This is the third in a series of posts started here (in particular, the notation is kept from there) intended to cover the basics of Verdier duality. Here, I will discuss the lower shriek functors needed even to state Verdier duality (in the most general form, at least); as we will see, the class of soft sheaves will be acyclic with respect to this functor. To see this, though, we shall need to prove some general facts on how push-forward behave with respect to base change, which are themselves of independent interest.

**1. The functors **

Let be a map of spaces. We have defined the functor

earlier, such that consists of the sections of whose support is proper over ; is always a subsheaf of , equal to it if is proper. When is a point, we get the functor

One can check that is in fact a sheaf. The observation here is that a map of topological spaces is proper if and only if there is an open cover of such that is proper for each . Now is a left-exact functor, as one easily sees. We now want to show that the class of soft sheaves is acyclic with respect to , and in particular so that one may use soft resolutions to compute the derived functors. To do this, we shall prove a general “base change” theorem that will compute the stalk of .

Definition 1As usual, denotes the th right derived functor of . There is also a total derived functor on the derived categories (and similarly ).

**2. Base change theorems **

To understand the functor and its derived functors , we will need to determine their stalks. We start by describing the situation for a *proper* map.

Theorem 2 (Proper base change)Let be a proper map of locally compact spaces. If , then there is a functorial isomorphism

for each .

Here is the restriction of to , which in turn is the fiber over .

*Proof:* Note that both are -functors from to the category of abelian groups. Let us first define the natural transformation. Indeed, we know that is the sheaf on associated to the presheaf

As a result, the stalk at is the direct limit \( \varinjlim_U H^i(f^{-1}(U), \mathcal{F}). \) Each maps naturally to , so we get the natural map (of -functors, even). Let us show that it is an isomorphism when . This equates to saying that the map

is an isomorphism.

But since is a closed map, and is compact, it follows that the sets for open and containing form a cofinal family in the set of open sets containing . The claim is now clear because a section over a compact set automatically extends to a small neighborhood.

Finally, we need to show that both functors are effaceable in positive dimensions. For , this is immediate. For the other functor, let us show that if is soft, then for . Since any sheaf can be imbedded in a soft (e.g. flabby) sheaf, this will be enough. But this in turn is clear because is itself then soft:

Lemma 3The restriction of a soft sheaf to a locally closed subspace is soft.

*Proof:* Indeed, this follows because if is soft and , then any compact subset of is a compact subset of . So a section of over can be extended all the way over , and a fortiori over .

Here is the analog for the functor:

Theorem 4Let be a continuous map of locally compact spaces. If , then there is a functorial isomorphism

for each .

*Proof:* This is now proved as before. Again, the key point is that a map of -functors can easily be defined, which is an isomorphism in degree zero, and both functors are effaceable in positive degrees.

But there are some subtleties! For one thing, it is *not* true that is the sheaf associated to the presheaf . This fails even if (take the identity map, for instance). However, by general nonsense, if we can define an isomorphism in degree zero, and show that both functors are effaceable, then we will be done. Both functors are effaceable since is effaceable (being a derived functor) while vanishes for soft sheaves.

So we just have to check that . Now there is a natural that sends a section of , with support proper over , to the restriction to . This map is injective; for if a section with proper support was zero on , then the image of in would not contain . But by properness this image is closed, so we can find a smaller neighborhood containing such that . Now we need to check surjectivity.

This is a bit tricky to do directly, but fortunately a trick helps out here. We know that is surjective when is soft: this is easy to see (for an element of can be extended to all of with compact, and certainly proper support). Now given , we find an exact sequence

with soft (e.g. injective). Since the map from the stalk to is an isomorphism for , and since both functors are obviously left exact, a diagram chase shows that the map is an isomorphism for .

**3. Applications **

We start by proving a result that provides some content to the phrase “base change.”

Theorem 5Consider a cartesian diagram of locally compact Hausdorff spaces:

Then there is a natural isomorphism, for any ,

Taking cohomology, one sees that for a sheaf , one has natural isomorphisms

*Proof:*One may define this map by the universal property, on the level of complexes. Namely, let be a complex of sheaves. We will define a map

To do this, we may as well assume is a single sheaf (by naturality). So we are reduced to defining a map

This is equivalent to defining a map . Given a section of over an open set , or equivalently a section of with proper support over , we can consider this as a section of over with proper support over , and thus over . This is equivalently a section of over , or a section of over .

So we can get the base change morphism, which is clearly natural. From here it is easy to see that it can be defined even in the derived category. To check that it is an isomorphism, we reduce by general facts on “way-out” functors (proved in Hartshorne’s Residues and Duality, for instance) to showing that the map is an isomorphism for a single sheaf , or equivalently that

But now this follows by taking the stalks at some ; on the left, we get , and on the right we get , which are both the same since these are isomorphic spaces. (By abuse of notation we have written for the restrictions to various subspaces.)

As another example of these base change theorems, we prove the promised result that soft sheaves are acyclic with respect to the lower shriek.

Proposition 6Let be an exact sequence of sheaves on . Suppose is soft. Then the sequence is also exact, and is cohomologically concentrated in degree zero (or, what is the same thing, vanish for ).

*Proof:* It follows that the sequence is exact if . But more generally because taking stalks at each gives . Thus, *soft* resolutions will suffice to compute , which will be very convenient.

As another example of base change, consider an open immersion . By looking at stalks (and defining a natural map), we find that for is just “extension by zero.”

**4. The Leray spectral sequence for **

In fact, even preserves soft sheaves, which leads to a Leray spectral sequence for .

Proposition 7If is continuous and soft, then is soft too.

*Proof:* Indeed, let be a compact set, and suppose is a section, so we know that extends to a small neighborhood of ; call the extension . Then this extension of becomes a section of with proper support.

Restricting to a *compact* neighborhood of , we get a compactly supported section of , which we can extend to all of so as to have compact support (and thus get a global section of ).

Corollary 8 (Leray spectral sequence)Given maps of locally compact Hausdorff spaces, there is a natural isomorphism , and a spectral sequence for any ,

*Proof:* This is now clear, because maps injective sheaves (which are flasque, hence soft) to soft sheaves, which are -acyclic, and we can apply the general theorem.

June 13, 2011 at 5:50 pm

[…] 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 […]