Étale Cohomology

Report on Meeting 8

We are finally ready to start using the second word in the title of this series. (It took us four weeks to define the first word, and another four to define the second. It’s a good thing I chose a short title.)

Theorem 1. Let X_{E} be a site. Then the category SH(X_{E}) of sheaves of abelian groups on X_{E} is an abelian category with enough injectives.

Proof: There is no profit in pursuing the proof of this theorem at this time. Look it up.

Remark 2. The point of the theorem, of course, is that left exact (covariant) functors on SH(X_{E}) give rise to right derived functors. We use this general machinery to produce cohomology groups (and some other things). The following diagram shows some left exact functors and their derived functors.

P(X_{E})
I↑_{H^{i}}
AB ⟵_{Ext_{SH}^{i}(F_{0},·)}^{Hom_{SH}(F_{0},·)} SH(X_{E}) ⟶_{H^{i}(X_{E},·)}^{Γ(X_{E},·)} AB
Hom(F_{0},·)↓_{Ext(F_{0},·)}
SH(X_{E})

Remark 3. As a matter of fact, there are several more sets of derived functors that I should mention. First, let π : X_{E} → X′_{E′} be a continuous morphism of sites. Since the direct image functor is left exact, there are derived functors R^{i}π_{∗} : SH(X_{E}) → SH(X′_{E′}). In the geometric setting, we know that π^{∗} is exact (and it preserves injectives). Although π^{∗} doesn’t have interesting derived functors, this observation can be used to show that the cohomology groups are contravariantly functorial in the first variable. To be more precise, we have the following result.

Lemma 4. For any sheaf F′ ∈ SH(X′_{E′}), there is a natural homomorphism

H^{i}(X′_{E′}, F′) → H^{i}(X_{E}, π^{∗}F′).

Proof: Since π^{∗} is exact, both sides are δ-functors. The left side is universal, so the obvious homomorphism when i = 0 induces the desired maps.

Remark 5. Let j : U → X be a morphism in C/X. The functor Γ(U, ·) is a left exact functor from the category of sheaves to the category of abelian groups. Its derived functors will (for the moment) be denoted H^{i}(U, ·). Now given a sheaf F on X_{E}, we have at least two ways to think about computing its cohomology on U. Fortunately, these two ways are canonically isomorphic.

Lemma 6. Let j : U → X be a morphism in C/X. Then

H^{i}(U, F) ≈ H^{i}(U_{E}, j^{∗}F).

Proof: In this case, j^{∗} is exact and preserves injectives.

Being particularly (if not polymorphically) perverse, we will use yet a third notation for these cohomology groups: H^{i}(U, F|_{U}).

Remark 7. Combining the two previous lemmas, we see that there are functors from C/X to AB given by the assignment U ↦ H^{i}(U, F|_{U}). Another name for such a functor is “presheaf of abelian groups” on X_{E}. In the diagram illustrating the myriad derived functors we knew how to construct, we included the derived functors of I, which produced presheaves. In fact, it is relatively easy to show that

H^{i}(F)(U) = H^{i}(U, F_{U}).

(Use the fact that both sides are universal δ-functors, which agree when i = 0 for trivial reasons.)

Proposition 8. There are natural isomorphisms

H^{i}(X_{E}, F) ≈ Ext_{SH}^{i}(Z, F).

Proof: Start by observing that there is a natural isomorphism

Hom_{SH}(Z, F)⟶^{≈}Γ(X, F).

The natural map arises as follows. Start with a sheaf homomorphism φ from the constant sheaf Z of integers to an arbitrary sheaf F. Taking global sections gives a homomorphism of abelian groups Z → Γ(X, F). This homomorphism of abelian groups is completely determined by the image s = φ(1), which is a global section of F. For the general case, note that an isomorphism between left exact functors induces an isomorphism between their derived functors. The result follows.

Remark 9. We use Z because we’re working in the category of all sheaves of abelian groups. This is different from the category of sheaves of O_{X}-modules. In the latter category, we would use O_{X} instead of Z for the basic sheaf.

Proposition 10. Let π : X_{E} → X′_{E′} be a continuous morphism of sites. Then there are natural isomorphisms

R^{i}π_{∗}(F) ≈ aπ_{pre}H^{i}(F).

Proof: When i = 0, this isomorphism is the statement π_{∗} = aπ_{pre}I. Since both a and π_{pre} are exact functors, the result follows by the usual appeal to the theory of universal δ-functors.

Before describing the next relation between the various derived functors, we need yet another interlude on stalks, fibres, local rings and such. Let X be a scheme, let x ∈ X be an (ordinary) point, and let x̄ = Spec(k̄(x)) be a geometric point lying over x. We know that the stalk of O_{X} at x for the Zariski topology is a local ring. What is the stalk of G_{a} at x̄ for theétale topology? The short answer is: the strict henselization of the usual local ring. Since that answer is singularly unenlightening, we need to expand upon it.

Definition 11. Let R be a local ring, and let R̂ be its completion. The henselization of R is defined as the set R^{h} of elements of R̂ that are algebraic over R.

Remark 12. Actually, this is a completely nonstandard definition—it is, however, a consequence of the usual definitions. It also is the quickest way to see that there is something interesting living in between a local ring and its completion. Another way to approach the definition follows.

Definition 13. Let R be a local ring with residue field k. The natural map from R[t] to k[t] will be denoted by f ↦ f̄. Now R is called henselian if whenever a non-constant monic polynomial f ∈ R[t] has a reduction that factors as f̄ = g_{0}h_{0} ∈ k[t] with both g_{0} and h_{0} monic, then there exist monic polynomials g and h in R[t] such that ḡ = g_{0}, h̄ = h_{0}, and f = gh.

Remark 14. This definition is taken from the conclusion of Hensel’s Lemma (the proof of which is essentially Newton’s Method) that complete local rings are henselian. So, the map R → R̂ sends R to a henselian ring. You can now give an alternate definition of the henselization R^{h}; it is the smallest henselian ring in R̂ that contains R. It is reasonably straightforward to check that the set of algebraic elements in R̂ forms a henselian ring, and it is a short step from there to showing that this definition is equivalent to the earlier one.

Now, we still haven’t made a connection between henselian rings and local rings for the étale topology. So, we need yet a third characterization of the henselization. Consider all commutative diagrams

\begin{aligned}& & U \\ & ↗ & ↓^{φ} \\ x & → & X \end{aligned}

where φ is étale of finite type. These diagrams describe “étale neighborhoods” of x in X; an essential part of the data is the lift of x into U. The henselization of the (Zariski) local ring O_{X,x} is the limit lim_{→} Γ(U, O_{U}) over all étale neighborhoods U of x. (If you believe the characterization of the henselization in terms of algebraic elements, this is relatively straightforward.)

This characterization of the henselization almost gets us to where we want to be. However, it is clear from the last construction that O_{X,x}/M_{x} = O_{X,x}^{h}/M_{x} = Õ_{X,x}/M_{x}. (The existence of the lift of x restricts the residue field.) To get to the generic point, we take the limit over all étale neighborhoods of x̄ rather than neighborhoods of x. This amounts to replacing x by x̄ in the previous diagram. The resulting ring is called the strict henselization, and it is denoted by either O_{X,x}^{sh} or O_{X,x̄}. Intuitively, we want to think of O_{X,x̄} as the étale analog of a small disk in the usual topology of a complex manifold. It is considerably smaller than a Zariski local ring, though not so small as its completion.

Theorem 15. Let π : Y → X be a quasicompact morphism of schemes. Let x̄ → X be a geometric point of X. Write X̃ = Spec(O_{X,x}^{sh}) for the spectrum of the strict henselization of the local ring. Define Ỹ via the pullback diagram

\begin{matrix}Ỹ & ⟶ & Y \\ ↓π̃ & & ↓^{π} \\ X̃ & ⟶ & X \end{matrix}

For every étale sheaf F on Y, there are isomorphisms

R^{i}π^{∗}(F)x̄⟶^{≈}H^{i}(Ỹ , F̃).

Proof: Note that the dependence of the right-hand side on x̄ is hidden in the definition of Ỹ. The stalk is a direct limit, and can be computed using a cofinal system. Choose affines U such that X̃ = lim_{←} U. Then Ỹ = (lim_{←} U) × Y) = lim_{←}(U × Y). Moreover, each map in the inverse system is a affine morphism of quasicompact schemes. The result reduces to the (extremely technical and unpleasant) assertion [EGA.IV.17] that étale cohomology commutes with such limits.