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

Abstract

Grothendieck proved that if f : X ⟶ Y f:X\longrightarrow Y is a proper morphism of nice schemes, then R f ∗ 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.