathSciNet review:
1308405

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

*Théorie des intersections et théorème de Riemann-Roch*, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR**0354655**- Marcel Bökstedt and Amnon Neeman,
*Homotopy limits in triangulated categories*, Compositio Math.**86**(1993), no. 2, 209–234. MR**1214458** - A. K. Bousfield,
*The localization of spaces with respect to homology*, Topology**14**(1975), 133–150. MR**380779**, DOI 10.1016/0040-9383(75)90023-3 - A. K. Bousfield,
*The localization of spectra with respect to homology*, Topology**18**(1979), no. 4, 257–281. MR**551009**, DOI 10.1016/0040-9383(79)90018-1 - A. K. Bousfield,
*The Boolean algebra of spectra*, Comment. Math. Helv.**54**(1979), no. 3, 368–377. MR**543337**, DOI 10.1007/BF02566281 - Edgar H. Brown Jr.,
*Abstract homotopy theory*, Trans. Amer. Math. Soc.**119**(1965), 79–85. MR**182970**, DOI 10.1090/S0002-9947-1965-0182970-6 - Robin Hartshorne,
*Residues and duality*, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR**0222093**, DOI 10.1007/BFb0080482 - Robin Hartshorne,
*Residues and duality*, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR**0222093**, DOI 10.1007/BFb0080482 - Reinhardt Kiehl,
*Ein “Descente”-Lemma und Grothendiecks Projektionssatz für nichtnoethersche Schemata*, Math. Ann.**198**(1972), 287–316 (German). MR**382280**, DOI 10.1007/BF01419561 - J. Lipman,
*Notes on derived categories and derived functors*, preprint. - —,
*Non–noetherian Grothendieck duality*, preprint. - Amnon Neeman,
*Stable homotopy as a triangulated functor*, Invent. Math.**109**(1992), no. 1, 17–40. MR**1168363**, DOI 10.1007/BF01232016 - Amnon Neeman,
*The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel*, Ann. Sci. École Norm. Sup. (4)**25**(1992), no. 5, 547–566. MR**1191736**, DOI 10.24033/asens.1659 - A. Neeman and V.Voevodsky,
*Triangulated categories*, incomplete preprint. - N. Spaltenstein,
*Resolutions of unbounded complexes*, Compositio Math.**65**(1988), no. 2, 121–154. MR**932640** - R. W. Thomason and Thomas Trobaugh,
*Higher algebraic $K$-theory of schemes and of derived categories*, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR**1106918**, DOI 10.1007/978-0-8176-4576-2_{1}0 - Jean-Louis Verdier,
*Base change for twisted inverse image of coherent sheaves*, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, pp. 393–408. MR**0274464**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
14F05,
55P42

Retrieve articles in all journals with MSC (1991): 14F05, 55P42

Additional Information

**Amnon Neeman**

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

MR Author ID:
129970

ORCID:
0000-0001-6225-3415

Email:
an3r@virginia.edu

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