Soft sheaves

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 $f^!$ 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 ${X}$ be a locally compact space. If ${\mathcal{F} \in \mathbf{Sh}(X)}$ and ${Z \subset X}$ is closed, we define ${\Gamma(Z, \mathcal{F}) = \Gamma(Z, i^*\mathcal{F})}$ for ${i: Z \rightarrow X}$ 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 ${\mathfrak{X} \rightarrow X}$ is the espace étale of ${\mathcal{F}}$ , then a section over ${Z}$ is the same thing as a section of the projection ${\mathfrak{X} \rightarrow X}$ over ${Z}$ . It follows from the latter interpretation that if we have sections ${s, t}$ on closed subsets ${F, G}$ that agree on ${F \cap G}$ , one obtains uniquely a closed subset on ${F \cup G}$ . Now we want to show that we can recover this notion from the familiar idea of sections over an open set.

Lemma 1 If ${Z \subset X}$ is compact, we have

$\displaystyle \Gamma(Z, \mathcal{F}) \simeq \varinjlim \Gamma(U, \mathcal{F}),$

where ${U}$ ranges over open sets containing ${Z}$ .

Proof: There is an obvious map ${\varinjlim \Gamma(U, \mathcal{F}) \rightarrow \Gamma(Z, \mathcal{F})}$ . It is injective: if ${s \in \Gamma(U, \mathcal{F})}$ restricts to zero over ${Z}$ , then the stalk ${s_z, z \in Z}$ is zero. It follows that there is a neighborhood ${U_z \subset U}$ for each ${z \in Z}$ such that ${s|_{U_z} = 0}$ . But then ${s|_{\bigcup U_z} = 0}$ , so that ${s}$ maps to zero in the colimit. (We have not yet used the compactness of ${Z}$ .) Surjectivity is the hard part. Suppose ${t \in \Gamma(Z, \mathcal{F})}$ is a section over the compact set ${Z}$ . Then we can cover ${Z}$ by neighborhoods ${\left\{N_\alpha\right\}}$ together with sections ${s_\alpha \in \mathcal{F}(N_\alpha)}$ such that

$\displaystyle s_\alpha|_{Z \cap N_\alpha} = t|_{Z \cap N_\alpha}.$

By choosing the ${N_\alpha}$ appropriately and in view of local compactness, we can assume that there are compact subsets ${Y_\alpha \subset N_\alpha}$ , each of which contains an open subset of ${Z}$ , and whose interiors (with respect to ${Z}$ !) cover ${Z}$ . It follows that a finitenumber of the ${\left\{Y_\alpha\right\}}$ cover ${Z}$ . So we are in the following situation. We have a section ${t \in \Gamma(Z, \mathcal{F})}$ which we want to extend to an open neighborhood; a finite set ${Y_1, \dots, Y_k}$ of closed subsets of ${Z}$ that cover, and open sets ${N_1, \dots, N_k}$ with sections ${s_i \in \mathcal{F}(N_i)}$ that agree with ${t}$ on ${Y_i}$ . We want to find an ${s}$ defined over some open neighborhood that looks locally like the ${s_i}$ , near ${Z}$ . To do this, we can assume inductively that ${k=2}$ , extending piece by piece. So then we have two compact subsets ${Y_1, Y_2}$ , and sections ${s_1, s_2}$ defined in a neighborhood of each. We are given that ${s_1|_{Y_1 \cap Y_2} = s_2|_{Y_1 \cap Y_2}}$ ; we now wish to find a section defined on a neighborhood of ${Y_1 \cup Y_2}$ . To do this, choose a small neighborhood ${N}$ of ${Y_1 \cap Y_2}$ such that ${s_1 |_N = s_2|_N}$ , which we can do by the first part of the proof. Then choose disjoint neighborhoods ${N_1, N_2}$ such that ${N_1 \cup N}$ contains ${Y_1}$ , and ${N_2 \cup N}$ contains ${Y_2}$ . It is easy to see that ${s_1|_{N_1}, s_1|_{N}, s_2|_{N_2}}$ all patch appropriately and give the desired section. We shall need the following weaker form of flasqueness.

Definition 2 A sheaf ${\mathcal{F} \in \mathbf{Sh}(X)}$ is soft if ${\mathcal{F}(X) \rightarrow \mathcal{F}(Z)}$ is surjective whenever ${Z \subset X}$ 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 ${\mathcal{F}}$ on a locally compact space ${X}$ is soft if and only if any section of ${\mathcal{F}}$ over a compact subset ${K \subset X}$ can be extended to a compactly supported global section of ${X}$ . Even more precisely, if ${\mathcal{F}}$ is soft, ${K \subset U}$ is an inclusion of a compact set ${K}$ in an open set ${U}$ , then a section over ${K}$ can be extended to a compactly supported global section with support in ${U}$ .

Proof: The definition of softness is that any section over ${K}$ ( ${K}$ compact but arbitrary) can be extended to ${X}$ . So one direction is clear.

We have to show conversely that if ${\mathcal{F}}$ is soft, and ${s \in \Gamma(K, \mathcal{F})}$ is a section over the compact subset ${K}$ , 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 ${U}$ ). To do this, choose a compact set ${L}$ containing ${K}$ in its interior, and with ${L \subset U}$ . Consider the section on ${K \cup \partial L}$ given by ${s}$ on ${K}$ and zero on ${\partial L}$ . By softness, this can be extended to a section ${t}$ of ${\mathcal{F}}$ over ${L}$ that vanishes at the boundary. Now the sections given by ${t}$ on ${L}$ and ${0}$ on ${\overline{X - L}}$ glue and give a compactly supported extension of ${s}$ ; it is clear that it vanishes outside ${U}$ . 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 ${X}$ be a locally compact space, and suppose ${\mathcal{C}}$ is the sheaf of continuous functions on ${X}$ . Then ${\mathcal{C}}$ is a soft sheaf. Indeed, suppose given a section ${s}$ of ${\mathcal{C}}$ over a compact set ${K}$ , i.e. a continuous function defined in some neighborhood ${U}$ of ${K}$ . By multiplying ${s}$ by a globally defined continuous function ${t}$ supported in ${U}$ such that ${t|_K \equiv 1}$ , we get a section of ${\Gamma(U,\mathcal{C})}$ 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 ${M}$ is a manifold, then the sheaf of ${C^p}$ functions (where ${p \in \mathbb{N} \cup \left\{\infty\right\}}$ ) is soft.

Example 3 If ${\mathcal{A}}$ is a soft sheaf of rings, and ${\mathcal{F}}$ is a sheaf of ${\mathcal{A}}$ -modules, then ${\mathcal{F}}$ is a flasque sheaf. Indeed, let ${s \in \Gamma(K, \mathcal{F})}$ be a section. Then ${s}$ comes from a section of ${\mathcal{F}}$ over some open subset ${U}$ containing ${K}$ . We can find a global section ${i \in \Gamma(X, \mathcal{A})}$ with support in ${U}$ that is identically ${1}$ on ${K}$ by softness. (Indeed, this follows from \cref{strongsoft}.) Then ${si}$ extends to a global section extending ${s}$ on ${K}$ . It follows in particular that if ${M}$ is a manifold and ${\mathcal{E}}$ is any (say, smooth) vector bundle, the sheaf of sections of ${\mathcal{E}}$ is a soft sheaf, as it is a sheaf of modules over the sheaf of smooth functions. As an example, the sheaves of ${k}$ -forms are all soft.

2. Compactly supported cohomology

Let ${X}$ be a topological space.

Definition 4 If ${\mathcal{F} \in \mathbf{Sh}(X)}$ , and ${U \subset X}$ is open, we define ${\Gamma_c(U, \mathcal{F}) }$ to be the subgroup of ${\Gamma(U, \mathcal{F})}$ consisting of sections with compact support.

It is easy to see that if ${s, t \in \Gamma(U, \mathcal{F})}$ have compact support, so does ${s+t}$ . So this is indeed a subgroup. Taking ${U = X}$ , we thus get a functor ${\Gamma_c = \Gamma_c(X, \cdot): \mathbf{Sh}(X) \rightarrow \mathbf{Ab}}$ for ${\mathbf{Ab}}$ the category of abelian groups. Note similarly that ${\Gamma_c }$ takes ${\mathbf{Sh}(X, k)}$ to the category of ${k}$ -modules.

Proposition 5 ${\Gamma_c}$ is left-exact.

Proof: Indeed, we know that the ordinary global section functor ${\Gamma}$ is left-exact. Since ${\Gamma_c \subset \Gamma}$ , all that needs to be seen is that if one has a sequence ${0 \rightarrow \mathcal{F}' \rightarrow \mathcal{F} \rightarrow \mathcal{F}''}$ , and ${s \in \Gamma_c(X, \mathcal{F})}$ maps to zero in ${\Gamma_c(X, \mathcal{F}'')}$ , then ${s}$ comes from a compactly supported section in ${\Gamma(X, \mathcal{F}')}$ . But it comes from some (unique!) section of ${\mathcal{F}'}$ , and injectivity of ${\mathcal{F}' \rightarrow \mathcal{F}}$ shows that the section must be compactly supported.

Definition 6 We write ${H^i_c(X, \cdot)}$ for the ${i}$ th (right) derived functor of ${\Gamma_c(X, \cdot)}$ . We can also define a total derived functor ${{\mathbb R} \Gamma_c}$ on ${\mathbf{D}^+(X)}$ , taking values in the (bounded-below) derived category of abelian groups. More generally there is a functor ${{\mathbb R} \Gamma_c : \mathbf{D}^+(X, k) \rightarrow \mathbf{D}^+(k)}$ for ${\mathbf{D}^+(k)}$ the derived category of ${k}$ -modules.

In the classical theory of sheaf cohomology, flasque sheaves (recall that a sheaf ${\mathcal{F}}$ is flasque if ${\mathcal{F}(X) \rightarrow \mathcal{F}(U)}$ 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 ${0 \rightarrow \mathcal{F}' \rightarrow \mathcal{F} \rightarrow \mathcal{F}'' \rightarrow 0}$ be an exact sequence of sheaves on ${X}$ . Suppose ${\mathcal{F}'}$ is soft. Then the sequence ${0 \rightarrow \Gamma_c(X, \mathcal{F}') \rightarrow \Gamma_c(X, \mathcal{F}) \rightarrow \Gamma_c(X, \mathcal{F}'') \rightarrow 0}$ is also exact.

Proof: We only need to prove surjectivity at the end. We shall start by making a reduction. We can assume that ${X}$ is itself compact. Indeed, let ${s \in \mathcal{F}''(X)}$ have compact support ${Z}$ . Choose a compact subset ${Z' \subset X}$ containing ${Z}$ in its interior.

Then by applying the result (assumed for compact spaces) to ${Z'}$ , we get a section of ${\Gamma(Z', \mathcal{F})}$ lifting ${s}$ . Then ${s'|_{\partial Z'}}$ must live inside ${\Gamma(\partial Z', \mathcal{F}')}$ as it maps to zero in ${\mathcal{F}''}$ , and this restriction can be extended to some global section ${t \in \Gamma(X, \mathcal{F}')}$ . It follows that ${s' - t}$ is a section of ${\mathcal{F}}$ over ${Z'}$ that restricts to zero on the boundary, so it extends to a section of ${\mathcal{F}}$ over ${X}$ .

With this reduction made, let us assume ${X}$ compact. In this case, compactly supported global sections are just the same thing as ordinary global sections. So ${X}$ is a compact space, and there is thus a finite cover of ${X}$ by compact sets ${V_i, 1 \leq i \leq n}$ , such that there exist liftings ${t_i \in \mathcal{F}(V_i)}$ of ${s|_{V_i}}$ . We can choose these compact sets ${V_i}$ , moreover, such that their interiors cover ${X}$ . This follows easily from the surjectivity of ${\mathcal{F} \rightarrow \mathcal{F}''}$ .

Now we need to piece together the ${t_i}$ . To do this, we reduce by induction to the case ${n=2}$ , and there are two sections to piece together. Consider the sections ${t_1, t_2}$ . If they agreed on ${V_1 \cap V_2}$ , then we would be. In general, they need not; however, ${t_1|_{V_1 \cap V_2} - t_2|_{V_1 \cap V_2}}$ is necessarily a section ${e_1}$ of ${\mathcal{F}'(V_1 \cap V_2)}$ as it maps to zero in ${\mathcal{F}''(V_1 \cap V_2)}$ . By softness, we can extend ${e_1}$ to a global section ${e}$ of ${\mathcal{F}'}$ . Then it is clear that the pair ${t_1, t_2 + e}$ of sections of ${\mathcal{F}}$ glues to give a global section. We can now derive two useful properties of soft sheaves.

Corollary 8

If ${0 \rightarrow \mathcal{F}' \rightarrow \mathcal{F} \rightarrow \mathcal{F}'' \rightarrow 0}$ is an exact sequence in ${\mathbf{Sh}(X)}$ and ${\mathcal{F}', \mathcal{F}}$ are soft, then ${\mathcal{F}''}$ is soft too.

${H^i_c(X, \mathcal{F}) = 0}$ for ${i>1}$ if ${\mathcal{F} }$ is soft.

Proof:

Let ${Z \subset X}$ be a closed subset. We know by \cref{softexact} that ${\Gamma_c(X, \mathcal{F}) \rightarrow \Gamma_c(X, \mathcal{F}'')}$ is surjective. Similarly, ${\Gamma_c(Z, \mathcal{F}) \rightarrow \Gamma_c(Z, \mathcal{F}'')}$ is surjective. In the diagram we see that the horizontal maps and the left vertical map are all surjective. It follows that ${\Gamma_c(X, \mathcal{F}'') \rightarrow \Gamma_c(Z, \mathcal{F}'')}$ 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 ${H^1_c(X, \mathcal{F}) = 0}$ for ${\mathcal{F}}$ soft. To do this, imbed ${\mathcal{F}}$ in an injective sheaf ${\mathcal{I}}$ ; the cokernel ${\mathcal{G}}$ is soft by the previous part of the result. The exact sequence
$\displaystyle 0 \rightarrow \Gamma_c(X, \mathcal{F}) \rightarrow \Gamma_c(X, \mathcal{I}) \twoheadrightarrow \Gamma_c(X, \mathcal{G}) \rightarrow H^1_c(X, \mathcal{F}) \rightarrow H^1_c(X, \mathcal{I}) = 0$

shows that ${H^1_c(X, \mathcal{F}) = 0}$ for any soft sheaf ${\mathcal{F}}$ . Now assume inductively that ${H^n_c(X,\mathcal{F}) = 0}$ for any soft sheaf ${\mathcal{F}}$ . Then the isomorphisms ${H^n_c(X, \mathcal{G}) \simeq H^{n+1}_c(X, \mathcal{F})}$ and the inductive hypothesis applied to ${\mathcal{G}}$ (which is also soft) complete the inductive step.