namely a natural transformation , which is denoted by (Hartshorne) or (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, exists as an exact functor from a derived category of quasi-coherent sheaves on , to the analogous category on , 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 , 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 for a bounded above complex of -modules with quasi-coherent cohomology and a bounded below complex of -modules with coherent cohomology. Here the 's are sheaves of homomorphisms. Construction of the f! 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 (Shukla cohomology): Let be a commutative ring, and let be a commutative algebra. There is a functor which takes a cochain complex to an object in the derived category over . Asumming is noetherian, a rigid dualizing complex over relative to is by definition a pair where is a dualizing complex over which has finite flat dimension over , and where is an isomorphism in the derived category . If such a rigid dualizing complex exists, then it is unique in a strong sense. Assuming is a localization of a finite type -algebra, existence of a rigid dualizing complex over relative to was first proved by Yekutieli and Zhang assuming is a regular noetherian ring of finite Krull dimension, and by Avramov, Iyengar and Lipman assuming is a Gorenstein ring of finite Krull dimension and is of finite flat dimension over . If is a scheme of finite type over ,