Coherent duality

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

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

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 f : X → Y of noetherian schemes of finite dimension, found in Hartshorne (Residues and duality) is the following quasi-isomorphism

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.

Construction of the f ! {displaystyle f^{!}} pseudofunctor using rigid dualizing complexes

Over the years, several approaches for constructing the f ! pseudofunctor emerged. One quite recent successful approach is based on the notion of a rigid dualizing complex. This notion was first defined by Van den Bergh in a noncommutative context. The construction is based on a variant of derived Hochschild cohomology (Shukla cohomology): Let k be a commutative ring, and let A be a commutative k-algebra. There is a functor R H o m A ⊗ k L A ( A , M ⊗ k L M ) which takes a cochain complex M to an object R H o m A ⊗ k L A ( A , M ⊗ k L M ) in the derived category over A.

Asumming A is noetherian, a rigid dualizing complex over A relative to k is by definition a pair ( R , ρ ) where R is a dualizing complex over A which has finite flat dimension over k, and where ρ : R → R H o m A ⊗ k L A ( A , R ⊗ k L R ) is an isomorphism in the derived category D(A). If such a rigid dualizing complex exists, then it is unique in a strong sense.

Assuming A is a localization of a finite type k-algebra, existence of a rigid dualizing complex over A relative to k was first proved by Yekutieli and Zhang assuming k is a regular noetherian ring of finite Krull dimension, and by Avramov, Iyengar and Lipman assuming k is a Gorenstein ring of finite Krull dimension and A is of finite flat dimension over A.

If X is a scheme of finite type over k, one can glue the rigid dualizing complexes that its affine pieces have, and obtain a rigid dualizing complex R X . Once one establishes a global existence of a rigid dualizing complex, given a map f : X → Y of schemes over k, one can define f ! := D X ∘ L f ∗ D Y , where for a scheme X, we set D X := R H o m O X ( − , R X ) .