Bounded sets of sheaves: invariance properties
Let k be a field, and X a k-prescheme (which we take to be of finite type, for simplicity).
For every extension K/k, we obtain a K-prescheme X_K=X\otimes_k K.
If {\mathscr{F}} is a coherent sheaf on X_K, and if K' is an extension of K, then {\mathscr{F}}\otimes_K K'={\mathscr{F}}_{K'} is a quasi-coherent sheaf on X_K\otimes_KK'=X_{K'}.
So, if K and K' are arbitrary extensions of k, and {\mathscr{F}} a quasi-coherent sheaf on X_K and {\mathscr{F}}' a quasi-coherent sheaf on X_{K'}, then we say that {\mathscr{F}} and {\mathscr{F}}' are equivalent if there exists an extension K''/k as well as k-homomorphisms K\to K'' and K'\to K'' such that {\mathscr{F}}_{K''} and {\mathscr{F}}'_{K''} are isomorphic on X_{K''} (TO-DO: ? is this right?).
This defines an equivalence relation, and we are interested in the equivalence classes of sheaves under this relation, and of sets of such equivalence classes.
Note that, if X_0 is of finite type over k, then every class of coherent sheaves can be defined by a coherent sheaf on X_K, where K is some extension of k of finite type.
We can thus, in the definition of classes of coherent sheaves, restrict ourselves to algebraically closed extensions of k, and we can also restrict ourselves to a fixed algebraically closed extension \Omega of k, of infinite transcendence degree;
two coherent sheaves {\mathscr{F}} and {\mathscr{F}}' on X_\Omega are then equivalent if and only if there exists a K-automorphism \sigma of \Omega such that {\mathscr{F}}\otimes_K(\Omega,\sigma) is isomorphic to {\mathscr{F}}'.
Note that there is a bijective correspondence between classes of coherent sheaves under the first definition and under the second.
Let E and E' be two sets of classes of coherent sheaves on X.
Consider the classes of all sheaves of the form {\mathscr{F}}\otimes{\mathscr{F}}', where {\mathscr{F}} and {\mathscr{F}}' are coherent sheaves on the same X_K, with the class of {\mathscr{F}} being in E and the class of {\mathscr{F}}' being in E'.
We thus define a set of classes of coherent sheaves that we denote by E\otimes E'.
We can similarly define \mathscr{T}\kern -.5pt or_i(E,E'), etc.
Generally, to every function {\mathscr{U}} that sends each sequence {\mathscr{F}}_1,\ldots,scr{F}_n of n coherent sheaves on one single X_K to a set {\mathscr{U}}({\mathscr{F}}_1,\ldots,{\mathscr{F}}_n) of coherent sheaves on X_K, and that has the evident property of compatibility with isomorphisms of sheaves and inverse images under change of base, we associate a function, denoted by the same notation {\mathscr{U}}, that sends each sequence E_1,\ldots,E_n of n sets of classes of coherent sheaves to a set {\mathscr{U}}(E_1,\ldots,E_n) of classes of coherent sheaves.
Our aim in this section is to give a definition of certain sets of classes of sheaves, said to be bounded, and to show that the most standard operations {\mathscr{U}}, applied to bounded sets, give sets that are again bounded.
Let X be a prescheme of finite type over S, with S Noetherian.
For all s\in S, the fibre X_s is a prescheme of finite type over k(s), and we will consider the classes of coherent sheaves on X_s, in the above sense.
This gives meaning to the phrase “class of coherent sheaves on a fibre of X/S”, as well as to analogous phrases.
Similarly, proceeding separately on each fibre, we can again consider operations such as E\otimes E' etc. that send systems of sets of classes of coherent sheaves on the fibres of X/S to sets of classes of coherent sheaves on the fibres of X/S.
Let E be a set of classes of coherent sheaves on the fibres of X/S.
We say that E is bounded if there exists a prescheme S' of finite type over S, along with a coherent sheaf {\mathscr{F}}' on X'=X\times_S S', such that E is contained in the set of classes of sheaves on the fibres of X/S defined by {\mathscr{F}}'.
This construction, by definition, sends s\in S to the classes of sheaves {\mathscr{F}}'\otimes_{S'}k(s'), where s' runs over the points of S' over s (so that k(s') is an extension of k(s), and X\otimes (X_s)_{k(s')} can be identified with the fibre X'\otimes_{S'}k(s')=X'_{s'} of X' at s').
We can say that the bounded sets are those that are contained in an algebraic family of coherent sheaves, parametrised by some S' of finite type over S.
A finite union of bounded sets is bounded (take the prescheme given by the sum of the parametrising preschemes S_i defining the bounding algebraic families).
A base change T\to S sends a family which is bounded with respect to X/S to a family which is bounded with respect to X_T/T, and the converse is true if T\to S is surjective (or, more generally, if its image contains the s which appear in the given family E for X/S).
This theoretically leads us to determine the bounded families only in the case where S is the spectrum of an algebra of finite type over the ring of integers \mathbb{Z}.
If E and E' are bounded families of classes of sheaves with respect to X/S, then E\otimes E' is also bounded:
indeed, if E (resp. E') is bounded by the algebraic families defined by T\to S and {\mathscr{F}} on X_T (resp. T'\to S and {\mathscr{F}}' on X_{T'}), then E\otimes E' is bounded by the algebraic family defined by T''=T\times_S T'\to S and the sheaf {\mathscr{F}}'' on X_{T''} given by the tensor product of the inverse images of {\mathscr{F}} and {\mathscr{F}}' on X_{T''}.
This argument is correct only because the functor {\mathscr{F}}\otimes{\mathscr{F}}' is right exact in both {\mathscr{F}} and {\mathscr{F}}', and thus commutates with base extension (and, in particular, with passing to the fibres).
It is not applicable as-is to local operations, such as \mathscr{T}\kern -.5pt or_i(E,E'), \mathscr{H}\kern -.5pt om(E,E'), \mathscr{E}\kern -.5pt xt^i(E,E').
We can, however, show that these operations also send bounded sets to bounded sets, by proceeding as for E\otimes E', but by also using results of the following type (all contained in [16, IV 6.11]):
a bounded family E is always bounded by an algebraic family defined by a coherent sheaf {\mathscr{F}} on some X_T (with T of finite type over S) that is flat with respect to T.
(We “cut into bits” the initial space of parameters).
Such flatness properties on suitable sheaves indeed ensure the commutativity of operations such as \mathscr{T}\kern -.5pt or_i({\mathscr{F}},{\mathscr{F}}') with arbitrary base change.
The same method applies to operations of a global nature:
direct images and derived direct images of coherent sheaves under proper morphisms, global \operatorname{Ext} with respect to proper morphisms (cf. [18, III §6]), etc.;
all these operations send bounded families of sheaves to bounded families of sheaves (N.B. here the preschemes over which we are taking the various sheaves can change under the operations in question).
The two claims that follow can be proven by essentially the same flatness technique;
for the primary decomposition on the fibres of a morphisms of finite type, see, in particular, [18, IV].
Let E and E' be bounded sets of classes of sheaves on the fibres of X/S, with X assumed to be proper over S.
Then
- the family of kernels, cokernels, and images of homomorphisms {\mathscr{F}}\to{\mathscr{F}}' (where the class of {\mathscr{F}} is in E and the class of {\mathscr{F}}' is in E') is bounded;
- the family of extensions {\mathscr{F}}'' of {\mathscr{F}} by {\mathscr{F}}' (where the class of {\mathscr{F}} is in E and the class of {\mathscr{F}}' is in E') is bounded.
Proof. After potentially applying a suitable base change, we can suppose that E and E' are defined by coherent sheaves {\mathscr{G}} and {\mathscr{G}}' (respectively) on some X_T/T, with T of finite type over S.
Further, we can suppose that certain flatness hypotheses are satisfied, implying that constructing the sheaves f_T(\mathscr{H}\kern -.5pt om_{{\mathcal{O}}_{X_T}}({\mathscr{G}},{\mathscr{G}}')) and \mathscr{E}\kern -.5pt xt_{f_T}^1({\mathscr{G}},{\mathscr{G}}') commutes with base change by an arbitrary morphism T'\to T.
Further, we can suppose that the coherent sheaves above are locally free on T.
So let T_0 and T_1 be the vector bundles on T whose sheaves of germs of sections are (respectively) the above sheaves.
We can then canonically define a homomorphism {\mathscr{G}}_{T_0}\to{\mathscr{G}}'_{T_0} of coherent sheaves on X_{T_0}, along with an extension
0 \to {\mathscr{G}}_{T_1} \to {\mathscr{G}}'' \to {\mathscr{G}}'_{T_1} \to 0
of coherent sheaves on X_{T_1} that has the evident universal property.
This second sheaf defines an algebraic family that bounds the family in question in (ii).
This is also true for the kernel, cokernel, and image of the above homomorphism, and the consideration of this proves (i) (provided that we assume the cokernel to be flat with respect to T_0, in which case we can again reduce to cutting T_0 into pieces…).
Let E be a bounded family of classes of sheaves on the fibres of X/S.
Then the classes of the structure sheaves of the (\operatorname{supp}{\mathscr{F}})_\mathrm{red}, where {\mathscr{F}} is a coherent sheaf on some X_K, with K algebraically closed, and whose class is in E, form a bounded family.
Here, (\operatorname{supp}{\mathscr{F}})_\mathrm{red} denotes the support of {\mathscr{F}}, endowed with the induced reduced structure, i.e. its structure sheaf is the quotient of {\mathcal{O}}_{X_K} by the largest sheaf of ideals that defines \operatorname{supp}{\mathscr{F}}.
We can prove the analogous result to (1.3) for the sheaves canonically induced from {\mathscr{F}} by the theory of primary decomposition;
for example, the {\mathscr{F}}/{\mathscr{F}}_i, where the {\mathscr{F}}_i are the primary subsheaves of {\mathscr{F}} for the components of the support of {\mathscr{F}}, and minimal with respect to this property;
or the {\mathcal{O}}_{X_K}/{\mathfrak{p}}, where {\mathfrak{p}} is a prime sheaf of ideals associated to {\mathscr{F}}, or the {\mathcal{O}}_{X_K}/{\mathfrak{q}}, where the {\mathscr{q}} is a primary sheaf of ideals associated to a component of the support of {\mathscr{F}} (the reference field being algebraically closed).
Bounded families and the Hilbert polynomial
In the following, we assume that X is projective over S, and endowed with a very ample sheaf, denoted by {\mathcal{O}}_X(1).
For every extension K of a residue extension k(s) of a point s of S, we consider the corresponding sheaf {\mathcal{O}}_{X_K}(1) on X_K, which will again be very ample.
To each coherent sheaf {\mathscr{F}} on X_K, we associate the function
P_{\mathscr{F}}(n) = \text{the Euler--Poincaré characteristic of }{\mathscr{F}}(n)\text{ on }X_k
which is a polynomial in the integer n, and called the Hilbert polynomial of {\mathscr{F}}.
For large values of n, P(n) is exactly the dimension of \operatorname{H}^0(X_k,{\mathscr{F}}(n)) over K, since the \operatorname{H}^i(X_k,{\mathscr{F}}(n)) are zero for i>0 and large enough n.
Now, if {\mathscr{F}} is a coherent sheaf on X which is flat with respect to S, then the Hilbert polynomials of the sheaves {\mathscr{F}}_S induced on the fibres X_S (with respect to one single connected component of S) are all equal [18, III, §7].
It thus follows (without any flatness hypothesis) that the set of Hilbert polynomials of the sheaves {\mathscr{F}}_s, for s\in S, is finite for every coherent sheaf {\mathscr{F}} on X.
Recall also that, if {\mathscr{F}} is a coherent sheaf on X, then it is isomorphic to a quotient sheaf of a sheaf of the form {\mathscr{O}}_X(-n)^N, for some large enough n,N.
So the sheaves {\mathscr{F}}_s induced on the fibres are also quotients of the sheaf {\mathscr{O}}(-n) on the fibre.
From these two remarks, we reduce the “necessary” part of the following theorem:
Let X be projective over S, with S Noetherian, and {\mathscr{O}}_X(1) very ample over X with respect to S.
Let E be a set of classes of sheaves on the fibres of X/S.
For E to be bounded, it is necessary and sufficient for it to satisfy the following conditions:
- There exists a coherent sheaf {\mathscr{L}} on X (which we can suppose to be of the form {\mathscr{O}}_X(-n)^N) such that E is contained in the family of classes of coherent sheaves given by quotients of sheaves of the form {\mathscr{L}}_K;
- The Hilbert polynomials P_{\mathscr{F}} of the sheaves {\mathscr{F}} whose class is in E are elements of a single finite set of polynomials.
It remains to prove the “sufficient” part, which will be a particular case of a more precise result.
For every coherent module {\mathscr{F}} on a prescheme of finite type over a field K, and for every integer r, let N_r be the submodule of {\mathscr{F}} whose sections over an open subset are the sections of {\mathscr{F}} over the same subset whose support is of dimension <r.
We thus have that N_r={\mathscr{F}} for r>\dim\operatorname{supp}{\mathscr{F}}, and N_r=0 for r\leqslant 0, and we thus obtain a finite increasing filtration of {\mathscr{F}} whose factors N_r/N_{r+1} are such that their associated prime cycles are exactly the associated prime cycles of {\mathscr{F}} that are of dimension r.
We set
{\mathscr{F}}_{(r)} = {\mathscr{F}}/N_r
so that the associated prime cycles of {\mathscr{F}}_{(r)} are exactly the associated prime cycles of {\mathscr{F}} that are of dimension \geqslant r, and, in particular, {\mathscr{F}}_{(r)} is equal to {\mathscr{F}} if and only if the associated prime cycles of {\mathscr{F}} are of dimension \geqslant r.
With this, we have:
Under the conditions of Theorem 2.1, let s be an integer, and suppose that E satisfies condition (a), as well as the following weakened form of (b):
- (bs) The Poincaré polynomials P_{\mathscr{F}} of the sheaves {\mathscr{F}} whose class is in E have coefficients in degrees \leqslant(s-1) that are bounded.
Under these conditions, the sheaves {\mathscr{F}}_{(s)} (for the {\mathscr{F}} whose class is in E) form a bounded family.
Furthermore, the coefficients in degree (s-2) of the P_{\mathscr{F}} are bounded below.
Thus:
Suppose that the sheaves {\mathscr{F}} whose class is in E are such that all their associated prime cycles are of dimension d, with s\leqslant d\leqslant r.
Then, in condition (b) of Theorem 2.1, we can restrict to the coefficients of P_{\mathscr{F}} between degree (s-1) and r, inclusive.
The end of this section is dedicated to a sketch proof of Theorem 2.2.
The key lemmas are the two following lemmas, of which the first is well known (and summarises the useful mathematical content of Chow coordinates):
Consider the structure sheaves of the subschemes Y with fibre X_K (where K is an algebraically closed extension of the residue field of S), where Y is reduced, and all its components are of the same dimension r (and with {\mathscr{O}}_Y being thought of as a quotient sheaf of {\mathscr{O}}_X).
If the degrees of Y are bounded, then the Y form a bounded family.
Here, the degree a of Y can be most conveniently defined as the coefficient of the dominant term of P_{{\mathscr{O}}_Y}=an^r/r!+\ldots.
Let {\mathscr{L}} be a coherent sheaf on X, and E a set of classes of the quotient sheaves {\mathscr{F}} of the sheaf {\mathscr{L}}_K (where K is a residue extension of S).
Suppose that the fibres of X over S are of dimension \leqslant r, and set
P_{\mathscr{F}}(n) = a_{\mathscr{F}}n^r/r! + b_{\mathscr{F}}n^{r-1}/(r-1)! + \text{terms of degree }<r-1.
Then the coefficient a_{\mathscr{F}} is bounded (above), and b_{\mathscr{F}} is bounded below.
If b_{\mathscr{F}} is bounded, then the family {\mathscr{F}}_{(r)} is bounded.
Proof. By replacing S by a union of subschemes of S that cover S, we can suppose that there exists a finite morphism f\colon X\to\mathbb{P}_S^r such that {\mathscr{O}}_X(1) is isomorphic to the inverse image of {\mathscr{O}}_{\mathbb{P}_S^r}(1), and thus, for every coherent sheaf {\mathscr{F}} on X, we have that P_{\mathscr{F}}=P_{f_*({\mathscr{F}})}.
We can also easily show (by the technique of the previous section) that a set of sheaves {\mathscr{F}} on X is bounded if and only if the set of f_*({\mathscr{F}}) is bounded.
Finally, we have that
f_*({\mathscr{F}})_{(r)} = f_*({\mathscr{F}}_{(r)}).
This thus allows us to reduce to the case where X=\mathbb{P}_S^r.
Furthermore, we can suppose that {\mathscr{L}}={\mathscr{O}}_{\mathbb{P}_S^r}(k)^s, for some suitable k and s.
The coefficient a_{\mathscr{F}} satisfies
0\leqslant a_{\mathscr{F}} \leqslant s
and is thus bounded.
With this in mind, saying that the n^{r-1} coefficient P_{\mathscr{F}}(n) is bounded below (resp. bounded) is equivalent to saying the same thing for the P_{\mathscr{F}}(n-k)=P_{{\mathscr{F}}(-k)}(n).
This leads us to the case where
{\mathscr{L}} = {\mathscr{O}}_{\mathbb{P}_S^r}^s.
Consider the exact sequence
0 \to N_r \to {\mathscr{F}} \to {\mathscr{F}}_{(r)} \to 0
whence
P_{\mathscr{F}} = P_{{\mathscr{F}}_{(r)}} + P_{N_r}
and, since the n^{r-1} coefficient of P_{N_r} is positive (since \dim\operatorname{supp}N_r\leqslant r-1), we have that
b_{{\mathscr{F}}_{(r)}} \leqslant b_{\mathscr{F}}.
This allows us, in proving the lemma, to replace {\mathscr{F}} by {\mathscr{F}}_{(r)}, i.e. to suppose that the quotients {\mathscr{F}} of {\mathscr{L}} in question are torsion free.
Since \mathbb{P}_K^r is normal, it follows that {\mathscr{F}} is locally free of rank a=a_{\mathscr{F}} on an open U=\mathbb{P}_K^r\setminus Y, where Y is of codimension \geqslant 2.
Thus \bigwedge^a{\mathscr{F}} is a sheaf on \mathbb{P}_K^r whose restriction to U is invertible, and thus (since \mathbb{P}_K^r is regular, and Y is of codimension \geqslant 2) isomorphism to the restriction of an invertible sheaf on \mathbb{P}_K^r, defined up to isomorphism.
This latter sheaf is of the form {\mathscr{O}}_{\mathbb{P}_K^r}(d) for some well defined integer d.
Since \bigwedge^a{\mathscr{F}} is a quotient of \bigwedge^a{\mathscr{O}}_{\mathbb{P}_K^r}^n\simeq{\mathscr{O}}_{\mathbb{P}_K^r}^N with N=\binom{n}{a}, it admits N canonical sections, which thus define sections of {\mathscr{O}}_{\mathbb{P}_K^r}(d) over U, which are restrictions of sections s_i (for 1\leqslant i\leqslant N) of {\mathscr{O}}_{\mathbb{P}_K^r}(d) (since \mathbb{P}_K^r is normal, and Y is of codimension \geqslant 2).
These s_i generate {\mathscr{O}}_{\mathbb{P}_K^r}(d) at the points of U, and are thus not all zero, which implies that d\geqslant 0.
An easy calculation also shows that
b_{\mathscr{F}} = a_{\mathscr{F}}(r+1)/2 + d.
This shows, in particular, that b_{\mathscr{F}}\geqslant 0, and so b_{\mathscr{F}} is bounded below.
It is bounded if and only if d is bounded;
we will show that {\mathscr{F}} then remains in a bounded family.
We can fix a_{\mathscr{F}} and b_{\mathscr{F}}, as well as a and b (and thus d).
The data of the N sections s_i of {\mathscr{O}}_{\mathbb{P}_K^r}(d), i.e. of a homomorphism s\colon\bigwedge^a{\mathscr{L}}_K\to{\mathscr{O}}_{\mathbb{P}_K^r}(d), allows us to recover {\mathscr{F}} as the co-image of the corresponding composite homomorphism:
{\mathscr{L}}_K
\to \mathscr{H}\kern -.5pt om\left( \bigwedge^{a-1}{\mathscr{L}}_{K'}, \bigwedge^a{\mathscr{L}}_K \right)
\to \mathscr{H}\kern -.5pt om\left( \bigwedge^{a-1}{\mathscr{L}}_{K'}, {\mathscr{O}}_{\mathbb{P}_K^r}(d) \right)
where the first arrow is the canonical homomorphism coming from the exterior product, and the second comes from s.
We then conclude by part (i) of (1.2).
The combination of the two lemmas above allows us to prove the following:
Suppose, under the preliminary conditions of (2.1), that, for all {\mathscr{F}}, we have
P_{\mathscr{F}}(n) = a_{\mathscr{F}} n^r/r! + b_{\mathscr{F}} n^{r-1}/(r-1)! + \text{terms of degree }<r-1
and that the coefficients a_{\mathscr{F}} are bounded.
Then the coefficients b_{\mathscr{F}} are bounded below..
Furthermore, if the b_{\mathscr{F}} are bounded, then the {\mathscr{F}}_{(r)} are bounded.
Proof. We can suppose that the base field K of the sheaves {\mathscr{F}} is algebraically closed.
We endow each \operatorname{supp}{\mathscr{F}}_{(r)}(the union of the components of degree r) with the induced reduced structure.
Then the degrees of the \operatorname{supp}{\mathscr{F}}_{(r)} are bounded above by a, and so, by (2.4), the \operatorname{supp}{\mathscr{F}}_{(r)} form a bounded set.
Furthermore, for each component of \operatorname{supp}{\mathscr{F}}_{(r)}, the length of {\mathscr{F}}_{(r)} for this component is \leqslant a, and so, if {\mathscr{I}}_{\mathscr{F}} is the ideal that defines \operatorname{supp}{\mathscr{F}}_{(r)}, then {\mathscr{F}}_{(r)} can be considered as a module on the subscheme Y_{\mathscr{F}} of X defined by {\mathscr{I}}_{\mathscr{F}}^a.
As in the previous lemma, we can also reduce to the case where {\mathscr{F}}={\mathscr{F}}_{(r)}, so that {\mathscr{F}} comes from a module on Y_{\mathscr{F}}.
The Y_{\mathscr{F}} correspond to a bounded family of quotient modules of the {\mathscr{O}}_{X_K}, and thus come from a closed subscheme Y of some scheme X\times_S T.
We can then apply (2.5) to Y/T and {\mathscr{L}}\otimes_X Y, whence the conclusion.
Proof. We can now prove (2.2) by induction on the upper bound r of the \dim\operatorname{supp}{\mathscr{F}}.
The statement is trivial for r<0, so suppose that r\geqslant 0 and that the statement has been proven for r'<r.
By (2.6), the {\mathscr{F}}_{(r)} form a bounded family, and so too, by part (i) of [(1.2)], do the kernels of the homomorphisms {\mathscr{L}}_K\to{\mathscr{F}}_{(r)};
there thus exists a coherent module {\mathscr{L}}' on X such that kernels in question, and thus also the N_r({\mathscr{F}})=\operatorname{Ker}({\mathscr{F}}\to{\mathscr{F}}_{(r)}), are quotients of modules {\mathscr{L}}'_K.
Since the {\mathscr{F}}_{(r)} are bounded, the P_{{\mathscr{F}}_{(r)}} are bounded, and the formula
P_{\mathscr{F}} = P_{{\mathscr{F}}_{(r)}} + P_{N_r}
then shows that the P_{N_r} satisfy the same condition (bs) as the P_{\mathscr{F}}.
Thus the N_r satisfy conditions (a) and (bs), and so, by the induction hypothesis, the (N_r)_{(s)} are bounded.
But {\mathscr{F}}_{(s)} is an extension of {\mathscr{F}}_{(r)} by (N_r)_{(s)}, and so, by part (ii) of (1.2), the {\mathscr{F}}_{(s)} are bounded.
For the last claim of (2.2), we note that the kernels N_s of {\mathscr{F}}\to{\mathscr{F}}_{(s)} are bounded, by part (i) of (1.2), and that the coefficient of the n^{s-1} term in P_{N_s} is bounded;
then (2.6) proves that the coefficient of the following term is bounded below.
This finishes the proof
Hilbert schemes: definition, existence theorem
Let X be a prescheme over another prescheme S, and {\mathscr{F}} a quasi-coherent module on X.
We denote by
\operatorname{Quot}({{\mathscr{F}}/X/S})
the set of quasi-coherent modules given by quotients of {\mathscr{F}} that are flat over S.
Now let S'\to S be a base change morphisms, and set X'=X\times_S S', and {\mathscr{F}}'={\mathscr{F}}\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_{S'}, so that X' is a prescheme over S' endowed with a quasi coherent module {\mathscr{F}}', and we can consider \operatorname{Quot}({{\mathscr{F}}'/X'/S'}).
We set
\mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}}(S') = \operatorname{Quot}({{\mathscr{F}}'/X'/S'})
(where X'=X\times_S S', as above).
Now, if S''\to S' is an S-morphism, then X''=X\times_S S'' is isomorphic to X'\times_{S'}S'', and {\mathscr{F}}'' is isomorphic to {\mathscr{F}}'\otimes_{{\mathscr{O}}_{S'}}{\mathscr{O}}_{S''}, and, since the inverse image functor {\mathscr{G}}'\mapsto{\mathscr{G}}'\otimes_{{\mathscr{O}}_{S'}}{\mathscr{O}}_{S''} from the category of quasi-coherent modules on X'' is right exact, and sends S'-flat modules to S''-flat modules, we obtain a natural map
\operatorname{Quot}({{\mathscr{F}}'/X'/S'}) \to \operatorname{Quot}({\mathscr{F}}''/X''/S'')
and so \mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}}(S') is a contravariant functor in S' (where S' is a prescheme over S), with values in the categories of sets.
In what follows, we suppose that X is projective over a Noetherian S, with {\mathscr{F}} coherent;
for simplicity, we will limit ourselves to considering those S' which are locally Noetherian over S.
Under these conditions, the contravariant functor \mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}} on the category of locally Noetherian S-preschemes is representable by an S-prescheme \underline{\operatorname{Quot}}_{{{\mathscr{F}}/X/S}}, given by the sum of a sequence of projective S-schemes (and a fortiori \underline{\operatorname{Quot}}_{{{\mathscr{F}}/X/S}} is locally of finite type over S).
We will obtain such a decomposition in the following way.
Let {\mathscr{O}}_X(1) be an invertible sheaf on X that is very ample with respect to S.
For every polynomial P(n) with rational coefficients, let \operatorname{Quot}^P({{\mathscr{F}}/X/S}) be the subset of \operatorname{Quot}({{\mathscr{F}}/X/S}) consisting of coherent quotients {\mathscr{G}} of {\mathscr{F}} that are flat over S and whose Hilbert polynomial at each s\in S is equal to P.
We then set
\mathscr{Q}\kern -.5pt out_{{\mathscr{F}}/X/S}^P(S') = \operatorname{Quot}^P({{\mathscr{F}}'/X'/S'})
and thus obtain a subfunctor of \mathscr{Q}\kern -.5pt out_{{\mathscr{F}}/X/S}.
The invariance property of Hilbert polynomials (recalled in §2) implies the following:
For \mathscr{Q}\kern -.5pt out_{{\mathscr{F}}/X/S} to be representable, it is necessary and sufficient for the \mathscr{Q}\kern -.5pt out_{{\mathscr{F}}/X/S}^P to be representable, and then the S-prescheme \underline{\operatorname{Quot}}_{{{\mathscr{F}}/X/S}} which represents it is isomorphic to the prescheme given by the sum of the \underline{\operatorname{Quot}}_{{{\mathscr{F}}/X/S}}^P that represent the functors \mathscr{Q}\kern -.5pt out_{{\mathscr{F}}/X/S}^P.
With this, Theorem 3.1 will be a consequence of the following theorem:
With the above notation, the functor \mathscr{Q}\kern -.5pt out_{{\mathscr{F}}/X/S}^P is representable by a projective S-prescheme \operatorname{Quot}_{{\mathscr{F}}/X/S}^P.
The rest of this section is dedicated to the proof of Theorem 3.2.
Let \nu be an integer.
For every S' over S, we denote by A_\nu(S') the set of quotients {\mathscr{G}}={\mathscr{F}}'/{\mathscr{H}} of {\mathscr{F}}'={\mathscr{F}}\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_{S'} that are coherent, flat over S', and satisfy the following conditions:
- \operatorname{R}^i f'_*({\mathscr{G}}(n))=0 for i>0 and n\geqslant\nu;
- \operatorname{R}^i f'_*({\mathscr{H}}(n))=0 for i>0 and n\geqslant\nu;
- f'_*({\mathscr{H}}(\nu+k))={\mathcal{S}}'_k f'_*({\mathscr{H}}(\nu)) for k\geqslant 0.
For this last condition, we suppose that X is written as the homogeneous prime spectrum of a quasi-coherent positively-graded algebra {\mathcal{S}}_\bullet over S that is generated by {\mathcal{S}}_1, and we set {\mathcal{S}}'={\mathcal{S}}\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_{S'}, so that X' is the homogeneous prime spectrum of {\mathcal{S}}'.
To prove Theorem 3.2, we can easily reduce to the case where X=\mathbb{P}_S^r (since S is a union of open subsets U such that X|U is a closed subscheme of \mathbb{P}_U^r, and {\mathscr{O}}_X(1) is induced by {\mathscr{O}}_{\mathbb{P}_U^r}(1)), and where {\mathscr{F}} is of the form ({\mathscr{O}}_{\mathbb{P}_S^r})^N, and thus flat over S.
Then, in the above, the sheaves {\mathscr{H}} are also flat over S'.
It then follows from the Künneth relations [18, III, §7] and from (b) that the conditions (a) and (b) are stable under base change, and imply that, for n\geqslant\nu, forming f'_*({\mathscr{G}}(n)) and F'_*({\mathscr{H}}(n)) commutes with extension of the base (loc. cit.).
In other words, A_\nu(S') is a contravariant functor in S', and in a precise sense a subfunctor of A(S')=\mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}}^P(S').
For varying \nu, we thus obtain an increasing sequence of subsets A_\nu(S') of A(S'), whose union is A(S') by a well known theorem of Serre [18, III, §2].
Note now that, if {\mathscr{G}} is a coherent quotient of {\mathscr{F}}' which is flat over S, and s an element of S such that the base change \operatorname{Spec}(k(s))\to S gives rise to a quotient {\mathscr{G}}_s of {\mathscr{F}}_s satisfying conditions (a), (b), and (c), i.e. is in A_\nu\operatorname{Spec}(k(s)), then there exists an open neighbourhood U of s such that these same conditions are satisfied by {\mathscr{G}}|(f')^{-1}(U), i.e. this quotient is in A_\nu(U);
for (a) and (b), this follows in fact from the “Theorem of holomorphic functions” [18, III, §7], and (c) follows from the Nakayama lemma and the fact that we know that f'_*({\mathscr{H}}(n+k))=S'_kf'_*({\mathscr{H}}(n)) anyway for n large enough and k\geqslant 0 [18, III,§2].
From these remarks, we conclude the following (compare with [17, IV]).
For the functor A to be representable, it is necessary and sufficient for the functors A_\nu to be representable, and then the S-prescheme Q that represents A is the increasing union of the opens Q_\nu that represent the A_\nu.
Let
M_\bullet
= \sum_{n\geqslant 0} f_*({\mathscr{F}}(n))
= {\mathcal{S}}_\bullet^N
so that we have
M'_\bullet
= M_\bullet\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_{S'}
= \sum_{n\geqslant 0} f'_*({\mathscr{F}}(n))
= {{\mathcal{S}}'_\bullet}^N.
It follows from (a) that we have
a’. f'_*({\mathscr{G}}(n)) is locally free of rank P(n) for n\geqslant\nu [18, III, §7]
and it follows from (b),for i=1, that we have
a’’. f'_*({\mathscr{G}}(n)) is a quotient module of M'_n.
Also, the knowledge of this quotient module, for n=\nu, implies, by (c), that the knowledge of the submodules f'_*({\mathscr{H}}(n)) of M_n for n\geqslant\nu, and thus the knowledge of {\mathscr{H}} and consequently of {\mathscr{G}}.
We thus obtain an injective map
A_\nu(S') \to \mathscr{G}\kern -.5pt rass_{P(\nu)}(M'_\nu)
from A_\nu(S') to the set of locally free quotient modules of M' of rank P(\nu), whence a functorial homomorphism
i_\nu\colon A_\nu(S') \to \mathscr{G}\kern -.5pt rass_{P(\nu)}(M_\nu)(S')
where the functor on the right hand side is representable by the Grassmannian scheme \operatorname{Grass}_{P(\nu)}(M_\nu) (compare with [17, V]), which is projective over S.
Then
A_\nu(S') is a representable functor, and the morphism Q_\nu\to\operatorname{Grass}_{P(\nu)}(M_\nu) that represents the homomorphism i_\nu is an immersion (which implies that Q_\nu is quasi-projective over S).
This claim is equivalent to the following (compare with [17, IV]):
If we have a quotient module N of M'_\nu that is locally free of rank P(\nu), then there exists a subprescheme Z of S' such that, for every locally Noetherian prescheme T' over S', the inverse image of N over T' is in \Im A_\nu(T') if and only if T'\to S' is bounded by the subprescheme Z.
Changing notation, we can suppose that S'=S, i.e. we have a quotient N_\nu of M_\nu by a submodule R_\nu.
For it to come from an element of A(s), it is necessary and sufficient for it to satisfy the following two conditions:
- M_{\nu+k}/{\mathcal{S}}_k R_\nu is locally free of rank P(\nu+k) for k\geqslant 0.
- Both the subsheaf {\mathscr{H}} of {\mathscr{F}} defined by the graded submodule R_\bullet=\sum_{k\geqslant 0}{\mathscr{S}}_kR_\nu of M_\bullet (cf. [18, II, §3]) and the quotient {\mathscr{G}}={\mathscr{F}}/{\mathscr{H}} satisfy conditions (a) and (b) above.
These conditions are clearly necessary, and if they are satisfied then the sheaf {\mathscr{G}} defined in (ii) above, being isomorphic to the sheaf associated to the graded {\mathscr{S}}_\bullet-module N_\bullet given by the sum of the M_{n+k}/{\mathscr{S}}_kR is flat over S (since its fibres are direct factors of localisations of N for homogeneous prime ideals of {\mathcal{S}}_\bullet), and correspond to the Hilbert polynomial P by virtue of (i).
Taking (ii) into account, we then see that (a’) and (a’’) are satisfied, and thus, for n\geqslant\nu, the natural homomorphism N_n\to f_*({\mathscr{G}}(n)) is a surjective homomorphism of locally free modules of equal rank, and thus an isomorphism, and thus f_*({\mathscr{H}}(\nu+k))={\mathcal{S}}_kR_\nu for all k\geqslant 0, which proves that {\mathscr{G}}\in A_\nu(S) and that R_\nu is the element of \operatorname{Grass}_{P(\nu)}(M_\nu) defined by {\mathscr{G}}.
Criteria (i) and (ii) above apply equally to the situation obtained after a change of base S'\to S.
We will prove first of all the fact that condition (i) is satisfied after the change of base S'\to S can be expressed by saying that S'\to S is bounded by a certain subprescheme Z of S;
once we have shown this result, we are led (replacing S with Z) to the case where condition (i) is already satisfied on S, and since it is stable under change of base, it remains to express condition (ii).
But then, if U denotes the set of s\in S such that the cohomology of the sheaves induced on the fibre X_s by {\mathscr{G}}(n) and {\mathscr{H}}(n) is zero in dimension >0 for n\geqslant\nu, then we have already shown that U is open, and condition (ii) will be satisfied after a change of base S'\to S if and only if S'\to S is bounded by U, which proves Lemma 3.3.
It thus remains to prove the following lemma:
Let S be a locally Noetherian prescheme, endowed with a quasi-coherent positively-graded algebra {\mathcal{S}}_\bullet generated by {\mathcal{S}}_1, and let M_\bullet be a quasi-coherent graded {\mathcal{S}}-module of finite type, P a polynomial with rational coefficients, and \nu and integer.
Then there exists a (clearly uniquet) subprescheme Z of S that has the following property:
for every prescheme S' over S, for M_n\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_{S'} to be locally free of rank P(n) for all n\geqslant\nu, it is necessary and sufficient for S'\to S to be bounded by Z.
We can evidently suppose that S is affine, and thus Noetherian.
Then:
For every integer N\geqslant\nu, let U_N be the open subset of S consisting of s\in S such that \operatorname{rank}_{k(s)}M_{ns}\otimes_{{\mathscr{O}}_{S,s}}k(s)\leqslant P(n) for all \nu\leqslant n\leqslant N.
Then the decreasing sequence of open subsets U_N stabilises.
Proof. We know [16, IV] that S admits a finite partition into reduced subschemes S_i such that each M\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_{S_i} is flat over S.
We can thus suppose that M is flat, and thus that the M_n are flat.
Finally, we can evidently suppose that S is connected.
But then [18, III, §7] there exists an integer n_0 and a polynomial Q such that
\operatorname{rank}_{k(s)}M_{ns}\otimes_{{\mathscr{O}}_{S,s}}k(s) = Q(n)
\qquad\text{for }n\geqslant n_0.
Suppose first of all that P\neq Q, and so P(n)\neq Q(n) for large n.
Then we evidently have U_N=\varnothing for large enough N, and thus a fortiori the sequence of U_N stabilises.
If instead P=Q then U_N=U_{n_0} for N\geqslant n_0, and the sequence of the U_N again stabilises.
In particular, the set U_\infty of s\in S such that
\operatorname{rank}_{k(s)}M_{ns}\otimes_{{\mathscr{O}}_{S,s}}k(s) \leqslant P(n)
\qquad\text{for all }n\geqslant\nu
\tag{$*$}
is open, since it is the intersection of the U_N.
We can then, for the proof of Lemma 3.4 replace S with the open subset U, which leads us to the case where the inequality in (*) is satisfied at all s\in S.
Let M be a module on a locally Noetherian prescheme S, and r an integer.
Then there exists a (clearly unique) subprescheme Z of S that has the following property:
for all S' over S, for M\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_{S'} to be locally free of rank r, it is necessary and sufficient for S'\to S to be bounded by Z.
If \operatorname{rank}_{k(s)}M_s\otimes_{{\mathscr{O}}_{S,s}}k(s)\leqslant r for all s, then Z is a closed subprescheme of S (supposing that M is coherent).
Proof. Indeed, the above reasoning leads us to the case where we have the inequality (*) for all s\in S (by replacing, if necessary, S with the open subset consisting of the s where the inequality is satisfied).
We can then suppose that M fits into an exact sequence
{\mathscr{O}}_S^q \to {\mathscr{O}}_S^r \to M \to 0
and the condition in question on the S' over S also implies that, in the corresponding exact sequence {\mathscr{O}}_{S'}^q\to{\mathscr{O}}_{S'}^r\to M'\to 0, the second arrow is an isomorphism, i.e. the first is zero.
We then see that the closed subprescheme Z of S defined by the ideal generated by the coefficients of the matrix defining the homomorphism {\mathscr{O}}_S^q\to{\mathscr{O}}_S^r satisfies the desired condition.
Proof. Returning to the proof of (3.4) where we left off, we denote by Z_n the closed subprescheme of S associated, by (3.6), to the module M_n and the integer r=P(n), and by Z'_N the infimum of the Z_n for \nu\leqslant n\leqslant N.
Then the Z_N form a decreasing sequence of closed subpreschemes of Z, which is thus necessarily stationary.
Let Z be the constant value of the Z_W for large N.
This is the desired Z in (3.4).
This finishes the proof of (3.4), and thus also of (3.3)
We have thus proven that \mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}}^P is representable by an S-prescheme Q that is an increasing union of open quasi-projective subpreschemes Q_\nu over S.
To go further, we need to invoke Theorem 2.1, whence we easily conclude that Q is quasi-compact (since it is the image of a prescheme S' of finite type over S that parametrises the family of quotient sheaves of the {\mathscr{F}}_K whose Hilbert polynomial is P).
Thus Q is equal to one of the Q_\nu, and thus quasi-projective over S.
To prove that it is projective over S, it thus remains to prove that it is proper over S, and for this it suffices to invoke the valuative criteria of properness in the form given in [18, II, 7.3.8].
It suffices to verify the following:
Let S be the spectrum of a discrete valuation ring, s its generic point, X a prescheme over S, {\mathscr{F}} a quasi-coherent module over X, and {\mathscr{G}}_s a quasi-coherent quotient module of {\mathscr{F}}_s={\mathscr{F}}\otimes_{{\mathscr{O}}_S}k(s) over X_s.
Then there exists a unique quasi-coherent quotient module {\mathscr{G}} of {\mathscr{F}} that is flat over S and whose restriction to X_s is {\mathscr{G}}_s.
Proof. Indeed, if {\mathscr{G}}_s={\mathscr{F}}_s/{\mathscr{H}}_s, it suffices to consider the largest subsheaf {\mathscr{H}} of {\mathscr{F}} that induces {\mathscr{H}}_s ([18, I, 9.4.2]) and to take {\mathscr{G}}={\mathscr{F}}/{\mathscr{H}}.
We easily verify that this sheaf works.
Theorem 3.2, and thus Theorem 3.1, is now completely proven.
The proof also shows, at the same time, the following:
Under the conditions of (3.2), let Q=\underline{\operatorname{Quot}}_{{{\mathscr{F}}/X/S}}^P, X_Q=X\times_s Q, {\mathscr{F}}_Q={\mathscr{F}}\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_Q, and let {\mathscr{G}} be the coherent quotient of {\mathscr{F}}_Q, which is flat over Q, that has P as its relative Hilbert polynomial, so that (Q,{\mathscr{G}}) represents the functor \mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}}^P.
Then there exists an integer \nu such that, for n\geqslant\nu, (f_Q)_*({\mathscr{G}}(n)) is a locally free module over Q of rank P(n), and is very ample with respect to S, i.e. it defines an immersion of Q into a Grassmannian scheme \underline{\operatorname{Grass}}_{P(n)}(M) over S.
A fortiori, for n\geqslant\nu, the sheaf \bigwedge^{P(n)}(f_Q)_*({\mathscr{G}}(n)) over Q is invertible and very ample with respect to S.
Proof. Indeed, we can reduce, as in (3.2), to the case where {\mathscr{F}} is flat over S, and then it suffices to take an integer \nu such that A_\nu=A (with the notation above).
The most important application of (3.2) is in the case where {\mathscr{F}}={\mathscr{O}}_X.
We then write
\begin{aligned}
\underline{\operatorname{Quot}}_{{\mathscr{O}}_X/X/S}
&= \underline{\operatorname{Hilb}}_{X/S}
\\\underline{\operatorname{Quot}}_{{\mathscr{O}}_X/X/S}^P
&= \underline{\operatorname{Hilb}}_{X/S}^P
\end{aligned}
and so we have a decomposition
\underline{\operatorname{Hilb}}_{X/S} = \coprod_P \underline{\operatorname{Hilb}}_{X/S}^P.
By definition, \underline{\operatorname{Hilb}}_{X/S} represents the functor \mathscr{H}\kern -.5pt ilb_{X/S}(S') which is given by the set of closed subpreschemes of X'=X\times_S S' that are flat over S;
and \underline{\operatorname{Hilb}}_{X/S}^P represents the subfunctor corresponding to the closed subpreschemes that admit a given Hilbert polynomial P.
These preschemes are also called the Hilbert prescheme of X over S and the Hilbert prescheme of index P, respectively.
The terminology is justified by the role played in the theory by the Hilbert polynomials.
Their difference in nature with the classical Chow varieties (meant to parametrise cycles, not varieties) is analogous to that between the Chow ring of classes of cycles of a variety and the ring of classes of sheaves of the variety (as is introduced in the Riemann–Roch theorem [1]);
we note that, when X=\mathbb{P}_S^r, with S the spectrum of a field, the knowledge of the Hilbert polynomial of a coherent module {\mathscr{F}} over X is equivalent to that of the Chern classes of {\mathscr{F}}, or even of the class of {\mathscr{F}} in the ring of classes of coherent sheaves on X.