Grothendieck duality

Grothendieck duality


Given a homomorphism ff of schemes, one says that it satisfies Grothendieck duality if the (derived) direct image functor f *f_\ast on quasicoherent sheaves has a (derived) right adjoint f !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:XYf\colon X \to Y is a quasi-compact and quasi-separated morphism of schemes; then the triangulated functor Rf *:D qc(X)D(Y)\mathbf{R}f_*\colon D_{qc}(X)\to D(Y) has a bounded below right adjoint. In other words, RHom X(,f ×𝒢)RHom Y(Rf *,𝒢)\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 XX be a noetherian scheme. A dualizing complex on XX is a complex D(ModX)\mathcal{R} \in \mathsf{D}(\mathsf{Mod} X) that has these three properties:

  • D c b(ModX)\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 𝒪 XRℋℴ𝓂 X(,)\mathcal{O}_X \to \mathrm{R} \mathcal{Hom}_{X}(\mathcal{R}, \mathcal{R}) in D(ModX)\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 S\mathsf{S}:

  • A psudofunctor ff !f \mapsto f^!, called the twisted inverse image, that assigns a functor

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

    to each map of schemes f:XYf : X \to Y in S\mathsf{S}, and has several known properties.

  • A dualizing complex X\mathcal{R}_X for every scheme XX in the category S\mathsf{S}, with several known functorial properties.

The relation between these two structures is demonstrated in the following Example.

Example. Suppose KK is a regular finite dimensional noetherian ring, and let S\mathsf{S} be the category of finite type KK-schemes. Given a twisted inverse image psudofunctor ff !f \mapsto f^!, we define dualizing complexes as follows: on any XSX \in \mathsf{S} with structural morphism π X:XSpecK\pi_X : X \to \operatorname{Spec} K, we let X:=π X !(K)\mathcal{R}_X := \pi_X^!(K).

Conversely, suppose we are given a dualizing complex X\mathcal{R}_X on each XSX \in \mathsf{S}. This gives rise to a duality (contrvariant equivalence) D XD_X of D c (ModX)\mathsf{D}^{}_{\mathrm{c}}(\mathsf{Mod} X), exchanging D c +(ModX)\mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} X) with D c (ModX)\mathsf{D}^{-}_{\mathrm{c}}(\mathsf{Mod} X), with formula

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

We then define

(3)f !():=D Y(Lf *(D X())). 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 AA over a base field KK.

Yekutieli and Zhang have shown how to define a rigid dualizing complex R A/KR_{A/K}, when KK is a regular finite dimensional noetherian ring, and AA is an essentially finite type KK-ring (both commutative). A refined variant of the rigid dualizing complex, namely the rigid residue complex 𝒦 A/K\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 AAA \to A' (via the rigid localization homomorphism), and contravariant (as graded modules) for all ring homomorphisms ABA \to B (via the ind-rigid trace homomorphism).

The rigid localization homomorphism permits the gluing of the rigid residue complexes 𝒦 A/K\mathcal{K}_{A/K} on affine open sets U=SpecAU = \operatorname{Spec} A of a scheme XX into a rigid residue complex 𝒦 X/K\mathcal{K}_{X/K} on XX. In this way one obtains a collection of dualizing complexes 𝒦 X/K\mathcal{K}_{X/K} on all essentially finite type KK-schemes XX, consisting of quasi-coherent injective sheaves. For any map of scheme f:XYf : X \to Y there is the ind-rigid trace homomoprhism

(4)Tr f:f *(𝒦 Y/K)𝒦 X/K, \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 ff is proper, Tr f\mathrm{Tr}_f is a homomorphism of complexes. The Duality Theorem says that when ff is proper, Tr f\mathrm{Tr}_f induces global duality. As explained above, there is a corresponding functor f !f^!; and the Duality Theorem says that Rf *\mathrm{R} f_* and f !f^! are adjoint functors.

Moreover, the rigidity method works also for finite type Deligne-Mumford stacks over KK. The key observation is that the rigid residue complexes are complexes of quasi-coherent sheaves in the etale site over KK. Details of this extension of the theory are still under preparation.

Noncommutative Grothendieck Duality

(To be added later)


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