Étale Cohomology

Report on Meeting 2

We talked about faithfully flat modules and descent theory. As before, let R be a commutative ring with 1, and let M be an R-module. We begin by completing the proof of the following result.

Lemma 1. Let 0 → K⟶^{ι}F → M → 0 be an exact sequence of modules over an arbitrary ring, with F free and M flat. Let x_{1}, …, x_{n} be any finite set of elements of K. Then there exists a partial splitting h : F → K such that hι(x_{j}) = x_{j} for all j.

Proof: By induction on the number of specified elements in K. When only one element x ∈ K is given, we begin by choosing a basis {e_{i}} for F. Write x = ∑_{1}^{m} a_{i}e_{i} as a finite linear combination of basis elements, with a_{i} ∈ R. Let I be the ideal of R generated by the elements {a_{i}}. There is an exact commutative diagram

0  0
↓  ↓
K ⊗ I ⟶ F ⊗ I ⟶ M ⊗ I ⟶ 0
↓  ↓  ↓
0 ⟶ K  ⟶ F  ⟶  M  ⟶ 0
↓  ↓  ↓
K/IK ⟶ F/IF ⟶ M/IM ⟶ 0
↓  ↓  ↓
0  0  0

(Note: the zeros at the top use flatness.) Using the snake lemma, we see that K/IK injects into F/IF. Since x ∈ K maps to zero in F/IF by definition, the injectivity tells us that it also maps to zero in K/IK. So, there exist elements y_{i} ∈ K such that x = ∑ a_{i}y_{i}. We define the desired partial splitting by sending the basis elements of F to K via e_{i} ↦ y_{i}.

Now suppose n > 1. Begin by choosing a map f : F → K such that f(x_{n}) = x_{n}. Then define new elements of K by setting y_{i} = x_{i} − f(x_{i}). By induction, we can find another homomorphism that satisfies g(y_{i}) = y_{i} for y = 1, …, n − 1. Now the desired map is defined by

h(x) = f(x) + g(x − f(x)).

One verifies this by computing

h(x_{n}) = f(x_{n}) + g(x_{n} − f(x_{n})) = x_{n} + g(0) = x_{n}

and

h(x_{i}) = f(x_{i}) + g(x_{i} − f(x_{i})) = f(x_{i}) + g(y_{i}) = f(x_{i}) + x_{i} − f(x_{i}) = x_{i}.

Remark 2. You might recall from installment one that we needed this lemma to prove that for a finitely generated module over a noetherian ring, the notions of flat, projective, and locally free are all equivalent.

Definition 3. Let A → B be a homomorphism of commutative rings with 1. We say that this morphism is flat (or that B is a flat A-algebra) if B is flat as an A-module. We will also say that the morphism is faithfully flat if, for every nonzero A-module M, the tensor product B ⊗_{A} M is a nonzero B-module.

Theorem 4. Let f : A → B be a flat ring homomorphism. Then the following are equivalent

i. A → B is faithfully flat. ii. The induced map Spec(B) → Spec(A) is surjective. iii. For every maximal ideal M in A, one has f(M)B ≠ B. iv. A sequence X → Y → Z of A-modules is exact if and only if the corresponding sequence B ⊗_{A} X → B ⊗_{A} Y → B ⊗_{A} Z is exact.

Proof: (i ⇒ ii) Let P be a prime ideal in A. Let k(P) denote the field of fractions of the integral domain A/P. By faithful flatness, the tensor product B ⊗ k(P) is nonzero. We have a commutative diagram of rings

A ⟶  B
↓  ↓
k(P) ⟶ B ⊗ k(P)

which is dual to the diagram of schemes

Spec(B ⊗ k(P)) ⟶ Spec(B)
↓  ↓
P  ⟶ Spec(A).

Tracing down the left side and across the bottom, one only needs to know that the upper lefthand corner contains a nonempty set (because the ring is nonzero) in order to check that there are points mapping to P. Since the diagram commutes, the result follows.

(ii ⇒ iii) We merely point out that B/f(M)B ≈ B ⊗ A/M = B ⊗ k(M) is the ring whose spectrum is the fibre over the point M ∈ Spec(A).

(iii ⇒ i) Let N be a nonzero A-module. Choose a nonzero element x ∈ N and let I be its annihilator ideal. By the definition of the annihilator, multiplication by x defines an injection A/I → N. By flatness, we can tensor this with B to get an injection B/IB → N ⊗ B. Thus, it suffices to show that B/IB is nonzero. To see this, take a maximal ideal M containing I. One has a surjection B/IB → B/MB, and the latter was nonzero by hypothesis.

(i ⇒ iv) Since we have assumed flatness throughout, every exact sequence of A-modules stays exact when tensored with B. So, we can assume that we have a composable pair of A-module homomorphisms X → Y → Z with the property that the induced sequence B ⊗_{A} X → B ⊗_{A} Y → B ⊗_{A} Z is exact. Introduce notations for the kernels and cokernels, as follows:

0 → K_{1} → X → Y → C_{1} → 0,
0 → K_{2} → Y → Z → C_{2} → 0,

and

0 → K_{3} → X → Z → C_{3} → 0,

After tensoring with B, these sequences remain exact. Thus, we have B-module isomorphisms B ⊗ C_{1} ≈ B ⊗ K_{2}, B ⊗ K_{3} ≈ B ⊗ X, and B ⊗ Z ≈ B ⊗ C_{3}. It is now enough to show that any A-module map M → N that induces an isomorphism is already an isomorphism. To see this, just note that the kernel and cokernel both become zero after tensoring, so they must already be zero.

(iv ⇒ i) Suppose B ⊗ M = 0. Consider the sequence 0 → M → 0 of A-modules. After tensoring with B, this sequence becomes exact. So, it’s already exact. Therefore, M = 0.

Remark 5. There is probably a less cicuituous proof of the previous equivalences, if I’d only been smart enough to list the four different properties in the correct order.

Remark 6. It’s time to turn our attention away from rings and modules toward the category of schemes and morphisms. The key is to pick the correct form of the equivalent properties in order to generalize these notions to schemes. Since sheaves are basically local creatures, we use the fact that flatness is local to make the first definition. We then use the only geometric version of faithful flatness (surjectivity on spectra) to generalize its definition.

Definition 7. Let X be a scheme and let F be a sheaf of O_{X}-modules. We say that F is a flat sheaf if each of the stalks F_{x} is a flat module over the local ring O_{X,x}.

Definition 8. Let f : Y → X be a morphism of schemes. We say that f is a flat morphism if f_{∗}O_{Y} is a flat sheaf of O_{X}-modules. (Equivalently, if each local homomorphism O_{X,f(y)} → O_{Y,y} is a flat ring homomorphism.)

Definition 9. A morphism f : Y → X is said to be faithfully flat if it is flat and surjective.

Remark 10. We have some idea why flatness is useful; it is a generalization of projective. So far, we have no justification for introducing the notion of faithful flatness. It’s time to remedy that situation. For the moment, let’s work with rings and modules.

Lemma 11. Suppose f : A → B is a faithfully flat ring homomorphism. Then there is an exact sequence

0 → A⟶^{f}B⟶^{φ}B ⊗_{A} B

where φ(b) = 1 ⊗ b − b ⊗ 1.

Proof: Suppose first that we have a section s : B → A of f. Then the injectivity of f follows automatically from the relation sf = 1_{A}. Now define t : B ⊗ B → B by t(b ⊗ c) = s(b)c. For any b ∈ B, one has b = t(φ(b)) + f(s(b)). Thus, if φ(b) = 0, we have b = f(s(b)) ∈ f(A).

Now we just need to produce a section. Since B is faithfully flat, we can tensor the given sequence with B to obtain

0 → B⟶^{g}B ⊗ B → B ⊗ B ⊗ B

The map g has an obvious section, given by multiplication. So this sequence is exact by the earlier argument, and the original sequence is exact by faithful flatness.

Corollary 12. Let C be any ring. Then there is an exact sequence

0 → Hom(C, A) → Hom(C, B) → Hom(C, B ⊗ B)

Proof: Hom(C, −) is a left exact functor; apply it to the sequence in the lemma.

Remark 13. The exact sequence of the lemma can be extended to the right. The most general statement of the result is: If B is a faithfully flat A-algebra and if M is any A-module, then there is an exact sequence

0 → M → M ⊗_{A} B⟶^{d_{0}}M ⊗_{A} B ⊗_{A} B⟶^{d_{1}}M ⊗_{A} B ⊗_{A} B ⊗_{A} B⟶^{d_{2}} ⋯

where

d_{n}(m ⊗ b_{0} ⊗ ⋯ ⊗ b_{n}) = ∑_{i=0}^{n}(−1)^{i}m ⊗ b_{0} ⊗ ⋯ ⊗ b_{i−1} ⊗ 1 ⊗ b_{i} ⊗ ⋯ ⊗ b_{n}.

Remark 14. Let’s return to the world of schemes. Suppose f : Y → X is a faithfully flat, quasi-compact morphism of schemes. Let Z be any other scheme. Consider the diagram of sets

Hom(X, Z) → Hom(Y, Z)\overrightarrow{→}Hom(Y × Y, Z)

The two maps on the right are given by composition with the two different projections from the fibred product. This sequence of sets is exact in the following sense: The first map is injective, and its image is exactly the set of homomorphisms where the two right hand maps agree. Notice that we just proved this result when X, Y, and Z are all affine. The main work remaining is to use quasi-compactness to reduce to the affine case. The injectivity (and hence uniqueness of the maps on X) allows you easily to do the case when Z is arbitrary. In the general case, you take a finite affine cover of Y, separate it into a disjoint union of those affines, and use the affine case and uniqueness to glue things together. (The short form of the statement about exactness of those sequences for all Z is to say: If f : Y → X is faithfully flat and quasi-compact, then f is a strict epimorphism.)

Remark 15. (Descent Theory for Modules) Let B be a faithfully flat A-algebra and let M be an A-module. Write N = M ⊗_{A} B. If we tensor N up to B ⊗_{A} B via the two different maps corresponding to the two projections (b ↦ 1 ⊗ b and b ↦ b ⊗ 1), we get two different B ⊗_{A} B-modules, N ⊗_{A} B and B ⊗_{A} N. Because the two restrictions to A are the same, these modules are isomorphic. Explicitly, the isomorphism is given by the map

φ : N ⊗_{A} B = M ⊗_{A} B ⊗_{A} B → B ⊗_{A} M ⊗_{A} B = B ⊗_{A} N

which satisfies

φ : m ⊗ b ⊗ c ↦ b ⊗ m ⊗ c.

There is an additional naturality to this isomorphism. The map φ induces three different maps

φ_{1} : B ⊗_{A} N ⊗_{A} B → B ⊗_{A} B ⊗_{A} N b ⊗ (m ⊗ c) ⊗ d ↦ b ⊗ c ⊗ (m ⊗ d)
φ_{2} : N ⊗_{A} B ⊗_{A} B → B ⊗_{A} B ⊗_{A} N (m ⊗ b) ⊗ c ⊗ d ↦ b ⊗ c ⊗ (m ⊗ d)
φ_{3} : N ⊗_{A} B ⊗_{A} B → B ⊗_{A} N ⊗_{A} B (m ⊗ b) ⊗ c ⊗ d ↦ b ⊗ (m ⊗ c) ⊗ d

by tensoring with the identity in the first, second, or third positions. Now check that φ_{2} = φ_{1}φ_{3}.

Now for the descent theory: Suppose N is a B-module for which you can find a B ⊗ B-isomorphism φ : N ⊗ B → B ⊗ N. Suppose the induced maps satisfy the identity φ_{2} = φ_{1}φ_{3}. Then there exists an A-module M such that N ≈ M ⊗_{A} B. To see this, you can explicitly identify M = {n ∈ M|1⊗n = φ(n⊗1)}. You need to check that the obvious map ψ : M ⊗_{A} B → N by ψ(m ⊗ b) = mb is an isomorphism. here is where we need to use the “strict epimorphism” result for faithfully flat algebras. Look at the following diagram

N ⊗_{A} B \overrightarrow{⟶}_{β⊗1}^{α⊗1} B ⊗_{A} N ⊗_{A} B
↓^{φ}  ↓^{φ_{1}}
B ⊗_{A} N \overrightarrow{⟶}_{p_{2}⊗1}^{p_{1}⊗1} B ⊗_{A} B ⊗_{A} N

where α(n) = 1 ⊗ n, β(n) = φ(n ⊗ 1), p_{1}(b) = 1 ⊗ b, and p_{2}(b) = b ⊗ 1. This diagram commutes with either the upper or lower maps (the lower maps use the identity on φ_{i}’s). By flatness and the definition of M, the kernel of the pair at the top of the diagram is M ⊗ B. By the earlier lemma, the kernel of the pair at the bottom is N. Since the vertical arrows are isomorphisms, the result follows.

Remark 16. Essentially the same argument, with a few flourishes, proves the following two propositions.

Proposition 17. (Descent Theory for Sheaves) Let f : Y → X be a faithfully flat, quasi-compact morphism of schemes. To give a quasi-coherent O_{X}-module F is the same as to give a quasi-coherent O_{Y} -module G together with an isomorphism φ : p_{1}^{∗}(G) → p_{2}^{∗}(G) satisfying

p_{31}^{∗}(φ) = p_{32}^{∗}(φ)p_{21}^{∗}(φ)

(Here the p_{i} and p_{ij} denote the obvious projections on fibre products of Y with itself over X.)

Proposition 18. (Descent Theory for Sheaves) Let f : Y → X be a faithfully flat, quasi-compact morphism of schemes. To give a scheme Z that is affine over X is the same as to give a scheme W that is affine over Y together with an isomorphism φ : p_{1}^{∗}(W) → p_{2}^{∗}(W) satisfying

p_{31}^{∗}(φ) = p_{32}^{∗}(φ)p_{21}^{∗}(φ)

Remark 19. If you stare at these descent theoretic computations long enough, you start to realize that you are computing some sort of Cech cohomology groups. The fundamental picture to keep in mind is the diagram

⋯ Y × Y × Y \overrightarrow{\overrightarrow{→}}Y × Y \overrightarrow{→}Y → X

Taking morphisms to a scheme Z turns this picture around and replaces the products with Hom-groups. Think of those group as “sections” of the “sheaf” Z, and think of the fibred product as “intersection” of the “open sets” in the “cover” Y → X. Everything we’ve done has been to check some sort of exactness in this complex, thereby verifying the vanishing of some sort of Cech cohomology group.