[Corrected to fix some embarrassing omissions — 6/12]
I have found lately that many of the foundational theorems in etale cohomology (for instance, the proper base change theorem) are analogs of usually much easier results in sheaf theory for nice (e.g. locally compact Hausdorff) spaces. It turns out that the present topic, duality, has its analog for etale cohomology, though I have not currently studied it. As a warm-up, I thought it would be instructive to blog about the duality theory for cohomology on spaces. This theory, known as Verdier duality, is stated as the existence of an adjoint functor to the derived push-forward. However, from this one can actually recover classical Poincare duality, as I hope to eventually explain.
For a space , we let be the category of sheaves of abelian groups. More generally, if is a ring, we let be the category of sheaves of -modules.
Consider a map of locally compact spaces. There is induced a push-forward functor , that sends a sheaf (of abelian groups) on to the push-forward on . It is well-known that this functor admits a left adjoint , which can be geometrically described in terms of the espace étale as follows: if is the espace étale of a sheaf on , then is the espace étale of .
Now, under some situations, the functor is very well-behaved. For instance, if is the inclusion of a closed subset, then is an exact functor. It turns out that it admits a right adjoint . This functor can be described as follows. Let be a sheaf on . We can define by saying that if is an open subset, such that for open, then is the subset of the sections of with support in . One can check that this does not depend on the choice of open subset .
Proposition 1 One has the adjoint relation:
Proof: Suppose given a map ; we need to produce a map . Indeed, we are given that for each open, there is a map
It is easy to see (because has support in ) that only sections of with support in are in the image. As a result, if is any open set, say , then we can produce the required map
In the reverse direction, if we are given maps as above, it is easy to produce a map . From this the result follows. Indeed, the two constructions are clearly inverses.
2. Verdier duality
In general, unlike the case of a closed immersion, the functor induced by a morphism does not have a right adjoint. However, in many nice situations it will have a right adjoint on the derived category. Namely, if is a ring, let be the bounded-below derived categories of sheaves of -modules on . Before this we had worked with , but there is no need to avoid this bit of generality. Since these categories have enough injectives, the functor induces a derived functor
To compute on a bounded-below complex of sheaves , one finds a quasi-isomorphism
where is a complex of injective sheaves (of -modules) and applies to .
Theorem 2 (Verdier duality, first form) If is a proper map of manifolds, then the derived functor admits a right adjoint .
We will actually state and prove this fact in more generality later. For now, let us consider a special case: when is a point. In this case, the derived functor is just , in other words the derived functors of the global sections (a.k.a. sheaf cohomology on the derived category). is the derived category of the category of -modules. The result states that there is an adjoint . Consider the complex with in degree zero and nothing anywhere else; then let be . It follows then that we have an isomorphism
3. The statement for non-proper maps
It will be convenient to have a statement of Verdier duality that holds for non-proper maps as well. To do this, given a map of spaces
we shall define a new functor
This will coincide with when is proper. Namely, if and is open, we let
We will describe the properties of below, but in any event one sees that there is induced a derived functor . The more general form of Verdier duality will state that has a right adjoint (even if is not proper).
Before doing this, it will be necessary to develop some preliminaries on sheaf theory in general.