Verdier duality

In mathematics, Verdier duality is a duality in sheaf theory that generalizes Poincaré duality for manifolds. Verdier duality was introduced by Jean-Louis Verdier (1967, 1995) 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 states that certain image functors for sheaves are actually adjoint functors. There are two versions.

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.

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.

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

By repeating this argument with the sheaf kX replaced with the same sheaf placed in degree i we get the classical Poincaré duality