Verdier duality

In mathematics, Verdier duality is a duality in sheaf theory that generalizes Poincaré duality for manifolds. Verdier duality was introduced by as an analog for locally compact spaces of the coherent duality for schemes due to Alexander Grothendieck. It is commonly encountered when studying constructible or perverse sheaves.
Verdier duality.
Verdier duality states that certain image functors for sheaves are actually adjoint functors. There are two versions.
Global Verdier duality states that for a continuous map formula_1, the derived functor of the direct image with proper supports formula_2 has a right adjoint formula_3 in the derived category of sheaves, in other words, for a sheaf formula_4 on formula_5 and formula_6 on formula_7 we have 
The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "formula_9 shriek" or ""formula_9" lower shriek" and ""f" upper shriek" – see also shriek map.
Local Verdier duality states that
in the derived category of sheaves of "k" modules over "Y". 
It is important to note that the distinction between the global and local versions is that the former relates maps between sheaves, whereas the latter relates (complexes of) sheaves directly and so can be evaluated locally. Taking global sections of both sides in the local statement gives global Verdier duality.
The dualizing complex formula_12 on formula_5 is defined to be
where "p" is the map from formula_5 to a point. Part of what makes Verdier duality interesting in the singular setting is that when formula_5 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.
If formula_5 is a finite-dimensional locally compact space, and formula_18 the bounded derived category of sheaves of abelian groups over formula_5, then the Verdier dual is a contravariant functor
defined by
It has the following properties:
Poincaré duality.
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 formula_22 is the constant sheaf on "X" with coefficients in "k". Let formula_23 be the constant map. 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 (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 "k"X replaced with the same sheaf placed in degree "i" we get the classical Poincaré duality