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 be a projective scheme and ample. Let be a coherent sheaf on . For every polynomial there is a locally closed subscheme with the following property: given any morphism the pullback on is flat over with Hilbert Polynomial if and only if can be factored as .
What this says, roughly, is that if we have a family , then the base change of is flat with given Hilbert polynomial if and only if the family was really . 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 of locally closed subschemes such that every point of is in exactly one . We say that a stratification is flattening if for all morphisms we have that is flat over if and only if the morphism factors , where is defined by taking the sheaf on and pulling back by to .
Proof: First we look at the case . Then is a coherent sheaf on . So then is just , and it’ll be flat over if and only if it is locally free over . So then for all we define . Fix a point and let and choose whose images give a basis. Then they extend to sections of in a neighborhood of , and using the , we get a homomorphism in .
Now, the generate , so what we need is Nakayama’s lemma. In its traditional form, Nakayama’s lemma states that if is a finitely generated module and is an ideal contained in every maximal ideal of the ring, then implies that . 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 generate itself, and so the homomorphism is surjective in a possibly smaller neighborhood, which we’ll call . If we shrink the neighborhood even more, to , we can assume that the kernel is generated by its sections over . Thus, we have an exact sequence for some . We’ll now denote .
Now we need to see that is generated by sections everywhere in , and so if , we have that . Thus, the set is locally closed. Even better, if , then we have if and only if the homomorphism is zero. Thus, if this map is expressed by an matrix with elements of functions on , then the subscheme of defined by the has support . In fact, it has the property that if is any morphism, noetherian, then is locally free of rank if and only if factors through the closed subscheme .
So now, is uniquely determined in a neighborhood of a point of , so the subschemes all patch together to give a locally closed subscheme on . The collection will then be a stratification of . 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 which give us which is locally free of rank .
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 be a morphism of finite type of noetherian schemes, and let be a coherent sheaf on . Assume that is reduced and irreducible. Then there is a non-empty open subset such that the restriction of to is flat over .
So now on to the general case. Let be a coherent sheaf on . Let be the projection to and set . We begin by noting that there is a finite set of locall closed subsets of such that and such that if is given its reduced subscheme structure, we have is flat over .
The following two statements follow from the lemma fairly quickly:
- There is a uniform such that if , then for all , we have for and is isomorphic to .
- Only a finite number of polynomials appear as Hilbert polynomials of the sheaves on the fibers over .
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 obtained from 1 and we take be any base extension where is noetherian. Now we suppose that is flat over . Then for , we get a canonical map where is the projection. In fact, this canonical map must be na isomorphism, and is locally free on . Now, if we instead assume that is flat for all , then will be flat over because a coherent sheaf on projective is flat over a base if and only if it can be twisted (by tensoring with for large enough) to something that pushes forward to a locally free sheaf.
Back to stratifications, if we have two stratifications and , we can define a new one by 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 gives an associated flattening stratification. So now we take the flattening stratifications for all and combine them as above. This will be the flattening stratification we want for .
Let be the component of the flattening stratification of where it has rank . Then let be the finitely many Hilbert polynomials from 2. Then for all , we can look at . So each 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 are finite intersections!
So now, 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 . Next up, we’ll use the Hilbert scheme of projective space to prove the existence of some more moduli spaces.