Étale Cohomology

Report on Meeting 7

The principal theme of this meeting was the study of sundry functors and their adjoints. To begin, take a continuous morphism of sites π : (C′/X′)_{E′} → (C/X)_{E}. At the last meeting, we explained how to build direct images and pullbacks of presheaves, obtaining a diagram

P((C′/X′)_{E′})\begin{matrix}π^{pre} \\ ⟵ \\ ⟶ \\ π_{pre} \end{matrix}P((C/X)_{E})

The two functors in the diagram are not unrelated; the top arrow (pullback) is the left adjoint of the bottom arrow (direct image). Just as a reminder, this means that for all sheaves P ∈ P((C/X)_{E}) and P′ ∈ P((C′/X′)_{E′}), there is a natural bijection

Hom(π^{pre}(P), P′) = Hom(P, π_{pre}(P′)).

Remark 1. It is always the case that π_{pre} is an exact functor, since exactness of a sequence of presheaves is measured by exactness of every sequence of sections. The pullback functor π^{pre} is always left exact. In the geometric cases of interest, it is actually exact. The categorical condition that is needed for this result is that the underlying categories ((Et)/X, (LF T)/X, etc) contain all finite inverse limits.

Remark 2. Our next task is to investigate the stalks of an étale presheaf. As we saw last time, points for the étale topology are given by spectra of separably closed fields. So, let x̄ = Spec(F̄) be such a spectrum. Let X be a scheme, u : x̄ → X a geometric point, and let P be an étale presheaf of abelian groups on X_{et}. Then the pullback u^{pre}(P) is a presheaf on x̄. In particular, we get an abelian group (u^{pre}(P))(x̄).

Definition 3. The stalk of the étale presheaf P on X_{et} at the geometric point x̄ is defined to be Px̄ = (u^{pre}(P))(x̄).

Remark 4. The functor P ↦ Px̄ is an exact functor from the category of étale presheaves to the category of abelian groups. This follows because u^{pre} is an exact functor. It is also worth pointing out that the composite functor

SH(x̄_{et})⟶^{I}P(x̄_{et}) ⟶^{Γ(x̄,·)}AB

is the equivalence of categories that we used to recognize separably closed fields as points in the étale topology. Thus, the stalk of a sheaf on x̄ “is” the sheaf.

Lemma 5. Let x̄ be the spectrum of a separably closed field. Then the inclusion functor I : SH(x̄_{et}) → P(x̄_{et}) has a left adjoint, which we denote by a.

Proof: Suppose U → x̄ is an étale morphism of finite type. Then U can be written uniquely as a finite disjoint union of schemes, ∐_{i=1}^{n} U_{i}, where each U_{i} = x̄. The sheaf associated to a presheaf P on x̄ is defined to be

(aP)(U) = ∏_{i} P (U_{i}) = P (x̄)^{n}.

It is easy to check that this is a sheaf, that there is a natural, universal morphism of presheaves P → aP, and (as a bonus) that the stalk of the associated sheaf aP is the same as the stalk of P.

Theorem 6. Let X be any scheme. Then the inclusion functor I : SH(X_{et}) → P(X_{et}) has a left adjoint, which we also denote by a.

Proof: For each ordinary point x ∈ X, choose a geometric point x̄ lying over it. (We can, of course, make this choice in such a way that the residue field k(x̄) is the separable closure of k(x).) Given a presheaf P on X_{et}, define P ∗ ; x̄ = a(upre ; x̄(P)). The functor a in this definition is the one defined in the lemma for presheaves on x̄. In any event, each P ∗ ; x̄ is a sheaf. Next, define P^{∗} = ∏_{x∈X}(ux̄)_{pre}(P ∗ ; x̄). Since the direct image of a sheaf remains a sheaf, and direct products of sheaves are sheaves, we know that P^{∗} is a sheaf on X_{et}. By mapping sections to their stalks, we can build a morphism φ : P → P^{∗}. (More formally, we ought to describe this as the composite

P → (ux̄)_{pre}(upre ; x̄(P)) → (ux̄)_{pre}(P ∗ ; x̄).

The first map exists because of adjunction, and the second is the map from a presheaf to its associated sheaf at the level of points.) In any event, this shows that we have at least one morphism from P to a sheaf. We can take aP to be the subsheaf of P^{∗} generated by the image of φ.

To finish the proof, you need to verify that aP has the requisite universal property. This reduces, by the usual argument, to showing that F = aF ↪ F^{∗} if F is already a sheaf.

Remark 7. The theorem generlaizes to sheaves and presheaves on an arbitrary site, but you need an argument that doesn’t use stalks. You replace the stalks by taking direct limits over “zeroth Cech cohomology group of covers.”

Remark 8. One of the facts used in the previous proof is that the direct image functor on presheaves takes sheaves to sheaves. This is easy to prove, since sections of a direct image presheaf can be computed explicitly in terms of sections of the original presheaf. This fact can be restated, using the associated sheaf functor, as a definition of the direct image functor π_{∗} : SH((C′/X′)_{E′} → SH(C/X)_{E}) by π_{∗} = aπ_{pre}I.

Theorem 9. The direct image functor on sheaves has a left adjoint, π^{∗}.

Proof: Consider the following diagram (over which I labored for hours in order to get the arrows running in both directions):

SH((C′/X′)_{E′}) \begin{matrix}π^{∗} \\ ⟵ \\ ⟶ \\ π_{∗} \end{matrix} SH((C/X)_{E})
a↑↓^{I}  ^{a}↑↓^{I}
P((C′/X′)_{E′}) \begin{matrix}π^{pre} \\ ⟵ \\ ⟶ \\ π_{pre} \end{matrix} P((C/X)_{E})

We have already noticed that π_{∗} can be computed by tracing around the other three sides of the diagram (using the arrows that go the correct direction, of course). Being optimistic, we try defining π^{∗} = aπ^{pre}I by going around the other three sides in the reverse direction. Now just check that the adjunction properties already proved work correctly to give the adjunction desired in the statement of the theorem.

Remark 10. In plainer English, the definition inside the proof of that theorem says that the pullback of a sheaf is given by taking the sheaf associated to the presheaf pullback.

Remark 11. It’s time now to look at some exact sequence of sheaves. One point of the construction of the associated sheaf functor is that you can use it to verify that the category of sheaves of abelian groups on a site is an abelian category. It is reasonably easy to check that the category of presheaves is an abelian category, since you can carry out all your constructions (on the level of sections) inside the category of abelian groups. To form inverse limits (like kernels) in the category of sheaves, you just construct them in the category of presheaves and check that they automatically turn out to be sheaves. To construct direct limits (like cokernels), you build them in the category of presheaves and then take the associated sheaf.

Remark 12. As our first example, define μ_{n}(U) to be group of n^{th} roots of unity in Γ(U, O_{U}). This is a subsheaf of the multiplicative group G_{m}. Now consider the Kummer sequence

0 → μ_{n} → G_{m}⟶^{n}G_{m} → 0.

Without the 0 on the right end, you get an exact sequence in the Zariski topology (and hence in the étale or flat topology) without any further effort. You almost never get surjectivity in the Zariski topology, since that would mean you could take n^{th} roots in all the local rings. To proceed, we need a definition.

Definition 13. Let X be a scheme. Define the characteristic of X, denoted char(X), to be the set {p|p = char(k(x)) for some x ∈ X}. If n is any integer, write (n, char(X)) = 1 to mean that none of the primes in the characteristic of X divide n.

Proposition 14. If (n, char(X)) = 1, then the Kummer sequence for n is exact in the étale topology (and hence in the flat topology).

Proof: We need to show surjectivity. So, take an étale map π : U → X of finite type, and a section s ∈ G_{m}(U) = Γ(U, O_{U}^{∗}). To show surjectivity, it is enough to show that we can find an n^{t}h root of s locally for the étale topology. So, we can start by replacing U by an open affine subset V = Spec(A) ⊂ U, and view s as an element of the ring A. Now let W = Spec(A[t]/(t^{n} − s)). There is an obvious map W → V, which is clearly finite. Moreover, a straightforward computation of the module of differentials shows that this map is étale whenever (n, char(X)) = 1. So, W is an étale neighborhood where s is an n^{th} power.

Corollary 15. The Kummer sequence is always exact in the flat topology.

Proof: The previous argument clearly works, since we only need to know that the map W → V in the proof is flat and locally of finte type.

Remark 16. Now let p be a prime, and restrict your attention to schemes X of characteristic p. The assignment x ↦ x^{p} is a ring endomorphism on any F_{p}-algebra, so it defines an endomorphism F : G_{a} → G_{a} of the additive group scheme on X. We define α_{p} = Ker(F), and we also note that there is an isomorphism Z/p ≈ Ker(1 − F).

Proposition 17. Let X be a scheme of positive characteristic p. Then the Artin-Schreier sequence

0 → Z/p → G_{a} → G_{a}⟶^{1−F}0

is exact in the étale topology (and hence in the flat topology).

Proof: Everything except surjectivity follows immediately from he above remarks. Surjectivity clearly fails in the Zariski topology. In the étale topology, we start with an étale map U → X and a section s ∈ G_{a}(U) = Γ(U, O_{U}). We shrink U to an open affine V = Spec(A), and define W = Spec(A[t]/(t^{p} − t − s)), and this shows that we can locally (for the étale topology) write s as (1 − F)(t).

Proposition 18. Let X be a scheme of positive characteristic p. Then the sequence

0 → α_{p} → G_{a} → G_{a}⟶^{F}0

is exact in the flat topology.

Proof: Note that this cannot possibly work in the étale topology, since it requires taking p^{th} roots of sections locally. It’s okay in the flat topology, since W = Spec(A[t]/(t^{p} − s)) is flat and finite over V = Spec(A).

Remark 19. While we’re in a mood to discuss examples, let’s return to the study of sheaves. This time we’ll shift from one-point spaces to something really challenging: twopoint spaces. The interesting case is the spectrum of a discrete valuation ring, which contains one open point and one closed point. We’re going to start out lsightly more generally. Let X be a scheme, i : Z ↪ X a closed immersion, and j : U ↪ X the open immersion of the complement. We have the following picture:

SH(Z_{et})\begin{matrix}i^{∗} \\ ⟵ \\ ⟶ \\ i_{∗} \end{matrix}SH(X_{et})\begin{matrix}j_{∗} \\ ⟵ \\ ⟶ \\ j^{∗} \end{matrix}SH(U_{et}).

So, given a sheaf F on X_{et}, we can pull it back to the subschemes to get sheaves F_{1} = i^{∗}(F) and F_{2} = j^{∗}(F). However, given any pair of sheaves F_{1} and F_{2}, there is no reason, in general, to believe that we can stick them together to build a sheaf on X. There must be some more data that explains how the pieces are connected.

To collect that data, start with the adjunction between j_{∗} and j^{∗}. We can use this adjunction to produce an element

ψ ∈ Hom(F, j_{∗}j^{∗}F) = Hom(j^{∗}F, j^{∗}F)

corresponding to the identity map on the right side of the bijection. Pulling ψ back to the closed subscheme gives a morphism

φ : F_{1} = i^{∗}F → i^{∗}j_{∗}j^{∗}F = i^{∗}j_{∗}F_{2}.

Definition 20. Let X be a sheaf, i : Z → X a closed immersion, and j : U → X an open immersion such that U = X − Z. Define a category T (X) whose objects are all triples (F_{1}, F_{2}, φ) with F_{1} ∈ SH(Z_{et}), F_{2} ∈ SH(U_{et}), and φ : F_{1} → i^{∗}j_{∗}F_{2}. A morphism in T (X) from one triple (F_{1}, F_{2}, φ) to another triple (G_{1}, G_{2}, ψ) is a pair of sheaf morphisms α_{i} : F_{i} → G_{i} that make the obvious square commute.

Remark 21. Even though the notation doesn’t show it, the category T (X) depends on the closed subscheme, its open complement, and the topology we’re using (in this case, étale).

Proposition 22. There is an equivalence of categories T : SH(X_{et}) → T (X).

Proof: The computations preceeding the definition of T (X) explain how to define the functor. To define the functor S that goes the other direction, form the fibre product

S(F_{1}, F_{2}, φ) ⟶ j_{∗}F_{2}
↓  ↓^{canon}
i_{∗}F_{1}  ⟶^{i_{∗}(φ)} i_{∗}i^{∗}j_{∗}F_{2}

in the category of étale sheaves on X. (Note: this category has all finite inverse limits, as remarked earlier, so in particular, it has fibre products.)

It’s celar that S is a functor, by the naturality of the canonical maps coming from the adjunctions, and the universal property of pullbacks. The same lements go into the verification that the composites of S with T (in either order) are isomorphisms.

Remark 23. Now let A be a discrete valuation ring with residue field k and field of fractions K. We can apply the previous proposition with X = Spec(A), Z = Spec(k), and U = Spec(K). Write G_{K} = Gal(K̄/K) and G_{k} = Gal(k̄/k) for the Galois groups. (Of course, the bar denotes separable closure.) In the usual way, one has the inertia and decomposition groups I < D < G_{K}, and the isomorphism D/I = G_{k}. (Irrelevant aside: K̄^{D} is the field of fractions of the henselization of A, and K̄^{I} is the field of fractions of the strict henselization.) We have previously identified SH(U_{et}) with the category of discrete G_{K}-modules and SH(Z_{et}) with the category of discrete G_{k}-modules. By the proposition, SH(X_{et}) can be identified with the set of triples (N, M, φ), where N is a G_{k}-module, M is a G_{K}-module, and φ : N → i^{∗}j_{∗}M is a homomorphism of G_{k}-modules.

To complete our understanding of this category, we only need to get a better handle on the operation i^{∗}j_{∗}. The simplest way to understand this operation is to realize that we have to turn a G_{K}-module M into a G_{k}-module. We can start by restricting the G_{K}-action to a D-action. But this isn’t enough, since there may still be an I-action. The easiest way to get rid of the I-action is to pass to the invariants M^{I}. Then the D-action on this module passes to an action of G_{k} = D/I. So, the third pieceof our triple is just a homomorphism N → M^{I}.

In fact, we have the following picture

SH(Z_{et})\begin{matrix}i^{∗} \\ ⟵ \\ i_{∗} \\ ⟶ \\ i^{!} \\ ⟵ \end{matrix}SH(X_{et})\begin{matrix}j_{!} \\ ⟵ \\ j^{∗} \\ ⟶ \\ j_{∗} \\ ⟵ \end{matrix}SH(U_{et})

Each functor is left adjoint to the one directly beneath it. We can describe them all explicitly in terms of triples by

\begin{matrix}N ←| (N, M, φ) & (0, M, 0) ←| M \\ N |→ (N, 0, 0) & (N, M, φ) |→ M \\ Ker(φ) ←| (N, M, φ) & (N, M^{I}, φ) ←| M \end{matrix}

As a final remark, we point out that all six functors are left exact, and the functors i^{∗}, i_{∗}, j^{∗}, and j_{!} are actually exact.