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…)

(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. (more…)

This is the third in a series of posts started here (in particular, the notation is kept from there) intended to cover the basics of Verdier duality. Here, I will discuss the lower shriek functors needed even to state Verdier duality (in the most general form, at least); as we will see, the class of soft sheaves will be acyclic with respect to this functor. To see this, though, we shall need to prove some general facts on how push-forward behave with respect to base change, which are themselves of independent interest.

1. The ${f_!}$ functors

Let ${f: X \rightarrow Y}$ be a map of spaces. We have defined the functor

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

earlier, such that ${f_!(U) }$ consists of the sections of ${\mathcal{F}(f^{-1}(U))}$ whose support is proper over ${U}$ ; ${f_!\mathcal{F} }$ is always a subsheaf of ${f_*\mathcal{F}}$, equal to it if ${f}$ is proper. When ${Y}$ is a point, we get the functor

$\displaystyle \mathcal{F} \mapsto \Gamma_c(X, \mathcal{F}) = \left\{\text{global sections with proper support}\right\}.$

One can check that ${f_! \mathcal{F}}$ is in fact a sheaf. The observation here is that a map ${A \rightarrow B}$ of topological spaces is proper if and only if there is an open cover ${\left\{B_i\right\}}$ of ${B}$ such that ${A \times_B B_i \rightarrow B_i}$ is proper for each ${i}$. Now ${f_!}$ is a left-exact functor, as one easily sees. We now want to show that the class of soft sheaves is acyclic with respect to ${f_!}$, and in particular so that one may use soft resolutions to compute the derived functors. To do this, we shall prove a general “base change” theorem that will compute the stalk of ${f_! \mathcal{F}}$. (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…)

As of late, I’ve been trying unsuccessfully to learn about Hilbert and Quot schemes. Nitin Nitsure’s notes (which appear in the book FGA Explained) are a good source, though they are also quite technical. I’ve been trying to read them, and the relevant parts of Mumford’s Lectures on curves on an algebraic surface. While doing so, I took a bunch of my own notes. They are right now unfinished, but they do cover the semicontinuity theorem and a small piece of the cohomology and base-change story. In addition, there is a little additional material (drafted) on applications of this to line bundles, which is more or less why Mumford develops all this in Abelian varieties.

I probably won’t get a chance to revise them further now. In fact, I’m going to take a long break from the “write lots of notes” mentality that has recently gripped me to focus on short, and ideally more-or-less self-contained posts in the future, not least because of limitations of time. This is something I’ve always had difficulty with: even my high school English teachers always had to tell me to shrink my essays. In a sense, though, the blog medium really is about pithy bites of profundity. Being short may take less time, but it’s also intellectually harder than just listing the main theorems in some small subsubsubfield and listing all the proofs in detail.

Last time, we proved an important theorem. Namely, for a proper morphism of noetherian schemes $X \to \mathrm{Spec} A$, we showed that the cohomology of a coherent sheaf $\mathcal{F}$ on $X$, flat over the base, could be described as the cohomology of a finite complex $K$ of flat, finitely generated modules, and moreover that if we base-changed to some other scheme $\mathrm{Spec B}$, we just had to compute the cohomology of $K \otimes_A B$ to get the cohomology of the base extension of the initial sheaf.

With this, it is not too hard to believe that as we vary over the fibers $X_y$, for $y$ in the base, the cohomologies will have somewhat comparable dimensions. Or, at least, their dimensions will vary somewhat reasonably. The precise statement is provided by the semicontinuity theorem.

Theorem 6 (The Semicontinuity theorem) Let ${f: X \rightarrow Y}$ be a proper morphism of noetherian schemes. Let ${\mathcal{F}}$ be a coherent sheaf on ${X}$, flat over ${Y}$. Then the function ${y \rightarrow \mathrm{dim} H^p(X_y, \mathcal{F}_y)}$ is upper semi-continuous on ${Y}$. Further, the function ${y \rightarrow \sum_p (-1)^p H^p(X_y, \mathcal{F}_y)}$ is locally constant on ${Y}$. (more…)

Yes, I’m still here. I just haven’t been in a blogging mood. I’ve been distracted a bit with the CRing project. I’ve also been writing a bunch of half-finished notes on Zariski’s Main Theorem and some of its applications, which I’ll eventually post.

I would now like to begin talking about the semicontinuity theorem in algebraic geometry, following Mumford’s Abelian Varieties. This result is used constantly throughout the book, mainly in showing that certain line bundles are trivial. Eventually, I’ll try to say something about this.

Let ${f: X \rightarrow Y}$ be a proper morphism of noetherian schemes, ${\mathcal{F}}$ a coherent sheaf on ${X}$. Suppose furthermore that ${\mathcal{F}}$ is flat over ${Y}$; intuitively this means that the fibers ${\mathcal{F}_y = \mathcal{F} \otimes_Y k(y)}$ form a “nice”  family of sheaves. In this case, we are interested in how the cohomology ${H^p(X_y, \mathcal{F}_y) = H^p(X_y, \mathcal{F} \otimes k(y))}$ behaves as a function of ${y}$. We shall see that it is upper semi-continuous and, under nice circumstances, its constancy can be used to conclude that the higher direct-images are locally free.

1. The Grothendieck complex

Let us keep the hypotheses as above, but assume in addition that ${Y = \mathrm{Spec} A}$ is affine, for some noetherian ring ${A}$. Consider an open affine cover ${\left\{U_i\right\}}$ of ${X}$; we know, as ${X}$ is separated, that the cohomology of ${\mathcal{F}}$ on ${X}$ can be computed using Cech cohomology. That is, there is a cochain complex ${C^*(\mathcal{F})}$ of ${A}$-modules, associated functorially to the sheaf ${\mathcal{F}}$, such that

$\displaystyle H^p(X, \mathcal{F}) = H^p(C^*(\mathcal{F})),$

that is, sheaf cohomology is the cohomology of this cochain complex. Furthermore, since the Cech complex is defined by taking sections over the ${U_i}$, we see that each term in ${C^*(\mathcal{F})}$ is a flat ${A}$-module as ${\mathcal{F}}$ is flat. Thus, we have represented the cohomology of ${\mathcal{F}}$ in a manageable form. We now want to generalize this to affine base-changes:

Proposition 1 Hypotheses as above, there exists a cochain complex ${C^*(\mathcal{F})}$ of flat ${A}$-modules, associated functorially to ${\mathcal{F}}$, such that for any ${A}$-algebra ${B}$ with associated morphism ${f: \mathrm{Spec} B \rightarrow \mathrm{Spec} A}$, we have$\displaystyle H^p(X \times_A B, \mathcal{F} \otimes_A B) = H^p(C^*(\mathcal{F}) \otimes_A B).$

Here, of course, we have abbreviated ${X \times_A B}$ for the base-change ${X \times_{\mathrm{Spec} A} \mathrm{Spec} B}$, and ${\mathcal{F} \otimes_A B}$ for the pull-back sheaf.

Proof: We have already given most of the argument. Now if ${\left\{U_i\right\}}$ is an affine cover of ${X}$, then ${\left\{U_i \times_A B \right\}}$ is an affine cover of the scheme ${X \times_A B}$. Furthermore, we have that

$\displaystyle \Gamma(U_i \times_A B, \mathcal{F}\otimes_A B) = \Gamma(U_i , \mathcal{F}) \otimes_A B$

by definition of how the pull-backs are defined. Since taking intersections of the ${U_i}$ commutes with the base-change ${\times_A B}$, we see more generally that for any finite set ${I}$,

$\displaystyle \Gamma\left( \bigcap_{i \in I} U_i \times_A B, \mathcal{F}\otimes_A B\right) = \Gamma\left( \bigcap_{i \in I} U_i, \mathcal{F}\right) \otimes_A B.$ (more…)