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.

Base change for semiorthogonal decompositions

“…: D qc (Y ) → D qc (X) the right adjoint functor of f * (usually it is referred to as the twisted pullback functor). It exists by [Nee96] (see also [KS06]). If the morphism f has finite Tor-dimension then [Nee96].…”

the stable derived category of a noetherian scheme

“…Let f * : Qcoh X → Qcoh Y denote the direct image functor. Note that the right derived functor Rf * : D(Qcoh X) → D(Qcoh Y) preserves coproducts [Nee96,Lemma 1.4]. Thus, Rf * and its right adjoint Grothendieck duality functor f !…”

“…For instance, an object in D(Qcoh X) is compact if and only if it is isomorphic to a perfect complex. It is well known that the derived category D(Qcoh X) is compactly generated, that is, there is a set of compact objects which generate D(Qcoh X) [Nee96]. To formulate our main result, let us denote by K c (Inj X) and S c (Qcoh X) the full subcategories of compact objects in K(Inj X) and S(Qcoh X), respectively.…”

The telescope conjecture for algebraic stacks

“…Brown representability [, Theorem 4.1] says that for a compactly generated triangulated category a triangulated functor preserves small coproducts if and only if admits a right adjoint.…”