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 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 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, 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
certain derived image functors for sheaves are actually adjoint functors. There are two versions.
Global Verdier duality states that for a continuous map of locally compact Hausdorff spaces, the derived functor of the direct image with compact supports has a right adjoint in the derived category of
sheaves, in other words, for sheaves on and on we have
Local Verdier duality states that
in the derived category of sheaves on Y.
It is important to note that the distinction between the global and local versions is that the former relates morphisms between
complexes of sheaves in the derived categories, whereas the latter relates internal Hom-complexes and so can be evaluated locally. Taking global sections of both sides in the local statement gives the global Verdier duality.
These results hold subject to the compactly supported direct image functor having finite cohomological dimension.
This is the case if there is a bound such that the compactly supported cohomology
vanishes for all fibres
and. This holds if all the fibres are at most -dimensional manifolds or more generally at most -dimensional CW-complexes.
The discussion above is about derived categories of sheaves of abelian groups. It is instead possible to consider a ring
and sheaves of -modules; the case above corresponds to
.
The dualizing complex on is defined to be
where p is the map from to a point. Part of what makes Verdier duality interesting in the singular setting is that when is not a manifold then the dualizing complex is not quasi-isomorphic to a sheaf concentrated in a single degree. From this perspective the derived category is necessary in the study of singular spaces.
If is a finite-dimensional locally compact space, and the bounded derived category of sheaves of abelian groups over, then the Verdier dual is a contravariant functor
defined by
It has the following properties:

Relation to classical Poincaré duality

can be derived as a special case of Verdier duality. Here one explicitly calculates cohomology of a space using the machinery of sheaf cohomology.
Suppose X is a compact orientable n-dimensional manifold, k is a field and is the constant sheaf on X with coefficients in k. Let be the constant map to a point. Global Verdier duality then states
To understand how Poincaré duality is obtained from this statement, it is perhaps easiest to understand both sides piece by piece. Let
be an injective resolution of the constant sheaf. Then by standard facts on right derived functors
is a complex whose cohomology is the compactly supported cohomology of X. Since morphisms between complexes of sheaves themselves form a complex we find that
where the last non-zero term is in degree 0 and the ones to the left are in negative degree. Morphisms in the derived category are obtained from the homotopy category of chain complexes of sheaves by taking the zeroth cohomology of the complex, i.e.
For the other side of the Verdier duality statement above, we have to take for granted the fact that when X is a compact orientable n-dimensional manifold
which is the dualizing complex for a manifold. Now we can re-express the right hand side as
We finally have obtained the statement that
By repeating this argument with the sheaf k_X replaced with the same sheaf placed in degree i we get the classical Poincaré duality