Étale Cohomology

Report on Meeting 5

With this meeting, we began formal consideration of the machinery needed to define general cohomology theories. Throughout the preliminary discussion, we need some common data. The following notations will be used. First, let S be a category which has a terminal object and has all finite fibred products. Next, let E be a class of morphisms in S such that

(i) Every isomorphism in S is in E. (ii) E is closed under composition. (iii) E is closed under base change along morphisms in S.

Remark 1. To see that examples of such things really exist, we first consider the case of an ordinary topological space T. Take S to be the category whose objects are the open subsets of T, and whose morphisms are the inclusion maps. Take E to contain all the morphisms in S. These satisfy the properties demanded above. (The terminal object of S is T itself; fibre products are just intersections.)

A second collection of examples comes from algebraic geometry. Let X be a scheme, and take S to be the category SCH/X whose objects are all X-schemes. Then X is a terminal object in this category, and fibre products of schemes exist (and are locally given by tensor products). Three interesting choices for E are (Zar) : A morphism is in this class if and only if it is an open immersion. (et) : A morphism is in this class if and only if it is an étale morphism of finite type. (fl) : A morphism is in this class if and only if it is flat and locally of finite type.

Definition 2. Let S and E be as above, and let X be an object of S. A site is a full subcategory C/X inside S/X which is closed under fibre products and satisfies the following condition: Whenever Y → X is a morphism in C/X and U → Y is an E-morphism in S, then the composite morphism U → X also lies in C/X. The site determined by all this data will be denoted by (C/X)_{E}.

Definition 3. Let (C/X)_{E} be a site, and let Y be an object of C/X. An E-covering of Y in C/X is defined to be a collection {U_{i}⟶^{g_{i}}Y }_{i∈I} where each g_{i} is an E-morphism and the induced map ∪U_{i} → Y is an epimorphism. The class of all E-coverings of objects in C/X is called the E-topology on the site (C/X)_{E}.

Remark 4. The definition of E-covering helps explain why we want C-morphisms composed with E-morphisms to lie in C: We want everything that can be part of an E-covering of a C-object to inherit the properties of C-objects.

Remark 5. Let’s return to our earlier examples. In the topological example, we can take X = T to be our original topological space, and alse take C/X = S/X = S. Then the E-coverings in this site are just the ordinary coveriongs in the sense of topology. In particular, every topological space gives rise to a site in a canonical way.

The examples from algebraic geometry allow us a little more latitude. Once again, we let X be a scheme and let S = SCH/X be the category of X-schemes. Any of the choices that we listed for E give rise to two natural sites. The small E-site is given by (E/X)_{E}. That is, we only take E-morphisms to X as the objects in the small site. This is like taking an ordinary topological space. (In the case that E = Zar, this is precisely true. In the étale and flat cases, it is only true by analogy.) At the opposite extreme, we have the big étale site (LF T/X)_{E}, whose objects are all schemes over X that are locally of finite type. This site is more like the category of all topological spaces. (Locally of finite type is a weak condition. When X = Spec(F) is the spectrum of a field, then the category LF T/X contains all F -varieties (along with lots of other stuff).)

Definition 6. We define X_{Zar} = (Zar/X)_{Zar}, the (small) Zariski site;

X_{et} = (et/X)_{et}, the (small) étale site; and
X_{fl} = (LF T/X)_{fl}, the (big) flat site.

Proposition 7. Let (C/X)_{E} be a site.

(i) If φ : U → U is an isomorphism in C/X, then {U⟶^{φ}U} is an E-covering. (ii) If {U_{i} → U}_{i∈I} is an E-covering and if each of {V_{ij} → U_{i}}_{j∈J_{i}} is an E-covering, then

the collection {V_{ij} → U}_{i∈I,j∈J_{i}} is also an E-covering. (iii) If {U_{i} → U}_{i∈I} is an E-covering, and if V → U is any morphism in C, then the collection {V ×_{U} U_{i} → V } is also an E-covering.

Proof: (i) is completely trivial; (ii) only uses that E is closed under composition; and (iii) uses closure under base change.

Definition 8. A presheaf of abelian groups on the site (C/X)_{E} is a contravariant functor from C/X to Ab (the category of abelian groups). A morphism of presheaves is just a natural transformation of functors (in disguise).

Remark 9. Note: The definition of a presheaf does not use the topology on the site. It only uses the underlying category C/X. We won’t need the topology until we try to define sheaves.

Remark 10. Let P be a presheaf of abelian groups on (C/X)_{E}. Because P is a functor, whenever we have a commutative diagram

\begin{matrix}U & ⟶ & X \\ ↑ & & ↑^{=} \\ V & ⟶ & X \end{matrix}

we get a restriction map P (U) → P (V). The image of an element s ∈ P (U) under this restriction homomorphism will either be denoted res_{U,V} (s) or s|_{V}^{U} or (when we’re feeling really reckless) s|_{V}.

Definition 11. Let P((C/X)_{E}) denote the category of presheaves of abelian groups on this site. (In the case of our three sites from algebraic geometry, we’ll also write P(X_{E}).)

Proposition 12. For any site (C/X)_{E}, the category P((C/X)_{E}) is an abelian category.

Proof: I’m not going to prove this; I’m merely going to remind you that this means: the Hom-sets in P are abelian groups, composition is bilinear (that much says that P is an AB-category), P has a zero object and finite direct sums (well, technically, biproducts; if we get this far, then P is called an additive category), and P has kernels and cokernels; every monic is a kernel, and every epi is a cokernel.

All of the properties are inherited from Ab; you get them in P by working “element-wise”.

Remark 13. As a first example of a presheaf, let X_{E} be one of the sites coming from algebraic geometry. For each object U ∈ X_{E} (and so, in utmost generality, each X-scheme U that is locally of finite type), define G_{a}(U) = Γ(U, O_{U}) to be the sections of the ordinary Zariski structure sheaf on U. (Isn’t it weird to have arrived at the point where we can talk about ordinary-old-Zariski sheaves without batting an eyelash?) This is a presheaf because every morphism of schemes U⟶^{f}V includes as part of its structure a morphism O_{V} → f_{∗}(O_{U}) of ordinary-Zariski-sheaves on V, and therefore there is a natural ring homomorphism

G_{a}(V) = Γ(V, O_{V}) → Γ(V, f_{∗}O_{U}) = Γ(U, O_{U}) = G_{a}(U).

Remark 14. The fact that the previous map is actually a ring homomorphism means that we can replace the additive group by the multiplicative group and get another presheaf G_{m}(U) = Γ(U, O_{U}^{∗}).

Remark 15. We can generalize the construction of G_{a}, by making it apply to any ordinary-Zariski-sheaf F of O_{X}-modules. Define a presheaf on any of our algebraicgeometric sites by taking

W (F)(U) = Γ(U, F ⊗_{O_{X}} O_{U}).

(Note: The tensor product really means to pull F back along the canonical map U → X that is part of the structure of an X-scheme.) It is easy to check that G_{a} = W (O_{X}).

An interesting application of this construction occurs when X → S displays X as an Sscheme for some base scheme S. We can then take the sheaf Ω_{X/S} of relative differentials on X, and produce a presheaf W (Ω_{X/S}). Why is this interesting? Well, consider the related presheaf

Ω_{/S}(U) = Γ(U, Ω_{U/S}).

In general, these two sheaves are different. To see that, note first that the natural maps Ω_{X/S} ⊗_{O_{X}} O_{U} → Ω_{U/S} from the First Exact Sequence can be used to define a morphism of presheaves W (Ω_{X/S}) → Ω_{/S}. That same sequence shows that this map is not surjective in the flat topology (because the quotient measures relative differential of U over X) but it is, in fact, an isomorphism in the étale topology.

Definition 16. A presheaf P on a site (C/X)_{E} is separated if for every s ∈ P (U), the existence of an E-covering {U_{i} → U} such that s|_{U_{i}} = 0 for all i forces s itself to be zero.

Definition 17. A separated presheaf P on a site (C/X)_{E} is called a sheaf (in the Etopology) if for every E-covering {U_{i} → U} and for every collection of sections s_{i} ∈ P (U_{i}) that satisfy s_{i}|_{U_{i}×_{U} U_{j}} = s_{j}|_{U_{i}×_{U} U_{j}} for all i, j, there exists a section s ∈ P (U) such that s_{i} = s|_{U_{i}} for all i.

Remark 18. We let SH((C/X)_{E}) denote the category of all sheaves of abelian groups on the site (C/X)_{E} with respect to the E-topology. Like the category of presheaves, the category of sheaves on a site is an abelian category. (This fact is much harder to prove; maybe we’ll get around to it next time.) It is also the fundamental example of a topos. I won’t really define topos, other than to quote the following categorical nonsense: A topos is a Cartesian closed category with a subobject classifier. (I have no idea how that translates back into English.)

Remark 19. Let’s do one more example. This example has two points. First, it points out that the sheaf condition is interesting even in the case when the cover contains just one morphism. (One might have expected this already when we noticed that X automorphisms of an X-scheme give coverings.) Second, it allows us to identify the category of étale sheaves on the spectrum of a field. To give the example, we need some preliminaries.

Definition 20. Let G be a finite group and let X be any scheme. Let G_{X} denote the X-groupscheme induced by G.

In more down-to-earth terms, that means that G_{X} = ∐_{g∈G} X_{g}, where each factor in

the disjoint union is a copy of X. Notice that G_{X} comes equipped with a natural map to X (that is the identity on each piece) and with an obvious G-action.

Definition 21. A morphism Y → X will be called Galois if it is finite and étale, and if the cardinality of the “Galois” group Aut(Y/X) is equal to the degree of the cover.

Lemma 22. If Y → X is Galois with group G, then there exists a faithfully flat morphism U → X that is locally of finite type and a U-isomorphism between G_{U} and Y ×_{X} U.

Proof: This is straightforward. In fact, under these hypotheses, we can take U = Y. At some later time, we might prefer to drop the étale part of the definition of a Galois cover, and replace it with the condition of the lemma.

Now suppose that P is an presheaf on X_{et} and that Y → X is a Galois cover with group G. By functoriality, G acts on P (Y). Suppose in addition that P takes (finite) disjoint unions of schemes to cartesian products of sets. Then the sheaf condition applied to the singleton covering Y → X (This is a covering since finite and étale imply faithfully flat, and hence surjective.) gives an exact diagram

P (X) → P (Y) \overrightarrow{⟶}_{(g_{1},...,g_{2})}^{(1,...,1)}P (Y)^{n}.

Note: the identification of the right half of the diagram uses the fact that P does nice things to disjoint unions and on the fact that Y ×_{X} Y = G_{Y} = ∐ Y_{g}. In any event, we can conclude from the exactness of the diagram that P (X) = P (Y)^{G}. It is a short step from this conclusion to the following result.

Proposition 23. Let F be a field and let X = Spec(F). Then the category of étale sheaves on X is naturally equivalent to the category of discrete G-modules, where G = Gal(F̄/F).

Proof: Just do it.