duality

# Grothendieck duality

## Idea

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.

## Statement

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.

## Dualizing Complexes

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

(1)$f^! : \mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} Y) \to \mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} X)$

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

(2)$D_X(\mathcal{M}) := \mathrm{R} \mathcal{Hom}_{X}(\mathcal{M}, \mathcal{R}_X) .$

We then define

(3)$f^!(\mathcal{M}) := D_Y( \mathrm{L} f^* ( D_X(\mathcal{M}) )) .$

### Rigid Dualizing Complexes

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

(4)$\mathrm{Tr}_f : f_*(\mathcal{K}_{Y/K}) \to \mathcal{K}_{X/K} ,$

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.

## References

• 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

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• 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

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• 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.

Revised on February 18, 2015 12:30:18 by Noam Zeilberger (80.215.196.192)