Download 0.1 A lemma of Kempf

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).
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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
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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.
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