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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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Additional Information

Amnon Neeman
Affiliation: Department of Mathematics University of Virginia Charlottesville, Virginia 22903
MR Author ID: 129970
ORCID: 0000-0001-6225-3415

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