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 ${Z}$. (more…)

I’ve been away from this blog for longer than I should have. I got stuck in my series on the cotangent complex, partially because I’ve been busy doing other things–namely, trying to learn about the foundations of etale cohomology. As I learn more I might write a few posts. And someday the cotangent complex thing will get finished as a short expository note on my website.

One thing I’ve discovered as of late is that many concepts that I learned earlier in life were in fact shadows or special cases of more powerful and general ones. I’ve consequently had to un-learn many such concepts, to replace them with the newer ones.

Sheaves

An example is basic sheaf theory: like many people, I learned this from Hartshorne chapter II, working out the exercises there. But as I have more recently discovered, many of the methods there are not the appropriate ones for the general theory of sheaves on a site. As an example, Hartshorne defines sheafification (and many other things) on a topological space using stalks. However, on a site this is meaningless because there is no analogous notion in general.

The stalk of a sheaf (or presheaf) on a space $X$ at a point corresponds to the inverse image functor via the inclusion $\{\ast\} \to X$. The analogy in the theory of sites would be the inverse image via a morphism from the site with one point (or something equivalent to this). It turns out, fortunately, in etale cohomology this more general notion does make sense, if $\{\ast\}$ is taken to be the spectrum of a separably closed field. So, if $X$ is a scheme, it is not topological points $\{\ast \} \to X$ that lead to the stalk functors in etale cohomology, but the morphisms $\mathrm{Spec} K \to X$ for $K$ a separably closed field (e.g. the separable closure of the residue fields of the topological points).

It is a curious story that there is an even more general theory of points of a (Grothendieck) topos. A point is a geometric morphism (that is, an adjunction where the left adjoint is exact) between the category of sets and the given topos. The direct and inverse image functors obtained from maps $\mathrm{Spec} K \to X$ show that there are lots of “points” in the etale topos. In fact, on general so-called “coherent” topoi there is a general theorem of Deligne that there are always enough points to detect isomorphisms of sheaves. Apparently this is a topos-theoretic reformulation of the completeness theorem in first-order logic! I’m far from understanding the story here though. (more…)

In the past, I said a few words about Grothendieck topologies and fpqc descent. Well, strictly speaking, I didn’t get very far into the descent bit. I described a topology on the category of schemes (the fpqc topology) and showed that it was a subcanonical topology, that is, any representable presheaf was a sheaf in this topology.

This amounted to saying that if ${X' \rightarrow X}$ was a fpqc morphism of schemes, then to hom out of ${X}$ was the same thing as homming out of ${X'}$ such that the two pull-backs to ${X' \times_X X'}$ were the same. If I had gotten further, I would have shown that to give a quasi-coherent sheaf on ${X}$ (among other things) is the same as giving “descent data” of a quasi-coherent sheaf on ${X'}$ together with an isomorphism between the two pull-backs to ${X' \times_X X'}$ satisfying the cocycle condition. Maybe I’ll do that later. But there is a more basic “toy” example that I now want to describe of a site (that is, category with a Grothendieck topology) and the associated category of sheaves on it.

1. ${G}$-sets

Our category ${\mathcal{C}}$ is going to be the category of left ${G}$-sets for a fixed group ${G}$; morphisms will be equivariant morphisms of ${G}$-sets. We are now going to define a Grothendieck topology on this category. For this, we need to axiomatize the notion of “cover.” We can do this very simply: a collection of maps ${\left\{U_i \rightarrow U\right\}}$ is called a cover if the images cover ${U}$. Now, fiber products of ${G}$-sets are calculated in the category of sets, or in other words the forgetful functor

$\displaystyle G-\mathbf{set} \rightarrow \mathbf{Sets}$

commutes with limits (as it has an adjoint, the functor ${S \mapsto G \times S}$). Thus, taking pull-backs preserve the notion of covering, and it is easy to see the other axioms are satisfied too: if we have a cover of each of the ${U_i}$ (which cover ${U}$), then collecting them gives a cover of ${U}$. Similarly, an isomorphism is a cover. This is obvious from the definitions.

2. Representable presheaves

So we indeed do have a perfectly good site. Now, we want a characterization of all the sheaves of sets on it. To start with, let us show that any representable functor forms a sheaf; that is, the topology is subcanonical. (In fact, this topology is the canonical topology, in that it is the finest possible that makes representable functors into sheaves.) (more…)

We continue in the quest towards descent theory. Today, we discuss the fpqc topology and prove the fundamental fact that representable functors are sheaves.

We now describe another topology on the category of schemes. First, we need the notion of an fpqc morphism.

Definition 1 A morphism of schemes ${f: X \rightarrow Y}$ is called fpqc if the following conditions are satisfied:

1. ${f}$ is faithfully flat (i.e., flat and surjective)
2. ${f}$ is quasi-compact.

Indeed, “fpqc” is an abbreviation for “fidelement plat et quasi-compact.” It is possible to carry out faithfully flat descent with a weaker notion of fpqc morphism, for which I refer you to Vistoli’s part of FGA explained.

As with many interesting classes of morphisms of schemes, we have a standard list of properties.

Proposition 2

1. Fpqc morphisms are closed under base-change and composition.
2. If ${f: X \rightarrow Y, g: X' \rightarrow Y'}$ are fpqc morphisms of ${S}$-schemes, then ${f \times_S f': X \times_S X' \rightarrow Y \times_S Y'}$ is fpqc.

Proof: We shall omit the proof, since the properties of flatness, quasi-compactness, and surjectivity are all (as is well-known) preserved under base-change, composition, and products. This can be looked up in EGA 1 (except for flatness, for which you need to go to EGA 4 or Hartshorne III). $\Box$

So we have the notion of fpqc morphism. Next, we use this to define a topology.

Definition 3 Consider the category ${\mathfrak{C}}$ of ${S}$-schemes, for ${S}$ a fixed base-scheme. The fpqc topology on ${\mathfrak{C}}$ is defined as follows: A collection of arrows ${\left\{U_i \rightarrow U\right\}}$ is said to be a cover of ${U}$ if the map ${\coprod U_i \rightarrow U}$ is an fpqc morphism.

This implies in particular that each ${U_i \rightarrow U}$ is a flat morphism. We need now to check that this is indeed a topology.

1. An isomorphism is obviously an fpqc morphism, so an isomorphism is indeed a cover.
2. If ${\left\{U_i \rightarrow U\right\}}$ is a fpqc cover and ${V \rightarrow U}$, then the morphism ${\coprod( U_i \times_U V )\rightarrow V }$ is equal to the base-change ${(\coprod U_i) \times_U V \rightarrow V}$, hence is fpqc.
3. Suppose ${\left\{U^i_j \rightarrow U_i\right\}}$ is a cover for each ${i}$ and ${\left\{U_i \rightarrow U\right\}}$ is a cover, I claim that ${\left\{U_j^i \rightarrow U\right\}}$ is a cover. Indeed, we have that$\displaystyle \coprod_{i,j} U^{j}_i \rightarrow U$factors through$\displaystyle \coprod_{i,j} U^{i}_j \rightarrow \coprod_i {U_i} \rightarrow U$and we know that each morphism in the composition is flat (since the coproduct of flat morphisms is flat) and quasi-compact (since the coproduct of quasi-compact morphisms is quasi-compact). Similarly for surjectivity. It follows that ${\left\{U^i_j \rightarrow U \right\}}$ is an fpqc cover.

So we have another topology on the category of schemes, which is very fine in that it is finer than many other topologies of interest (e.g. the fppf and etale topologies, which I will discuss at some other point). (more…)

It is possible to define sheaves on a Grothendieck topology. Before doing so, let us recall the definition of a sheaf of sets on a topological space ${X}$.

Definition 1 A sheaf of sets ${\mathcal{F}}$ assigns to each open set ${U \subset X}$ a set ${\mathcal{F}(U)}$ (called the set of sections over ${U}$) together with “restriction” maps ${\mathrm{res}^U_V: \mathcal{F}(U) \rightarrow \mathcal{F}(V)}$ for inclusions ${V \subset U}$ such that the following conditions are satisfied:

• ${\mathrm{res}^U_U = \mathrm{id}}$ and for a tower ${W \subset V \subset U}$, the composite ${\mathrm{res}^V_W \circ \mathrm{res}^U_V }$ equals ${\mathrm{res}^U_W}$.
• If ${\left\{U_i\right\}}$ is a cover of ${U \subset X}$, then the map $\displaystyle \mathcal{F}(U) \rightarrow \prod \mathcal{F}(U_i)$is injective, and the image consists of those families ${f_i \in \mathcal{F}(U_i)}$ such that the restrictions to the intersections are equal $\displaystyle \mathrm{res}^{U_i}_{U_i \cap U_j} f_i = \mathrm{res}^{U_j}_{U_i \cap U_j}$

In particular, this says that if we have a family of elements ${f_i \in \mathcal{F}(U_i)}$ that satisfy the above gluing condition, then there is a unique ${f \in \mathcal{F}(U)}$ which restricts to each of them.

While the material on the nonabelian Livsic theorem remains on hold, I have been writing up some notes on algebraic geometry. Here they are. So far, there is no actual algebraic geometry in it, however; they cover elementary facts about sheaves and ringed spaces. At some point in the future, I will certainly post an updated and presumably much longer version.

Today’s main goal is the Leray theorem (though at the end I have to ask a question):

Theorem 1 Let ${\mathcal{F}}$ be a sheaf on ${X}$, and ${\mathfrak{U} = \{ U_i, i \in I\}}$ an open cover of ${X}$. Suppose$\displaystyle H^n( U_{i_1} \cap \dots \cap U_{i_k}, \mathcal{F}|_{ U_{i_1} \cap \dots \cap U_{i_k}}) = 0$

for all ${k}$-tuples ${i_1, \dots , i_k \in I}$, and all ${n>0}$. Then the canonical morphism$\displaystyle H^n( \mathfrak{U}, \mathcal{F}) \rightarrow H^n( X, \mathcal{F})$

is an isomorphism for all ${n}$

This seems rather useless, because the theorem presupposes the vanishing of (regular) cohomology on the covering. However, in many cases it turns out to be helpful. If ${X}$ is a separated scheme, ${U_i}$ an open affine cover of ${X}$, and ${\mathcal{F}}$ quasi-coherent, it applies. The reason is that each of the intersections ${ U_{i_1} \cap \dots \cap U_{i_k}}$ are all affine by separatedness, so ${\mathcal{F}}$ has no cohomology on them by a basic property of quasi-coherent sheaves. This gives a practical way of computing sheaf cohomology in algebraic geometry. Hartshorne uses it to compute the cohomology of line bundles on projective space.

Another instance arises when ${\mathcal{O}}$ is the sheaf of holomorphic functions over some Riemann surface ${X}$. In this case ${\{U_i\}}$ is a covering of charts. It is a theorem (which I will eventually prove) that for any open subset of ${\mathbb{C}}$ (which any intersection of the ${U_i}$‘s is isomorphic to), the sheaf ${\mathcal{O}}$ has trivial cohomology. (more…)