We have spent a while in the past few days going through the rather categorical formalism of the upper shriek functor obtained from a map between locally compact Hausdorff spaces of finite cohomological dimension. That is, we showed that the upper shriek must exist on the derived category; this was Verdier duality. However, so far we have not seen any concrete applications of this formalism. I actually feel a bit guilty about having not indicated better some of these in the introductory post and having essentially plunged into the abstract nonsense.

Now we shall apply the existence of to questions involving manifolds. Once we know that exists, we will be able to describe it using the adjoint property rather simply (for manifolds). This will lead to clean statements of theorems in algebraic topology. For instance, Poincaré duality will be a direct consequence of the fact that, on an -dimensional oriented manifold, the dualizing sheaf (see below) is just .

**1. The dualizing complex**

After wading through the details of the proof of Verdier duality, let us now consider the simpler case where . is still a locally compact space of finite dimension, and remains a noetherian ring. Then Verdier duality gives a right adjoint to the functor . In other words, for each and each complex of -modules, we have an isomorphism

Of course, the category is likely to be much simpler than , especially if, say, is a field.

Definition 1is called thedualizing complexon the space . is an element of the derived category , and is well-defined there. We will always assume that is a bounded-below complex ofinjectivesheaves. (more…)