nLab Verdier duality

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 $X$ (e.g. a locally compact topological space) satisfies Verdier duality if there exists a derived right adjoint $\mathbb{L}p^!$ (exceptional inverse image) to the operation $\mathbb{R}p_!$ of direct image with compact support for the terminal map $X \to \ast$. More generally, in the relative situation, for $f \colon X \to Y$ a map, Verdier duality of $f$ means that $\mathbb{R}f_!$ has a derived right adjoint. If here $X$ is a scheme and $f_! \simeq f_\ast$ 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) $\omega \coloneqq \mathbb{L}p^! k$ (the image under the extra right adjoint of the unit object) and for every abelian sheaf $V$ on $X$ can define the dual object in a closed category

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

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

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 $\mathbb{D} \circ f_\ast \simeq f_! \circ \mathbb{D}$ (Joshua, corollary 5.4).

Related concepts

• Verdier-Grothendieck context

• Grothendieck duality

• Poincaré duality

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 (2011 pdf, 2013 pdf, 2014 pdf)

A general abstract formalization is in

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

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

• Roy Joshua, Generalised Verdier duality for presheaves of spectra—I, Journal of Pure and Applied Algebra Volume 70, Issue 3, 29 March 1991, Pages 273–289

• Jonathan Block and Andrey Lazarev, Homotopy Theory and Generalized Duality for Spectral Sheaves, IMRN International Mathematics Research Notices

  1996, No. 20 (pdf)

• Vladimir Drinfeld, Dennis Gaitsgory, section 5.3 On some finiteness questions for algebraic stacks (arXiv:1108.5351)

• Jacob Lurie, section 4.2 of Representability theorems

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

• Peter Schneider, Verdier duality on the building, Journal reine angew. Math. 494, 1998

• Justin Curry, Sheaves, Cosheaves and Applications (arXiv:1303.3255)

• Jacob Lurie, section 5.5.5 of Higher Algebra

Discussion in the context of prestacks and diagrams of schemes

• Dennis Gaitsgory, The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a curve (arXiv:1505.02331)

• Alexey Kalugin, On categorical approach to Verdier Duality (arXiv:1505.06922)