Coherent Duality - Adjoint Functor Point of View

Adjoint Functor Point of View

While Serre duality uses a line bundle or invertible sheaf as a dualizing sheaf, the general theory (it turns out) cannot be quite so simple. (More precisely, it can, but at the cost of the Gorenstein ring condition.) In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a right adjoint functor f !, called twisted or exceptional inverse image functor, to a higher direct image with compact support functor Rf_!.

Higher direct images are a sheafified form of sheaf cohomology in this case with proper (compact) support; they are bundled up into a single functor by means of the derived category formulation of homological algebra (introduced with this case in mind). In case f is proper Rf_! = Rf_∗ is itself a right adjoint, to the inverse image functor f ∗. The existence theorem for the twisted inverse image is the name given to the proof of the existence for what would be the counit for the comonad of the sought-for adjunction, namely a natural transformation

Rf_!f ! → id,

which is denoted by Tr_f (Hartshorne) or ∫_f (Verdier). It is the aspect of the theory closest to the classical meaning, as the notation suggests, that duality is defined by integration.

To be more precise, f ! exists as an exact functor from a derived category of quasi-coherent sheaves on Y, to the analogous category on X, whenever

f: X → Y

is a proper or quasi projective morphism of noetherian schemes, of finite Krull dimension. From this the rest of the theory can be derived: dualizing complexes pull back via f !, the Grothendieck residue symbol, the dualizing sheaf in the Cohen-Macaulay case.

In order to get a statement in more classical language, but still wider than Serre duality, Hartshorne (Algebraic Geometry) uses the Ext functor of sheaves; this is a kind of stepping stone to the derived category.

The classical statement of Grothendieck duality for a projective or proper morphism of noetherian schemes of finite dimension, found in Hartshorne (Residues and duality) is the following quasi-isomorphism

Rf_∗RHom_X(F⋅, f ! G⋅) → R Hom_Y(Rf_∗ F⋅, G⋅)

for F⋅ a bounded above complex of O_X-modules with quasi-coherent cohomology and G⋅ a bounded below complex of O_Y-modules with coherent cohomology. Here the Hom's are the sheaf of homomorphisms.