We have spent a while in the past few days going through the rather categorical formalism of the upper shriek functor $f^!$ obtained from a map $f: X \to Y$ between locally compact Hausdorff spaces of finite cohomological dimension. That is, we showed that the upper shriek must exist on the derived category; this was Verdier duality. However, so far we have not seen any concrete applications of this formalism. I actually feel a bit guilty about having not indicated better some of these in the introductory post and having essentially plunged into the abstract nonsense.

Now we shall apply the existence of ${f^!}$ to questions involving manifolds. Once we know that ${f^!}$ exists, we will be able to describe it using the adjoint property rather simply (for manifolds). This will lead to clean statements of theorems in algebraic topology. For instance, Poincaré duality will be a direct consequence of the fact that, on an ${n}$-dimensional oriented manifold, the dualizing sheaf (see below) is just ${k[n]}$.

1. The dualizing complex

After wading through the details of the proof of Verdier duality, let us now consider the simpler case where ${Y = \left\{\ast\right\}}$. ${X}$ is still a locally compact space of finite dimension, and ${k}$ remains a noetherian ring. Then Verdier duality gives a right adjoint ${f^!}$ to the functor ${\mathbf{R} \Gamma_c: \mathbf{D}^+(X, k) \rightarrow \mathbf{D}^+(k)}$. In other words, for each ${\mathcal{F}^\bullet \in \mathbf{D}^+(X, k)}$ and each complex ${G^\bullet}$ of ${k}$-modules, we have an isomorphism

$\displaystyle \hom_{\mathbf{D}^+(k)}(\mathbf{R} \Gamma_c (\mathcal{F}^\bullet), G^\bullet) \simeq \hom_{\mathbf{D}^+(X, k)}(\mathcal{F}^\bullet, f^!(G^\bullet)).$

Of course, the category ${\mathbf{D}^+(k)}$ is likely to be much simpler than ${\mathbf{D}^+(X, k)}$, especially if, say, ${k}$ is a field.

Definition 1 ${\mathcal{D}^\bullet = f^!(k)}$ is called the dualizing complex on the space ${X}$. ${\mathcal{D}^\bullet}$ is an element of the derived category ${\mathbf{D}^+(X, k)}$, and is well-defined there. We will always assume that ${\mathcal{D}^\bullet}$ is a bounded-below complex of injective sheaves.

In fact, ${\mathcal{D}^\bullet}$ can always taken to be a bounded complex of injective sheaves, though we shall not need this. We now want to determine the properties of this dualizing complex, and show in particular that we can recover Poincaré duality when ${X}$ is a manifold. To do this, let us try to compute the cohomology ${H^\bullet(\mathcal{D}^\bullet)}$ of the dualizing complex (which will be a collection of sheaves). The ${i}$th cohomology can be obtained as the sheaf associated to the presheaf

$\displaystyle U \mapsto \hom_{\mathbf{D}^+(X, k)}(k_U, \mathcal{D}^\bullet[i]).$

Here, as usual ${k_U = j_!(k)}$ is the extension by zero of the constant sheaf ${k}$ from ${U}$ to ${X}$. Indeed, to check this relation we recall that we assumed ${\mathcal{D}^\bullet}$ a complex of injectives (without loss of generality), so that maps ${k_U \rightarrow \mathcal{D}^\bullet[i]}$ are just homotopy classes of maps ${k_U \rightarrow \mathcal{D}^\bullet[i]}$, or equivalently (by the universal property of ${k_U}$) elements of ${H^0(\mathcal{D}(U)^\bullet[i])}$.

But the sheaf associated to this presheaf is clearly the homology ${H^i(\mathcal{D}^\bullet)}$. We have proved in fact:

Proposition 2 If ${\mathcal{F}^\bullet \in \mathbf{D}^+(X, k)}$, the cohomology ${H^i(\mathcal{F}^\bullet) \in \mathbf{Sh}(X, k)}$ is the sheaf associated to the presheaf ${\hom_{\mathbf{D}^+(X, k)}(k_U, \mathcal{F}^\bullet[i])}$.

So we need to compute ${\hom_{\mathbf{D}^+(X, k)}(k_U, \mathcal{D}^\bullet[i]) = \hom_{\mathbf{D}^+(X, k)}(k_U[-i], \mathcal{D}^\bullet)}$. By taking ${U}$ small, we may assume that ${U}$ is a ball in ${\mathbb{R}^n}$. From the adjoint property, however, this is feasible: such maps are in natural bijection with maps ${\mathbf{R} \Gamma_c(k_U[-i]) \rightarrow k}$ in ${\mathbf{D}^+(k)}$. So we need to compute ${\hom_{\mathbf{D}^+(k)}(\mathbf{R} \Gamma_c(k_U[-i]), k)}$. Here ${\mathbf{R} \Gamma_c(k_U[-i])}$ is represented by a bounded complex. One might hope that this is somehow related to the compactly supported cohomology of ${U}$.

When ${k}$ is a field, every complex is quasi-isomorphic to its cohomology, and it is true. We get:

Proposition 3 If ${k}$ is a field, then the ${i}$th cohomology of the dualizing complex ${H^i(\mathcal{D}^\bullet)}$ is the sheaf associated to the presheaf ${U \mapsto H^i_c(U, k)^\vee}$.

Chasing through the definitions, one sees that the restriction maps are the duals to the maps ${H^i_c(U, k) \rightarrow H^i_c(U', k)}$ for ${U \subset U'}$. (This goes the opposite way as in ordinary cohomology.) We in particular see that ${\mathcal{D}^\bullet}$ lives in the bounded derived category, at least when ${k}$ is a field (because ${\mathcal{D}^\bullet}$ is now quasi-isomorphic to a suitable truncation).

The next goal will be to compute the cohomology of ${\mathcal{D}^\bullet}$ for a manifold. For a field at least, this will require nothing more than a computation of the cohomology of suitable open sets in ${\mathbb{R}^n}$, by the previous result.

2. The cohomology of ${\mathbb{R}^n}$

The point of Poincaré duality lies in the cohomology of ${\mathbb{R}^n}$ and the fact that any manifold is locally homeomorphic to ${\mathbb{R}^n}$. Of course, we mean compactly supported cohomology here.

Lemma 4 Let ${k}$ be any ring. Then we have ${H_c^i(\mathbb{R}^n, k) \simeq k}$ if ${i = n}$, and ${H_c^i(\mathbb{R}^n, k) = 0}$ otherwise.

Proof: We refer the reader to Iversen’s book on sheaf cohomology. The strategy, in rough outline, is as follows:

1. It is sufficient to handle the case ${k= \mathbb{Z}}$, because then it is clear for any free abelian group, and one can use the exact sequences to deduce it in general (together with the fact that cohomology commutes with filtered colimits).
2. One shows that ${H_c^\bullet([0,1], \mathbb{R}) = H^\bullet([0,1], \mathbb{R})}$ is ${\mathbb{R}}$ in dimension zero and zero otherwise. This follows, for instance, by use of the soft de Rham resolution

$\displaystyle 0 \rightarrow \mathbb{R} \rightarrow \mathcal{C}^\infty \rightarrow \mathcal{C}^{\infty} \rightarrow 0,$

where ${\mathcal{C}^{\infty}}$ denotes the sheaf of smooth functions and the last map is one-variable differentiation. In particular, ${H^1_c([0,1], \mathbb{R})}$ can be computed as the cokernel of differentiation

$\displaystyle C^{\infty}([0,1]) \stackrel{f \mapsto f'}{\rightarrow}C^{\infty}([0,1]),$

which is clearly trivial.

3. One computes ${H_c^\bullet([0,1], \mathbb{Z})}$ using the soft(!) resolution

$\displaystyle 0 \rightarrow \mathbb{Z} \rightarrow \mathcal{C} \rightarrow \mathcal{C}^{S^1} \rightarrow 0,$

where ${\mathcal{C}}$ is the sheaf of real-valued continuous functions and ${\mathcal{C}^{S^1}}$ is the sheaf of continuous functions into the circle group. One can deduce from this (and the so-called Vietoris-Bergle mapping theorem) that sheaf cohomology is a homotopy invariant.

4. In the end, one can show that sheaf cohomology with coefficients in the constant sheaf ${\mathbb{Z}}$ is a cohomology theory (satisfying, that is, the usual Eilenberg-Steenrod axioms) on suitably nice spaces. The analog of relative cohomology is local cohomology. Because of the normalization of the cohomology of a point, it follows that this is ordinary cohomology and ${H^\bullet(S^n, \mathbb{Z}) }$ is ${\mathbb{Z}}$ in dimensions ${0}$ and ${n}$, and zero otherwise.
5. Finally, to compute ${H^\bullet_c(\mathbb{R}^n, \mathbb{Z})}$, one uses the fact that the one-point compactification of ${\mathbb{R}^n}$ is ${S^n}$ and the long exact sequence.

It follows that the same is true when ${\mathbb{R}^n}$ is replaced by any space homeomorphic to it, e.g. an open ball. Using this, we make the following observation: if ${X}$ is an ${n}$-dimensional manifold, then the sheaf associated to the presheaf ${U \mapsto H^i_c(U, k)^{\vee}}$ is zero unless ${i = n}$. It follows that the dualizing complex ${\mathcal{D}_X^\bullet}$ is cohomologically concentrated in one degree, namely ${n}$. It follows (by the use of truncation functors) that the dualizing complex is quasi-isomorphic to a translate of a single sheaf.

On a ${n}$-dimensional manifold ${X}$, we define the orientation sheaf ${\omega_X}$ as the sheaf associated to the presheaf ${U \mapsto H^n_c(U, k)}$. (This is actually already a sheaf, though we do not need this.)

Corollary 5 Let ${k}$ be a field, and let ${X}$ be an ${n}$-dimensional manifold. Then the dualizing complex ${\mathcal{D}^\bullet}$ on ${X}$ is isomorphic to ${\omega_X[n]}$.

3. Poincaré duality

Fix a field ${k}$. Let ${X}$ be an ${n}$-dimensional manifold with orientation sheaf ${\omega_X}$. We know that the dualizing sheaf is ${\omega_X[n]}$, which implies for any complex ${\mathcal{F}^\bullet \in \mathbf{D}^+(X, k)}$, there is a natural isomorphism ${\hom_{\mathbf{D}^+(X, k)}( \mathcal{F}^\bullet, \omega_X[n]) \simeq \hom_{\mathbf{D}^+(k)}(\mathbf{R} \Gamma_c (\mathcal{F}^\bullet), k)}$. Take in particular ${\mathcal{F}^\bullet = \mathcal{H}[r]}$ for some ${r \in \mathbb{Z}}$ and ${\mathcal{H} \in \mathbf{Sh}(X, k)}$. On the left, we get ${\mathrm{Ext}^{n-r}_k(\mathcal{H}, \omega_X)}$; on the right, we get (since ${k}$ is a field) ${H^r_c(X, \mathcal{H})^{\vee}}$.

Theorem 7 (Poincaré duality) There is a map

$\displaystyle \int_c: H^n_c(X, \omega_X) \rightarrow k$

such that the pairing

$\displaystyle H_c^r(X, \mathcal{H}) \times \mathrm{Ext}_k^{n-r}(\mathcal{H}, \omega_X) \mapsto H^{n}_c(X, \omega_X) \stackrel{\int_c}{\rightarrow} k$

identifies ${ H_c^r(X, \mathcal{H})^{\vee} \simeq \mathrm{Ext}^{n-r}_k(\mathcal{H}, \omega_X)}$.

When ${\mathcal{H}}$ is the constant sheaf ${k}$, then ${\mathrm{Ext}_k^{n-r}(\mathcal{H}, \omega_X) = H^{n-r}(X, \omega_X)}$ (this is not compactly supported cohomology!) and consequently one finds that there is a natural isomorphism ${H_c^r(X, k)^{\vee} \simeq H^{n-r}(X, \omega_X)}$. When ${X}$ is orientable and ${\omega_X}$ is the constant sheaf ${k}$, then we have recovered the usual form of Poincaré duality.

Notice also how, in this statement of Poincare duality, the $\mathrm{Ext}$ groups appear.

Corollary 8 Let ${X}$ be an ${n}$-dimensional manifold. The functor ${\mathcal{F} \mapsto H^n(X, \mathcal{F})^{\vee}}$ on ${\mathbf{Sh}(X, k)}$ is representable.

Proof: Indeed, the representing object is ${\omega_X}$, as follows from Poincaré duality above.

4. Relative cohomology

Let ${X}$ be a topological space, and ${Z \subset X}$ a closed subspace. Sheaf cohomology of the constant sheaf ${k}$ on ${X}$ is to be thought of as an analog to the singular cohomology ${H_{\mathrm{sing}}^\bullet(X; k)}$; in fact, these coincide for a nice space ${X}$. The analog of the relative singular cohomology ${H_{\mathrm{sing}}^\bullet(X, Z; k)}$ are the local cohomology groups ${H^\bullet_Z(X, k)}$ for ${k}$ the constant sheaf. Namely, consider the functor that sends ${\mathcal{F} \in \mathbf{Sh}(X)}$ to ${\Gamma_Z(\mathcal{F})}$, the group of global sections with support in ${Z}$. The derived functors ${H^i_Z(X, \mathcal{F})}$ are called thelocal cohomology groups of ${\mathcal{F}}$. We recall that if ${i: Z \rightarrow X}$ is the inclusion, then we showed much earlier that the push-forward ${i_*}$ had a right adjoint ${i^!}$. Since (as is easy to see),

$\displaystyle \Gamma_Z(\mathcal{F}) = \hom_{\mathbf{Sh}(X)}(i_* \mathbb{Z}, \mathcal{F}),$

$\displaystyle \Gamma_Z(\mathcal{F}) = \hom_{\mathbf{Sh}(Z)}(\mathbb{Z}, i^! \mathcal{F}).$

In other words, the global sections of ${i^! \mathcal{F}}$ are precisely ${\Gamma_Z(\mathcal{F})}$; this is also immediate from the actual construction of ${i^!}$ we gave.

Our notation is, however, slightly confusing! We have defined ${i^!: \mathbf{Sh}(X, k) \rightarrow \mathbf{Sh}(Z, k)}$ as the right adjoint to ${i_*}$. However, we also used the notation ${i^!}$ for the right adjoint to the derived functor ${\mathbf{R} i_! = \mathbf{R} i_*: \mathbf{D}^+(Z, k) \rightarrow \mathbf{D}^+(X, k)}$. The next lemma will show that our abuse of notation is not as bad as it may seem.

Lemma 9 Let ${i^!: \mathbf{Sh}(X, k) \rightarrow \mathbf{Sh}(Z, k)}$ be right adjoint to ${i_*}$. Then ${\mathbf{R} i^!: \mathbf{D}^+(X, k) \rightarrow \mathbf{D}^+(Z, k) }$ is right adjoint to ${\mathbf{R} i_*}$ (so it is the “upper shriek” of before).

Proof: Now ${i^!: \mathbf{Sh}(X, k) \rightarrow \mathbf{Sh}(Z, k)}$ is a left-exact functor (as a right adjoint). Since its left adjoint ${i_*}$ is exact, a simple formal argument shows that ${i^!}$ preserves injectives. From this the argument is straightforward. Namely, let ${\mathcal{F}^\bullet \in \mathbf{D}^+(X, k), \mathcal{G}^\bullet \in \mathbf{D}^+(Z, k)}$. We may assume that these are bounded-below complexes of injective sheaves. Then we have that ${\mathbf{R} i_*( \mathcal{F}^\bullet) = i_* (\mathcal{F}^\bullet)}$ and ${\mathbf{R} i^!( \mathcal{G}^\bullet) = i^!(\mathcal{G}^\bullet)}$. We have that

$\displaystyle \hom_{\mathbf{D}^+(Y, k)}( \mathbf{R} i_* (\mathcal{F}^\bullet), \mathcal{G}^\bullet) = \hom_{\mathbf{D}^+(Y, k)}( i_*(\mathcal{F}^\bullet), \mathcal{G}^\bullet) = [i_*(\mathcal{F}^\bullet), \mathcal{G}^\bullet)],$

where the last symbol denotes homotopy classes; this follows because ${\mathcal{G}^\bullet}$ is injective. Similarly we get

$\displaystyle \hom_{\mathbf{D}^+(X, k)}( \mathcal{F}^\bullet, \mathbf{R} i^!(\mathcal{G}^\bullet)) = \hom_{\mathbf{D}^+(X, k)}( \mathcal{F}^\bullet, i^!(\mathcal{G}^\bullet)) = [ \mathcal{F}^\bullet, i^!(\mathcal{G}^\bullet) ]$

because ${i^!}$ preserves injectives. Now by adjointness on the ordinary categories, we see the natural isomorphism.

Now we want to bring in Verdier duality.

Lemma 10 Let ${X, Y, Z}$ be locally compact spaces of finite dimension, and suppose ${f: X \rightarrow Y, g: Y \rightarrow Z}$ are continuous maps. Then ${(g \circ f)^{!} = f^! \circ g^!: \mathbf{D}^+(Z, k) \rightarrow \mathbf{D}^+(X, k)}$.

To be precise, we should say “up to a natural isomorphism.”

Proof: This is immediate from the adjointness definition of the upper shriek and the fact that

$\displaystyle \mathbf{R}(g \circ f)_! = \mathbf{R} g_! \circ \mathbf{R} f_!$

(which is the Leray spectral sequence). We can deduce the following. If ${\mathcal{D}_X^\bullet}$ is the dualizing complex on ${X}$, then ${\mathbf{R} i^!(\mathcal{D}_X^\bullet)}$ is the dualizing complex ${\mathcal{D}_Z^\bullet}$ on ${Z}$. This follows because the dualizing complex is the upper shriek of the constant complex ${k}$. Suppose now ${X}$ is a manifold of dimension ${n}$, with orientation sheaf ${\omega_X}$. Then it follows that

$\displaystyle \mathcal{D}_Z^\bullet = \mathbf{R} i^!(\omega_X)[n].$

From this we will get:

Theorem 11 (Alexander duality) Let ${k}$ be a field. Suppose ${Z \subset X}$ is a closed subset of an ${n}$-dimensional manifold ${X}$ with ${i: Z \rightarrow X}$ the inclusion. Then, for a sheaf ${\mathcal{F} \in \mathbf{Sh}(Z, k)}$, we have natural isomorphisms:

$\displaystyle H^r_c(Z, \mathcal{F})^{\vee} \simeq \mathrm{Ext}_k^{n-r}(i_* \mathcal{F}, \omega_X[n-r]).$

The most important case is when ${\mathcal{F} = k}$, in which case we find:

$\displaystyle H^r_c(Z, k)^{\vee} \simeq H_Z^{n-r}(X, \omega_X).$

Proof: Indeed, this follows purely formally now. Let ${\mathcal{F} \in \mathbf{Sh}(Z)}$ be any sheaf. In the following, we shall interchange ${i_*}$ and ${\mathbf{R} i_*}$, which is no matter as ${i_*}$ is an exact functor. Then:

Take now ${\mathcal{F} = k}$. We then get at the last term ${\mathrm{Ext}^{n-r}_k(i_* k, \omega_X)}$. But recall that these are the local cohomology groups ${H^{n-r}_Z(X, \omega_X)}$ because ${\hom_{\mathbf{Sh}(X, k)}(i_* k, \cdot)}$ is the same functor as ${\Gamma_Z(\cdot)}$.

The classical statement about closed subsets of the sphere now follows because the sphere is cohomologically rather simple, and orientable (so the orientation sheaf is trivial).