Étale Cohomology

Report on Meeting 4

We’re almost ready to define the first word in the title of this series. Before that, however, we need one more result about unramified morphisms.

Proposition 1. A morphism f : Y → X of schemes is unramified if and only if the diagonal morphism Δ : Y → Y ×_{X} Y is an open immersion.

Proof: We already know that Δ is locally closed. So, shrink Y ×_{X} Y to an open subscheme U in which Y is a closed subscheme defined by an ideal sheaf I. If f is unramified, then the sheaf of differentials vanishes, so I = I^{2} at the points of Y. By Nakayama’s Lemma applied to the local rings, we see that I itself is 0 on Y. Thus, I must be the zero sheaf on an open neighborhood V ⊂ U that contains Y. But then Y is a closed subscheme of V defined by the 0 sheaf, hence Y = V is an open subscheme of the product.

Definition 2. Let f : Y → X be a morphism of schemes. We say that f is étale if it is flat, unramified, and locally of finite type.

Lemma 3. An open immersion is étale.

Proof: The local rings are the same; all the properties are local.

Proposition 4. The composition of étale morphisms is étale; the base change of an étale morphism is étale.

Proof: We have already checked that flatness and LFT are preserved under composition and base change. For unramified, the result about composition follows from the first exact sequence, and the result about base change follows from the computation last time of the behavior of differentials under base change.

Proposition 5. Let Y ⟶^{f}X⟶^{g}S be morphisms of schemes. Assume that g is unramified and gf is étale. Then f is étale.

Proof: The graph of f gives a morphism Y ⟶^{Γ}Y ×_{S} X. Composing with the projection Y ×_{S} X⟶^{p_{2}}X allows us to recover f. Now Γ is a pullback of the open immersion Δ : X → X ×_{S} X; hence, Γ is étale. However, p_{2} is the pullback of the étale structure map Y → S, so it is also étale. Now f is the composite of étale morphisms, hence étale.

Remark 6. There is a canonical example of an étale morphism. Let A be a ring, and consider finitely generated algebras of the form C = A[t_{1}, …, t_{n}]/(P_{1}, …, P_{n}). If the Jacobian J = det(∂P_{i}/∂t_{j}) is a unit in C, then C is an étale A-algebra. (This is an easy computation.) In fact, every étale morphism is locally of this form. (This is rather harder, so I won’t prove it.)

Remark 7. The remainder of this week’s meeting was devoted to an affine approach to étale cohomology, and consisted primarily of assertions unencumbered by proofs. Here’s a synopsis.

Definition 8. Given a ring A, let Et(A) denote the category of étale A-algebras.

Remark 9. The last proposition (in the affine case) shows that every morphism in the category Et(A) is an étale morphism.

Definition 10. Let B ∈ Et(A). An étale covering of B is a collection B = {B → B_{i}} of étale B-algebras with the property that B → ∏ B_{i} is faithfully flat.

Remark 11. The introduction of the faithfully flat hypothesis is a red herring. It’s designed to allow us to state the conditions in a purely affine setting. If we switch to geometry, then the crucial condition is just that ∪φ_{i}(Spec(B_{i})) covers Spec(B) in the usual sense.

Definition 12. An étale presheaf on A is a covariant functor P : Et(A) → Sets (or to Ab, etc). An étale presheaf is a sheaf if it satisfies the following two properties.

(i) P (B × C) = P (B) × P (C) (ii) If B → C is a faithfully flat morphism in Et(A), then

P (B) → P (C)\overrightarrow{→}P (C ⊗_{B} C)

is exact.

Remark 13. This condition can be reinterpreted as the usual set of sheaf theoretic conditions (a section that is locally zero is actually zero, and a collection of compatible local sectyions can be patched to give a global section). Given any presheaf P, you can build an associated sheaf that is universal for morphisms of presheaves from P to a sheaf. Each étale A-algebra B gives rise to a presheaf C ↦ Hom_{A}(B, C). The descent theory from the previous two weeks shows that this is actually a sheaf. Any sheaf isomorphic to one of these is called a representable sheaf.

Finally, we note that the category of sheaves of abelian groups is an abelian category with enough injectives, and the global section functor is left exact. Therefore, we can define cohomology by taking right exact functors

H_{et}^{i}(A, S) = R^{i}(Γ(A, S)).

These have the standard set of nice properties.

Finally, I tried without success to show that the category of étale sheaves on Spec(F) with F a field is equivalent to the category of discrete modules with an action of G = Gal(F̄/F). It would follow from this fact that the étale cohomology of Spec(F) is isomorphic to the Galois cohomology of F. But that will have to wait until next time.