Étale Cohomology

Report on Meeting 1

We talked about flat modules. If R is any commutative ring with 1 and M is an R-module, then we say that M is flat if M ⊗_{R} − is an exact functor on the category of R-modules.

Proposition 1. For arbitrary modules over arbitrary rings, free ⇒ projective ⇒ flat.

Proof: Since projective is equivalent to direct summand of free, it is easy to see that free implies projective. Now suppose that M is projective. Find another module N so that M ⊕ N = F is free. Given an exact sequence of modules

0 → X → Y → Z → 0,

we can tensor it with both M and N separately. Since tensor commutes with direct sums, the sum of the tensored sequences is the same thing you would get by tensoring with F. Since free clearly implies flat, we know that the map (M ⊗X)⊕(N ⊗X) = F ⊗X → F ⊗Y is injective. Hence, the restriction to M ⊗ X must also be injective, as desired.

Remark 2. If R is any ring and S is any multiplicative set in R, then the localization S^{−1}R is a flat R-module. This follows directly from the definition of localization. It can be used to give examples of flat modules that are not projective.

Proposition 3. Flatness is preserved under base extension.

Proof: More precisely, this means that if M is a flat R-module and if S is any R-algebra, then M ⊗_{R} S is a flat S-module. The result now follows from the equality of the functors (M ⊗_{R} S) ⊗_{S} − and M ⊗_{R} −).

Proposition 4. For arbitrary modules over arbitrary rings, flatness is a local property (which can be checked by localizing at all maximal ideals).

Proof: Since localization of modules is the same as base extension to the local ring, we see immediately that flat implies locally flat. Now suppose M is a locally flat R-module and let

0 → X → Y → Z → 0

be any exact sequence of R-modules. Tensoring with M, we get a new exact sequence

0 → K → M ⊗ X → M ⊗ Y → M ⊗ Z → 0.

This sequence defines the kernel K. The problem is to show that K = 0.

Since localization is an exact functor, we can transport the four-term exact sequence to any local ring of R. Since M is locally flat, we know that all the localizations K_{p} are trivial. In particular, if x ∈ K is any element and if p is any prime ideal of R, then there exists an element s ∈ R−p such that sx = 0. Now let I be the annihilator ideal in the ring R of the element x. We have just seen that no prime ideal of R can contain I. Therefore, I = R. Since the element 1 lies in the annihilator of x, we must have x = 0, and therefore K = 0. The result follows.

Proposition 5. Let R be a noetherian local ring, and let M be a finitely generated flat R-module. Then M is free.

Proof: Let I be the maximal ideal of R and let k = R/I be the residue field. Write M̄ = M ⊗ k = M/IM for the reduction of M to a module over the residue field. This is a finitely generated vector space; choose a basis and lift it arbitrarily to x_{1}, x_{2}, …, x_{n} ∈ M. By Nakayama’s Lemma, the x_{i} generate M. Now let F be free on n-generators, mapping to M in the obvious way. We get an exact sequence

0 → K → F → M → 0.

Because we chose the generators by lifting a basis, we know that the induced map F̄ → M̄ is an isomorphism. Tensoring the above exact sequence with the canonical exact sequence

0 → I → R → k → 0,

we get an exact commutative diagram

0  0
↓  ↓
K ⊗ I ⟶ F ⊗ I ⟶ M ⊗ I ⟶ 0
↓  ↓  ↓
0 ⟶ K ⟶ F  ⟶ M ⟶ 0
↓  ↓  ↓
K̄ ⟶ F̄ ⟶ M̄ ⟶ 0
↓  ↓  ↓
0  0  0

By chasing the diagram, we see that the left map on the bottom row is injective. By our earlier observation, we now know that K̄ = 0. Because R is noetherian and F is finitely generated, K is also finitely generated. So, we can apply Nakayama’s Lemma again to conclude that K = 0, and thus F ≈ M.

Remark 6. Putting together the results we have obtained so far, we see that for finitely generated modules over local rings, the concepts of flat, free, and projective are equivalent. We certainly know examples of finitely generated projective modules over noetherian rings that are not free. The earlier examples (using localization) give infinitely generated flat modules over noetherian rings that are not projective. That’s the best we can do, however, because of the following result.

Proposition 7. Let M be a finitely generated module over a noetherian ring. Then M is flat ⟺ M is projective ⟺ M is locally free.

Proof: We already know, under these hypotheses, that flat ⟺ locally flat ⟺ locally free, so we only need to check their equivalence with projective. First, choose a finite set of generators for M, so that we have an exact sequence

0 → K → F → M → 0,

where F is free. The goal is to try to split this exact sequence. If we use the noetherian hypothesis, we can conclude that K is finitely generated. Now the problem reduces to the following lemma.

Lemma 8. 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 g : F → K such that gι(x_{j}) = x_{j} for all j.

Proof: We don’t know how to prove the lemma, but we expect to use induction.

Another possible approach is to apply the functor Hom(−, K) to get an exact sequence

0 → Hom(M, K) → Hom(F, K) → Hom(K, K) → Ext(M, K).

To split the sequence, we need to find g ∈ Hom(F, K) that maps to the identity element in Hom(K, K). We see that the obstruction lies in Ext(M, K). We already know that the obstruction vanishes if we do all our computations over one of the local rings of R. So, we could finish the proof if we knew how to show the following lemma.

Lemma 9. Ext_{R}(M, K)_{p} = Ext_{R_{p}}(M_{p}, K_{p}) for all prime ideals of R.

Proof: We don’t know how to do this, either.

Remark 10. Assignment for next time: First, figure out how to finish off the proof of the last result. (Use whichever of the two methods seems most congenial.) Then do some reading. The goal is to learn something about the following concepts:

Flat morphisms of schemes Flat families of schemes Faithfully flat ring extensions (and relations with descent theory) Faithfully flat morphisms of schemes (and descent theory)

Some places to read include:

Hartshorne, Chapter III, Section 9

Milne, Chapter I, Section 2

Freitag and Kiehl, Chaper I, Section 1

Matsumura (some unknown chapter, because somebody has my copy)