Coherent duality

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Coherent duality in mathematics refers to 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 an accessible 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 1992 Greenlees-May duality is part of the continuing consideration of this area.

[edit] 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

$f^!\;,$

now called twisted inverse image, to a higher direct image with proper 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). It should be noted that 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

$Rf_!f^! \to I\,$

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. (Verdier, Bombay 1968 paper; an elegant and more general approach was found by Amnon Neeman, by using methods from algebraic topology notably Brown representability). 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 $f : X \rightarrow Y$ of noetherian schemes of finite dimension, found in Hartshorne (Residues and duality) is the following quasi-isomorphism

$R f_* Hom_X(F^\bullet, f^! G^\bullet)\rightarrow R Hom_Y(R f_* F^\bullet, G^\bullet)\$

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 $H o m$ 's are the sheaf of homomorphisms.