abstract duality: opposite category,
Given a homomorphism of schemes, one says that it satisfies Grothendieck duality if the (derived) direct image functor on quasicoherent sheaves has a (derived) right adjoint . This is Verdier duality in a “Grothendieck context” of six operations.
Grothendieck duality is intimately connected to dualizing complexes. This was the original approach of Grothendieck in the book Residues and Duality.
Let be a noetherian scheme. A dualizing complex on is a complex that has these three properties:
(i.e. has bounded coherent cohomology sheaves).
has finite injective dimension.
The canonical morphism in is an isomorphism.
The following two structures are basically equivalent to each other, for a given category of noetherian schemes :
A psudofunctor , called the twisted inverse image, that assigns a functor
to each map of schemes in , and has several known properties.
A dualizing complex for every scheme in the category , with several known functorial properties.
The relation between these two structures is demonstrated in the following Example.
Example. Suppose is a regular finite dimensional noetherian ring, and let be the category of finite type -schemes. Given a twisted inverse image psudofunctor , we define dualizing complexes as follows: on any with structural morphism , we let .
Conversely, suppose we are given a dualizing complex on each . This gives rise to a duality (contrvariant equivalence) of , exchanging with , with formula
We then define
The notion of rigid dualizing complex was introduced by Van den Bergh in 1997, for a noncommutative ring over a base field .
Yekutieli and Zhang have shown how to define a rigid dualizing complex , when is a regular finite dimensional noetherian ring, and is an essentially finite type -ring (both commutative). A refined variant of the rigid dualizing complex, namely the rigid residue complex , was shown to exist, and to be unique (up to a unique isomorphism of complexes).
These rigid residue complexes have all the good functorial properties alluded to above, and even more. Specifically, they are covariant for essentially etale ring homomorphisms (via the rigid localization homomorphism), and contravariant (as graded modules) for all ring homomorphisms (via the ind-rigid trace homomorphism).
The rigid localization homomorphism permits the gluing of the rigid residue complexes on affine open sets of a scheme into a rigid residue complex on . In this way one obtains a collection of dualizing complexes on all essentially finite type -schemes , consisting of quasi-coherent injective sheaves. For any map of scheme there is the ind-rigid trace homomoprhism
which is a homomorphism of graded quasi-coherent sheaves. The Residue Theorem says that when is proper, is a homomorphism of complexes. The Duality Theorem says that when is proper, induces global duality. As explained above, there is a corresponding functor ; and the Duality Theorem says that and are adjoint functors.
Moreover, the rigidity method works also for finite type Deligne-Mumford stacks over . The key observation is that the rigid residue complexes are complexes of quasi-coherent sheaves in the etale site over . Details of this extension of the theory are still under preparation.
(To be added later)
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