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.
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 states that (subject to suitable finiteness conditions discussed below) certain derived image functors for sheaves are actually adjoint functors.
Global Verdier duality states that for a continuous map
of locally compact Hausdorff spaces, the derived functor of the direct image with compact (or proper) supports
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.
These results hold subject to the compactly supported direct image functor
The discussion above is about derived categories of sheaves of abelian groups.
Part of what makes Verdier duality interesting in the singular setting is that when
is not a manifold (a graph or singular algebraic variety for example) 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.
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: Poincaré duality can be derived as a special case of Verdier duality.
Suppose X is a compact orientable n-dimensional manifold, k is a field and
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.
Since morphisms between complexes of sheaves (or vector spaces) 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 kX replaced with the same sheaf placed in degree i we get the classical Poincaré duality