(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 F is a left-exact functor on some abelian category with enough injectives and T^\bullet is an acyclic complex consisting of F-acyclic objects, then F(T^\bullet) 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 {X} has cohomological dimension {n} if {H^k_c(X, \mathcal{F}) =0} for any sheaf {\mathcal{F} \in \mathbf{Sh}(X)} and {k > n}, and {n} is the smallest integer with these properties. We shall write {\dim X} for the cohomological dimension of {X}.

A point, for instance, has cohomological dimension zero. For here the global section functor is an equivalence of categories between {\mathbf{Sh}(\left\{\ast\right\})} and the category of abelian groups. Our first major goal will be to show that any interval in {\mathbb{R}} has cohomological dimension one.

If we had used ordinary (not compactly supported) cohomology, this would follow from the fact that the topological dimension of {\mathbb{R}} is one: namely, one has a cofinal set of coverings of {\mathbb{R}} 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 {H^n(\mathbb{R}, \mathcal{F}) = 0} if {n \geq 2} and {\mathcal{F} \in \mathbf{Sh}(\mathbb{R})} 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, {k} will be a noetherian ring.

Lemma 3 A sheaf {\mathcal{F} \in \mathbf{Sh}(X)} is soft if and only if {H^1_c(U, \mathcal{F}) = 0} whenever {U \subset X} is open.

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 {Z \subset X} is any closed set, then there is a short exact sequence

\displaystyle 0 \rightarrow j_! j^* \mathcal{F} \rightarrow \mathcal{F} \rightarrow i_* i^* \mathcal{F} \rightarrow 0

where {j: X- Z \rightarrow X} is the inclusion, {i: Z \rightarrow X} is the inclusion, and {j_!} denotes extension by zero. If we apply {\Gamma_c}, we get a piece of the long exact sequence

\displaystyle \Gamma_c(X, \mathcal{F}) \rightarrow \Gamma_c(Z, \mathcal{F}) \rightarrow H^1_c(X, j_! j^* \mathcal{F}).

The claim is that {H^1_c(X, j_! j^* \mathcal{F}) \simeq H^1_c(U, \mathcal{F})}. If we have proved this, it will follow that {\Gamma_c(X, \mathcal{F}) \rightarrow \Gamma_c(Z, \mathcal{F})} is always surjective, implying softness. But this follows from the Leray spectral sequence

\displaystyle H^{p}_c(X, R^q j_! \mathcal{G}) \implies H_c^{p+q}(U, \mathcal{G})

valid for any {\mathcal{G} \in \mathbf{Sh}(U)}, and the fact that {j_!} is exact. We saw as a corollary of the above proof that whenever {Z \subset X} was closed, {U = X - Z}, then there is a long exact sequence

\displaystyle \dots \rightarrow H^i_c(U, \mathcal{F}) \rightarrow H^i_c(X,\mathcal{F}) \rightarrow H^i_c(Z, \mathcal{F}) \rightarrow H^{i+1}_c(U, \mathcal{F}) \rightarrow \dots.

This long exact sequence does not make sense in ordinary cohomology. In fact, there is no reason to expect maps {\Gamma(U, \mathcal{F}) \rightarrow \Gamma(X, \mathcal{F})} in the first place. This does, however, make perfect sense in compactly supported cohomology, because one simply extends by zero.

Proposition 4 Let {X} be a locally compact space, and {n = \dim X}, then in any sequence

\displaystyle 0 \rightarrow \mathcal{F}_0 \rightarrow \mathcal{F}_1 \rightarrow \dots \rightarrow \mathcal{F}_{n+1} \rightarrow 0, \ \ \ \ \ (1)

if {\mathcal{F}_1, \dots, \mathcal{F}_{n}} are all soft, so is {\mathcal{F}_{n+1}}. Conversely, if in every exact sequence (1), the softness of {\mathcal{F}_1, \dots, \mathcal{F}_n} implies that of {\mathcal{F}_0}, it follows that {\dim X \leq n}.

Notice that no hypothesis is made on {\mathcal{F}_0}Proof: We know that each of the {\mathcal{F}_i, 1 \leq i \leq n}, has no (compactly supported) cohomology above dimension one. By a standard “dimension-shifting” argument, it follows that {H^1_c(U, \mathcal{F}_{n+1}) \simeq H^{n+1}_c(U, \mathcal{F}_0) =0} for each {U \subset X} open. This implies softness by \cref{whensoft}. Let us now prove the last claim. If {\mathcal{F} \in \mathbf{Sh}(X)} is any sheaf, then there is an injective resolution

\displaystyle 0 \rightarrow \mathcal{F} \rightarrow \mathcal{I}^0 \rightarrow \mathcal{I}^1 \rightarrow \dots.

We truncate this resolution at the length {n}; that is, we consider the sequence

\displaystyle 0 \rightarrow \mathcal{F} \rightarrow \mathcal{I}^0 \rightarrow \mathcal{I}^1 \rightarrow \dots \mathcal{I}^{n-1} \rightarrow \mathrm{im}(\mathcal{I}^{n-1} \rightarrow \mathcal{I}^n) \rightarrow 0.

By assumption, the last term is soft; it follows that any sheaf has a soft resolution of length at most {n}. Since soft resolutions can be used to compute compactly supported cohomology, it follows that {H^i_c(X, \mathcal{F}) =0} if {i > n}, proving the claim.

Corollary 5 If {X} is a finite-dimensional space and {\mathcal{F}} is a sheaf admitting a soft resolution from the left, then {\mathcal{F}} is soft.

We shall use the following fact in the proof of Verdier duality.

Corollary 6 If {X} is a finite-dimensional space, {\mathcal{F}, \mathcal{G} \in \mathbf{Sh}(X,k)}, and {\mathcal{G}} is soft and {k}-flat, then {\mathcal{F} \otimes_k \mathcal{G}} is soft as well.

Here “{k}-flat” means that the stalks are flat {k}-modules. Proof: Indeed, we can find a resolution of {\mathcal{F}},

\displaystyle \dots \rightarrow \mathcal{F}_3 \rightarrow \mathcal{F}_2 \rightarrow \mathcal{F}_1 \rightarrow \mathcal{F} \rightarrow 0,

where each {\mathcal{F}_i} is a direct sum of sheaves of the form {j_!(k)} for {j} an inclusion of an open subset of {X}. (Indeed, we can use such sheaves to hit each section of {\mathcal{F}}.) Since {\mathcal{G}} is {k}-flat, we get a resolution of {\mathcal{F} \otimes_k \mathcal{G}},

\displaystyle \dots \rightarrow \mathcal{F}_3 \otimes_k \mathcal{G} \rightarrow \mathcal{F}_2 \otimes_k \mathcal{G} \rightarrow \mathcal{F}_1 \otimes_k \mathcal{G} \rightarrow \mathcal{F} \otimes_k \mathcal{G} \rightarrow 0,

where each of the {\mathcal{F}_i \otimes_k \mathcal{G}} is soft. Indeed, any sheaf of the form {j_!(k) \otimes_k \mathcal{G} \simeq j_! (\mathcal{G}|_U)} (if {j: U \rightarrow X} is the inclusion) is soft because {\mathcal{G}} is soft. We saw this in the proof of the Leray spectral sequence for the lower shriek. Now by the softness of {\mathcal{F} \otimes_k \mathcal{G}} is clear.

This shows that if {k \rightarrow \mathcal{L}^\bullet} is a soft, flat resolution of the constant sheaf {k}, then there is a functorial soft resolution {\mathcal{F} \rightarrow \mathcal{F} \otimes_k \mathcal{L}^\bullet } of any {\mathcal{F} \in \mathbf{Sh}(X, k)}. Of course, we should check that {k} 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 {\mathcal{F} \in \mathbf{Sh}(X, k)} into a soft (even flasque) sheaf {C(\mathcal{F}) = \prod_{x \in X} (i_x)_* \mathcal{F}_x}, where {i_x: \left\{x\right\} \rightarrow X} is the inclusion. The Godement resolution of a given sheaf is a quasi-isomorphism

\displaystyle \mathcal{F} \rightarrow C^\bullet(\mathcal{F}),

where {C^0(\mathcal{F}) = C(\mathcal{F}), C^1(\mathcal{F}) = C(\mathrm{coker}(\mathcal{F} \rightarrow C^0(\mathcal{F}))}, 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 {C(\mathcal{F})} is always flasque. We now need to show that the sheaves are flat. If {\mathcal{F}} is flat, then {C(\mathcal{F})} is flat (as a {k}-sheaf) because, for a noetherian ring, the product (even infinite!) of flat modules is flat. Moreover, because the map {\mathcal{F} \rightarrow C(\mathcal{F})} is a split injection on the level of stalks, the cokernel is also flat, and we can see that {C^\bullet(\mathcal{F})} is flat by induction.

We shall use the Godement resolution of {k} 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 {Z \subset X} is a locally closed subspace, then {\dim Z \leq \dim X}.

Proof: We need to check this for open subspaces and for closed subspaces. If {Z \subset X} is closed, then for any {\mathcal{F} \in \mathbf{Sh}(Z)}, we have that

\displaystyle H^i_c(Z, \mathcal{F}) \simeq H^i_c(Z, i_* \mathcal{F})

for {i: Z \rightarrow X} the inclusion, as one may see from (for instance) the Leray spectral sequence. Thus the result {\dim Z \leq \dim X} is clear. If {U \subset X} is open with {j: U \rightarrow X} the inclusion, then we know that

\displaystyle \Gamma_c(X, j_! \mathcal{G}) \simeq \Gamma(U, \mathcal{G}), \quad \mathcal{G} \in \mathbf{Sh}(U).

The corresponding identity on derived functors (which follows from the Leray spectral sequence, as {j_!} is exact, for instance) yields

\displaystyle H^i_c(X, j_! \mathcal{G}) \simeq H^i_c(U, \mathcal{G}) ,

which easily implies that {\dim U \leq \dim X}. Nonetheless, dimension is a “local” invariant by the following fact:

Proposition 9 If {X} is a locally compact space and {\left\{U_\alpha\right\}} a covering of {X} by open sets, then

\displaystyle \dim X = \sup \dim U_\alpha.

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 {\mathcal{F} \in \mathbf{Sh}(X)} and there is a cover {\left\{U_\alpha\right\}} of {X} (which is either open or a finite cover by compact sets) such that {\mathcal{F}|_{U_\alpha}} is soft for each {U_\alpha}, then {\mathcal{F}} is itself soft.

This may be seen as follows. We first treat the compact case, so that {X} is compact. Let {Z \subset X} be a compact set and {s \in \Gamma(Z, \mathcal{F})} be a section. Let {U_1, \dots, U_k} be compact sets covering {X} such that {\mathcal{F}|_{U_i}} is soft for each {i}. Then we want to show that {\mathcal{F}} is soft. We will construct extensions {s_i} over {Z \cup U_1 \cup \dots \cup U_i}. When {i=0}, there is nothing to do. If {s_{i-1}} is constructed, then we may extend {s_{i-1}|_{(Z \cup U_1 \cup \dots \cup U_i) \cap U_{i+1}}} to {U_{i+1}} and glue this extension with {s_{i-1}} to get {s_i}. This procedure stops and eventually gives a section {s_k} over {X} extending {s}.

Now suppose {X} is arbitrary and the {\left\{U_\alpha\right\}} are an open covering. Let {s} be a section over a compact set {Z}, and let {Z'} be a compact set containing {Z} in its interior. Note that there is a finite compact covering of {Z'} on which {\mathcal{F}} is soft, by refining the {\left\{U_\alpha\right\}}. We can extend the section {\widetilde{s} \in \Gamma(Z \cup \partial Z', \mathcal{F})} given by {s} on {Z} and {0} on {\partial Z'} to {Z'} by what has already been proved; this extends automatically by zero to all of {X}.

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 {H^i_c(X, \cdot)} commute with filtered colimits of sheaves.

Note that this already fails with {i=0} if ordinary cohomology is used. For instance, the global section functor does not commute with arbitrary direct sums: take for instance {X = \mathbb{R}}, and the inclusions {(i_n)_* \mathbb{Z}} where {i_n: \left\{n\right\} \rightarrow \mathbb{R}} is the inclusion. Then

\displaystyle \Gamma(X, \bigoplus_{\mathbb{Z}} (i_n)_* \mathbb{Z}) = \prod_{\mathbb{Z}} \mathbb{Z} \neq \bigoplus_{\mathbb{Z}} \mathbb{Z}.

Proof: Let us first prove this form {i=0}. We need to show that if {\left\{\mathcal{F}_\alpha\right\}} is a filtered system of sheaves, then

\displaystyle \varinjlim \Gamma_c(X, \mathcal{F}_\alpha) \rightarrow \Gamma_c(X, \varinjlim \mathcal{F}_\alpha)

is an isomorphism. Let {t_{\beta \gamma}, \beta \leq \gamma}, be the transition maps of the directed system.

  1. (Injectivity) Suppose {s \in \Gamma_c(X, \mathcal{F}_\alpha)} maps to zero in the filtered colimit. We need to show that it maps to zero in some {\mathcal{F}_\beta}. By assumption (since {\varinjlim} commutes with stalks), for each {x \in X} there is a neighborhood {N_x} of {x} and a {\beta_x} such that {t_{\alpha \beta_x} (s)|_{N_x} = 0}. We can find a finite collection of the {\left\{N_x\right\}} that cover {\mathrm{supp}(s)}, and find a {\beta} majoring them all. Then {t_{\alpha \beta}(s) = 0}.
  2. (Surjectivity) Suppose given {\sigma \in \Gamma_c(X, \varinjlim \mathcal{F}_\alpha)}. We must show that it comes from some {\mathcal{F}_\beta}. Let {Z} be a compact set containing the support of {\sigma} in its interior. For each {x \in Z} there is a neighborhood {N_x} of {x}, a {\beta_x}, and a section {s_x \in \mathcal{F}_{\beta_x}(N_x)} mapping to {\sigma} over {N_x}. We can find finitely many (say {x_1, \dots, x_N}) that cover {Z}. Moreover, choosing {\beta} larger than all the {\beta_{x_i}}, and even larger to make the {s_{x_i}} glue on the finitely many intersections {N_{x_i} \cap N_{x_j}}, we can obtain a section {s} over {Z} that maps to {\sigma}. Choosing {\beta} even larger, we can assume {s|_{\partial Z} = 0}, so that {s} 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 {\left\{F_\alpha\right\}} be a directed system of soft sheaves. We know then that for each {\alpha} and {Z \subset X} closed,

\displaystyle \Gamma_c(X, \mathcal{F}_\alpha) \rightarrow \Gamma_c(Z, \mathcal{F}_\alpha)

is a surjection. Taking the colimit and using the {i=0} case of the above result, we find that

\displaystyle \Gamma_c(X, \varinjlim \mathcal{F}_\alpha) \rightarrow \Gamma_c(Z, \varinjlim \mathcal{F}_\alpha)

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 {\mathbf{Sh}(X)}. On this we have two {\delta}-functors given by

\displaystyle \left\{\mathcal{F}_\alpha\right\} \mapsto H^\bullet_c(X, \varinjlim \mathcal{F}_\alpha), \quad \left\{\mathcal{F}_\alpha\right\} \mapsto \varinjlim H^\bullet(X, \mathcal{F}_\alpha).

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 {\delta}-functors are naturally isomorphic.

Corollary 12 Let {\mathcal{C}} be a filtered system of closed subsets of a topological space {X}. Let {F = \bigcap_{Z \in \mathcal{C}} F}. Then for any {\mathcal{F} \in \mathbf{Sh}(X)}, the map

\displaystyle \varinjlim_{\mathcal{C}} H^i_c(Z, \mathcal{F}) \rightarrow H^i_c(F, \mathcal{F})

is an isomorphism.

Proof: For each closed subset {Z \subset X}, let {i_Z: Z \rightarrow X} be the imbedding. We consider the sheaves {(i_Z)_* (i_Z)^* \mathcal{F}} for {Z \in \mathcal{C}}. These form a direct system on {X}, whose colimit is {(i_F)_* (i_F)^* \mathcal{F}}. For whenever {Z_1 \subset Z_2}, there is a natural map

\displaystyle (i_{Z_2})_* (i_{Z_2})^* \mathcal{F} \rightarrow (i_{Z_1})_* (i_{Z_1})^* \mathcal{F}

obtained from the adjointness relation, for instance. The colimit of these is {(i_F)_* (i_F)^* \mathcal{F}}. If one applies {H^\bullet_c} 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 {\mathbb{R}}: namely, the fact that {\mathbb{R}} can be covered in many ways as {\mathbb{R} = F_1 \cup F_2} where {F_1, F_2} 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 {\mathcal{F} \in \mathbf{Sh}(X)} be a sheaf, and let {A, B \subset X} be closed subsets. Then there is a short exact sequence of sheaves (with the “{i}‘s” the inclusions)

\displaystyle 0 \rightarrow (i_{A \cup B})_* (i_{A \cup B})^*\mathcal{F} \rightarrow (i_{A })_* (i_{A })^*\mathcal{F} \oplus (i_{B })_* (i_{B })^*\mathcal{F} \rightarrow (i_{A \cap B })_* (i_{A \cap B})^*\mathcal{F} \rightarrow 0.

Exactness can simply be checked on stalks, while the maps themselves are defined using the adjointness of {i_*} and {i^*}. There is consequently a long exact sequence in cohomology

\displaystyle \dots \rightarrow H_c^i(A \cup B, \mathcal{F}) \rightarrow H_c^i(A, \mathcal{F}) \oplus H_c^i(B, \mathcal{F}) \rightarrow H_c^i(A \cap B, \mathcal{F}) \rightarrow H^{i+1}(A \cup B, \mathcal{F} ) \rightarrow \dots.

Theorem 13 If {X} is a locally compact space, then {\dim X \times \mathbb{R} \leq \dim X + 1}.

Proof: This is of interest only when {\dim X < \infty}. Suppose conversely that there existed a sheaf {\mathcal{F} \in \mathbf{Sh}(X)}, and a nonzero cohomology class {\gamma \in H^{m}_c(X, \mathcal{F})}, where {m > \dim X + 1}.

Consider the class of all closed subsets {F \subset \mathbb{R}}. For each of these, there is a space {Y_F = X \times F \subset X \times \mathbb{R}}. We can restrict {\gamma} to each {Y_F}, obtaining a family of classes {\gamma_F \in H^{m}_c(Y_F, \mathcal{F})}. We know that {\gamma_{\mathbb{R}}} is not zero. The claim is that there is a minimal {F \subset \mathbb{R}} such that {\gamma_F} is not zero. If this is true, then we will obtain the result easily. For then choose a splitting {F = F_1 \cup F_2} where {F_1, F_2 \subsetneq F} and {F_1 \cap F_2} is a point (e.g. {F_1 = [t, \infty) \cap F, F_2 = (-\infty, t] \cap F}). Then there is a Mayer-Vietoris sequence

\displaystyle H^{m-1}_c(Y_{F_1 \cap F_2}, \mathcal{F}) \rightarrow H_c^m(Y_{F_1 \cup F_2}, \mathcal{F}) \rightarrow H_c^m(Y_{F_1}, \mathcal{F}) \oplus H_c^m(Y_{F_2}, \mathcal{F}) \rightarrow \dots.

The first term is zero since {Y_{F_1 \cap F_2} \simeq X} and {m-1> \dim X}. It follows that one of the restrictions of {\gamma} to {F_1} or {F_2} must be nonzero, contradicting minimality.

But now we need to see that there is such a minimal {F}. By Zorn’s lemma, we need to show that every totally ordered collection of closed subsets {F \subset \mathbb{R}} with {\gamma_F \neq 0} has an intersection on which {\gamma} 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 {\mathbb{R}^n}. But {\dim \mathbb{R}^n \leq n} by the above result. For {\dim \left\{\ast\right\} = 0} clearly, and by induction we find {\dim \mathbb{R}^n \leq 1 + \dim \mathbb{R}^{n-1}}.

The dimension of {\mathbb{R}^n} is in fact {n}. This follows because {H^n_c(\mathbb{R}^n, \mathbb{R}) = \mathbb{R}}, for instance; this can be seen using de Rham cohomology.

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