While Serre duality uses a line bundle or invertible sheaf as a dualizing sheaf, the general theory cannot be quite so simple. In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a right adjoint functorf !, 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 support; they are bundled up into a single functor by means of the derived category formulation of homological algebra. 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 Trf or ∫f. 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 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 is the following quasi-isomorphism for F⋅ a bounded above complex of OX-modules with quasi-coherent cohomology and G⋅ a bounded below complex of OY-modules with coherent cohomology. Here the Hom's are the sheaf of homomorphisms.
Construction of the pseudofunctor using rigid dualizing complexes
Over the years, several approaches for constructing the 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 : Let k be a commutative ring, and let A be a commutative k-algebra. There is a functor which takes a cochain complexM to an object in the derived category over A. Asumming A is noetherian, a rigid dualizing complex over A relative to k is by definition a pair where R is a dualizing complex over A which has finite flat dimension over k, and where is an isomorphism in the derived category D. If such a rigid dualizing complex exists, then it is unique in a strong sense. Assuming A is a localization of a finite typek-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. Once one establishes a global existence of a rigid dualizing complex, given a map of schemes over k, one can define, where for a scheme X, we set.
Consider the projective variety We can compute using a resolution by locally free sheaves. This is given by the complex Since we have that This is the complex
