Coherent duality

In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory.
The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent. Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism; Serre duality was recovered as the case of the morphism of a non-singular projective variety (or complete variety) to a point. The resulting theory is now sometimes called Serre–Grothendieck–Verdier duality, and is a basic tool in algebraic geometry. A treatment of this theory, "Residues and Duality" (1966) by Robin Hartshorne, became a reference. One concrete spin-off was the Grothendieck residue.
To go beyond proper morphisms, as for the versions of Poincaré duality that are not for closed manifolds, requires some version of the "compact support" concept. This was addressed in SGA2 in terms of local cohomology, and Grothendieck local duality; and subsequently. The Greenlees–May duality, first formulated in 1976 by Ralf Strebel and in 1978 by Eben Matlis, is part of the continuing consideration of this area.
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 imposing the Gorenstein ring condition.) In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a right adjoint functor formula_1, called "twisted" or "exceptional inverse image functor", to a higher direct image with compact support functor formula_2.
"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). If formula_3 is proper, then
formula_4 is a right adjoint to the "inverse image" functor formula_5. 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 formula_7 (Hartshorne) or formula_8 (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, formula_1 exists as an exact functor from a derived category of quasi-coherent sheaves on formula_10, to the analogous category on formula_11, 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 formula_1, 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 formula_12 of noetherian schemes of finite dimension, found in Hartshorne ("Residues and duality") is the following quasi-isomorphism
for formula_16 a bounded above complex of formula_17-modules with quasi-coherent cohomology and formula_18 a bounded below complex of formula_19-modules with coherent cohomology. Here the formula_20's are sheaves of homomorphisms.
Construction of the formula_1 pseudofunctor using rigid dualizing complexes.
Over the years, several approaches for constructing the formula_1 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 formula_23 be a commutative ring, and let 
formula_24 be a commutative formula_25algebra. There is a functor formula_26 which takes a cochain complex formula_27 to an object formula_26 in the derived category over formula_24.
Asumming formula_24 is noetherian, a rigid dualizing complex over formula_24 relative to formula_23 is by definition a pair formula_33 where formula_34 is a dualizing complex over formula_24 which has finite flat dimension over formula_23, and where
formula_37 is an isomorphism in the derived category formula_38. If such a rigid dualizing complex exists, then it is unique in a strong sense.
Assuming formula_24 is a localization of a finite type formula_23-algebra, existence of a rigid dualizing complex over formula_24 relative to formula_23 was first proved by Yekutieli and Zhang assuming formula_23 is a regular noetherian ring of finite Krull dimension, and by Avramov, Iyengar and Lipman assuming formula_23 is a Gorenstein ring of finite Krull dimension and formula_24 is of finite flat dimension over formula_23.
If formula_11 is a scheme of finite type over formula_23, one can glue the rigid dualizing complexes that its affine pieces have, and obtain a rigid dualizing complex formula_49. Once one establishes a global existence of a rigid dualizing complex, given a map formula_50 of schemes over formula_23, one can define formula_52, where for a scheme formula_11, we set formula_54.
Dualizing Complex Examples.
Dualizing Complex for a Projective Variety.
The dualizing complex for a projective variety formula_55 is given by the complex
Plane Intersecting a Line.
Consider the projective variety
We can compute formula_58 using a resolution formula_59 by locally free sheaves. This is given by the complex
Since formula_61 we have that
This is the complex