Verdier duality

In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by Jean-Louis Verdier (1965) as an analog for locally compact topological spaces of

Alexander Grothendieck's theory of

Poincaré duality in étale cohomology

for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism.

Verdier duality generalises the classical Poincaré duality of manifolds in two directions: it applies to continuous maps from one space to another (reducing to the classical case for the unique map from a manifold to a one-point space), and it applies to spaces that fail to be manifolds due to the presence of singularities. It is commonly encountered when studying constructible or perverse sheaves.

Verdier duality

Verdier duality states that (subject to suitable finiteness conditions discussed below)

certain derived image functors for sheaves are actually adjoint functors. There are two versions.

Global Verdier duality states that for a continuous map

f

:

X

→

Y

{\displaystyle f\colon X\to Y}

of locally compact Hausdorff spaces, the derived functor of the direct image with compact (or proper) supports

R

f

!

{\displaystyle Rf_{!}}

has a right adjoint

f

!

{\displaystyle f^{!}}

in the derived category of

sheaves, in other words, for (complexes of) sheaves (of abelian groups)

F

{\displaystyle {\mathcal {F}}}

on

X

{\displaystyle X}

and

G

{\displaystyle {\mathcal {G}}}

on

Y

{\displaystyle Y}

we have

R

H

o

m

(

R

f

!