(This is the fourth in a series of posts intended to cover the basics of Verdier duality, starting here.)
One of the features of derived categories that seems to require the most bookkeeping is the size. Many results apply specifically to the bounded-below or bounded-above derived categories, for instance; the problem is, in general, with statements like the following. If is a left-exact functor on some abelian category with enough injectives and
is an acyclic complex consisting of
-acyclic objects, then
is not necessarily acyclic (though it is if the complex is bounded below). Dimensionality bounds will, for apparently similar reasons, play a crucial role in the proof of Verdier duality, and it will be necessary to show that the spaces in question are fairly nice. I will try to explain the necessary tools in this post, after which we can actually start the proof.
1. Cohomological dimension
The Verdier duality theorem will apply not only to manifolds, but more generally to locally compact spaces of finite cohomological dimension, and it will thus be useful to show that simple spaces (e.g. finite-dimensional CW complexes) satisfy this condition. The resulting theory will also show that much of basic algebraic topology can be done entirely using sheaf cohomology.
Definition 1 A locally compact space
has cohomological dimension
if
for any sheaf
and
, and
is the smallest integer with these properties. We shall write
for the cohomological dimension of
.
A point, for instance, has cohomological dimension zero. For here the global section functor is an equivalence of categories between and the category of abelian groups. Our first major goal will be to show that any interval in
has cohomological dimension one.
If we had used ordinary (not compactly supported) cohomology, this would follow from the fact that the topological dimension of is one: namely, one has a cofinal set of coverings of
such that any intersection of three distinct elements is trivial (e.g. using open intervals). Since Cech cohomology suffices to compute sheaf cohomology on a paracompact space, it follows that:
Theorem 2
if
and
is any sheaf.
It will be a bit trickier to obtain this for compactly supported cohomology. Before this, we shall develop a number of elementary ideas.
2. Softness and dimension
To start with, we shall characterize cohomological dimension using softness. Throughout, will be a noetherian ring.
There is an analog of this result for flasque sheaves that uses local cohomology. Proof: We already know that softness implies this. Conversely, suppose the vanishing hypothesis satisfied. If is any closed set, then there is a short exact sequence
where is the inclusion,
is the inclusion, and
denotes extension by zero. If we apply
, we get a piece of the long exact sequence
The claim is that . If we have proved this, it will follow that
is always surjective, implying softness. But this follows from the Leray spectral sequence
valid for any , and the fact that
is exact. We saw as a corollary of the above proof that whenever
was closed,
, then there is a long exact sequence
This long exact sequence does not make sense in ordinary cohomology. In fact, there is no reason to expect maps in the first place. This does, however, make perfect sense in compactly supported cohomology, because one simply extends by zero.
Proposition 4 Let
be a locally compact space, and
, then in any sequence
if
are all soft, so is
. Conversely, if in every exact sequence (1), the softness of
implies that of
, it follows that
.
Notice that no hypothesis is made on . Proof: We know that each of the
, has no (compactly supported) cohomology above dimension one. By a standard “dimension-shifting” argument, it follows that
for each
open. This implies softness by \cref{whensoft}. Let us now prove the last claim. If
is any sheaf, then there is an injective resolution
We truncate this resolution at the length ; that is, we consider the sequence
By assumption, the last term is soft; it follows that any sheaf has a soft resolution of length at most . Since soft resolutions can be used to compute compactly supported cohomology, it follows that
if
, proving the claim.
Corollary 5 If
is a finite-dimensional space and
is a sheaf admitting a soft resolution from the left, then
is soft.
We shall use the following fact in the proof of Verdier duality.
Corollary 6 If
is a finite-dimensional space,
, and
is soft and
-flat, then
is soft as well.
Here “-flat” means that the stalks are flat
-modules. Proof: Indeed, we can find a resolution of
,
where each is a direct sum of sheaves of the form
for
an inclusion of an open subset of
. (Indeed, we can use such sheaves to hit each section of
.) Since
is
-flat, we get a resolution of
,
where each of the is soft. Indeed, any sheaf of the form
(if
is the inclusion) is soft because
is soft. We saw this in the proof of the Leray spectral sequence for the lower shriek. Now by the softness of
is clear.
This shows that if is a soft, flat resolution of the constant sheaf
, then there is a functorial soft resolution
of any
. Of course, we should check that
in fact has a soft, flat resolution.
To get this, we can use the Godement resolution, which is based on an imbedding of a sheaf into a soft (even flasque) sheaf
, where
is the inclusion. The Godement resolution of a given sheaf is a quasi-isomorphism
where , and so on.
Lemma 7 The Godement resolution of a flat sheaf is a complex of soft, flat sheaves.
Proof: It is clear that the Godement resolution consists of soft (even flasque) sheaves because is always flasque. We now need to show that the sheaves are flat. If
is flat, then
is flat (as a
-sheaf) because, for a noetherian ring, the product (even infinite!) of flat modules is flat. Moreover, because the map
is a split injection on the level of stalks, the cokernel is also flat, and we can see that
is flat by induction.
We shall use the Godement resolution of to get a functorial “soft replacement” in the derived category in the proof of Verdier duality.
2. Subspaces
Verdier duality applies to spaces of finite (cohomological) dimension, so we shall need some criteria to establish that a space is indeed of this form. Here we shall handle some of the simplest ones. We now want to show:
Proposition 8 If
is a locally closed subspace, then
.
Proof: We need to check this for open subspaces and for closed subspaces. If is closed, then for any
, we have that
for the inclusion, as one may see from (for instance) the Leray spectral sequence. Thus the result
is clear. If
is open with
the inclusion, then we know that
The corresponding identity on derived functors (which follows from the Leray spectral sequence, as is exact, for instance) yields
which easily implies that . Nonetheless, dimension is a “local” invariant by the following fact:
Proposition 9 If
is a locally compact space and
a covering of
by open sets, then
The same holds for a finite covering by compact sets.
Proof: This now follows from the above criteria and the technical fact that a sheaf which is locally soft is soft. That is, if and there is a cover
of
(which is either open or a finite cover by compact sets) such that
is soft for each
, then
is itself soft.
This may be seen as follows. We first treat the compact case, so that is compact. Let
be a compact set and
be a section. Let
be compact sets covering
such that
is soft for each
. Then we want to show that
is soft. We will construct extensions
over
. When
, there is nothing to do. If
is constructed, then we may extend
to
and glue this extension with
to get
. This procedure stops and eventually gives a section
over
extending
.
Now suppose is arbitrary and the
are an open covering. Let
be a section over a compact set
, and let
be a compact set containing
in its interior. Note that there is a finite compact covering of
on which
is soft, by refining the
. We can extend the section
given by
on
and
on
to
by what has already been proved; this extends automatically by zero to all of
.
3. Cohomology and filtered colimits
It is a basic fact that sheaf cohomology commutes with filtered colimits on a noetherian space. Of course, the spaces of interest here (locally compact Hausdorff spaces) are anything but noetherian. Nonetheless, we shall find useful the following result:
Proposition 10 The functors
commute with filtered colimits of sheaves.
Note that this already fails with if ordinary cohomology is used. For instance, the global section functor does not commute with arbitrary direct sums: take for instance
, and the inclusions
where
is the inclusion. Then
Proof: Let us first prove this form . We need to show that if
is a filtered system of sheaves, then
is an isomorphism. Let , be the transition maps of the directed system.
- (Injectivity) Suppose
maps to zero in the filtered colimit. We need to show that it maps to zero in some
. By assumption (since
commutes with stalks), for each
there is a neighborhood
of
and a
such that
. We can find a finite collection of the
that cover
, and find a
majoring them all. Then
.
- (Surjectivity) Suppose given
. We must show that it comes from some
. Let
be a compact set containing the support of
in its interior. For each
there is a neighborhood
of
, a
, and a section
mapping to
over
. We can find finitely many (say
) that cover
. Moreover, choosing
larger than all the
, and even larger to make the
glue on the finitely many intersections
, we can obtain a section
over
that maps to
. Choosing
even larger, we can assume
, so that
extends to the entire space as a compactly supported section.
To prove it for higher dimensions, we shall use:
Lemma 11 The filtered colimit of soft sheaves is soft.
Proof: Let be a directed system of soft sheaves. We know then that for each
and
closed,
is a surjection. Taking the colimit and using the case of the above result, we find that
is a surjection as well. This implies the colimit is soft. The remainder of the proof is now formal, just as in Hartshorne’s Algebraic Geometry (for a noetherian space and ordinary cohomology). Namely, we can consider the category of filtered systems (by a given fixed filtering category) of elements of . On this we have two
-functors given by
They are isomorphic in degree zero, and moreover both are effaceable. The first is effaceable because we can imbed any filtered system into a filtered system of soft sheaves (e.g. using the Godement resolution), and the second is effaceable by the same effacement because the filtered colimit of soft sheaves is soft. Now general nonsense implies that the two -functors are naturally isomorphic.
Corollary 12 Let
be a filtered system of closed subsets of a topological space
. Let
. Then for any
, the map
is an isomorphism.
Proof: For each closed subset , let
be the imbedding. We consider the sheaves
for
. These form a direct system on
, whose colimit is
. For whenever
, there is a natural map
obtained from the adjointness relation, for instance. The colimit of these is . If one applies
to this relation, one gets the result.
4. Dimension bounds
We can now finally bound the dimensions of standard spaces. The proof will use the geometric nature of : namely, the fact that
can be covered in many ways as
where
intersect in only a point and are closed (e.g. via two intervals).
We shall also need the Mayer-Vietoris sequence in sheaf cohomology. Let be a sheaf, and let
be closed subsets. Then there is a short exact sequence of sheaves (with the “
‘s” the inclusions)
Exactness can simply be checked on stalks, while the maps themselves are defined using the adjointness of and
. There is consequently a long exact sequence in cohomology
Theorem 13 If
is a locally compact space, then
.
Proof: This is of interest only when . Suppose conversely that there existed a sheaf
, and a nonzero cohomology class
, where
.
Consider the class of all closed subsets . For each of these, there is a space
. We can restrict
to each
, obtaining a family of classes
. We know that
is not zero. The claim is that there is a minimal
such that
is not zero. If this is true, then we will obtain the result easily. For then choose a splitting
where
and
is a point (e.g.
). Then there is a Mayer-Vietoris sequence
The first term is zero since and
. It follows that one of the restrictions of
to
or
must be nonzero, contradicting minimality.
But now we need to see that there is such a minimal . By Zorn’s lemma, we need to show that every totally ordered collection of closed subsets
with
has an intersection on which
is nonzero. But this is clear from the general facts about filtered colimits above.
Corollary 14 Any manifold is of finite dimension.
Proof: Indeed, a manifold is covered by open sets homeomorphic to . But
by the above result. For
clearly, and by induction we find
.
The dimension of is in fact
. This follows because
, for instance; this can be seen using de Rham cohomology.
July 9, 2011 at 1:41 pm
[…] Mathew: The lower shriek and base change, Cohomological dimension, The proof of Verdier duality, Verdier duality on manifolds (How to get classical algebraic […]
August 23, 2012 at 10:00 pm
I thought quasi isomorphisms were between complexes, you’re taking a map from a sheaf (one object) to its godement resolution (complex) to be a quasi-isomorphism?
September 6, 2012 at 8:29 pm
Sorry I haven’t responded for so long — it’s been a hectic start to the semester! Any sheaf (or any object in an abelian category) defines a complex (that object in dimension zero and zero elsewhere).