(page 1) Chapter 10

The Semicontinuity Theorem

As usual, we let X ⊂ P^{n} be a smooth projective surface. Let T be any k-scheme and take an effective Cartier divisor D on X ×T relative to T. Write π : X ×T → T for the projection map. We need to understand the higher direct image sheaves R^{i}π_{∗}(O_{D}(m)). These sheaves are supposed to be thought of as “cohomology along the fibres”, but we need a better idea of what that really means.

In this chapter, we will generalize the above situation somewhat. Throughout the chapter, we will always consider a projective morphism f : X → Y between noetherian schemes, and let F be a coherent sheaf on X that is flat over Y. Our goal is to find reasonable criteria that will ensure that the sheaves R^{i}f_{∗}(F) are locally free. Since this question is obviously local on the base, we will start by assuming that Y = Spec(A) for some noetherian ring A. There are three basic facts:

• H^{i}(X_{y}, F_{y}) = H^{i}(X, F ⊗ k(y)).
• R^{i}f_{∗}(F) = H^{i}(X, F)˜ .
• R^{i}f_{∗}(F) ⊗ k(y) = H^{i}(X, F) ⊗ k(y).

So, to understand more precisely how the higher direct images are related to cohomology along the fibres, we really need to investigate the difference between tensoring with k(y) before we compute cohomology and after we compute cohomology. Let’s define functors

T^{i}(M) = H^{i}(X, F ⊗_{A} M).

Proposition 10.1. Each T^{i} for i ≥ 0 is an additive, covariant functor from the category of A-modules to itself, which is exact in the middle. The entire collection {T^{i}}_{i≥0} is a delta functor.

Proof: Given a short exact sequence of A-modules

0 → L → M → N → 0,

we can tensor with the flat sheaf F to get an exact sequence of sheaves

0 → F ⊗_{A} L → F ⊗_{A} M → F ⊗_{A} N → 0.

The result now follows from the long exact sequence of chomology.

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(page 2)

2 Chapter 10: The Semicontinuity Theorem

Proposition 10.2. There exists a complex L^{•} of finitely generated free A-modules that is bounded above and is such that

T^{i}(M) = h^{i}(L^{•} ⊗_{A} M).

Proof: Since the sheaves F ⊗_{A} M are quasicoherent, their cohomology groups can be computed using Čech cohomology. Take an affine open cover U of X, and define C^{•} = C^{•}(U, F) to be the associated Čech complex. Then

C^{•}(U, F ⊗_{A} M) = C^{•} ⊗_{A} M,

since the same result is true for global sections of any quasicoherent sheaf on X. Now the Čech complex isn’t quite good enough, since there is no reason in the world for the C^{•} to be finitely generated or free. Instead, we need the following result from homological algebra.

Lemma 10.3. Let A be a noetherian ring, and let C^{•} be a complex of A-modules that is bounded above, such that each h^{i}(C^{•}) is a finitely generated A-module. Then there exists a complex L^{•} of finitely generated free A-modules, which is bounded above, and a morphism L^{•} → C^{•} that is a homology isomorphism. Moreover, if each C^{•} is a flat A-module, then this map of complexes induces isomorphisms h^{i}(L^{•} ⊗_{A} M) = h^{i}(C^{•} ⊗_{A} M) for all A-modules M.

Proof: For any complex N^{•}, we define the cocycles

Z^{i}(N^{•}) = ker(d^{i} : N^{i} → N^{i+1}),

the coboundaries

B^{i}(N^{•}) = Im(d^{i−1} : N^{i−1} → N^{i}),

and the cohomology groups

h^{i}(N^{•}) = Z^{i}(N^{•})/B^{i}(N^{•}).

For n >> 0, one has C^{n} = 0. In this range, we can take L^{n} = 0 and proceed by decreasing induction on n. So, let’s assume that we have already defined L^{•} and the map g : L^{•} → C^{•} such that

h^{i}(L^{•})⟶^{≈}h^{i}(C^{•})

is an isomorphism for i > n + 1 and

Z^{n+1}⟶h^{n+1}(C^{•})

is a surjection.

Choose generators x̄_{1}, …, x̄_{r} of the finitely generated module h^{n}(C^{•}), and lift them to cocycles x_{1}, …, x_{r} ∈ Z^{n}(C^{•}). Next, choose generators y_{r+1}, …, y_{s} of g^{−1}(B^{n+1}(C^{•})) ⊂ L^{n+1}. (Note: we need to work in the inverse image because, being a submoduel of the finitely generated module L^{n+1} over a noetherian ring, we know it is also finitely generated.) Now define ȳ_{i} = g(y_{i}) ∈ B^{n+1}(C^{•}), and lift the elements y_{i} to elements x_{r+1}, …, x_{s} ∈ C^{n}.

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Now define L^{n} to be the free A-module on s generators e_{1}, …, e_{s}. The boundary map is defined by

de_{i} = \{ \begin{matrix}0 & if 1 ≤ i ≤ r, \\ y_{i} & if r + 1 ≤ i ≤ s. \end{matrix}

The map of complexes is defined by

ge_{i} = x_{i}

for all i. The first part of the propositon clearly follows immediately from these definitions.

Now suppose in addition that each C^{•} is a flat A-module. We will show that g induces an isomorphism on the cohomology of L^{•} ⊗_{A} M with that of C^{•} ⊗_{A} M by using descending induction on n. The result is true for n >> 0 since both complexes are bounded above, so the cohomology groups both vanish. Now both ⊗ and h^{i} commute with direct limits. Thus, it is enough to prove the result for finitely generated A-modules. If M is finitely generated, we can choose a surjection from a finitely generated moduel E, and we get an exact sequence

0 → R → E → M → 0.

Since both C^{•} and L^{•} are flat, we can tensor this short exact sequence with either complex to produce a short exact sequence of complexes. Taking the long exact sequence of cohomology, we get an exact commutative diagram

h^{i}(L^{•}) ⊗ E → h^{i}(L^{•}) ⊗ M → h^{i+1}(L^{•}) ⊗ R → h^{i+1}(L^{•}) ⊗ E
↓^{≈}  ↓  ↓^{≈}  ↓^{≈}
h^{i}(C^{•}) ⊗ E → h^{i}(C^{•}) ⊗ M → h^{i+1}(C^{•}) ⊗ R → h^{i+1}(C^{•}) ⊗ E

The two right vertical maps are isomorphisms by the inductive hypothesis. The left vertical map is an isomorphism because E is free, and cohomology commutes with direct sums. Now the map in the middle is a surjection by the Five Lemma. Extending the diagram one term to the left and using surjectivity of the resulting map, another application ofthe Five Lemma shows that the middle map is also injective, hence an isomorphism.

Remark 10.4. The cohomology groups of a complex N^{•} are usually computed as a cokernel, by taking cocycles modulo coboundaries. It is less well-known that the cohomology groups can also be computed as a kernel. We can define

W^{i}(N^{•}) = Cok(d^{i−1} : N^{i−1} → N^{i}).

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Now one has an exact commutative diagram

0 → N^{i−1}  ⟶^{=} N^{i−1}
↓  ↓_{d^{i−1}}
0 → Z^{i}(N^{•}) ⟶  N^{i}  ⟶^{d^{i}} N^{i+1}
↓  ↓  ↓^{=}
0 → h^{i}(N^{•}) ⟶ W^{i}(N^{•}) ⟶ N^{i+1}
↓  ↓
0  0

So, the bottom row shows that h^{i}(N^{•}) = ker(W^{i}(N^{.}) → N^{i+1}). Since tensor product is right exact, we also have that W^{i}(N^{•} ⊗_{A} M) = W^{i}(N^{•}) ⊗_{A} M for all A-modules M.

Proposition 10.5. The following are equivalent (i) T^{i} is a left exact functor. (ii) W^{i}(L^{•}) is a projective A-module. (iii) There exists a finitely generated A-module Q such that

T^{i}(M) = Hom_{A}(Q, M).

Moreover, under these conditions, Q is unique up to isomorphism.

Proof: We will start by showing that (i) is equivalent to (ii). Write W^{i} = W^{i}(L^{•}). Let 0 → N → M be any injection of A-modules. Comparing the sequences that identify the cohomology groups as a kernel, we get a commutative diagram

0 → T^{i}(N) → W^{i} ⊗ N → L^{i+1} ⊗ N
↓^{α}  ↓^{β}  ↓^{γ}
0 → T^{i}(M) → W^{i} ⊗ M → L^{i+1} ⊗ M

Since L^{i+1} is free, it is flat, and the map γ is injective. Chasing the diagram, we see that α is injective if and only if β is injective. But T^{i} is left exact if and only if every α is injective, and W^{i} is flat if and only if every β is injective. Since W^{i} is finitely generated over a noetherian ring, it is flat if and only if it is projective.

Now (iii) clearly implies (i), so we are left with the task of showing that (ii) implies (iii). Denote the dual of a projective A-module P by P̌. It is a standard fact that Hom(P̌, M) = P ⊗ M. Now we can define Q = Cok(Ľ^{i+1} → W̌^{i}). By the left exactness of Hom, we get an exact sequence

0 → Hom(Q, M) → W^{i} ⊗ M → L^{i+1} ⊗ M.

So, T^{i}(M) = Hom(Q, M), as desired.

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Proposition 10.6. For all A-modules M there exists a natural map ϕ : T^{i}(A) ⊗ M → T^{i}(M). Moreover, the following are equivalent.

(i) T^{i} is right exact.

(ii) ϕ is an isomorphism for all M.

(iii) ϕ is a surjection for all M.

Proof: The functoriality of T^{i} gives a map

ψ : Hom(A, M) → Hom(T^{i}(A), T^{i}(M)).

So, we can define ϕ by the rule

ϕ(α ⊗ m) = ψ(m)(α).

Now both T^{i} and tensor product commute with direct limits, so it is enough to show the equivalence of (i)–(iii) when M is finitely generated. Since A is noetherian, this actually means we can take a finite presentation

A^{r} → A^{s} → M → 0.

Now we get a commutative diagram

T^{i}(A) ⊗ A^{r} → T^{i}(A) ⊗ A^{s} → T^{i}(A) ⊗ M → 0
↓^{≈}  ↓^{≈}  ↓^{ϕ}
T^{i}(A^{r}) → T^{i}(A^{s}) → T^{i}(M).

Notice that the bottom row need not be exact. However, the first two vertical arrows are isomorphisms. therefore, we can read off immediately from the diagram that (i) implies (ii). Since (ii) trivially implies (iii), it only remains to check that (iii) implies (i). So, take any surjection M → N → 0 os A-modules. there is a commutative diagram

T^{i}(A) ⊗ M ⟶ T^{i}(A) ⊗ N → 0
↓  ↓^{ϕ}
T^{i}(M) ⟶ T^{i}(N)

Chasinfg the diagram, we see that (iii) implies (i).

Corollary 10.7. The following are equivalent.

(i) T^{i} is exact.

(ii) T^{i} is right exact and T^{i}(A) is a projective module.

Now we would like to localize the previous discussion. Let y ∈ Y = Spec(A) correspond to a prime ideal P. We let T_{y}^{i} denote the restriction of T^{i} to the full

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subcategory of A_{P} -modules. Consider the fibre product upper semicontinuous

X′  ⟶ X
↓^{f′}  ↓
Y′ = Spec(O_{y}) ⟶ Y

Since cohomology commutes with flat base change, we see that T_{y}^{i} is the same as the T^{i} functor associated to f′. Now we will say that T^{i} is left exact at y if T_{y}^{i} is left exact. (Similar usages will apply to “right exact” or “exact” at y.) One should note that T^{i} is a left exact functor if and only if it is left exact at each point y.

Proposition 10.8. If T^{i} is left exact (right exact, exact) at y_{0}, then there exists an open neighborhood U of y_{0} in Y such that T^{i} is left exact (right exact, exact) at all points y ∈ U.

Proof: If T^{i} is left exact at y^{0}, then W̃_{y_{0}}^{i} is projective, hence free. But then it is free in a neighborhood, and hence projective, and hence T^{i} must be left exact in a neighborhood.

Now T^{i} is right exact at y if and only if T^{i+1} is left exact at y, so this property also extends to neighborhoods. The result follows for exact by intersecting the neighborhoods wherein it is both left exact and right exact.

Definition 10.9. let Y be a topological space. We say that a function ϕ : Y → Z is upper semicontinuous if for all points y ∈ Y there exists an open neighborhood y ∈ U ⊂ Y such that for all u ∈ U one has ϕ(u) ≤ ϕ(y).

Remark 10.10. This definition says that the values of ϕ can get larger at special points, but it can’t get any smaller. An equivalent definition says that ϕ is upper semicontinuous if and only if for every n ∈ Z the set {y ∈ Y : ϕ(y) ≥ n} is closed in Y.

Example 10.11. The most basic example of an upper semicontinuous function in algebraic geometry arises as follows. Let Y be a noehterian scheme and let F be a coherent sheaf on Y. Define

ϕ(y) = dim_{k(y)}(F ⊗ k(y)).

Then ϕ is upper semicontinuous. To see this, note that Nakayama’s Lemma shows that ϕ(y) is equal to the minimal number of generators of the stalk F_{y} as a module over the local ring O_{y}. If s_{1}, …, s_{r} generate F_{y}, then we can extend then to sections defined in an open neighborhood U. By coherence, they must still generate F at each point of U. So, for u ∈ U, we have that

ϕ(u) = minimal number of generators ≤ rx = ϕ(y).

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Theorem 10.12. (Semicontinuity) Let ϕ : X → Y be a projective morphism between noetherian schemes. Let F be a coherent sheaf on X that is flat over Y. Then the function

h^{i}(y, F) = dim_{k(y)} H^{i}(X_{y}, F_{y})

is an upper semicontinuous function on Y.

Proof: Since the result is local on Y, we can assume Y = Spec(A) for some noetherian ring A. In this case, we have

h^{i}(y, F) = dim_{k(y)} H^{i}(X_{y}, F_{y})
= dim_{k(y)} H^{i}(X, F ⊗ k(y))
= dim_{k(y)} T^{i}(k(y)).

As before, we can write T^{i}(k(y)) = ker(W^{i} ⊗ k(y) → L^{i+1} ⊗ k(y)). However, since we have an exact sequence

W^{i} → L^{i+1} → W^{i+1} → 0,

we get a four term exact sequence

0 → T^{i}(k(y)) → W^{i} ⊗ k(y) → L^{i+1} ⊗ k(y)) → W^{i+1} ⊗ k(y)) → 0.

So, we have

h^{i}(y, F) = dim_{k(y)}(W^{i} ⊗ k(y)) + dim_{k(y)}(W^{i+1} ⊗ k(y)) − dim_{k(y)}(L^{i+1} ⊗ k(y)).

the fiTst two terms are upper semicontinuous, as a consequence of Nakayama’s Lemma. Since L^{i+1} is free, the last term is constant. The result follows.

Theorem 10.13. (Grauert) Let f : X → Y be a projective morphism of noetherian schemes, with Y integral. Suppose for some i that the function h^{i}(Y, F) is constant. Then the higher direct image sheaf R^{i}f_{∗}(F) is locally free, and the natural maps

R^{i}f_{∗}(F) ⊗ k(y) → H^{i}(X_{y}, F_{y})

are isomorphisms for all y.

Proof: The result is local on the base, so we can assume that Y = Spec(A). By the proof of the Semiocontinuity Theorem, we have

h^{i}(y, F) = dim_{k(y)}(W^{i} ⊗ k(y)) + dim_{k(y)}(W^{i+1} ⊗ k(y)) − dim_{k(y)}(L^{i+1} ⊗ k(y)).

Since all the pieces are upper semicontinuous, the assumption that h^{i}(y, F) is constant forces each piece to be constant. Thus, the coherent sheaves W̃^{i}, W̃^{i+1} are locally free on Y. Thus, both T^{i} and T^{i+1} are left exact functors. Hence, T^{i} is also right exact, thus exact. Therefore, T^{i}(A) isa projective A-module. However,

R^{i}f_{∗}(F) = H^{i}(X, F)˜ = T^{i}(A)˜ .

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So, as claimed, R^{i}f_{∗}(F) is locally free. Now we use the fact that T^{i}(A)⊗M = T^{i}(M) under these conditions to see that

R^{i}f_{∗}(F) ⊗ k(y) = H^{i}(X, F) ⊗ k(y) = H^{i}(X, F ⊗ k(y)) = H^{i}(X_{y}, F_{y}).

Example 10.14. let X_{t} be a flat family of integral curves in P_{k}^{n} with k an algebraically closed field. For all closed t ∈ T, we have H^{0}(X_{t}, O_{X_{t}}) = k. But the arithmetic genus p_{a} = 1 − χ(O_{X}) = 1 − P_{X}(0) is ocnstant in flat families. Hence, H^{1}(X_{t}, O_{X_{t}}) is also constant.

Example 10.15. Look at the family of curves in space defined by

x = t^{2} − 1, y = t^{3} − t, z = at.

When a ≠ 0, this isa twisted cubic curve. the ideal of the entire family is

{a^{2}(x + 1) − z^{2}, ax(x + 1) − yz, xz − ay, y^{2} − x^{2}(x + 1)}.

So, when a = 0, the family specializes to the curve

{z^{2}, yz, xz, y^{2} − x^{2}(x = 1)}.

The specialization is a nodal cubic curve with an embedded point at the node. We have

h^{0} = \{ \begin{matrix}1 & if t ≠ 0 \\ 2 & if t = 0, \end{matrix}

and

h^{1} = \{ \begin{matrix}0 & if t ≠ 0 \\ 1 & if t = 0, \end{matrix}

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