3. Castelnuovo-Mumford Regularity and Spectral Sequences. This section is devoted to the following refinement of Serre’s theorems A and B: Uniform Vanishing Theorem: Fix the following data:

an infinite field k and a projective k-scheme X,

a locally free coherent sheaf E on X and a polynomial P (d). Then there is a d_{0} depending only on X, E and P (d) such that for all coherent subsheaves F ↪ E with Hilbert polynomial χ(X, F(d)) = P (d) and for all d ≥ d_{0}, F(d) is generated by its sections and H^{i}(X, F(d)) = 0 for all i > 0.

As we will see, this is critical to the construction of the Hilbert scheme. Mumford’s proof of this theorem uses the notion of: Castelnuovo Regularity: A coherent sheaf F on P_{k}^{n} is m-regular if:

H^{i}(P_{k}^{n}, F(m − i)) = 0 for all i > 0

Examples: (a) O_{P_{k}^{n}} is m-regular for m ≥ 0, but not −1-regular.

(b) Suppose L is an invertible sheaf on a curve C such that:

(i) L is generated by it global sections,

(ii) H^{1}(C, L) = 0, and

(iii) the multiplication map Sym^{2}H^{0}(C, L) → H^{0}(C, L^{2}) is surjective. Let f : C ↪ P_{k}^{n} be the closed immersion given by the complete linear

series H^{0}(C, L). Then it follows from the exact sequences:

0 → I_{C}(d) → O_{P_{k}^{n}}(d) → f_{∗}L^{⊗d} → 0

and (ii)-(iii) that I_{C} is 3-regular. The Regularity Theorem: If F is m-regular, then:

(a) F is m + 1-regular, and

(b) the multiplication map

H^{0}(P_{k}^{n}, O(1)) ⊗ H^{0}(P_{k}^{n}, F(m)) → H^{0}(P_{k}^{n}, F(m + 1))

is surjective.

Corollary: If F is an m-regular coherent sheaf on P_{k}^{n}, then:

(a) H^{i}(P_{k}^{n}, F(d)) = 0 for all i > 0 and d ≥ m − 1 and

(b) F(d) is generated by global sections for all d ≥ m.

Proof: Induction gives (a) immediately, and implies that the maps:

Sym^{e}H^{0}(P_{k}^{n}, O_{P_{k}^{n}}(1)) ⊗ H^{0}(P_{k}^{n}, F(d)) → H^{0}(P_{k}^{n}, F(d + e))

are surjective for all d ≥ m and all e > 0. In particular, fix d and let e be large. By Serre’s theorem B, the sheaf F(d + e) is generated by its sections, and it follows that F(d) is also generated by global sections.

Example: If L is a line bundle on a projective curve C satisfying (i)-(iii) of the example above, then H^{1}(C, L^{d}) = 0 for all d ≥ 1 and all multiplication maps Sym^{d}H^{0}(C, L) → H^{0}(C, L^{d}) are surjective, i.e. the embedded curve is projectively normal. The corollary also tells us that I_{C}(3) is generated by its sections, hence C is an intersection of cubic hypersurfaces.

Mumford has a quick proof of the regularity theorem, but the funnylooking definition of regularity and the role of P_{k}^{n} seem to be better explained using spectral sequences, so we will prove it this way (following Mark Green).

Spectral Sequences: A double complex of abelian groups (or sheaves of abelian groups) is a collection K_{p,q} of such groups (we’ll assume p, q ≥ 0), together with maps

d′ : K_{p,q} → K_{p+1,q} and d′′ : K_{p,q} → K_{p,q+1}

satisfying d′ ∘ d′ = 0, d′′ ∘ d′′ = 0 and d′ ∘ d′′ + d′′ ∘ d′ = 0.

Given a double complex, one defines the associated total complex:

K_{n} := ⨁_{p+q=n} K_{p,q}, D = d′ + d′′ : K_{n} → K_{n+1}

The basic homological algebra result asserts that the homology of the total complex:

E_{n} := \frac{ker(D : K_{n} → K_{n+1})}{im(D : K_{n−1} → K_{n})}

filters in two different ways.

Starting from d′: Let E_{p,q}^{′1} be the homology of K_{p,q} with respect to d′. Then d′′ induces maps d′_{1} : E_{p,q}^{′1} → E_{p,q+1}^{′1} satisfying d′_{1} ∘ d′_{1} = 0. Inductively, if E_{p,q}^{′r+1} is the homology of E_{p,q}^{′r} with respect to d′_{r} : E_{p,q}^{′r} → E_{p−r+1,q+r}^{′r}, then there are induced maps d′_{r+1} : E_{p,q}^{′r+1} → E_{p−r,q+r+1}^{′r+1}.

Once E_{p,q}^{′r} = E_{p,q}^{′r+1} = ..., then we define E_{p,q}^{′∞} := E_{p,q}^{′r}.

Starting from d′′: The analog for d′′ produces d′′_{r} : E_{p,q}^{′′r} → E_{p+r,q−r+1}^{′′r} and we define E_{p,q}^{′′∞} similarly.

The Spectral Sequence Theorem: There are two filtrations:

0 ⊂ F′_{n,n} ⊂ ... ⊂ F′_{n,q} ⊂ ... ⊂ F′_{n,0} = E_{n} and 0 ⊂ ... ⊂ F′′_{p,n} ⊂ ... ⊂ E_{n}

with subquotients F′_{n,q}/F′_{n,q+1} ≅ E_{n−q,q}^{′∞} and F′′_{p,n}/F′′_{p+1,n} ≅ E_{p,n−p}^{′′∞}.

Corollary: If E_{p,q}^{′∞} = 0 for all p, q, then E_{p,q}^{′′∞} = 0 for all p, q.

Now suppose that 0 → F_{0} → F_{1} → ... → F_{m} → 0 is a long exact sequence of coherent sheaves on a Noetherian separated scheme X. Take an open cover U of X by affines U_{0}, ..., U_{n} and consider the double complex:

K_{p,q} = C^{q}(U, F_{p}) := ⨁_{i_{0}<...<i_{q}} F_{p}(U_{i_{0}} ∩ ... ∩ U_{i_{q}})

with (−1)^{p}d′′ the Cech coboundary map and d′ induced from the sheaf maps.

Because the sequence of sheaves is exact, it follows that E_{p,q}^{′∞} = E_{p,q}^{′1} = 0 for all p, q. So by the theorem, it must be true that E_{p,q}^{′′∞} = 0 as well. Since the terms E_{p,q}^{′′1} = H^{q}(X, F_{p}) are in general not zero, this gives us information about maps between cohomology groups of the F_{p}.

Exercise: Apply this to a short exact sequence of sheaves to obtain the connecting homomorphisms from the maps d′′_{2}.

The Koszul complex K_{∗} is the exact sequence:

0 → ∧^{n+1}V ⊗_{k} O(−n − 1) → ∧^{n}V ⊗_{k} O(−n) → ... → V ⊗_{k} O(−1) → O → 0

of locally free sheaves on P_{k}^{n}, where V is the vector space H^{0}(P_{k}^{n}, O_{P_{k}^{n}}(1)) with basis x_{0}, ..., x_{n} and ∧^{r}V ⊗_{k} O(−r) → ∧^{r−1}V ⊗_{k} O(−r + 1) are the contraction maps:

x_{i_{1}} ∧ ... ∧ x_{i_{r}} ⊗ s → ∑(−1)^{j−1}x_{i_{1}} ∧ ... ∧ x̂_{i_{j}} ∧ ... ∧ x_{i_{r}} ⊗ x_{i_{j}}s

Proof of the Regularity Theorem: Let K_{∗} ⊗_{OPn_{k}} F(m + 1) be the complex of sheaves obtained by tensoring the terms of the Koszul complex by F(m+1). Because the terms in the Koszul complex are locally free, it follows that this complex is exact. One now computes: E_{n,0}^{′′1} = V ⊗_{k} H^{0}(P_{k}^{n}, F(m)), E_{n+1,0}^{′′1} = H^{0}(P^{n}, F(m + 1)) and d′′_{1} is the multiplication map. Furthermore, each E_{n−r,r}^{′′1} = ∧^{r+1}V ⊗ H^{r}(P^{n}, F(m − r)) that might eventually produce an E_{n−r,r}^{′′r} mapping to E_{n+1,0}^{′′r} under d′′_{r} is already zero, since we assumed F was m-regular. Since all d′′_{r} maps out of E_{n+1,0}^{′′r} are obviously zero as well, the vanishing of E_{n+1,0}^{′′∞} forces the surjectivity of d′′_{1}, giving us (b).

Similarly, the vanishings H^{i}(P_{k}^{n}, O_{P_{k}^{n}}(m + 1 − i)) are seen by considering the long exact sequences K_{∗} ⊗_{OPn_{k}} F(m + 1 − i) for each 1 ≤ i ≤ n.

Proof of the Uniform Vanishing Theorem: We prove the theorem first in case X = P_{k}^{n}, F is locally free, and E = ⊕O_{P_{k}^{n}}(l) for some l. We’ll work by induction on n, with the case n = 0 being trivial.

An injective map i : F ↪ E of locally free sheaves need not be injective on all fibers, but it does need to be injective on fibers at all generic points, hence at all points x ∈ U of a dense open set, by Nakayama’s lemma. The converse also holds. A map of locally free sheaves on an integral scheme is injective if and only if it is injective on fibers at some point.

Now suppose f : H ↪ P_{k}^{n} is the hyperplane corresponding to the section s ∈ H^{0}(P_{k}^{n}, O_{P_{k}^{n}}(1)). Then the following sequences are exact for all d:

(∗) 0 → F(d − 1) →^{s} F(d) → f_{∗}F(d)|_{H} → 0

where F|_{H} := f^{∗}F. Moreover, if H passes through a point where i is injective on fibers, then i|_{H} : F|_{H} → E|_{H} ≅ ⊕O_{H}(l) is injective, as well. We will argue by induction using such an H.

Suppose P (d) = χ(P^{n}, F(d)) = ∑_{i=0}^{n} a_{i} \binom{d}{i}. Then

\begin{aligned}χ(H, F|_{H}(d)) & = & ∑_{i=0}^{n} a_{i} \binom{d}{i} − a_{i} \binom{d−1}{i} \\ & = & ∑_{i=0}^{n−1} a_{i+1} \binom{d−1}{i} \end{aligned}

Thus the restrictions F|_{H} all have the same Hilbert polynomial and all are subsheaves of the fixed locally free sheaf E|_{H} = ⊕O_{H}(l).

So we may assume that there is an N_{0} independent of F such that

H^{i}(H, F_{H}(d)) = 0 for all i > 0, d ≥ N_{0}

From the exact sequence on cohomology associated to (∗), this gives us:

H^{i}(P_{k}^{n}, F(N_{0} − 1)) = H^{i}(P_{k}^{n}, F(N_{0})) = ... forall i ≥ 2

and so by Serre’s theorem B, we get H^{i}(P_{k}^{n}, F(d)) = 0 for i ≥ 2, d ≥ N_{0} − 1. To get the H^{1} term, we use the cohomology sequences for d ≥ N_{0}:

0 → H^{0}(F(d − 1)) → H^{0}(F(d)) →^{ρ_{d}} H^{0}(F_{H}(d)) →

→ H^{1}(F(d − 1)) → H^{1}(F(d)) → 0

Since all the higher cohomology of F_{H}(N_{0}) and higher twists vanishes, it follows in particular that F_{H} is N_{1} := N_{0} +n−1 regular, so by the regularity theorem, the multiplication map:

H^{0}(O_{H}(1)) ⊗ H^{0}(F_{H}(d)) → H^{0}(F_{H}(d + 1))

is surjective for d ≥ N_{1}. Thus if the restriction map ρ_{d} is surjective for some d ≥ N_{1}, then the restriction map ρ_{d+1} must also be surjective, as is evident from the following diagram:

H^{0}(O_{P_{k}^{n}}(1)) ⊗ H^{0}(F(d)) → H^{0}(F(d + 1))
↓  ↓
H^{0}(O_{H}(1)) ⊗ H^{0}(F_{H}(d)) → H^{0}(F_{H}(d + 1))

On the other hand, dim_{k}H^{1}(F(d − 1)) > dim_{k}H^{1}(F(d)) whenever ρ_{d} is not surjective, so the dimensions of H^{1}(F(d)) are strictly decreasing down to zero for d ≥ N_{1} − 1. In particular, all the higher cohomology of F(d) vanishes as soon as d ≥ N_{2} := N_{1} + dim_{k}H^{1}(F(N_{1} − 1)). We now need to show that this can be bounded independent of F.

Since H^{i}(F(N_{1} − 1)) = 0 for i ≥ 2, we have:

dim_{k}H^{1}(F(N_{1} − 1)) = dim_{k}H^{0}(F(N_{1} − 1)) − χ(F(N_{1} − 1))
≤ dim_{k}H^{0}(E(N_{1} − 1)) − P (N_{1} − 1)

which indeed only depends upon P (d) and E.

Not only do we have vanishing of higher cohomology when d ≥ N_{2}, but we also see that F is N_{2} + 1 regular. Hence the F(d) are generated by their sections for all d ≥ d_{0} := N_{2} + 1 and we get uniform vanishing.

Next, suppose F is not assumed to be locally free. Then the proof above holds provided there is always a hyperplane H ⊂ P_{k}^{n} so that the sequences:

(∗∗) 0 → F(d − 1) → F(d) → F_{H}(d) → 0

are exact, and i_{H} : F_{H} → E_{H} is injective.

For this, we need a result from commutative algebra. If F is a sheaf and s ∈ F(U) is a section over an open set U, then the support of s in U is the set of points x ∈ U such that 0 ≠ s_{x} ∈ F_{x}. Note that this is always closed. If F is a coherent sheaf over a Noetherian scheme, then we say x ∈ X is an associated point of F if there exist an s and x ∈ U as above so that x is in the support of s and moreover, the support of s is the closure(!) of x in U.

For example, the generic point(s) are the only possible associated points of a subsheaf of a locally free sheaf on a reduced scheme. The extra associated points of O_{X} on a non-reduced scheme X are called the embedded primes.

If X = Spec(A) is affine and F = M̃, then it follows from the primary decomposition of the sub-module 0 ⊂ M (see Atiyah-MacDonald Ch. 4,7) that there are only finitely many associated points of F, and that if a ∈ A does not belong to the union of the corresponding prime ideals in A, then am ≠ 0 for all m ∈ M. What we glean from this in general is:

A(F) := {associated points of F} is a finite set

Returning to the proof, notice that since F is a subsheaf of a locally free sheaf, F has only the generic point for an associated point, hence (∗∗) is exact for any choice of H. If we choose H to be disjoint from the finite set A(E/F), then the equation for H is not a zero divisor in any stalk of E/F. Let O_{x} be the stalk of O_{P_{k}^{n}} at x and s_{x} be the equation for H at x. Then on stalks at x the map i_{H} is the first map of the sequence:

F_{x} ⊗_{O_{x}} O_{x}/s_{x} → E_{x} ⊗_{O_{x}} O_{x}/s_{x} → E_{x}/F_{x} ⊗_{O_{x}} O_{x}/s_{x} → 0

which has kernel Tor_{O_{x}}^{1}(E_{x}/F_{x}, O_{x}/s_{x}). But since our choice of H guarantees that multiplication of E_{x}/F_{x} by s_{x} has no kernel, it follows that this Tor is zero, and that i_{H} is injective.

Finally, to prove the theorem on any projective k-scheme f : X ↪ P_{k}^{n} and for any locally free E on X, we first note that by Serre’s theorem A, we can always inject E ↪ ⊕O_{X}(l) for some l. Thus, we may as well assume that E = ⊕O_{X}(l). Then for any F ↪ ⊕O_{X}(l), there is a diagram:

0  0
↓  ↓
K_{1} =  K_{1}
↓  ↓
0 → K_{2} → ⊕O_{P_{k}^{n}}(l) → f_{∗} (⊕O_{X}(l)/F) → 0
↓ ↓ ∥
0 → f_{∗}F → ⊕O_{X}(l) → f_{∗} (⊕O_{X}(l)/F) → 0
↓  ↓
0  0

Note that K_{1} and K_{2} are both subsheaves of ⊕O_{P_{k}^{n}}(l) with Hilbert polynomials only depending upon X, E and P (d). Thus uniform vanishing applies to both K_{1} and K_{2}. From the exact sequences:

→ H^{i}(P_{k}^{n}, K_{2}(d)) → H^{i}(X, F(d)) → H^{i+1}(P_{k}^{n}, K_{1}(d)) →

and the surjectivity:

K_{2}(d) → f_{∗}(F(d))

it follows that uniform vanishing holds for F with d_{0} equal to the maximum of the values for K_{1} and K_{2}. This completes the proof.

Remark: The only place we used the fact that E was locally free was in the assertion that there is an injection:

E ↪ ⊕O_{X}(l)

This injection is achieved by applying Serre’s theorem A to the dual locally free sheaf E^{∗} = Hom_{O_{X}}(E, O_{X}) to find a surjection ⊕O_{P_{k}^{n}} → E^{∗}(l), twisting by O_{P_{k}^{n}}(−l) and then dualizing back. Thus it certainly suffices to assume that E^{∗∗} ≅ E (i.e. E is reflexive), or even just that E injects into E^{∗∗} (i.e. E is torsion-free), since E^{∗∗} is always reflexive.