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Verdier duality

Context

Duality

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

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diagram chasing

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Idea

What is called Verdier duality is the refinement of Poincaré duality from ordinary cohomology to abelian sheaf cohomology. The Grothendieck six operations formalism is a formalization of aspects of Verdier duality.

In general abstract formulation one says that a locally compact site XX (e.g. a locally compact topological space) satisfies Verdier duality if there exists a derived right adjoint 𝕃p !\mathbb{L}p^! (exceptional inverse image) to the operation p !\mathbb{R}p_! of direct image with compact support for the terminal map X*X \to \ast. More generally, in the relative situation, for f:XYf \colon X \to Y a map, Verdier duality of ff means that f !\mathbb{R}f_! has a derived right adjoint. If here XX is a scheme and f !f *f_! \simeq f_\ast coincides with the direct image (“Grothendieck context”) then this is called Grothendieck duality. Therefore one often speaks of Verdier-Grothendieck duality.

If these adjoints exist, then one has the dualizing object in a closed category (dualizing module) ω𝕃p !k\omega \coloneqq \mathbb{L}p^! k (the image under the extra right adjoint of the unit object) and for every abelian sheaf VV on XX can define the dual object in a closed category

𝔻V[V,ω]. \mathbb{D}V \coloneqq [V,\omega] \,.

The analog of Poincaré duality is then the statement that the abelian sheaf cohomology with coefficients in VV is dual to the abelian sheaf cohomology with compact support with coefficients in 𝔻V\mathbb{D}V.

Implementations

A fairly general class of implementations of Verdier-Grothendieck duality is discussed in (Joshua, theorem 1.3).

With assumption (Joshua, 4.3) this derives that dualization intertwines the two pushforwards as 𝔻f *f !𝔻\mathbb{D} \circ f_\ast \simeq f_! \circ \mathbb{D} (Joshua, corollary 5.4).

References

The duality is named after Jean-Louis Verdier.

Accounts of the traditional notion include

  • Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, Jour. Amer. Math. Soc. 9 (1996), 205–236. (JSTOR)
  • Wikipedia, Verdier duality

  • Liviu Niculaescu, The derived category of sheaves and the Poincaré-Verdier duality (pdf)

  • Jacob Lurie, lecture notes on Verdier duality (pdf, pdf)

A general abstract formalization is in

  • Roy Joshua, Grothendieck-Verdier duality in enriched symmetric monoidal tt-categories (pdf)

Discussion in the context of higher algebra/higher geometry is in

Formulation in terms of an equivalence between sheaves and cosheaves is in

Discussion in the context of prestacks and diagrams of schemes

Last revised on July 1, 2015 at 05:38:10. See the history of this page for a list of all contributions to it.