The Grothendieck Duality Theorem via Bousfield’s Techniques and Brown Representability

Grothendieck proved that if f : X −→ Y is a proper morphism ofnice schemes, then Rf∗ has a right adjoint, which is given as tensor productwith the relative canonical bundle. The original proof was by patching localdata. Deligne proved the existence of the adjoint by a global argument, andVerdier showed that this global adjoint may be computed locally.In this article we show that the existence of the adjoint is an immediateconsequence of Brown’s representability theorem. 1It follows almost as imme-diately, by “smashing” arguments, that the adjoint is given by tensor productwith a dualising complex. Verdier’s base change theorem is an easy conse-quence. Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use