# nLab Verdier duality

Contents

duality

## In QFT and String theory

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## 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).

## 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)

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

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 February 18, 2020 at 12:20:07. See the history of this page for a list of all contributions to it.