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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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The Grothendieck duality theorem via Bousfield’s techniques and Brown representability
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by Amnon Neeman PDF
J. Amer. Math. Soc. 9 (1996), 205-236 Request permission


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
  • Email:
  • 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
  • © Copyright 1996 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 9 (1996), 205-236
  • MSC (1991): Primary 14F05, 55P42
  • DOI:
  • MathSciNet review: 1308405