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.

In fact, can always taken to be a *bounded* complex of injective sheaves, though we shall not need this. We now want to determine the properties of this dualizing complex, and show in particular that we can recover Poincaré duality when is a manifold. To do this, let us try to compute the cohomology of the dualizing complex (which will be a collection of sheaves). The th cohomology can be obtained as the sheaf associated to the presheaf

Here, as usual is the extension by zero of the constant sheaf from to . Indeed, to check this relation we recall that we assumed a complex of injectives (without loss of generality), so that maps are just homotopy classes of maps , or equivalently (by the universal property of ) elements of .

But the sheaf associated to this presheaf is clearly the homology . We have proved in fact:

Proposition 2If , the cohomology is the sheaf associated to the presheaf .

So we need to compute . By taking small, we may assume that is a ball in . From the adjoint property, however, this is feasible: such maps are in natural bijection with maps in . So we need to compute . Here is represented by a bounded complex. One might hope that this is somehow related to the compactly supported cohomology of .

When is a field, every complex is quasi-isomorphic to its cohomology, and it is true. We get:

Proposition 3If is a field, then the th cohomology of the dualizing complex is the sheaf associated to the presheaf .

Chasing through the definitions, one sees that the restriction maps are the duals to the maps for . (This goes the opposite way as in ordinary cohomology.) We in particular see that lives in the *bounded* derived category, at least when is a field (because is now quasi-isomorphic to a suitable truncation).

The next goal will be to compute the cohomology of for a manifold. For a field at least, this will require nothing more than a computation of the cohomology of suitable open sets in , by the previous result.

**2. The cohomology of **

The point of Poincaré duality lies in the cohomology of and the fact that any manifold is locally homeomorphic to . Of course, we mean *compactly* supported cohomology here.

Lemma 4Let be any ring. Then we have if , and otherwise.

*Proof:* We refer the reader to Iversen’s book on sheaf cohomology. The strategy, in rough outline, is as follows:

- It is sufficient to handle the case , because then it is clear for any free abelian group, and one can use the exact sequences to deduce it in general (together with the fact that cohomology commutes with filtered colimits).
- One shows that is in dimension zero and zero otherwise. This follows, for instance, by use of the soft
*de Rham resolution*where denotes the sheaf of smooth functions and the last map is one-variable differentiation. In particular, can be computed as the cokernel of differentiation

which is clearly trivial.

- One computes using the soft(!) resolution
where is the sheaf of real-valued continuous functions and is the sheaf of continuous functions into the circle group. One can deduce from this (and the so-called Vietoris-Bergle mapping theorem) that sheaf cohomology is a homotopy invariant.

- In the end, one can show that sheaf cohomology with coefficients in the constant sheaf is a
*cohomology theory*(satisfying, that is, the usual Eilenberg-Steenrod axioms) on suitably nice spaces. The analog of relative cohomology is local cohomology. Because of the normalization of the cohomology of a point, it follows that this is ordinary cohomology and is in dimensions and , and zero otherwise. - Finally, to compute , one uses the fact that the one-point compactification of is and the long exact sequence.

It follows that the same is true when is replaced by any space homeomorphic to it, e.g. an open ball. Using this, we make the following observation: if is an -dimensional manifold, then the sheaf associated to the presheaf is zero unless . It follows that the dualizing complex is cohomologically concentrated in *one* degree, namely . It follows (by the use of truncation functors) that the dualizing complex is quasi-isomorphic to a translate of a single sheaf.

On a -dimensional manifold , we define the **orientation sheaf** as the sheaf associated to the presheaf . (This is actually already a sheaf, though we do not need this.)

Corollary 5Let be a field, and let be an -dimensional manifold. Then the dualizing complex on is isomorphic to .

3. Poincaré duality

Fix a *field* . Let be an -dimensional manifold with orientation sheaf . We know that the dualizing sheaf is , which implies for any complex , there is a natural isomorphism . Take in particular for some and . On the left, we get ; on the right, we get (since is a field) .

Theorem 7 (Poincaré duality)There is a map

such that the pairing

identifies .

When is the *constant* sheaf , then (this is *not* compactly supported cohomology!) and consequently one finds that there is a natural isomorphism . When is orientable and is the constant sheaf , then we have recovered the usual form of Poincaré duality.

Notice also how, in this statement of Poincare duality, the groups appear.

Corollary 8Let be an -dimensional manifold. The functor on is representable.

*Proof:* Indeed, the representing object is , as follows from Poincaré duality above.

**4. Relative cohomology**

Let be a topological space, and a closed subspace. Sheaf cohomology of the constant sheaf on is to be thought of as an analog to the singular cohomology ; in fact, these coincide for a nice space . The analog of the *relative* singular cohomology are the *local cohomology* groups for the constant sheaf. Namely, consider the functor that sends to , the group of global sections with support in . The derived functors are called the**local cohomology groups** of . We recall that if is the inclusion, then we showed much earlier that the push-forward had a right adjoint . Since (as is easy to see),

we get by adjointness

In other words, the global sections of are precisely ; this is also immediate from the actual construction of we gave.

Our notation is, however, slightly confusing! We have defined as the right adjoint to . However, we also used the notation for the right adjoint to the *derived* functor . The next lemma will show that our abuse of notation is not as bad as it may seem.

Lemma 9Let be right adjoint to . Then is right adjoint to (so it is the “upper shriek” of before).

*Proof:* Now is a left-exact functor (as a right adjoint). Since its left adjoint is *exact*, a simple formal argument shows that preserves injectives. From this the argument is straightforward. Namely, let . We may assume that these are bounded-below complexes of injective sheaves. Then we have that and . We have that

where the last symbol denotes homotopy classes; this follows because is injective. Similarly we get

because preserves injectives. Now by adjointness on the ordinary categories, we see the natural isomorphism.

Now we want to bring in Verdier duality.

Lemma 10Let be locally compact spaces of finite dimension, and suppose are continuous maps. Then .

To be precise, we should say “up to a natural isomorphism.”

*Proof:* This is immediate from the adjointness definition of the upper shriek and the fact that

(which is the Leray spectral sequence). We can deduce the following. If is the dualizing complex on , then is the dualizing complex on . This follows because the dualizing complex is the upper shriek of the constant complex . Suppose now is a manifold of dimension , with orientation sheaf . Then it follows that

From this we will get:

Theorem 11 (Alexander duality)Let be a field. Suppose is a closed subset of an -dimensional manifold with the inclusion. Then, for a sheaf , we have natural isomorphisms:

The most important case is when , in which case we find:

*Proof:* Indeed, this follows purely formally now. Let be any sheaf. In the following, we shall interchange and , which is no matter as is an exact functor. Then:

Take now . We then get at the last term . But recall that these are the local cohomology groups because is the same functor as .

The classical statement about closed subsets of the sphere now follows because the sphere is cohomologically rather simple, and orientable (so the orientation sheaf is trivial).

## Leave a Reply