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Theorem 1 (Serre). Let X = SpecA be an affine scheme, F a quasi-coherent sheaf. Then H i (X, F) = 0 for i ≥ 1. We shall prove this result following [?]. The idea is that X has a very nice basis: namely, the family of all sets D(f ), f ∈ A. These are themselves affine, and moreover the intersection of any two elements in this basis is still in this basis. For D(f g) = D(f ) ∩ D(g). 0.1 A lemma of Kempf First, we set up some notation, following Kempf. Given V ⊂ X open and a sheaf F on X, define V F to be the sheaf i∗ (i−1 F) on X, where i : V → X is the inclusion. This is equivalently the sheaf U → F(V ∩ U ). There is a natural map F → V F, which of course induces maps on cohomology. The elementary result that Kempf proves is: Lemma 1 (Kempf). Let X be a topological space and let A be a basis for X which is closed under finite intersections. Suppose n ∈ N is fixed. Suppose F ∈ Sh(X) for X a topological space is such that H i (U, F|U ) = 0 for all 0 < i < n and U ∈ A. Suppose α ∈ H n (X, F). Then there is a covering of X by open sets V ∈ A such that the image of α in H n (X, V F) is zero for each V . Proof. We will prove this result by induction on n. First, suppose n > 1, and that the result is valid for n − 1. The base case will be handled next. We can embed F in a flabby sheaf G, and let H be the cokernel. There is an exact sequence 0→F →G→H→0 (1) and by the long exact sequence for cohomology (and since n > 1), it follows that 0 → F(U ) → G(U ) → H(U ) → 0 (2) is exact for every U in the open basis A. Now fix V ∈ A and consider the complex of sheaves 0 → V F → V G → V H → 0. (3) In general, we know that i−1 is exact, by looking at the stalks, but only that i∗ is left-exact. From this alone we get that (3) is exact except perhaps at the last step. But we also know that for any U ∈ A, we have that the sequence of sections 0 → V F(U ) → V G(U ) → V H(U ) → 0 (4) is exact in view of the definition of V and exactness of (2). Consequently, since A is a basis, we can pass to the direct limit to the stalks, and we see that (3) must be exact at the last step too. But V G is also flabby and consequently has trivial cohomology. As a result, we find that for any V ∈ A, there is an isomorphism H n−1 (X, V H) ' H n (X, V F). 1 Moreover, since G is flabby and thus has trivial cohomology on X, we get isomorphisms from the long exact sequence of (1): H n−1 (X, H) ' H n (X, F). This means that H satisfies the conditions of the proposition with n − 1, and we have assumed inductively that the result is valid for n − 1. So α maps to some β ∈ H n−1 (X, H); this means there is an open cover of X by various V ∈ A such that β maps to zero in H n−1 (X, V H). This means that α maps to zero in these H n (X, V F) by naturality. This completes the proof of the inductive step. The base case remains, i.e. n = 1. Fix α ∈ H 1 (X, F). We can still embed F in a flabby sheaf and obtain an exact sequence as in (1). So we get an exact sequence: 0 → Gm(X, F) → Gm(X, G) → Gm(X, H) → H 1 (X, F) → 0. However, exactness of (4) is now no longer valid, so we cannot conclude that (3) is exact. We do, however, have an exact sequence 0 → V F → V G → K(V ) → 0 exact for some cokernel K(V ) , and we can fit these into an exact commutative diagram /G /H /0 /F 0 0 / VF / VG / K(V ) /0 Let α ∈ H 1 (X, F). Then α lifts to some β ∈ Gm(X, H). We have a commutative diagram of exact sequences G(X) V G(X) / H(X) / H 1 (X, F) /0. / K(V ) (X) / H 1 (X, V F) /0 So to say that α is killed by the map to H 1 (X, V F) is the same as saying that the image of β in K(V ) lifts to something in V G(X). But if V is small, surjectivity of G → H implies that we can lift β to something in V G(X). So we can cover X by such sets V in A, completing the proof. 0.2 Proof of the vanishing theorem We now apply the lemma. Proof of the vanishing theorem. Induction on n. Let X = SpecA be an affine scheme. Consider the basis A of open sets D(f ) = SpecAf ; this is obviously closed under intersection, as D(f g) = D(f ) ∩ f is the (quasi-coherent) sheaf on X associated to an A-module D(g). Now if M M , then the pull-back to D(f ) is the sheaf associated to M ⊗A Af = Mf . So 2 f to A is just M gf . In particular, these are quasi-coherent the direct image D(f ) M on X. We will now apply the previous lemma. Suppose that n is fixed and H i (X, F) = f, 0 for any quasi-coherent sheaf on X and 0 < i < n. Then, this is true for M n f so the previous lemma says that given α ∈ H (X, M ), there is an open cover f) is zero. Now {D(fi )} of X such theLcohomology class of α in H n (X, D(fi ) M there is a map M → Mfi which is injective, since the fi generate the unit ideal; this induces a map of sheaves M f→ f 0→M D(fi ) M → K → 0 where K is also quasi-coherent. There is thus a long exact sequence, of which we write a piece: M f) → H n (X, f H n−1 (X, K) → H n (X, M D(fi ) M ). Since α is in the kernel of the second map, it is in the image of H n−1 (X, K). But, if n > 1, inductively we assumed H n−1 (X, K) was zero. This means α = 0. If n = 1, then we write out a bit more of the exact sequence: M M 1 1 f f Gm(X, D(fi ) M ) D(fi ) M ) → Gm(X, H) → H (X, F) → H (X, As before, we find that α is in the image of Gm(X, H). But since Gm is exact on quasi-coherent sheaves, it follows that the first map is surjective, and the map out of Gm(X, H) is zero. So again we find that α = 0. This proves Serre’s vanishing theorem. 3