## Flattening Stratifications

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.