6 Hilbert schemes

Grothendieck, A. “Technique de descente et théorèmes d’existence en géométrie algébrique, IV: Les schémas de Hilbert”. Séminaire Bourbaki 13 (1960–61), Talk no. 221.

TO-DO:

  • additif
  • errata

Introduction

The techniques described in [15] were, for the most part, independent of any projective hypotheses on the schemes in question. Unfortunately, they have not as of yet allowed us to solve the existence problems posed in [15]. In the current exposé, and the following, we will solve these problems by imposing projective hypotheses. The techniques used are typically projective, and practically make no use of any results from [15]. Here we will construct “Hilbert schemes”, which are meant to replace the use of Chow coordinates, as was mentioned in [15]. In the next exposé, the theory of passing to the quotient in schemes, developed in [15], combined with the theory of Hilbert schemes, will allow us, for example, to construct Picard schemes (defined in [15]) under rather general conditions.

In summary, we can say that we now have a more or less satisfying technique of projective constructions, apart from the fact that we are still missing11 a theory of passing to the quotient by groups such as the projective group, acting “without fixed points” (cf. [15]). The situation even seems slightly better in analytic geometry (if we restrict to the study of “projective” analytic spaces over a given analytic space), since, for analytic spaces, the difficulty of passing to the quotient by a group that acts nicely disappears. Either way, in algebraic geometry, as well as in analytic geometry, it remains to develop a construction technique that works without any projective hypotheses.

6.1 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 \underline{\operatorname{Tor}}_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 \underline{\operatorname{Tor}}_i(E,E'), \underline{\operatorname{Hom}}(E,E'), \underline{\operatorname{Ext}}^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]): 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 \underline{\operatorname{Tor}}_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]), 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].

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

  1. 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;

  2. 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(\underline{\operatorname{Hom}}_{{\mathcal{O}}_{X_T}}({\mathscr{G}},{\mathscr{G}}')) and \underline{\operatorname{Ext}}_{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).

6.2 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 [???, 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:

  1. 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;

  2. 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 \src{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):

  • The Poincaré (TO-DO: should this be Hilbert?) 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 TO-DO: minorés.

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

References

[15] A. Grothendieck. “Technique de descente et théorèmes d’existence en géométrie algébrique, I, II, III.” Séminaire Bourbaki. 12, 13 (n.d.), Talks no. 190, 195, and 212.

[16] A. Grothendieck. Séminaire de géométrie algébrique, I, II, III, IV. Paris, Institut des Hautes Études Scientifiques, n.d. (n.d.).

[18] A. Grothendieck, J. Dieudonné. “Eléments de géométrie algébrique, I, II, III, IV.” Publications Mathématiques de L’Institut Des Hautes Études Scientifiques. (n.d.).


  1. See the addendum at the end of this exposé.↩︎