This is the fifth in a series of posts on Verdier duality, started here. In this post, I will describe the proof of the duality theorem, which itself states the existence of an adjoint to the derived version of the lower shriek functor . This might not sound too exciting at first, but we will see that in fact, the dualizing functor will be computable in the important special case of a manifold, and Poincaré duality will fall out quickly. Moreover, the flexible interpretation of sheaf cohomology will allow other duality theorems (such as Alexander duality) to be derived very efficiently from the general formalism.
I will try to explain some of this story (namely, that using sheaf cohomology and Verdier duality one can re-derive much of the classical theory of homology and cohomology) next time. First, though, it will be good to prove the result.
We can now enunciate the result we shall prove in full generality.
Theorem 1 (Verdier duality) Let be a continuous map of locally compact spaces of finite dimension, and let be a noetherian ring. Then admits a right adjoint . In fact, we have an isomorphism in
Here is defined as follows. Recall that given chain complexes of sheaves, one may define a chain complex of -modules ; the elements in degree are given by the product , and the differential sends a collection of maps to . Then is the derived functor of , and lives in the derived category if . Since the cohomology in degree zero is given by , we see that the last statement of Verdier duality implies the adjointness relation. (more…)