abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
Given a homomorphism $f$ of schemes, one says that it satisfies Grothendieck duality if the (derived) direct image functor $f_\ast$ on quasicoherent sheaves has a (derived) right adjoint $f^!$. 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.
Suppose $f\colon X \to Y$ is a quasi-compact and quasi-separated morphism of schemes; then the triangulated functor $\mathbf{R}f_*\colon D_{qc}(X)\to D(Y)$ has a bounded below right adjoint. In other words, $\mathbf{R}Hom_X(\mathcal{F}, f^\times \mathcal{G})\stackrel{\sim}{\to} \mathbf{R}Hom_Y(\mathbf{R}f_*\mathcal{F}, \mathcal{G})$ is a natural isomorphism.
Let $X$ be a noetherian scheme. A dualizing complex on $X$ is a complex $\mathcal{R} \in \mathsf{D}(\mathsf{Mod} X)$ that has these three properties:
$\mathcal{R} \in \mathsf{D}^{\mathrm{b}}_{\mathrm{c}}(\mathsf{Mod} X)$ (i.e. $\mathcal{R}$ has bounded coherent cohomology sheaves).
$\mathcal{R}$ has finite injective dimension.
The canonical morphism $\mathcal{O}_X \to \mathrm{R} \mathcal{Hom}_{X}(\mathcal{R}, \mathcal{R})$ in $\mathsf{D}(\mathsf{Mod} X)$ is an isomorphism.
The following two structures are basically equivalent to each other, for a given category of noetherian schemes $\mathsf{S}$:
A psudofunctor $f \mapsto f^!$, called the twisted inverse image, that assigns a functor
to each map of schemes $f : X \to Y$ in $\mathsf{S}$, and has several known properties.
A dualizing complex $\mathcal{R}_X$ for every scheme $X$ in the category $\mathsf{S}$, with several known functorial properties.
The relation between these two structures is demonstrated in the following Example.
Example. Suppose $K$ is a regular finite dimensional noetherian ring, and let $\mathsf{S}$ be the category of finite type $K$-schemes. Given a twisted inverse image psudofunctor $f \mapsto f^!$, we define dualizing complexes as follows: on any $X \in \mathsf{S}$ with structural morphism $\pi_X : X \to \operatorname{Spec} K$, we let $\mathcal{R}_X := \pi_X^!(K)$.
Conversely, suppose we are given a dualizing complex $\mathcal{R}_X$ on each $X \in \mathsf{S}$. This gives rise to a duality (contrvariant equivalence) $D_X$ of $\mathsf{D}^{}_{\mathrm{c}}(\mathsf{Mod} X)$, exchanging $\mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} X)$ with $\mathsf{D}^{-}_{\mathrm{c}}(\mathsf{Mod} X)$, with formula
We then define
The notion of rigid dualizing complex was introduced by Van den Bergh in 1997, for a noncommutative ring $A$ over a base field $K$.
Yekutieli and Zhang have shown how to define a rigid dualizing complex $R_{A/K}$, when $K$ is a regular finite dimensional noetherian ring, and $A$ is an essentially finite type $K$-ring (both commutative). A refined variant of the rigid dualizing complex, namely the rigid residue complex $\mathcal{K}_{A/K}$, 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 $A \to A'$ (via the rigid localization homomorphism), and contravariant (as graded modules) for all ring homomorphisms $A \to B$ (via the ind-rigid trace homomorphism).
The rigid localization homomorphism permits the gluing of the rigid residue complexes $\mathcal{K}_{A/K}$ on affine open sets $U = \operatorname{Spec} A$ of a scheme $X$ into a rigid residue complex $\mathcal{K}_{X/K}$ on $X$. In this way one obtains a collection of dualizing complexes $\mathcal{K}_{X/K}$ on all essentially finite type $K$-schemes $X$, consisting of quasi-coherent injective sheaves. For any map of scheme $f : X \to Y$ there is the ind-rigid trace homomoprhism
which is a homomorphism of graded quasi-coherent sheaves. The Residue Theorem says that when $f$ is proper, $\mathrm{Tr}_f$ is a homomorphism of complexes. The Duality Theorem says that when $f$ is proper, $\mathrm{Tr}_f$ induces global duality. As explained above, there is a corresponding functor $f^!$; and the Duality Theorem says that $\mathrm{R} f_*$ and $f^!$ are adjoint functors.
Moreover, the rigidity method works also for finite type Deligne-Mumford stacks over $K$. The key observation is that the rigid residue complexes are complexes of quasi-coherent sheaves in the etale site over $K$. Details of this extension of the theory are still under preparation.
(To be added later)
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