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 1 If
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 finitenumber 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 2 A sheaf
is soft if
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 3 A 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 1 Let
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 2 Similarly, using suitable cutoff functions, it follows that if
is a manifold, then the sheaf of
functions (where
) is soft.
Example 3 If
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 4 If
, 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 5
is 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 6 We write
for the
th (right) derived functor of
. We can also define a total derived functor
on
, 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 7 Let
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
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