Verdier Duality - Verdier Duality

Verdier Duality

Verdier duality states that certain image functors for sheaves are actually adjoint functors. There are two versions.

Global Verdier duality states that the higher direct image functor with compact supports Rf_! has a right adjoint f! in the derived category of sheaves, in other words

The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "f shriek" or "f 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 X. 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 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 D_X on X is defined to be

where p is the map from X to a point. Part of what makes Verdier duality interesting in the singular setting is that when X 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 X is a finite dimensional locally compact space, and Db(X) the bounded derived category of sheaves of abelian groups over X, then the Verdier dual is a contravariant functor

defined by

It has the following properties:

• for sheaves with constructible cohomology.
• (Intertwining of functors f_* and f_!) If f is a continuous map from X to Y then there is an isomorphism

  .