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. (more…)

This is the fifth in a series of posts on Verdier duality, started here. In this post, I will describe the proof of the duality theorem, which itself states the existence of an adjoint to the derived version of the lower shriek functor $f_!$. This might not sound too exciting at first, but we will see that in fact, the dualizing functor will be computable in the important special case of a manifold, and Poincaré duality will fall out quickly. Moreover, the flexible interpretation of sheaf cohomology will allow other duality theorems (such as Alexander duality) to be derived very efficiently from the general formalism.

I will try to explain some of this story (namely, that using sheaf cohomology and Verdier duality one can re-derive much of the classical theory of homology and cohomology) next time. First, though, it will be good to prove the result.

1. Duality

We can now enunciate the result we shall prove in full generality.

Theorem 1 (Verdier duality) Let ${f: X \rightarrow Y}$ be a continuous map of locally compact spaces of finite dimension, and let ${k}$ be a noetherian ring. Then ${\mathbf{R} f_!: \mathbf{D}^+(X, k) \rightarrow \mathbf{D}^+(Y, k)}$ admits a right adjoint ${f^!}$. In fact, we have an isomorphism in ${\mathbf{D}^+(k)}$

$\displaystyle \mathbf{R}\mathrm{Hom}( \mathbf{R} f_! \mathcal{F}^\bullet , \mathcal{G}^\bullet) \simeq \mathbf{R}\mathrm{Hom}( \mathcal{F}^\bullet, f^! \mathcal{G}^\bullet)$

when ${\mathcal{F}^\bullet \in \mathbf{D}^+(X, k), \mathcal{G}^\bullet \in \mathbf{D}^+(Y, k)}$.

Here ${\mathbf{R}\mathrm{Hom}}$ is defined as follows. Recall that given chain complexes ${A^\bullet, B^\bullet}$ of sheaves, one may define a chain complex of ${k}$-modules ${\hom^\bullet(A^\bullet, B^\bullet)}$; the elements in degree ${n}$ are given by the product ${\prod_m \hom(A^m, B^{m+n})}$, and the differential sends a collection of maps ${\left\{f_m: A^m \rightarrow B^{m+n}\right\}}$ to ${ df_m + (-1)^{n+1} f_{m+1}d: A^m \rightarrow B^{m+n+1}}$. Then ${\mathbf{R}\mathrm{Hom}}$ is the derived functor of ${\hom^\bullet}$, and lives in the derived category ${\mathbf{D}^+(k)}$ if ${A^\bullet, B^\bullet \in \mathbf{D}^+(X, k)}$. Since the cohomology in degree zero is given by ${\hom_{\mathbf{D}^+(X, k)}(A^\bullet, B^\bullet)}$, we see that the last statement of Verdier duality implies the adjointness relation. (more…)

[Corrected to fix some embarrassing omissions — 6/12]

I have found lately that many of the foundational theorems in etale cohomology (for instance, the proper base change theorem) are analogs of usually much easier results in sheaf theory for nice (e.g. locally compact Hausdorff) spaces. It turns out that the present topic, duality, has its analog for etale cohomology, though I have not currently studied it. As a warm-up, I thought it would be instructive to blog about the duality theory for cohomology on spaces. This theory, known as Verdier duality, is stated as the existence of an adjoint functor to the derived push-forward. However, from this one can actually recover classical Poincare duality, as I hope to eventually explain.

For a space ${X}$, we let ${\mathbf{Sh}(X)}$ be the category of sheaves of abelian groups. More generally, if ${k}$ is a ring, we let ${\mathbf{Sh}(X, k)}$ be the category of sheaves of ${k}$-modules.

1. Preliminaries

Consider a map ${f: X \rightarrow Y}$ of locally compact spaces. There is induced a push-forward functor ${f_*: \mathbf{Sh}(X) \rightarrow \mathbf{Sh}(Y)}$, that sends a sheaf ${\mathcal{F}}$ (of abelian groups) on ${X}$ to the push-forward ${f_*\mathcal{F}}$ on ${Y}$. It is well-known that this functor admits a left adjoint ${f^*}$, which can be geometrically described in terms of the espace étale as follows: if ${\mathfrak{Y} \rightarrow Y}$ is the espace étale of a sheaf ${\mathcal{G}}$ on ${Y}$, then ${\mathfrak{Y} \times_Y X \rightarrow X}$ is the espace étale of ${f^* \mathcal{G}}$.

Now, under some situations, the functor ${i_*}$ is very well-behaved. For instance, if ${i: Z \rightarrow Y}$ is the inclusion of a closed subset, then ${i_*}$ is an exact functor. It turns out that it admits a right adjoint ${i^!: \mathbf{Sh}(Y) \rightarrow \mathbf{Sh}(Z)}$. This functor can be described as follows. Let ${\mathcal{G}}$ be a sheaf on ${Y}$. We can define ${i^! \mathcal{G} \in \mathbf{Sh}(Z)}$ by saying that if ${U \subset Z}$ is an open subset, such that ${U = V \cap Z}$ for ${V \subset Y}$ open, then ${i^!(\mathcal{G})(U) }$ is the subset of the sections of ${\mathcal{G}(V)}$ with support in ${Z}$. One can check that this does not depend on the choice of open subset ${V}$.

Proposition 1 One has the adjoint relation:

$\displaystyle \hom_{\mathbf{Sh}(Z)}(\mathcal{F}, i^!\mathcal{G}) = \hom_{\mathbf{Sh}(Y)}(i_*\mathcal{F}, \mathcal{G}).$ (more…)