(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 1A locally compact space hascohomological dimensionif 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 2if 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 4Let 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 5If 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 6If 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 7The 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 8If 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 9If is a locally compact space and a covering of by open sets, then

The same holds for afinitecovering 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 10The 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 11The 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 12Let 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 13If 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 14Any 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).