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)
Robin Hartshorne, Residues and duality (Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne.) Springer LNM 20, 1966 MR222093
Domingo Toledo, Yue Lin L. Tong, Duality and intersection theory in complex manifolds. I., Math. Ann. 237 (1978), no. 1, 41–77, MR80d:32008, doi
Mitya Boyarchenko, Vladimir Drinfeld, A duality formalism in the spirit of Grothendieck and Verdier, arxiv/1108.6020
Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les $\mathcal{D}_X$-modules cohérents, Travaux en Cours 35. Hermann, Paris, 1989. x+254 pp. MR90m:32026
Amnon Neeman, Derived categories and Grothendieck duality, in: Triangulated categories, 290–350, London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press 2010
Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236, MR96c:18006, doi
Brian Conrad, Grothendieck duality and base change, Springer Lec. Notes Math. 1750 (2000) vi+296 pp.
Joseph Lipman, Notes on derived functors and Grothendieck duality, in: Foundations of Grothendieck duality for diagrams of schemes, 1–259, Lecture Notes in Math. 1960, Springer 2009, doi, draft pdf
J. Lipman, Grothendieck operations and coherence in categories, conference slides, 2009, pdf
Alonso Tarrío, Leovigildo; Jeremías López, Ana; Joseph Lipman, Studies in duality on Noetherian formal schemes and non-Noetherian ordinary schemes, Contemporary Mathematics 244 Amer. Math. Soc. 1999. x+126L. MR2000h:14017; Duality and flat base change on formal schemes, Contemporary Math. 244 (1999), pp. 3–90.
J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I., Astérisque No. 314 (2007), x+466 pp. (2008) MR2009h:14032; II. Astérisque No. 315 (2007), vi+364 pp. (2008) MR2009m:14007; also a file at K-theory archive THESE.pdf
Amnon Yekutieli, James Zhang, Rings with Auslander Dualizing Complexes, J. Algebra 213 (1999), 1-51; Rigid dualizing complexes over commutative rings, Algebr. Represent. Theory 12 (2009), no. 1, 19–52, doi; Dualizing Complexes and Perverse Sheaves on Noncommutative Ringed Schemes, Selecta Math. 12 (2006), 137-177; Dualizing Complexes and Perverse Modules over Differential Algebras, Compositio Mathematica 141 (2005), 620-654.
Amnon Yekutieli, An Explicit Construction of the Grothendieck Residue Complex, Astérisque 208 (1992); The residue complex of a noncommutative graded algebra, J. Algebra 186 (1996), no. 2, 522–543; Smooth formal embeddings and the residue complex, Canad. J. Math. 50 (1998), no. 4, 863–896, MR99i:14004; Rigid dualizing complexes via differential graded algebras (survey), in: Triangulated categories, 452–463, London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press 2010, MR2011h:18015; Residues and Differential Operators on Schemes, Duke Math. J. 95 (1998), 305-341; Duality and Tilting for Commutative DG Rings, arXiv:1312.6411; Residues and Duality for Schemes and Stacks (lecture notes).
Roy Joshua, Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories (pdf)
Pieter Belmans, section 2.2 of Grothendieck duality: lecture 3, 2014 (pdf)
Amnon Neeman, An improvement on the base-change theorem and the functor $f^!$, arXiv.
Last revised on February 18, 2015 at 12:30:18. See the history of this page for a list of all contributions to it.