Étale Cohomology

Report on Meeting 6

In this meeting, we talked about sheaves, stalks, and points in the étale and flat topologies, those curious inventions of Grothendieck. The first result tells us, essentially, that the theory of sheaves in either of these topologies is an amalgamation of the theory of Zariski sheaves with Galois theory.

Theorem 1. Let P be a presheaf on X_{et} or X_{fl}. Then P is a sheaf if and only if

(i) for all U ∈ C_{X}, the restriction of P to U_{Zar} is a Zariski sheaf, and (ii) for every cover V → U where U and V are both affine schemes, the sequence

P (U) → P (V)\overrightarrow{⟶}P (V ×_{U} V)

is exact.

Proof: The implication (⇒) is trivial. To prove the converse, note first that if V = ∐ V_{i} is an object in C/X, then P (V) = ∏ P (V_{i}). (This property is true for Zariski sheaves, and so it follows in general from (i).) Next, suppose you have a covering U = {U_{i} → U} where the index set is finite. Then you can define V = ∐ U_{i} to get a cover involving just one object—but with the same sequence of sections as the one given by the cover U. If each U_{i} is affine, then so is V. So, you can use (ii) to conclude that the sheaf properties hold for the covering U.

Now let’s try to do the case of a general covering U = {U_{j} → U}. We’ll still define V = ∐ U_{j}, and let f : V → U be the canonical map. This may not have all the properties we want, so we’ll have to shrink it back down to get everything in the correct category. First, choose a covering of U by Zariski open affines W_{i}. Next, choose Zariski open affines V_{ik} covering each f^{−1}(W_{i}). Since W_{i} is affine, it is quasicompact. So, we only need to use finitely many pieces in the inverse image to get a covering {V_{ik} → W_{i}}. By increasing the number of pieces in the cover of U, we can arrange for the V_{ik} to cover V, even though only finitely many of them are used to cover each of the W_{i}. Now we have a megadiagram

P (U)  ⟶  P (V)  \overrightarrow{⟶}  P (V ×_{U} V)
↓  ↓  ↓
∏_{i} P (W_{i})  ⟶  ∏_{i,k} P (V_{ik})  \overrightarrow{⟶} ∏_{i,k,l} P (V_{ik} ×_{W_{i}} V_{il})
↓ ↓  ↓ ↓
∏_{ij} P (W_{i} ×_{U} W_{j}) ⟶ ∏_{i,j,k,l} P (V_{ik} ×_{V} V_{jl})

Now, the first column and middle column are exact, since they come from Zariski covers. The cental row is exact, since it comes from the earlier observation using (i) and (ii) to handle finite covers of one affine by other affines. Now an almost standard diagram chase shows that the top row is exact. (Actually there’s one more twist. The diagram chase easily gives injectivity in the top row. That means that P is separated. That means the bottom row is injective. You need this fact to finish the diagram chase.)

Corollary 2. Let F be a quasicoherent Zariski sheaf of O_{X}-modules. Then W (F) is a sheaf in the étale or flat topologies.

Proof: You just need to check the cases of the theorem. Property (i) holds since f^{∗}(F) is always a Zariski sheaf. Property (ii) is just descent theory for quasicoherent sheaves.

Remark 3. In particular, this corollary applies to the sheaves G_{a}, G_{m}, and W (Ω_{X/S}).

Corollary 4. Every commutative group scheme G defines a sheaf of abelian groups in the étale or flat topology.

Proof: This reduces to the case of descent theory that we actually worked out explicitly, since the sections of the presheaf are the groups Hom_{X}(U, G).

Remark 5. As our next task, we’re going to try to understand what it means to be a point for the étale topology. The first question to ask is: what do we mean by a point in more traditional topologies? The key point is to realize that the answer can be given in a purely sheaf-theoretic sense. Let SH(X) denote the category of sheaves of abelian groups on a topological space X. Let ∗ be a one-point space. Then SH(∗) = AB is naturally equivalent to the category of abelian groups. This same property holds true for points in the Zariski topology. A one-point space is just X = Spec(F) for some field F, and the category SH(X_{Zar}) of Zariski sheaves of abelian groups is naturally equivalent to the category of abelian groups. However, we know that this property does not hold in the étale topology. In fact, SH(X_{et}) is equivalent to the category of discrete G-modules, where G = Gal(X̄/X). The only time this reduces to the category of abelian groups is when F is separably closed (so G is trivial). Thus, points for the étale topology are the same things as Spec(F) where F is a separably closed field.

This claim is not as ridiculous as it might sound at first. Part of the problem when studying a Zariski sheaf X defined over a general field F is that the closed points of X might have residue fields that are arbitrary finite extensions of F. From the point of view of the separable closure of F, a closed point of X actually represents several distinct points that are related by the Galois action. In the étale topology, these distinct points are given equal weight, as though they were actual points of X.

Remark 6. Let’s also see if we can understand the stalks of sheaves in the étale topology. The place to look for stalks, of course, is at a point. So, let x̄ = Spec(F̄) be the sepctrum of an algebraically closed field, and let i : x̄ → X be a point (for the étale topology). Let F be an étale sheaf on X. Look at the pullback i^{∗}(F). This is an étale sheaf on x̄, and so it is just an abelian group. We can take this abelian group to be the stalk of F at x̄.

The first point to realize, of course, is that the pullback must be computed in the correct category. Consider the Zariski case, with a closed point i : x ↪ X. The pullback of a Zariski (pre)sheaf F on X is defined by taking the direct limit lim_{→} F(U) over open neighborhoods of i(x). This is the usual definition of the stalk of F. However, this is the pullback in the category of sheaves of abelian groups; it’s not the pullback in the category of sheaves of O_{X}-modules.

Of course, we now realize that there’s a problem with our argument in the étale case. We don’t know how to pull sheaves back in the étale topology. That’s our next task.

Definition 7. Let π : X′ → X be a morphism of schemes. We will say that π defines a continuous morphism of sites π : (C′/X′)_{E′} → (C/X)_{E} if the following properties hold:

(i) For every object Y ∈ C/X, the fibre product Y ×_{X} X′ is in C′/X′. (ii) If U → Y is an E-morphism in C/X, then its pullback U ×_{X} X′ → Y ×_{X} X′ is an E′-morphism.

Remark 8. This definition tells you what a morphism of sites is. It is not a functor between the categories (at least, not in the obvious direction indicated by the arrow). It’s a morphism of the underlying schemes that has some extra properties that reflect the topology. Roughly, those conditions should be thought of as the statement that the inverse image of an open set under a continuous function must be an open set.

Remark 9. There are two canonical examples. First, let Y → X be any morphism of schemes (in the usual sense). Then it induces continuous morphisms of sites

\begin{aligned}Y_{Zar} & → X_{Zar} \\ Y_{et} & → X_{et} \\ Y_{fl} & → X_{fl}. \end{aligned}

Second, the identity morphism of any scheme X defines continuous morphisms of sites

X_{fl} → X_{et} → X_{Zar}.

We can see from these examples why we want continuous morphism to go in the directions indicated by the arrows. The first set of examples says that sites are “just like schemes.” The second example says that the flat topology is finer than the étale topology, which is finer than the Zariski topology.

Definition 10. Let π : X′_{E′} → X_{E} be a continuous morphism of sites. Let P′ be any presheaf on X′_{E′}. Define the direct image presheaf on X_{E} by

π_{pre}(P)(U) = P (π^{−1}(U)) = P (U ×_{X} X′).

Remark 11. It is clear that π_{pre} is a functor from the category P(X′_{E′}) of presheaves of abelian groups to the category P(X_{E}).

Theorem 12. Let π : X′_{E′} → X_{E} be a continuous morphism of sites. Then π_{pre} has a left adjoint π^{pre} : P(X_{E}) → P(X′_{E′}), which is usually referred to as the pullback functor on presheaves.

Proof: There is a proof using categorical nonsense, since the underlying categories C and C′ can be assumed to be small, and the presheaf categories are just functor categories taking values in a category AB that has all direct limits.

The concrete interpretation of this categorical proof is as follows. Let U′ ∈ C′/X′ be any object, and let P ∈ P(X_{E}) be any presheaf. Then (π^{pre})(U′) = lim_{→} P (U), where the limit is taken over all commutative diagrams

\begin{matrix}U′ & ⟶^{g} & U \\ ↓ & & ↓ \\ X′ & ⟶^{π} & X \end{matrix}

with U → X in C/X.