The statement of Verdier duality was fairly fancy–it used the jazzed up language of derived categories, and was phrased in terms of the existence of an adjoint to a suitable derived functor. Though by the end of it hopefully it’ll become clear that, at least for manifolds, the mysterious adjoint functor has a fairly concrete interpretation that will turn out to be plain old Poincare duality.

But before we get there, we need to discuss some sheaf theory. I have usually thought of sheaves in the context of algebraic geometry, and I wanted to review (at least for my own benefit) some of the classical theory of sheaves on ordinary non-pathological topological spaces, like locally compact Hausdorff ones. In addition, it will at any rate be necessary to introduce the compactly supported cohomology functors to get Verdier duality. So I’ll do a couple of preparatory posts.

**1. Soft sheaves **

Let be a locally compact space. If and is closed, we define for the inclusion. This allows us to make sense of a section “over a closed subset.” The interpretation via the espace étale is helpful here: if is the espace étale of , then a section over is the same thing as a section of the projection over . It follows from the latter interpretation that if we have sections on closed subsets that agree on , one obtains uniquely a closed subset on . Now we want to show that we can recover this notion from the familiar idea of sections over an *open* set.

Lemma 1If is compact, we have

where ranges over open sets containing .

*Proof:* There is an obvious map . It is injective: if restricts to zero over , then the stalk is zero. It follows that there is a neighborhood for each such that . But then , so that maps to zero in the colimit. (We have not yet used the compactness of .) Surjectivity is the hard part. Suppose is a section over the compact set . Then we can cover by neighborhoods together with sections such that

By choosing the appropriately and in view of local compactness, we can assume that there are *compact* subsets , each of which contains an open subset of , and whose interiors (with respect to !) cover . It follows that a *finite*number of the cover . So we are in the following situation. We have a section which we want to extend to an open neighborhood; a finite set of closed subsets of that cover, and open sets with sections that agree with on . We want to find an defined over some open neighborhood that looks locally like the , near . To do this, we can assume inductively that , extending piece by piece. So then we have two compact subsets , and sections defined in a neighborhood of each. We are given that ; we now wish to find a section defined on a neighborhood of . To do this, choose a small neighborhood of such that , which we can do by the first part of the proof. Then choose disjoint neighborhoods such that contains , and contains . It is easy to see that all patch appropriately and give the desired section. We shall need the following weaker form of flasqueness.

Definition 2A sheaf issoftif is surjective whenever is compact.

A flasque sheaf is readily seen to be soft. We note that the restriction of a soft sheaf to a closed (or open!) subset is always still soft, as follows easily from the definition. We have the following property of soft sheaves:

Proposition 3A sheaf on a locally compact space is soft if and only if any section of over a compact subset can be extended to a compactly supported global section of . Even more precisely, if is soft, is an inclusion of a compact set in an open set , then a section over can be extended to a compactly supported global section with support in .

*Proof:* The definition of softness is that any section over ( compact but arbitrary) can be extended to . So one direction is clear.

We have to show conversely that if is soft, and is a section over the compact subset , then we can choose the extending global section in such a way that it is compactly supported (and that we can take the support inside ). To do this, choose a compact set containing in its interior, and with . Consider the section on given by on and zero on . By softness, this can be extended to a section of over that vanishes at the boundary. Now the sections given by on and on glue and give a compactly supported extension of ; it is clear that it vanishes outside . We have seen that flasque sheaves are soft; in particular, injective sheaves are soft, because injective sheaves are flasque. But soft sheaves are more general than flasque sheaves. We illustrate this below.

Example 1Let be a locally compact space, and suppose is the sheaf of continuous functions on . Then is a soft sheaf. Indeed, suppose given a section of over a compact set , i.e. a continuous function defined in some neighborhood of . By multiplying by a globally defined continuous function supported in such that , we get a section of which vanishes outside a compact set. We can thus extend this to a global continuous function.

Example 2Similarly, using suitable cutoff functions, it follows that if is a manifold, then the sheaf of functions (where ) is soft.

Example 3If is a soft sheaf of rings, and is a sheaf of -modules, then is a flasque sheaf. Indeed, let be a section. Then comes from a section of over some open subset containing . We can find a global section with support in that is identically on by softness. (Indeed, this follows from \cref{strongsoft}.) Then extends to a global section extending on . It follows in particular that if is a manifold and is any (say, smooth) vector bundle, the sheaf of sections of is a soft sheaf, as it is a sheaf of modules over the sheaf of smooth functions. As an example, the sheaves of -forms are all soft.

**2. Compactly supported cohomology **

Let be a topological space.

Definition 4If , and is open, we define to be the subgroup of consisting of sections with compact support.

It is easy to see that if have compact support, so does . So this is indeed a subgroup. Taking , we thus get a functor for the category of abelian groups. Note similarly that takes to the category of -modules.

Proposition 5is left-exact.

*Proof:* Indeed, we know that the ordinary global section functor is left-exact. Since , all that needs to be seen is that if one has a sequence , and maps to zero in , then comes from a *compactly supported* section in . But it comes from *some* (unique!) section of , and injectivity of shows that the section must be compactly supported.

Definition 6We write for the th (right) derived functor of . We can also define atotal derived functoron , taking values in the (bounded-below) derived category of abelian groups. More generally there is a functor for the derived category of -modules.

In the classical theory of sheaf cohomology, flasque sheaves (recall that a sheaf is *flasque* if is surjective for each open set) are the prototypical sheaves with no cohomology. For our purposes, it will be convenient to use the less restrictive notion of a *soft* sheaf, because for instance the de Rham resolution (see \cref{} below) is a soft but not flasque resolution. We will show this using:

Proposition 7Let be an exact sequence of sheaves on . Suppose is soft. Then the sequence is also exact.

*Proof:* We only need to prove surjectivity at the end. We shall start by making a reduction. We can assume that is itself compact. Indeed, let have compact support . Choose a compact subset containing in its interior.

Then by applying the result (assumed for compact spaces) to , we get a section of lifting . Then must live inside as it maps to zero in , and this restriction can be extended to some global section . It follows that is a section of over that restricts to zero on the boundary, so it extends to a section of over .

With this reduction made, let us assume *compact.* In this case, compactly supported global sections are just the same thing as ordinary global sections. So is a compact space, and there is thus a *finite* cover of by compact sets , such that there exist liftings of . We can choose these compact sets , moreover, such that their *interiors* cover . This follows easily from the surjectivity of .

Now we need to piece together the . To do this, we reduce by induction to the case , and there are two sections to piece together. Consider the sections . If they agreed on , then we would be. In general, they need not; however, is necessarily a section of as it maps to zero in . By softness, we can extend to a global section of . Then it is clear that the pair of sections of glues to give a global section. We can now derive two useful properties of soft sheaves.

Corollary 8

- If is an exact sequence in and are soft, then is soft too.
- for if is soft.

*Proof:*

- Let be a closed subset. We know by \cref{softexact} that is surjective. Similarly, is surjective. In the diagram we see that the horizontal maps and the left vertical map are all surjective. It follows that is surjective, which implies softness.
- This now follows by a general lemma on acyclicity, proved in \cite{Tohoku}. We can just repeat it for the special case of interest. Let us first prove that for soft. To do this, imbed in an injective sheaf ; the cokernel is soft by the previous part of the result. The exact sequence
shows that for

*any*soft sheaf . Now assume inductively that for any soft sheaf . Then the isomorphisms and the inductive hypothesis applied to (which is also soft) complete the inductive step.

June 11, 2011 at 2:59 pm

[…] Akhil Mathew: The nPOV, sheathes, and derived categories, Verdier duality, Soft sheaves […]