[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}).

Proof: Suppose given a map {i_*\mathcal{F} \rightarrow \mathcal{G}}; we need to produce a map {\mathcal{F} \rightarrow i^!\mathcal{G}}. Indeed, we are given that for each {V \subset Y} open, there is a map

\displaystyle \mathcal{F}(V \cap Z) \rightarrow \mathcal{G}(V).

It is easy to see (because {i_*\mathcal{F}} has support in {Z}) that only sections of {\mathcal{G}} with support in {Z} are in the image. As a result, if {U \subset Z} is any open set, say {U = V \cap Z}, then we can produce the required map

\displaystyle \mathcal{F}(U) \rightarrow i^!\mathcal{G}(U) \subset \mathcal{G}(V).

In the reverse direction, if we are given maps {\mathcal{F}(U) \rightarrow \mathcal{G}(V)} as above, it is easy to produce a map {i_*\mathcal{F} \rightarrow \mathcal{G}}. From this the result follows. Indeed, the two constructions are clearly inverses.

2. Verdier duality

In general, unlike the case of a closed immersion, the functor {f_*: \mathbf{Sh}(X) \rightarrow \mathbf{Sh}(Y)} induced by a morphism {f: X \rightarrow Y} does not have a right adjoint. However, in many nice situations it will have a right adjoint on the derived category. Namely, if {k} is a ring, let {\mathbf{D}^+(X,k), \mathbf{D}^+(Y,k)} be the bounded-below derived categories of sheaves of {k}-modules on {X, Y}. Before this we had worked with {k = \mathbb{Z}}, but there is no need to avoid this bit of generality. Since these categories have enough injectives, the functor {f_*} induces a derived functor

\displaystyle \mathbf{R} f_*: \mathbf{D}^+(X, k) \rightarrow \mathbf{D}^+(Y, k).

To compute {\mathbf{R} f _*} on a bounded-below complex of sheaves {\mathcal{F}^\bullet}, one finds a quasi-isomorphism

\displaystyle \mathcal{F}^\bullet \rightarrow \mathcal{I}^\bullet,

where {\mathcal{I}^\bullet} is a complex of injective sheaves (of {k}-modules) and applies f_* to \mathcal{I}^\bullet.

Theorem 2 (Verdier duality, first form) If {f: X \rightarrow Y} is a proper map of manifolds, then the derived functor {\mathbf{R} f_*: \mathbf{D}^+(X, k) \rightarrow \mathbf{D}^+(Y, k)} admits a right adjoint {f^!: \mathbf{D}^+(Y, k) \rightarrow \mathbf{D}^+(X, k)}.

We will actually state and prove this fact in more generality later. For now, let us consider a special case: when {Y = \ast} is a point. In this case, the derived functor {\mathbf{R} f_*} is just {\mathbf{R} \Gamma}, in other words the derived functors of the global sections (a.k.a. sheaf cohomology on the derived category). {\mathbf{D}^+(Y, k)} is the derived category {\mathbf{D}^+(k)} of the category of {k}-modules. The result states that there is an adjoint {f^!: \mathbf{D}^+(k) \rightarrow \mathbf{D}^+(X, k)}. Consider the complex with {k} in degree zero and nothing anywhere else; then let {\mathcal{D}^\bullet \in \mathbf{D}^+(X, k)} be {f^!(k)}. It follows then that we have an isomorphism

\displaystyle \hom_{\mathbf{D}^+(X, k)}( \mathcal{F}^\bullet, \mathcal{D}^\bullet) \simeq \hom_{\mathbf{D}^+( k)}( \mathbf{R} \Gamma(\mathcal{F}^\bullet), k).

3. The statement for non-proper maps

It will be convenient to have a statement of Verdier duality that holds for non-proper maps as well. To do this, given a map of spaces

\displaystyle f: X \rightarrow Y,

we shall define a new functor

\displaystyle f_!: \mathbf{Sh}(X) \rightarrow \mathbf{Sh}(Y).

This will coincide with {f_*} when {f} is proper. Namely, if {\mathcal{F} \in \mathbf{Sh}(X)} and {U \subset Y} is open, we let

\displaystyle f_!(\mathcal{F})(U) = \left\{s \in \Gamma(\mathcal{F}(f^{-1}(U)) \text{ such that } \mathrm{supp}(s) \text{ is proper over } U\right\}.

We will describe the properties of {f_!} below, but in any event one sees that there is induced a derived functor {\mathbf{R} f_!: \mathbf{D}^+(X, k) \rightarrow \mathbf{D}^+(Y, k)}. The more general form of Verdier duality will state that {\mathbf{R} f_!} has a right adjoint (even if {f} is not proper).

Before doing this, it will be necessary to develop some preliminaries on sheaf theory in general.

Advertisements