Now we move on to the next ingredient in the construction: flattening stratifications. We’ll start with just stating the theorem that Kollár used:

Theorem: Let X/S be a projective scheme and \mathscr{O}(1) ample. Let \mathscr{F} be a coherent sheaf on X. For every polynomial P there is a locally closed subscheme i_P:S_P\to S with the following property: given any morphism p:Z\to S the pullback p^*\mathscr{F} on Z\times_S X is flat over Z with Hilbert Polynomial P if and only if p can be factored as Z\to S_P\stackrel{i_P}{\to}S.

What this says, roughly, is that if we have a family Z\to S, then the base change of \mathscr{F} is flat with given Hilbert polynomial if and only if the family was really Z\to S_P. So there is a subscheme depending only on the Hilbert polynomial which all flat maps with fibers of that polynomial map into.

Now, we define a stratification to be a finite set S_1,\ldots,S_m of locally closed subschemes such that every point of S is in exactly one S_i. We say that a stratification is flattening if for all morphisms g:T\to S we have that \mathscr{F}_g is flat over T if and only if the morphism g factors T\to \coprod_{i=1}^m S_i\to S, where \mathscr{F}_g is defined by taking the sheaf \mathscr{F} on \mathbb{P}^n_S and pulling back by 1_{\mathbb{P}^n}\times g to \mathbb{P}^n_T.

Proof: First we look at the case n=0. Then \mathscr{F} is a coherent sheaf on S. So then \mathscr{F}_g is just g^*\mathscr{F}, and it’ll be flat over T if and only if it is locally free over T. So then for all s\in S we define e(s)=\dim_{K(s)} (\mathscr{F}_s\otimes_{\mathscr{O}_{S,s}}K(s)). Fix a point s\in S and let e=e(s) and choose a_1,\ldots,a_e\in \mathscr{F}_s whose images give a basis. Then they extend to sections of \mathscr{F} in a neighborhood U_1 of s, and using the a_i, we get a homomorphism \mathscr{O}^e_S\to \mathscr{F} in U_1.

Now, the a_i generate \mathscr{F}_s\otimes K(s), so what we need is Nakayama’s lemma. In its traditional form, Nakayama’s lemma states that if M is a finitely generated module and I is an ideal contained in every maximal ideal of the ring, then IM=M implies that M=0. Most often, this is used for local rings, and it’s going to get at least a minipost in the near future, because it’s so important.

Anyway, back on topic: Nakayama’s Lemma can be used to prove that a_i generate \mathscr{F}_s itself, and so the homomorphism is surjective in a possibly smaller neighborhood, which we’ll call U_2. If we shrink the neighborhood even more, to U_3, we can assume that the kernel is generated by its sections over U_3. Thus, we have an exact sequence \mathscr{O}^f_S\to \mathscr{O}^e_S\to\mathscr{F}\to 0 for some f. We’ll now denote U_3=U_s.

Now we need to see that \mathscr{F} is generated by e(s) sections everywhere in U_s, and so if s'\in U_s, we have that e(s')\leq e(s). Thus, the set Z_e=\{s\in S|e(s)=e\} is locally closed. Even better, if s'\in U_s, then we have e(s')=e(s) if and only if the homomorphism k(s')^f\to k(s')^e is zero. Thus, if this map is expressed by an e\times f matrix with elements \psi_{ij} of functions on U_s, then the subscheme Y_s of U_s defined by the \psi_{ij} has support Z_e\cap U_s. In fact, it has the property that if g:T\to U_s is any morphism, T noetherian, then g^*\mathscr{F} is locally free of rank e=e(s) if and only if g factors through the closed subscheme Y_s.

So now, Y_s is uniquely determined in a neighborhood of a point of Z_e\cap U_s, so the subschemes Y_s all patch together to give a locally closed subscheme Y_e on Z_e. The collection \{Y_e\} will then be a stratification of S. And now, by the property, it must be a flattening stratification. We’ve not just proved this, but we’ve in fact indexed the subschemes Y_e which give us \mathscr{F}\otimes_{\mathscr{O}_S}\mathscr{O}_{Y_e} which is locally free of rank e.

We’re now just going to state a lemma, and this won’t be proved, because it’s not that enlightening and becomes mostly an exercise in commutative algebra.

Lemma: Let f:X\to Y be a morphism of finite type of noetherian schemes, and let \mathscr{F} be a coherent sheaf on X. Assume that Y is reduced and irreducible. Then there is a non-empty open subset U\subset Y such that the restriction of \mathscr{F} to h^{-1}(U) is flat over U.

So now on to the general case. Let \mathscr{F} be a coherent sheaf on \mathbb{P}^n_S. Let p be the projection to S and set \mathscr{E}_m=p_*(\mathscr{F}(m)). We begin by noting that there is a finite set of locall closed subsets Y_1,\ldots,Y_k of S such that S=\cup Y_i and such that if Y_i is given its reduced subscheme structure, we have \mathscr{F}\otimes_{\mathscr{O}_S}\mathscr{O}_{Y_i} is flat over Y_i.

The following two statements follow from the lemma fairly quickly:

  1. There is a uniform m_0 such that if m\geq m_0, then for all s\in S, we have H^i(p^{-1}(s),\mathscr{F}(m)|_{p^{-1}(s)})=0 for i>0 and \mathscr{E}_m\otimes K(s) is isomorphic to H^0(p^{-1}(s),\mathscr{F}(m)|_{p^{-1}(s)}).
  2. Only a finite number of polynomials appear as Hilbert polynomials of the sheaves \mathscr{F}|_{p^{-1}(s)} on the fibers p^{-1}(s) over S.

That second one is the one that’s really useful to us. It lets us turn the statement we were going to get, which assumes that the stratification is finite into the one we wanted, which doesn’t, by saying that it had to have been finite in the first place.

So now we fix m_0 obtained from 1 and we take g:T\to S be any base extension where T is noetherian. Now we suppose that \mathscr{F}_g is flat over T. Then for m\geq m_0, we get a canonical map g^*(\mathscr{E}_m)\to q_*(\mathscr{F}_g(m)) where q:\mathbb{P}^n_T\to T is the projection. In fact, this canonical map must be na isomorphism, and g^*(\mathscr{E}_m) is locally free on T. Now, if we instead assume that g^*(\mathscr{E}_m) is flat for all m\geq m_0, then \mathscr{F}_g will be flat over T because a coherent sheaf on projective is flat over a base if and only if it can be twisted (by tensoring with \mathscr{O}(m) for m large enough) to something that pushes forward to a locally free sheaf.

Back to stratifications, if we have two stratifications Y_i and Z_j, we can define a new one by Y_i\cap Z_j as sets, and giving it the scheme structure by taking the sum of the defining ideal sheaves. From the base case, we have that each \mathscr{E}_m gives an associated flattening stratification. So now we take the flattening stratifications for all m\geq m_0 and combine them as above. This will be the flattening stratification we want for \mathscr{F}.

Let Y_e^{(m)} be the component of the flattening stratification of \mathscr{E}_m where it has rank e. Then let P_1,\ldots,P_k be the finitely many Hilbert polynomials from 2. Then for all i, we can look at Z_i=\cap_{m=m_0}^\infty Y_{P_i(m)}^{(m)}. So each Z_i is a limit of a descending chain of locally closed subschemes with fixed underlying topological space. Now, we made the hypothesis that things were noetherian. That means that descending chains terminate after finitely many sets, so the Z_i are finite intersections!

So now, Z_1,\ldots,Z_k give us a flattening stratification. QED

That completes the core of the proof of existence of the Hilbert scheme, at least, the existence of the Hilbert scheme of \mathbb{P}^n_S. Next up, we’ll use the Hilbert scheme of projective space to prove the existence of some more moduli spaces.