Moduli Stacks

Introduction

In this chapter we verify basic properties of moduli spaces and moduli stacks such as \(\mathit{Hom}\), \(\mathit{Isom}\), \(\Cohstack_{X/B}\), \(\Quotfunctor_{\mathcal{F}/X/B}\), \(\Hilbfunctor_{X/B}\), \(\Picardstack_{X/B}\), \(\Picardfunctor_{X/B}\), \(\mathit{Mor}_B(Z, X)\), \(\Spacesstack'_{fp, flat, proper}\), \(\Polarizedstack\), and \(\Complexesstack_{X/B}\). We have already shown these algebraic spaces or algebraic stacks under suitable hypotheses, see Quot, Sections 08JS, 08K7, 08KA, 08WB, 09TQ, 0CZX, 0D02, 0D24, 0D19, 0D1D, 0D1L, and 0DLB. The stack of curves, denoted \(\textit{Curves}\) and introduced in Quot, Section 0D4Y, is discussed in the chapter on moduli of curves, see Moduli of Curves, Section 0DMJ.

In some sense this chapter is following the footsteps of Grothendieck’s lectures [Gr-I], [Gr-II], [Gr-III], [Gr-IV], [Gr-V], and [Gr-VI].

Conventions and abuse of language

We continue to use the conventions and the abuse of language introduced in Properties of Stacks, Section 04XA. Unless otherwise mentioned our base scheme will be \(\Spec(\mathbf{Z})\).

Properties of Hom and Isom

Let \(f : X \to B\) be a morphism of algebraic spaces which is of finite presentation. Assume \(\mathcal{F}\) and \(\mathcal{G}\) are quasi-coherent \(\mathcal{O}_X\)-modules. If \(\mathcal{G}\) is of finite presentation, flat over \(B\) with support proper over \(B\), then the functor \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) defined by \[T/B \longmapsto \Hom_{\mathcal{O}_{X_T}}(\mathcal{F}_T, \mathcal{G}_T)\] is an algebraic space affine over \(B\). If \(\mathcal{F}\) is of finite presentation, then \(\mathit{Hom}(\mathcal{F}, \mathcal{G}) \to B\) is of finite presentation. See Quot, Proposition 08K6.

If both \(\mathcal{F}\) and \(\mathcal{G}\) are of finite presentation, flat over \(B\) with support proper over \(B\), then the subfunctor \[\mathit{Isom}(\mathcal{F}, \mathcal{G}) \subset \mathit{Hom}(\mathcal{F}, \mathcal{G})\] is an algebraic space affine of finite presentation over \(B\). See Quot, Proposition 08K9.

Properties of the stack of coherent sheaves

Let \(f : X \to B\) be a morphism of algebraic spaces which is separated and of finite presentation. Then the stack \(\Cohstack_{X/B}\) parametrizing flat families of coherent modules with proper support is algebraic. See Quot, Theorem 09DS.

Lemma

The diagonal of \(\Cohstack_{X/B}\) over \(B\) is affine and of finite presentation.

Proof

The representability of the diagonal by algebraic spaces was shown in Quot, Lemma 08W6. From the proof we find that we have to show \(\mathit{Isom}(\mathcal{F}, \mathcal{G}) \to T\) is affine and of finite presentation for a pair of finitely presented \(\mathcal{O}_{X_T}\)-modules \(\mathcal{F}\), \(\mathcal{G}\) flat over \(T\) with support proper over \(T\). This was discussed in Section 0DLW.

Lemma

The morphism \(\Cohstack_{X/B} \to B\) is quasi-separated and locally of finite presentation.

Proof

To check \(\Cohstack_{X/B} \to B\) is quasi-separated we have to show that its diagonal is quasi-compact and quasi-separated. This is immediate from Lemma 0DLY. To prove that \(\Cohstack_{X/B} \to B\) is locally of finite presentation, we have to show that \(\Cohstack_{X/B} \to B\) is limit preserving, see Limits of Stacks, Proposition 0CMY. This follows from Quot, Lemma 08KD (small detail omitted).

Lemma

Assume \(X \to B\) is proper as well as of finite presentation. Then \(\Cohstack_{X/B} \to B\) satisfies the existence part of the valuative criterion (Morphisms of Stacks, Definition 0CLK).

Proof

Taking base change, this immediately reduces to the following problem: given a valuation ring \(R\) with fraction field \(K\) and an algebraic space \(X\) proper over \(R\) and a coherent \(\mathcal{O}_{X_K}\)-module \(\mathcal{F}_K\), show there exists a finitely presented \(\mathcal{O}_X\)-module \(\mathcal{F}\) flat over \(R\) whose generic fibre is \(\mathcal{F}_K\). Observe that by Flatness on Spaces, Theorem 0DLR any finite type quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) flat over \(R\) is of finite presentation. Denote \(j : X_K \to X\) the embedding of the generic fibre. As a base change of the affine morphism \(\Spec(K) \to \Spec(R)\) the morphism \(j\) is affine. Thus \(j_*\mathcal{F}_K\) is quasi-coherent. Write \[j_*\mathcal{F}_K = \colim \mathcal{F}_i\] as a filtered colimit of its finite type quasi-coherent \(\mathcal{O}_X\)-submodules, see Limits of Spaces, Lemma 0829. Since \(j_*\mathcal{F}_K\) is a sheaf of \(K\)-vector spaces over \(X\), it is flat over \(\Spec(R)\). Thus each \(\mathcal{F}_i\) is flat over \(R\) as flatness over a valuation ring is the same as being torsion free (More on Algebra, Lemma 0539) and torsion freeness is inherited by submodules. Finally, we have to show that the map \(j^*\mathcal{F}_i \to \mathcal{F}_K\) is an isomorphism for some \(i\). Since \(j^*j_*\mathcal{F}_K = \mathcal{F}_K\) (small detail omitted) and since \(j^*\) is exact, we see that \(j^*\mathcal{F}_i \to \mathcal{F}_K\) is injective for all \(i\). Since \(j^*\) commutes with colimits, we have \(\mathcal{F}_K = j^*j_*\mathcal{F}_K = \colim j^*\mathcal{F}_i\). Since \(\mathcal{F}_K\) is coherent (i.e., finitely presented), there is an \(i\) such that \(j^*\mathcal{F}_i\) contains all the (finitely many) generators over an affine étale cover of \(X\). Thus we get surjectivity of \(j^*\mathcal{F}_i \to \mathcal{F}_K\) for \(i\) large enough.

Lemma

Let \(B\) be an algebraic space. Let \(\pi : X \to Y\) be a quasi-finite morphism of algebraic spaces which are separated and of finite presentation over \(B\). Then \(\pi_*\) induces a morphism \(\Cohstack_{X/B} \to \Cohstack_{Y/B}\).

Proof

Let \((T \to B, \mathcal{F})\) be an object of \(\Cohstack_{X/B}\). We claim

1. \((T \to B, \pi_{T, *}\mathcal{F})\) is an object of \(\Cohstack_{Y/B}\) and

2. for \(T' \to T\) we have \(\pi_{T', *}(X_{T'} \to X_T)^*\mathcal{F} = (Y_{T'} \to Y_T)^*\pi_{T, *}\mathcal{F}\).

Part (b) guarantees that this construction defines a functor \(\Cohstack_{X/B} \to \Cohstack_{Y/B}\) as desired.

Let \(i : Z \to X_T\) be the closed subspace cut out by the zeroth fitting ideal of \(\mathcal{F}\) (Divisors on Spaces, Section 0CZ3). Then \(Z \to B\) is proper by assumption (see Derived Categories of Spaces, Section 0CZB). On the other hand \(i\) is of finite presentation (Divisors on Spaces, Lemma 0CZ5 and Morphisms of Spaces, Lemma 084Q). There exists a quasi-coherent \(\mathcal{O}_Z\)-module \(\mathcal{G}\) of finite type with \(i_*\mathcal{G} = \mathcal{F}\) (Divisors on Spaces, Lemma 0CZ6). In fact \(\mathcal{G}\) is of finite presentation as an \(\mathcal{O}_Z\)-module by Descent on Spaces, Lemma 0610. Observe that \(\mathcal{G}\) is flat over \(B\), for example because the stalks of \(\mathcal{G}\) and \(\mathcal{F}\) agree (Morphisms of Spaces, Lemma 0DK1). Observe that \(\pi_T \circ i : Z \to Y_T\) is quasi-finite as a composition of quasi-finite morphisms and that \(\pi_{T, *}\mathcal{F} = (\pi_T \circ i)_*\mathcal{G})\). Since \(i\) is affine, formation of \(i_*\) commutes with base change (Cohomology of Spaces, Lemma 07U8). Therefore we may replace \(B\) by \(T\), \(X\) by \(Z\), \(\mathcal{F}\) by \(\mathcal{G}\), and \(Y\) by \(Y_T\) to reduce to the case discussed in the next paragraph.

Assume that \(X \to B\) is proper. Then \(\pi\) is proper by Morphisms of Spaces, Lemma 04NX and hence finite by More on Morphisms of Spaces, Lemma 0A4X. Since a finite morphism is affine we see that (b) holds by Cohomology of Spaces, Lemma 07U8. On the other hand, \(\pi\) is of finite presentation by Morphisms of Spaces, Lemma 05WT. Thus \(\pi_{T, *}\mathcal{F}\) is of finite presentation by Descent on Spaces, Lemma 0610. Finally, \(\pi_{T, *}\mathcal{F}\) is flat over \(B\) for example by looking at stalks using Cohomology of Spaces, Lemma 0DK3.

Lemma

Let \(B\) be an algebraic space. Let \(\pi : X \to Y\) be an open immersion of algebraic spaces which are separated and of finite presentation over \(B\). Then the morphism \(\Cohstack_{X/B} \to \Cohstack_{Y/B}\) of Lemma 0DN9 is an open immersion.

Proof

Omitted. Hint: If \(\mathcal{F}\) is an object of \(\Cohstack_{Y/B}\) over \(T\) and for \(t \in T\) we have \(\text{Supp}(\mathcal{F}_t) \subset |X_t|\), then the same is true for \(t' \in T\) in a neighbourhood of \(t\).

Lemma

Let \(B\) be an algebraic space. Let \(\pi : X \to Y\) be a closed immersion of algebraic spaces which are separated and of finite presentation over \(B\). Then the morphism \(\Cohstack_{X/B} \to \Cohstack_{Y/B}\) of Lemma 0DN9 is a closed immersion.

Proof

Let \(\mathcal{I} \subset \mathcal{O}_Y\) be the sheaf of ideals cutting out \(X\) as a closed subspace of \(Y\). Recall that \(\pi_*\) induces an equivalence between the category of quasi-coherent \(\mathcal{O}_X\)-modules and the category of quasi-coherent \(\mathcal{O}_Y\)-modules annihilated by \(\mathcal{I}\), see Morphisms of Spaces, Lemma 04CJ. The same, mutatis mutandis, is true after base by \(T \to B\) with \(\mathcal{I}\) replaced by the ideal sheaf \(\mathcal{I}_T = \Im((Y_T \to Y)^*\mathcal{I} \to \mathcal{O}_{Y_T})\). Analyzing the proof of Lemma 0DN9 we find that the essential image of \(\Cohstack_{X/B} \to \Cohstack_{Y/B}\) is exactly the objects \(\xi = (T \to B, \mathcal{F})\) where \(\mathcal{F}\) is annihilated by \(\mathcal{I}_T\). In other words, \(\xi\) is in the essential image if and only if the multiplication map \[\mathcal{F} \otimes_{\mathcal{O}_{Y_T}} (Y_T \to Y)^*\mathcal{I} \longrightarrow \mathcal{F}\] is zero and similarly after any further base change \(T' \to T\). Note that \[(Y_{T'} \to Y_T)^*( \mathcal{F} \otimes_{\mathcal{O}_{Y_T}} (Y_T \to Y)^*\mathcal{I}) = (Y_{T'} \to Y_T)^*\mathcal{F} \otimes_{\mathcal{O}_{Y_{T'}}} (Y_{T'} \to Y)^*\mathcal{I})\] Hence the vanishing of the multiplication map on \(T'\) is representable by a closed subspace of \(T\) by Flatness on Spaces, Lemma 083M.

Situation

Let \(f : X \to B\) be as in the introduction to this section. Let \(I\) be a set and for \(i \in I\) let \(E_i \in D(\mathcal{O}_X)\) be perfect. Given an object \((T \to B, \mathcal{F})\) of \(\Cohstack_{X/B}\) denote \(E_{i, T}\) the derived pullback of \(E_i\) to \(X_T\). The object \[K_i = Rf_{T, *}(E_{i, T} \otimes_{\mathcal{O}_{X_T}}^\mathbf{L} \mathcal{F})\] of \(D(\mathcal{O}_T)\) is perfect and its formation commutes with base change, see Derived Categories of Spaces, Lemma 0A1P. Thus the function \[\chi_i : |T| \longrightarrow \mathbf{Z},\quad \chi_i(t) = \chi(X_t, E_{i, t} \otimes_{\mathcal{O}_{X_t}}^\mathbf{L} \mathcal{F}_t) = \chi(K_i \otimes_{\mathcal{O}_T}^\mathbf{L} \kappa(t))\] is locally constant by Derived Categories of Spaces, Lemma 0D1Z. Let \(P : I \to \mathbf{Z}\) be a map. Consider the substack \[\Cohstack^P_{X/B} \subset \Cohstack_{X/B}\] consisting of flat families of coherent sheaves with proper support whose numerical invariants agree with \(P\). More precisely, an object \((T \to B, \mathcal{F})\) of \(\Cohstack_{X/B}\) is in \(\Cohstack^P_{X/B}\) if and only if \(\chi_i(t) = P(i)\) for all \(i \in I\) and \(t \in T\).

Lemma

In Situation 0DNC the stack \(\Cohstack^P_{X/B}\) is algebraic and \[\Cohstack^P_{X/B} \longrightarrow \Cohstack_{X/B}\] is a flat closed immersion. If \(I\) is finite or \(B\) is locally Noetherian, then \(\Cohstack^P_{X/B}\) is an open and closed substack of \(\Cohstack_{X/B}\).

Proof

This is immediately clear if \(I\) is finite, because the functions \(t \mapsto \chi_i(t)\) are locally constant. If \(I\) is infinite, then we write \[I = \bigcup\nolimits_{I' \subset I\text{ finite}} I'\] and we denote \(P' = P|_{I'}\). Then we have \[\Cohstack^P_{X/B} = \bigcap\nolimits_{I' \subset I\text{ finite}} \Cohstack^{P'}_{X/B}\] Therefore, \(\Cohstack^P_{X/B}\) is always an algebraic stack and the morphism \(\Cohstack^P_{X/B} \subset \Cohstack_{X/B}\) is always a flat closed immersion, but it may no longer be an open substack. (We leave it to the reader to make examples). However, if \(B\) is locally Noetherian, then so is \(\Cohstack_{X/B}\) by Lemma 0DLZ and Morphisms of Stacks, Lemma 06R6. Hence if \(U \to \Cohstack_{X/B}\) is a smooth surjective morphism where \(U\) is a locally Noetherian scheme, then the inverse images of the open and closed substacks \(\Cohstack^{P'}_{X/B}\) have an open intersection in \(U\) (because connected components of locally Noetherian topological spaces are open). Thus the result in this case.

Lemma

Let \(f : X \to B\) be as in the introduction to this section. Let \(E_1, \ldots, E_r \in D(\mathcal{O}_X)\) be perfect. Let \(I = \mathbf{Z}^{\oplus r}\) and consider the map \[I \longrightarrow D(\mathcal{O}_X),\quad (n_1, \ldots, n_r) \longmapsto E_1^{\otimes n_1} \otimes \ldots \otimes E_r^{\otimes n_r}\] Let \(P : I \to \mathbf{Z}\) be a map. Then \(\Cohstack^P_{X/B} \subset \Cohstack_{X/B}\) as defined in Situation 0DNC is an open and closed substack.

Proof

We may work étale locally on \(B\), hence we may assume that \(B\) is affine. In this case we may perform absolute Noetherian reduction; we suggest the reader skip the proof. Namely, say \(B = \Spec(\Lambda)\). Write \(\Lambda = \colim \Lambda_i\) as a filtered colimit with each \(\Lambda_i\) of finite type over \(\mathbf{Z}\). For some \(i\) we can find a morphism of algebraic spaces \(X_i \to \Spec(\Lambda_i)\) which is separated and of finite presentation and whose base change to \(\Lambda\) is \(X\). See Limits of Spaces, Lemmas 07SK and 0851. Then after increasing \(i\) we may assume there exist perfect objects \(E_{1, i}, \ldots, E_{r, i}\) in \(D(\mathcal{O}_{X_i})\) whose derived pullback to \(X\) are isomorphic to \(E_1, \ldots, E_r\), see Derived Categories of Spaces, Lemma 09RJ. Clearly we have a cartesian square \[\xymatrix{ \Cohstack^P_{X/B} \ar[r] \ar[d] & \Cohstack_{X/B} \ar[d] \\ \Cohstack^P_{X_i/\Spec(\Lambda_i)} \ar[r] & \Cohstack_{X_i/\Spec(\Lambda_i)} }\] and hence we may appeal to Lemma 0DND to finish the proof.

Example

Let \(f : X \to B\) be as in the introduction to this section. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(P : \mathbf{Z} \to \mathbf{Z}\) be a numerical polynomial. Then we can consider the open and closed algebraic substack \[\Cohstack^P_{X/B} = \Cohstack^{P, \mathcal{L}}_{X/B} \subset \Cohstack_{X/B}\] consisting of flat families of coherent sheaves with proper support whose numerical invariants agree with \(P\): an object \((T \to B, \mathcal{F})\) of \(\Cohstack_{X/B}\) lies in \(\Cohstack^P_{X/B}\) if and only if \[P(n) = \chi(X_t, \mathcal{F}_t \otimes_{\mathcal{O}_{X_t}} \mathcal{L}_t^{\otimes n})\] for all \(n \in \mathbf{Z}\) and \(t \in T\). Of course this is a special case of Situation 0DNC where \(I = \mathbf{Z} \to D(\mathcal{O}_X)\) is given by \(n \mapsto \mathcal{L}^{\otimes n}\). It follows from Lemma 0DNE that this is an open and closed substack. Since the functions \(n \mapsto \chi(X_t, \mathcal{F}_t \otimes_{\mathcal{O}_{X_t}} \mathcal{L}_t^{\otimes n})\) are always numerical polynomials (Spaces over Fields, Lemma 0DN4) we conclude that \[\Cohstack_{X/B} = \coprod\nolimits_{P\text{ numerical polynomial}} \Cohstack^P_{X/B}\] is a disjoint union decomposition.

Properties of Quot

Let \(f : X \to B\) be a morphism of algebraic spaces which is separated and of finite presentation. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then \(\Quotfunctor_{\mathcal{F}/X/B}\) is an algebraic space. If \(\mathcal{F}\) is of finite presentation, then \(\Quotfunctor_{\mathcal{F}/X/B} \to B\) is locally of finite presentation. See Quot, Proposition 09TU.

Lemma

The diagonal of \(\Quotfunctor_{\mathcal{F}/X/B} \to B\) is a closed immersion. If \(\mathcal{F}\) is of finite type, then the diagonal is a closed immersion of finite presentation.

Proof

Suppose we have a scheme \(T/B\) and two quotients \(\mathcal{F}_T \to \mathcal{Q}_i\), \(i = 1, 2\) corresponding to \(T\)-valued points of \(\Quotfunctor_{\mathcal{F}/X/B}\) over \(B\). Denote \(\mathcal{K}_1\) the kernel of the first one and set \(u : \mathcal{K}_1 \to \mathcal{Q}_2\) the composition. By Flatness on Spaces, Lemma 083M there is a closed subspace of \(T\) such that \(T' \to T\) factors through it if and only if the pullback \(u_{T'}\) is zero. This proves the diagonal is a closed immersion. Moreover, if \(\mathcal{F}\) is of finite type, then \(\mathcal{K}_1\) is of finite type (Modules on Sites, Lemma 082T) and we see that the diagonal is of finite presentation by the same lemma.

Lemma

The morphism \(\Quotfunctor_{\mathcal{F}/X/B} \to B\) is separated. If \(\mathcal{F}\) is of finite presentation, then it is also locally of finite presentation.

Proof

To check \(\Quotfunctor_{\mathcal{F}/X/B} \to B\) is separated we have to show that its diagonal is a closed immersion. This is true by Lemma 0DM2. The second statement is part of Quot, Proposition 09TU.

Lemma

Let \(R\) be a valuation ring with fraction field \(K\), let \(X\) be an algebraic space over \(R\), and let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Every quasi-coherent quotient \[\mathcal{F}_K \longrightarrow \mathcal{Q}_K\] extends uniquely, up to unique isomorphism compatible with the quotient, to a quasi-coherent quotient \[\mathcal{F} \longrightarrow \mathcal{Q}\] such that \(\mathcal{Q}\) is flat over \(R\).

Proof

Write \(j : X_K \to X\). Define \(\mathcal{L}\) to be the kernel of the composition \[\mathcal{F} \longrightarrow j_*\mathcal{F}_K \longrightarrow j_*\mathcal{Q}_K\] and set \(\mathcal{Q} = \mathcal{F}/\mathcal{L}\). These modules are quasi-coherent because \(j\) is affine. On an affine scheme over \(X\) this construction is the following. If \(M\) is the module corresponding to \(\mathcal{F}\) and \(M_K \to N_K\) is the given quotient, then \[L = \{x \in M \mid x/1 \text{ maps to }0\text{ in }N_K\}\] and \(\mathcal{Q}\) corresponds to \(M/L\). The module \(M/L\) embeds into \(N_K\) and hence is torsion free as an \(R\)-module. It is flat over \(R\) by More on Algebra, Lemma 0539.

Suppose \(\mathcal{F} \to \mathcal{Q}'\) is another extension flat over \(R\). On an affine scheme as above, flatness implies that the module corresponding to \(\mathcal{Q}'\) is \(R\)-torsion free. Its kernel is therefore exactly \(L\): if the image of \(x\) vanishes after tensoring with \(K\), then it is annihilated by a nonzero element of \(R\), hence is zero. This proves uniqueness and completes the proof.

Lemma

Assume \(X \to B\) is proper as well as of finite presentation and \(\mathcal{F}\) quasi-coherent of finite type. Then \(\Quotfunctor_{\mathcal{F}/X/B} \to B\) satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition 03IX).

Proof

Taking base change, this immediately reduces to the following problem: given a valuation ring \(R\) with fraction field \(K\), an algebraic space \(X\) proper over \(R\), a finite type quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\), and a coherent quotient \(\mathcal{F}_K \to \mathcal{Q}_K\), show there exists a quotient \(\mathcal{F} \to \mathcal{Q}\) where \(\mathcal{Q}\) is a finitely presented \(\mathcal{O}_X\)-module flat over \(R\) whose generic fibre is \(\mathcal{Q}_K\). Apply Lemma moduli-lemma-quotient-extension-valuation-ring. The resulting \(\mathcal{Q}\) is of finite type because it is a quotient of \(\mathcal{F}\). By Flatness on Spaces, Theorem 0DLR it is of finite presentation. Its support is closed in \(X\) and hence proper over \(R\). Thus this quotient is an object of the Quot functor and gives the required extension.

Lemma

Let \(B\) be an algebraic space. Let \(\pi : X \to Y\) be an affine quasi-finite morphism of algebraic spaces which are separated and of finite presentation over \(B\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then \(\pi_*\) induces a morphism \(\Quotfunctor_{\mathcal{F}/X/B} \to \Quotfunctor_{\pi_*\mathcal{F}/Y/B}\).

Proof

Set \(\mathcal{G} = \pi_*\mathcal{F}\). Since \(\pi\) is affine we see that for any scheme \(T\) over \(B\) we have \(\mathcal{G}_T = \pi_{T, *}\mathcal{F}_T\) by Cohomology of Spaces, Lemma 07U8. Moreover \(\pi_T\) is affine, hence \(\pi_{T, *}\) is exact and transforms quotients into quotients. Observe that a quasi-coherent quotient \(\mathcal{F}_T \to \mathcal{Q}\) defines a point of \(\Quotfunctor_{X/B}\) if and only if \(\mathcal{Q}\) defines an object of \(\Cohstack_{X/B}\) over \(T\) (similarly for \(\mathcal{G}\) and \(Y\)). Since we’ve seen in Lemma 0DN9 that \(\pi_*\) induces a morphism \(\Cohstack_{X/B} \to \Cohstack_{Y/B}\) we see that if \(\mathcal{F}_T \to \mathcal{Q}\) is in \(\Quotfunctor_{\mathcal{F}/X/B}(T)\), then \(\mathcal{G}_T \to \pi_{T, *}\mathcal{Q}\) is in \(\Quotfunctor_{\mathcal{G}/Y/B}(T)\).

Lemma

Let \(B\) be an algebraic space. Let \(\pi : X \to Y\) be an affine open immersion of algebraic spaces which are separated and of finite presentation over \(B\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then the morphism \(\Quotfunctor_{\mathcal{F}/X/B} \to \Quotfunctor_{\pi_*\mathcal{F}/Y/B}\) of Lemma 0DP1 is an open immersion.

Proof

Omitted. Hint: If \((\pi_*\mathcal{F})_T \to \mathcal{Q}\) is an element of \(\Quotfunctor_{\pi_*\mathcal{F}/Y/B}(T)\) and for \(t \in T\) we have \(\text{Supp}(\mathcal{Q}_t) \subset |X_t|\), then the same is true for \(t' \in T\) in a neighbourhood of \(t\).

Lemma

Let \(B\) be an algebraic space. Let \(j : X \to Y\) be an open immersion of algebraic spaces which are separated and of finite presentation over \(B\). Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module and set \(\mathcal{F} = j^*\mathcal{G}\). Then there is an open immersion \[\Quotfunctor_{\mathcal{F}/X/B} \longrightarrow \Quotfunctor_{\mathcal{G}/Y/B}\] of algebraic spaces over \(B\).

Proof

If \(\mathcal{F}_T \to \mathcal{Q}\) is an element of \(\Quotfunctor_{\mathcal{F}/X/B}(T)\) then we can consider \(\mathcal{G}_T \to j_{T, *}\mathcal{F}_T \to j_{T, *}\mathcal{Q}\). Looking at stalks one finds that this is surjective. By Lemma 0DN9 we see that \(j_{T, *}\mathcal{Q}\) is finitely presented, flat over \(B\) with support proper over \(B\). Thus we obtain a \(T\)-valued point of \(\Quotfunctor_{\mathcal{G}/Y/B}\). This defines the morphism of the lemma. We omit the proof that this is an open immersion. Hint: If \(\mathcal{G}_T \to \mathcal{Q}\) is an element of \(\Quotfunctor_{\mathcal{G}/Y/B}(T)\) and for \(t \in T\) we have \(\text{Supp}(\mathcal{Q}_t) \subset |X_t|\), then the same is true for \(t' \in T\) in a neighbourhood of \(t\).

Lemma

Let \(B\) be an algebraic space. Let \(\pi : X \to Y\) be a closed immersion of algebraic spaces which are separated and of finite presentation over \(B\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then the morphism \(\Quotfunctor_{\mathcal{F}/X/B} \to \Quotfunctor_{\pi_*\mathcal{F}/Y/B}\) of Lemma 0DP1 is an isomorphism.

Proof

For every scheme \(T\) over \(B\) the morphism \(\pi_T : X_T \to Y_T\) is a closed immersion. Then \(\pi_{T, *}\) is an equivalence of categories between \(\QCoh(\mathcal{O}_{X_T})\) and the full subcategory of \(\QCoh(\mathcal{O}_{Y_T})\) whose objects are those quasi-coherent modules annihilated by the ideal sheaf of \(X_T\), see Morphisms of Spaces, Lemma 04CJ. Since a qotient of \((\pi_*\mathcal{F})_T\) is annihilated by this ideal we obtain the bijectivity of the map \(\Quotfunctor_{\mathcal{F}/X/B}(T) \to \Quotfunctor_{\pi_*\mathcal{F}/Y/B}(T)\) for all \(T\) as desired.

Lemma

Let \(X \to B\) be as in the introduction to this section. Let \(\mathcal{F} \to \mathcal{G}\) be a surjection of quasi-coherent \(\mathcal{O}_X\)-modules. Then there is a canonical closed immersion \(\Quotfunctor_{\mathcal{G}/X/B} \to \Quotfunctor_{\mathcal{F}/X/B}\).

Proof

Let \(\mathcal{K} = \Ker(\mathcal{F} \to \mathcal{G})\). By right exactness of pullbacks we find that \(\mathcal{K}_T \to \mathcal{F}_T \to \mathcal{G}_T \to 0\) is an exact sequecnce for all schemes \(T\) over \(B\). In particular, a quotient of \(\mathcal{G}_T\) determines a quotient of \(\mathcal{F}_T\) and we obtain our transformation of functors \(\Quotfunctor_{\mathcal{G}/X/B} \to \Quotfunctor_{\mathcal{F}/X/B}\). This transformation is a closed immersion by Flatness on Spaces, Lemma 083M. Namely, given an element \(\mathcal{F}_T \to \mathcal{Q}\) of \(\Quotfunctor_{\mathcal{F}/X/B}(T)\), then we see that the pull back to \(T'/T\) is in the image of the transformation if and only if \(\mathcal{K}_{T'} \to \mathcal{Q}_{T'}\) is zero.

Remark

Let \(f : X \to B\) and \(\mathcal{F}\) be as in the introduction to this section. Let \(I\) be a set and for \(i \in I\) let \(E_i \in D(\mathcal{O}_X)\) be perfect. Let \(P : I \to \mathbf{Z}\) be a function. Recall that we have a morphism \[\Quotfunctor_{\mathcal{F}/X/B} \longrightarrow \Cohstack_{X/B}\] which sends the element \(\mathcal{F}_T \to \mathcal{Q}\) of \(\Quotfunctor_{\mathcal{F}/X/B}(T)\) to the object \(\mathcal{Q}\) of \(\Cohstack_{X/B}\) over \(T\), see proof of Quot, Proposition 09TU. Hence we can form the fibre product diagram \[\xymatrix{ \Quotfunctor^P_{\mathcal{F}/X/B} \ar[r] \ar[d] & \Cohstack^P_{X/B} \ar[d] \\ \Quotfunctor_{\mathcal{F}/X/B} \ar[r] & \Cohstack_{X/B} }\] This is the defining diagram for the algebraic space in the upper left corner. The left vertical arrow is a flat closed immersion which is an open and closed immersion for example if \(I\) is finite, or \(B\) is locally Noetherian, or \(I = \mathbf{Z}\) and \(E_i = \mathcal{L}^{\otimes i}\) for some invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) (in the last case we sometimes use the notation \(\Quotfunctor^{P, \mathcal{L}}_{\mathcal{F}/X/B}\)). See Situation 0DNC and Lemmas 0DND and 0DNE and Example 0DNF.

Lemma

Let \(f : X \to B\) and \(\mathcal{F}\) be as in the introduction to this section. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Then tensoring with \(\mathcal{L}\) defines an isomorphism \[\Quotfunctor_{\mathcal{F}/X/B} \longrightarrow \Quotfunctor_{\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}/X/B}\] Given a numerical polynomial \(P(t)\), then setting \(P'(t) = P(t + 1)\) this map induces an isomorphism \(\Quotfunctor^P_{\mathcal{F}/X/B} \longrightarrow \Quotfunctor^{P'}_{\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}/X/B}\) of open and closed substacks.

Proof

Set \(\mathcal{G} = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}\). Observe that \(\mathcal{G}_T = \mathcal{F}_T \otimes_{\mathcal{O}_{X_T}} \mathcal{L}_T\). If \(\mathcal{F}_T \to \mathcal{Q}\) is an element of \(\Quotfunctor_{\mathcal{F}/X/B}(T)\), then we send it to the element \(\mathcal{G}_T \to \mathcal{Q} \otimes_{\mathcal{O}_{X_T}} \mathcal{L}_T\) of \(\Quotfunctor_{\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}/X/B}(T)\). This is compatible with pullbacks and hence defines a transformation of functors as desired. Since there is an obvious inverse transformation, it is an isomorphism. We omit the proof of the final statement.

Lemma

Let \(f : X \to B\) and \(\mathcal{F}\) be as in the introduction to this section. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Then \[\Quotfunctor^{P, \mathcal{L}}_{\mathcal{F}/X/B} = \Quotfunctor^{P', \mathcal{L}^{\otimes n}}_{\mathcal{F}/X/B}\] where \(P'(t) = P(nt)\).

Proof

Follows immediately after unwinding all the definitions.

Bounded families of coherent sheaves

Let \(S\) be a Noetherian scheme and let \(X \to S\) be a morphism of finite type. We introduce a fibrewise notion of boundedness which is useful when the sheaves in question do not come with a fixed parameter scheme.

Definition

A fibrewise coherent sheaf over \(X/S\) is a triple \[(s, K, \mathcal{F})\] where \(s \in S\), where \(K/\kappa(s)\) is a field extension, and where \(\mathcal{F}\) is a coherent \(\mathcal{O}_{X_K}\)-module. Two such triples \((s, K, \mathcal{F})\) and \((s', K', \mathcal{F}')\) are equivalent if \(s=s'\) and there are \(\kappa(s)\)-embeddings of \(K\) and \(K'\) into a field \(L\) such that \[\mathcal{F}_L \cong \mathcal{F}'_L\] on \(X_L\). A family of fibrewise coherent sheaves is a set of equivalence classes of such triples.

A family \(E\) is bounded if there are a scheme \(T\) of finite type over \(S\) and a coherent \(\mathcal{O}_{X_T}\)-module \(\mathcal{G}\) with the following property. For every \((s, K, \mathcal{F})\) whose class belongs to \(E\), there are a point \(t \in T\) above \(s\), a field \(L\), and \(\kappa(s)\)-embeddings \[K \longrightarrow L \longleftarrow \kappa(t)\] such that \(\mathcal{F}_L \cong \mathcal{G}_t \otimes_{\kappa(t)} L\). We say that \((T, \mathcal{G})\) bounds \(E\).

Lemma

Let \(E\) and \(E'\) be bounded families of fibrewise coherent sheaves over \(X/S\).

1. A finite union of bounded families is bounded.

2. After any base change \(S' \to S\), the family obtained from \(E\) by base change is bounded over \(X_{S'}/S'\).

3. The family of all tensor products \(\mathcal{F} \otimes_{\mathcal{O}_{X_K}} \mathcal{F}'\), where the two classes belong to \(E\) and \(E'\) over the same field-valued point, is bounded.

4. There is a pair \((T, \mathcal{G})\) bounding \(E\) for which \(\mathcal{G}\) is flat over \(T\).

Proof

For (1), take the disjoint union of finitely many parameter schemes and the sheaf which restricts to the given sheaf on each component. For (2), if \((T, \mathcal{G})\) bounds \(E\), then \[(T \times_S S', \mathcal{G}_{S'})\] bounds the base-changed family.

Choose \((T, \mathcal{G})\) and \((T', \mathcal{G}')\) bounding \(E\) and \(E'\). On \(X_{T \times_S T'}\) form the tensor product of the pullbacks of \(\mathcal{G}\) and \(\mathcal{G}'\). It is coherent, and its formation commutes with arbitrary base change. Thus this sheaf, with parameter scheme \(T \times_S T'\), bounds the family in (3).

Finally, \(T\) is Noetherian. More on Morphisms, Lemma 0ASY gives a finite stratification of \(T\) by locally closed subschemes such that the restriction of \(\mathcal{G}\) to each stratum is flat over that stratum. The disjoint union of the strata is still of finite type over \(S\), meets every point of \(T\), and carries a flat coherent sheaf whose fibre classes are the same. This proves (4).

Proposition

Assume in addition that \(X \to S\) is proper. Let \(E\) and \(E'\) be bounded families of fibrewise coherent sheaves over \(X/S\).

1. The kernels, images, and cokernels of all homomorphisms \[\mathcal{F} \longrightarrow \mathcal{F}'\] between members of \(E\) and \(E'\) form bounded families.

2. The middle terms of all short exact sequences \[0 \longrightarrow \mathcal{F}' \longrightarrow \mathcal{H} \longrightarrow \mathcal{F} \longrightarrow 0\] between members of \(E\) and \(E'\) form a bounded family. The same holds with \(E\) and \(E'\) interchanged.

Proof

By Lemma moduli-lemma-bounded-coherent-elementary, after replacing the two parameter schemes by finite stratifications on which their sheaves are flat and taking their fibre product over \(S\), we may work with one scheme \(T\) of finite type over \(S\) and two coherent modules \(\mathcal{G}\) and \(\mathcal{G}'\) on \(X_T\) which are flat over \(T\) and bound \(E\) and \(E'\).

The functor \[U/T \longmapsto \Hom_{\mathcal{O}_{X_U}}(\mathcal{G}_U, \mathcal{G}'_U)\] is represented by an algebraic space \(H\) affine and of finite presentation over \(T\); see Quot, Proposition 08K6. In particular, \(H\) is a scheme of finite type over \(S\). On \(X_H\) there is a universal homomorphism \[u : \mathcal{G}_H \longrightarrow \mathcal{G}'_H.\] Its kernel, image, and cokernel are coherent. Every homomorphism between a member of \(E\) and a member of \(E'\) becomes a fibre of \(u\) after passage to a common field extension. There is one base-change point to address. Apply More on Morphisms, Lemma 0ASY to \(\Coker(u)\) and replace \(H\) by the finite disjoint union of the resulting strata. Then \(\Coker(u)\) is flat over \(H\). Since \(\mathcal{G}'_H\) is flat, \(\Im(u)\) is flat over \(H\) as well. The two exact sequences defining \(\Im(u)\) and \(\Ker(u)\) therefore remain exact after every base change. Thus the fibres of the three coherent modules are respectively the kernel, image, and cokernel of the fibre homomorphism, and they bound the three families in (1).

For completeness, we recall the additional base-change step needed for extensions. Noetherian generic flatness and cohomology and base change, applied to the relative derived Hom of \(\mathcal{G}'\) into \(\mathcal{G}\), give a finite stratification of \(T\) such that on each stratum the relative \(\Ext^1\) functor commutes with arbitrary base change and is the functor of sections of a finite locally free module. This is the Ext stratification used in the proof of the cited result. Replacing \(T\) by the finite disjoint union of these strata, let \(V \to T\) be the corresponding vector bundle. Its tautological section is a relative extension class and hence gives a short exact sequence \[0 \longrightarrow \mathcal{G}_V \longrightarrow \mathcal{H} \longrightarrow \mathcal{G}'_V \longrightarrow 0\] on \(X_V\). The base-change assertion says that every extension of a geometric fibre of \(\mathcal{G}'\) by the corresponding fibre of \(\mathcal{G}\) occurs as a fibre of this sequence after a field extension. Since \(V\) is of finite type over \(S\) and \(\mathcal{H}\) is coherent, it bounds the required middle terms. Interchanging \(\mathcal{G}\) and \(\mathcal{G}'\) proves the final assertion.

Lemma

Let \(T\) be a scheme of finite type over \(S\), and let \(Z \subset X_T\) be a closed subscheme. The structure sheaves of the reduced schemes \[(Z_{\overline{t}})_{red} \subset X_{\overline{t}},\] where \(\overline{t}\) ranges over geometric points of \(T\), form a bounded family of fibrewise coherent sheaves over \(X/S\).

Proof

We prove the assertion by Noetherian induction on \(T\). If \(T\) is reducible, each of its finitely many irreducible components is a proper closed subscheme and the induction hypothesis together with Lemma moduli-lemma-bounded-coherent-elementary applies. Thus we may assume that \(T\) is irreducible.

Apply More on Morphisms, Lemma 0550 to \(Z \to T\). After replacing \(T\) by a nonempty open \(V\), it gives a finite universal homeomorphism \(T' \to V\) and \[Z'=(Z \times_T T')_{red}.\] The scheme \(T'\) is integral and the generic fibre of \(Z' \to T'\) is geometrically reduced. By More on Morphisms, Lemma 0578, after shrinking \(T'\) and \(V\) compatibly, every fibre of \(Z' \to T'\) is geometrically reduced.

The closed immersion \(Z' \to Z \times_T T'\) is a universal homeomorphism. The same is true on every geometric fibre. Since the source fibre is reduced, it is the unique reduced closed subscheme with the underlying topological space of the target fibre. Hence \[Z'_{\overline{t}'} = (Z_{\overline{t}'})_{red}\] for every geometric point \(\overline{t}'\) of \(T'\). The pushforward of \(\mathcal{O}_{Z'}\) to \(X_{T'}\) is coherent, and \(T'\) is of finite type over \(S\), so this one family bounds all reduced fibres over \(V\). The complement \(T \setminus V\), with its reduced induced scheme structure, is a proper closed subscheme of \(T\) and is handled by the induction hypothesis. A finite union finishes the proof.

Proposition

Let \(E\) be a bounded family of fibrewise coherent sheaves over \(X/S\). The structure sheaves \[\mathcal{O}_{(\operatorname{Supp}(\mathcal{F}))_{red}}\] for members \(\mathcal{F}\) of \(E\) over algebraically closed fields form a bounded family.

Proof

Choose a pair \((T, \mathcal{G})\) bounding \(E\). Let \(Z \subset X_T\) be the closed subscheme cut out by \(\operatorname{Fit}_0(\mathcal{G})\). Formation of this Fitting ideal commutes with arbitrary base change by Divisors, Lemma 0C3D. Moreover, Divisors, Lemma 0CYX gives \[|Z_{\overline{t}}|=\operatorname{Supp}(\mathcal{G}_{\overline{t}})\] for every geometric point \(\overline{t}\) of \(T\). Consequently \[(Z_{\overline{t}})_{red}= (\operatorname{Supp}(\mathcal{G}_{\overline{t}}))_{red}.\] Lemma moduli-lemma-bounded-fibrewise-reductions bounds the structure sheaves of the schemes on the left. If a member of \(E\) is represented by \(\mathcal{F}\) over an algebraically closed field, then after passage to a common extension it is isomorphic to a geometric fibre of \(\mathcal{G}\). The same is therefore true of the reduced supports. This proves the proposition.

Lemma

Let \(T\) be a scheme of finite type over \(S\), and let \(\mathcal{H}\) be a coherent \(\mathcal{O}_{X_T}\)-module. There is a finite stratification \[T = \coprod T_a\] by locally closed subschemes such that, for every \(a\) and every morphism \(T' \to T_a\), one has \[\mathop{\rm Ann}(\mathcal{H}_{T'}) = \mathop{\rm Ann}(\mathcal{H}_{T_a})\mathcal{O}_{X_{T'}}.\] In particular, the structure sheaves \[\mathcal{O}_{X_{\overline{t}}}/ \mathop{\rm Ann}(\mathcal{H}_{\overline{t}})\] of the scheme theoretic supports of the geometric fibres form a bounded family.

Proof

Choose a finite affine open covering \(X_T = \bigcup U_j\). On \(U_j\) choose generators \(h_{j1}, \ldots, h_{jr_j}\) of \(\mathcal{H}|_{U_j}\) and consider the map \[\mathcal{O}_{U_j} \longrightarrow (\mathcal{H}|_{U_j})^{\oplus r_j},\qquad a \longmapsto (ah_{j1},\ldots,ah_{jr_j}).\] Its kernel is \(\mathop{\rm Ann}(\mathcal{H}|_{U_j})\). Apply More on Morphisms, Lemma 0ASY, first to \(\mathcal{H}|_{U_j}\) and then to the cokernel of the displayed map, and take a common finite refinement of the resulting stratifications of \(T\).

On a stratum both the target and the cokernel of the displayed map are flat over the stratum. Its image is therefore flat as well. The two short exact sequences obtained by factoring the map through its image remain exact after arbitrary base change. Hence its kernel commutes with arbitrary base change. The chosen sections continue to generate after base change, so the base-changed kernel is the annihilator of the base-changed module. This proves the asserted equality on every \(U_j\), and hence on \(X_{T'}\). The quotients by these annihilator ideals over the finitely many strata give the bounded family in the last assertion.

Proposition

Let \(E\) be a bounded family of fibrewise coherent sheaves over \(X/S\). Let \((s,K,\mathcal{F})\) represent a member of \(E\), with \(K\) algebraically closed. The following families are bounded.

1. For every generic point \(x\) of an irreducible component of \(\operatorname{Supp}(\mathcal{F})\), the canonical primary quotient \[\mathcal{P}_x(\mathcal{F}) = \mathop{\rm Im}\bigl( \mathcal{F} \longrightarrow (j_x)_*j_x^*\mathcal{F} \bigr),\] where \(j_x : \Spec(\mathcal{O}_{X_K,x}) \to X_K\) is the canonical morphism.

2. For every \(x \in \operatorname{Ass}(\mathcal{F})\), the structure sheaf \(\mathcal{O}_{(\overline{\{x\}})_{red}}\).

3. For every \(x\) as in (1), the structure sheaf \[\mathcal{O}_{X_K}/\mathfrak q_x, \qquad \mathfrak q_x = \mathop{\rm Ann}(\mathcal{P}_x(\mathcal{F})).\] Here \(\mathfrak q_x\) is primary with radical the prime ideal defining \((\overline{\{x\}})_{red}\).

Remark

In the last sentence of [EGA, IV, 3.2.6], \(\Spec(\kappa(x))\) is a typographical error for \(\Spec(\mathcal{O}_{X,x})\). The localization appearing in (1) is also the construction used in the proof there. For example, if \(X = \Spec(k[\epsilon]/(\epsilon^2))\) and \(\mathcal{F}=\mathcal{O}_X\), then \(x\) is the unique associated point and the unique primary quotient is \(\mathcal{F}\) itself, whereas the map to the residue-field fibre has nonzero kernel.

Proof

Choose a pair \((T,\mathcal{G})\) bounding \(E\). It is enough to bound the three constructions for every geometric fibre of \(\mathcal{G}\). We do this by Noetherian induction on \(T\). Passing to the reduction and then to the finitely many irreducible components reduces the induction step to the case where \(T\) is integral.

We recall the generic-fibre primary-decomposition argument. After a finite dominant base change \(T' \to V\), where \(V\) is a nonempty open of \(T\), a reduced irredundant primary decomposition of the generic fibre can be spread out to finitely many coherent quotients \[\mathcal{G}_{T'} \longrightarrow \mathcal{Q}_i.\] Let \(Z_i\) be the closure of the associated point of the generic fibre of \(\mathcal{Q}_i\). The base change and the quotient list may be chosen so that the generic fibres of all \(Z_i \to T'\) are geometrically irreducible. Indeed, this is the finite field-extension and spreading-out construction of More on Morphisms, Lemma 0551, applied to the finitely many closures, followed by a refinement of the primary decomposition. After shrinking once more, More on Morphisms, Lemma 0559 makes every geometric fibre of every \(Z_i\) irreducible.

The primary-decomposition theorem near a generic fibre [EGA, IV, 9.8.2–9.8.4] now gives, after a further shrinking, the following facts for every geometric point \(\overline{t}'\) of \(T'\): the map \[\mathcal{G}_{\overline{t}'} \longrightarrow \bigoplus_i \mathcal{Q}_{i,\overline{t}'}\] is injective; each \(\mathcal{Q}_{i,\overline{t}'}\) has no embedded associated point; its unique associated point is the generic point of \(Z_{i,\overline{t}'}\); these points are pairwise distinct and exhaust \(\operatorname{Ass}(\mathcal{G}_{\overline{t}'})\); and the associated point is maximal in the support precisely for the indices for which it was maximal on the generic fibre.

For a maximal associated point the corresponding primary quotient is unique; equivalently it is the image of localization displayed in (1), see [EGA, IV, 3.2.6]. Thus the finitely many \(\mathcal{Q}_i\) whose generic associated point is maximal bound the family in (1) over \(V\). For all indices \(i\), Lemma moduli-lemma-bounded-fibrewise-reductions, applied to \(Z_i \to T'\), bounds the reduced closures of all associated points and hence the family in (2). Finally apply Lemma moduli-lemma-annihilator-stratification to the maximal \(\mathcal{Q}_i\). On its strata the quotient by the relative annihilator has geometric fibres \[\mathcal{O}_{X_{\overline{t}'}}/ \mathop{\rm Ann}(\mathcal{Q}_{i,\overline{t}'}),\] which are exactly the sheaves in (3).

The cover \(T' \to V\) is surjective, so every geometric fibre over \(V\) occurs after a field extension among the fibres just constructed. The closed complement \(T \setminus V\), with its reduced induced structure, is a proper closed subscheme and is handled by the induction hypothesis. Taking a finite union completes the proof.

Boundedness for Quot

Contrary to what happens classically, we already know the Quot functor is an algebraic space, but we don’t know that it is ever represented by a finite type algebraic space.

Lemma

Let \(n \geq 0\), \(r \geq 1\), \(P \in \mathbf{Q}[t]\). The algebraic space \[X = \Quotfunctor^P_{\mathcal{O}^{\oplus r}_{\mathbf{P}^n_\mathbf{Z}}/ \mathbf{P}^n_\mathbf{Z}/\mathbf{Z}}\] parametrizing quotients of \(\mathcal{O}_{\mathbf{P}^n_\mathbf{Z}}^{\oplus r}\) with Hilbert polynomial \(P\) is proper over \(\Spec(\mathbf{Z})\).

Proof

We already know that \(X \to \Spec(\mathbf{Z})\) is separated and locally of finite presentation (Lemma 0DM3). We also know that \(X \to \Spec(\mathbf{Z})\) satisfies the existence part of the valuative criterion, see Lemma 0DM4. By the valuative criterion for properness, it suffices to prove our Quot space is quasi-compact, see Morphisms of Spaces, Lemma 0A40. Thus it suffices to find a quasi-compact scheme \(T\) and a surjective morphism \(T \to X\). Let \(m\) be the integer found in Varieties, Lemma 08AG. Let \[N = r{m + n \choose n} - P(m)\] We will write \(\mathbf{P}^n\) for \(\mathbf{P}^n_\mathbf{Z} = \text{Proj}(\mathbf{Z}[T_0, \ldots, T_n])\) and unadorned products will mean products over \(\Spec(\mathbf{Z})\). The idea of the proof is to construct a “universal” map \[\Psi : \mathcal{O}_{T \times \mathbf{P}^n}(-m)^{\oplus N} \longrightarrow \mathcal{O}_{T \times \mathbf{P}^n}^{\oplus r}\] over an affine scheme \(T\) and show that every point of \(X\) corresponds to a cokernel of this in some point of \(T\).

Definition of \(T\) and \(\Psi\). We take \(T = \Spec(A)\) where \[A = \mathbf{Z}[a_{i, j, E}]\] where \(i \in \{1, \ldots, r\}\), \(j \in \{1, \ldots, N\}\) and \(E = (e_0, \ldots, e_n)\) runs through the multi-indices of total degree \(|E| = \sum_{k = 0, \ldots n} e_k = m\). Then we define \(\Psi\) to be the map whose \((i, j)\) matrix entry is the map \[\sum\nolimits_{E = (e_0, \ldots, e_n)} a_{i, j, E} T_0^{e_0} \ldots T_n^{e_n} : \mathcal{O}_{T \times \mathbf{P}^n}(-m) \longrightarrow \mathcal{O}_{T \times \mathbf{P}^n}\] where the sum is over \(E\) as above (but \(i\) and \(j\) are fixed of course).

Consider the quotient \(\mathcal{Q} = \Coker(\Psi)\) on \(T \times \mathbf{P}^n\). By More on Morphisms, Lemma 0ASY there exists a \(t \geq 0\) and closed subschemes \[T = T_0 \supset T_1 \supset \ldots \supset T_t = \emptyset\] such that the pullback \(\mathcal{Q}_p\) of \(\mathcal{Q}\) to \((T_p \setminus T_{p + 1}) \times \mathbf{P}^n\) is flat over \(T_p \setminus T_{p + 1}\). Observe that we have an exact sequence \[\mathcal{O}_{(T_p \setminus T_{p + 1}) \times \mathbf{P}^n}(-m)^{\oplus N} \to \mathcal{O}_{(T_p \setminus T_{p + 1}) \times \mathbf{P}^n}^{\oplus r} \to \mathcal{Q}_p \to 0\] by pulling back the exact sequence defining \(\mathcal{Q} = \Coker(\Psi)\). Therefore we obtain a morphism \[\coprod (T_p \setminus T_{p + 1}) \longrightarrow \Quotfunctor_{\mathcal{O}^{\oplus r}/\mathbf{P}/\mathbf{Z}} \supset \Quotfunctor^P_{\mathcal{O}^{\oplus r}/\mathbf{P}/\mathbf{Z}} = X\] Since the left hand side is a Noetherian scheme and the inclusion on the right hand side is open, it suffices to show that any point of \(X\) is in the image of this morphism.

Let \(k\) be a field and let \(x \in X(k)\). Then \(x\) corresponds to a surjection \(\mathcal{O}_{\mathbf{P}^n_k}^{\oplus r} \to \mathcal{F}\) of coherent \(\mathcal{O}_{\mathbf{P}^n_k}\)-modules such that the Hilbert polynomial of \(\mathcal{F}\) is \(P\). Consider the short exact sequence \[0 \to \mathcal{K} \to \mathcal{O}_{\mathbf{P}^n_k}^{\oplus r} \to \mathcal{F} \to 0\] By Varieties, Lemma 08AG and our choice of \(m\) we see that \(\mathcal{K}\) is \(m\)-regular. By Varieties, Lemma 08A8 we see that \(\mathcal{K}(m)\) is globally generated. By Varieties, Lemma 08A6 and the definition of \(m\)-regularity we see that \(H^i(\mathbf{P}^n_k, \mathcal{K}(m)) = 0\) for \(i > 0\). Hence we see that \[\dim_k H^0(\mathbf{P}^n_k, \mathcal{K}(m)) = \chi(\mathcal{K}(m)) = \chi(\mathcal{O}_{\mathbf{P}^n_k}(m)^{\oplus r}) - \chi(\mathcal{F}(m)) = N\] by our choice of \(N\). This gives a surjection \[\mathcal{O}_{\mathbf{P}^n_k}^{\oplus N} \longrightarrow \mathcal{K}(m)\] Twisting back down and using the short exact sequence above we see that \(\mathcal{F}\) is the cokernel of a map \[\Psi_x : \mathcal{O}_{\mathbf{P}^n_k}(-m)^{\oplus N} \to \mathcal{O}_{\mathbf{P}^n_k}^{\oplus r}\] There is a unique ring map \(\tau : A \to k\) such that the base change of \(\Psi\) by the corresponding morphism \(t = \Spec(\tau) : \Spec(k) \to T\) is \(\Psi_x\). This is true because the entries of the \(N \times r\) matrix defining \(\Psi_x\) are homogeneous polynomials \(\sum \lambda_{i, j, E} T_0^{e_0} \ldots T_n^{e_n}\) of degree \(m\) in \(T_0, \ldots, T_n\) with coefficients \(\lambda_{i, j, E} \in k\) and we can set \(\tau(a_{i, j, E}) = \lambda_{i, j, E}\). Then \(t \in T_p \setminus T_{p + 1}\) for some \(p\) and the image of \(t\) under the morphism above is \(x\) as desired.

Lemma

Let \(n \geq 0\), \(r \geq 1\), \(P \in \mathbf{Q}[t]\). The algebraic space \[X = \Quotfunctor^P_{\mathcal{O}^{\oplus r}_{\mathbf{P}^n_\mathbf{Z}}/ \mathbf{P}^n_\mathbf{Z}/\mathbf{Z}}\] is a projective scheme over \(\mathbf{Z}\). If \(X\) is nonempty, then for all sufficiently large \(m\) the universal quotient \(\mathcal{G}\) has the following properties. The module \[\mathcal{E}_m = p_*\mathcal{G}(m),\qquad p : \mathbf{P}^n_X \longrightarrow X,\] is finite locally free of rank \(P(m)\) and its formation commutes with arbitrary base change. The quotient \[H^0(\mathbf{P}^n_\mathbf{Z}, \mathcal{O}(m)^{\oplus r}) \otimes_\mathbf{Z} \mathcal{O}_X \longrightarrow \mathcal{E}_m\] defines a closed immersion into the scheme of locally free quotients of the indicated rank, and the pullback of its Plücker invertible module is \(\det(\mathcal{E}_m)\).

Proof

By Lemma 0DPA, \(X\) is proper over \(\mathbf{Z}\); in particular it is quasi-compact. If \(X\) is empty, the assertion is clear, so assume it is nonempty. On \(\mathbf{P}^n_X\) write the universal sequence as \[0 \longrightarrow \mathcal{K} \longrightarrow \mathcal{O}^{\oplus r} \longrightarrow \mathcal{G} \longrightarrow 0.\] The modules \(\mathcal{K}\) and \(\mathcal{G}\) are of finite presentation and flat over \(X\). Choose \(m_0 \geq 0\) as in Varieties, Lemma 08AG. For every geometric point \(x\) of \(X\), the module \(\mathcal{K}_x\) is \(m_0\)-regular. Hence for \(m \geq m_0\) the higher cohomology of \(\mathcal{K}_x(m)\) and \(\mathcal{G}_x(m)\) vanishes, and \(\mathcal{K}_x(m)\) is globally generated; see Varieties, Lemmas 08A6 and 08A8.

By Derived Categories of Spaces, Lemmas 0CTM and 0E6A, the modules \[\mathcal{W}_m = p_*\mathcal{K}(m) \quad\text{and}\quad \mathcal{E}_m = p_*\mathcal{G}(m)\] are finite locally free, of ranks \[r{n + m \choose n} - P(m)\quad\text{and}\quad P(m),\] respectively, their formation commutes with arbitrary base change, and there is an exact sequence \[0 \longrightarrow \mathcal{W}_m \longrightarrow H^0(\mathbf{P}^n_\mathbf{Z}, \mathcal{O}(m)^{\oplus r}) \otimes_\mathbf{Z} \mathcal{O}_X \longrightarrow \mathcal{E}_m \longrightarrow 0.\] Indeed, the cited open loci contain every geometric point of \(X\) and therefore equal \(X\).

Put \(M = r{n + m \choose n}\) and \(e = P(m)\). The last quotient defines a morphism \[\varphi_m : X \longrightarrow H,\] where \(H = \mathbf{G}(M - e, M)\) if \(0 < e < M\), and where \(H\) is \(\Spec(\mathbf{Z})\) in the two extreme cases. The evaluation map \(p^*\mathcal{W}_m \to \mathcal{K}(m)\) is surjective: this can be checked on geometric fibres, where it follows from regularity. Thus after any base change the quotient of global sections determines \(\mathcal{K}(m)\), and hence determines the universal quotient \(\mathcal{G}\). It follows that \(\varphi_m\) is a monomorphism.

The morphism \(\varphi_m\) is locally of finite type. It is also proper: its graph is a closed immersion because \(H\) is separated over \(\mathbf{Z}\), and the projection \(X \times_\mathbf{Z} H \to H\) is proper. Hence \(\varphi_m\) is a closed immersion by More on Morphisms of Spaces, Lemma 05W8. If \(0 < e < M\), Constructions, Lemma constructions-lemma-grassmannian-pluecker gives a closed immersion of \(H\) into projective space and identifies the pullback of \(\mathcal{O}(1)\) with the determinant of its universal quotient. The same conclusions are immediate in the extreme cases. Thus \(X\) is a projective scheme and the final assertion follows.

Lemma

Let \(B\) be an algebraic space. Let \(X = B \times \mathbf{P}^n_\mathbf{Z}\). Let \(\mathcal{L}\) be the pullback of \(\mathcal{O}_{\mathbf{P}^n}(1)\) to \(X\). Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation. The algebraic space \(\Quotfunctor^P_{\mathcal{F}/X/B}\) parametrizing quotients of \(\mathcal{F}\) having Hilbert polynomial \(P\) with respect to \(\mathcal{L}\) is proper over \(B\).

Proof

The question is étale local over \(B\), see Morphisms of Spaces, Lemma 083R. Thus we may assume \(B\) is an affine scheme. In this case \(\mathcal{L}\) is an ample invertible module on \(X\) (by Constructions, Lemma 01MW and the definition of ample invertible modules in Properties, Definition 01PS). Thus we can find \(r' \geq 0\) and \(r \geq 0\) and a surjection \[\mathcal{O}_X^{\oplus r} \longrightarrow \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes r'}\] by Properties, Proposition 01Q3. By Lemma 0DP7 we may replace \(\mathcal{F}\) by \(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes r'}\) and \(P(t)\) by \(P(t + r')\). By Lemma 0DP5 we obtain a closed immersion \[\Quotfunctor^P_{\mathcal{F}/X/B} \longrightarrow \Quotfunctor^P_{\mathcal{O}_X^{\oplus r}/X/B}\] Since we’ve shown that \(\Quotfunctor^P_{\mathcal{O}_X^{\oplus r}/X/B} \to B\) is proper in Lemma 0DPA we conclude.

Lemma

Let \(f : X \to B\) be a proper morphism of finite presentation of algebraic spaces. Let \(\mathcal{F}\) be a finitely presented \(\mathcal{O}_X\)-module. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module ample on \(X/B\), see Divisors on Spaces, Definition 0D31. The algebraic space \(\Quotfunctor^P_{\mathcal{F}/X/B}\) parametrizing quotients of \(\mathcal{F}\) having Hilbert polynomial \(P\) with respect to \(\mathcal{L}\) is proper over \(B\).

Proof

The question is étale local over \(B\), see Morphisms of Spaces, Lemma 083R. Thus we may assume \(B\) is an affine scheme. Then we can find a closed immersion \(i : X \to \mathbf{P}^n_B\) such that \(i^*\mathcal{O}_{\mathbf{P}^n_B}(1) \cong \mathcal{L}^{\otimes d}\) for some \(d \geq 1\). See Morphisms, Lemma 01VS. Changing \(\mathcal{L}\) into \(\mathcal{L}^{\otimes d}\) and the numerical polynomial \(P(t)\) into \(P(dt)\) leaves \(\Quotfunctor^P_{\mathcal{F}/X/B}\) unaffected; some details omitted. Hence we may assume \(\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_B}(1)\). Then the isomorphism \(\Quotfunctor_{\mathcal{F}/X/B} \to \Quotfunctor_{i_*\mathcal{F}/\mathbf{P}^n_B/B}\) of Lemma 0DP4 induces an isomorphism \(\Quotfunctor^P_{\mathcal{F}/X/B} \cong \Quotfunctor^P_{i_*\mathcal{F}/\mathbf{P}^n_B/B}\). Since \(\Quotfunctor^P_{i_*\mathcal{F}/\mathbf{P}^n_B/B}\) is proper over \(B\) by Lemma 0DPB we conclude.

Proposition

Let \(S\) be a Noetherian scheme, let \(f : X \to S\) be a projective morphism of schemes, let \(\mathcal{L}\) be an \(f\)-very ample invertible module, and let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. For \(P \in \mathbf{Q}[t]\) the algebraic space \[Q = \Quotfunctor^P_{\mathcal{F}/X/S}\] is a projective scheme over \(S\).

Let \(\mathcal{G}\) be the universal quotient on \(X_Q\), and write \(p : X_Q \to Q\). If \(Q\) is nonempty, then for all sufficiently large \(m\) the module \[\mathcal{E}_m = p_*(\mathcal{G} \otimes \mathcal{L}^{\otimes m})\] is finite locally free of rank \(P(m)\) and its formation commutes with arbitrary base change. Zariski locally on \(S\), a quotient of a finite free module onto \(\mathcal{E}_m\) defines a closed immersion of \(Q\) into the corresponding Grassmannian. In particular, \(\det(\mathcal{E}_m)\) is very ample on \(Q/S\).

Proof

The morphism \(Q \to S\) is proper by Lemma 0DPC. Since \(S\) is Noetherian, it has a finite affine open covering \(S = \bigcup U_i\). After refining this covering, Morphisms, Lemma 02NP and properness give closed immersions \[\iota_i : X_{U_i} \longrightarrow \mathbf{P}^{n_i}_{U_i}\] such that \(\mathcal{L}|_{X_{U_i}} = \iota_i^*\mathcal{O}(1)\). After choosing \(a_i \geq 0\) sufficiently large, there is a surjection \[\mathcal{O}_{\mathbf{P}^{n_i}_{U_i}}^{\oplus r_i} \longrightarrow (\iota_i)_*(\mathcal{F}|_{X_{U_i}})(a_i).\] Here we use Properties, Proposition 01Q3.

Put \(P_i(t) = P(t + a_i)\). Lemmas 0DP7, 0DP4, and 0DP5 identify \(Q_{U_i}\) with a closed subspace of \[\Quotfunctor^{P_i}_{ \mathcal{O}^{\oplus r_i}_{\mathbf{P}^{n_i}_{U_i}}/ \mathbf{P}^{n_i}_{U_i}/U_i}.\] The latter is the base change to \(U_i\) of the projective scheme over \(\mathbf{Z}\) constructed in Lemma moduli-lemma-quot-Pn-projective. Consequently \(Q_{U_i}\) is a projective scheme over \(U_i\). Since being a scheme is local on the base, \(Q\) is a scheme.

If \(Q\) is empty, this also proves that it is projective and there is nothing further to show. Assume \(Q\) is nonempty.

The final part of Lemma moduli-lemma-quot-Pn-projective, restricted to the displayed closed subspace, shows that for all sufficiently large \(d\) the module \[p_*\mathcal{G}(a_i + d)\] is finite locally free of rank \(P(a_i + d)\), commutes with arbitrary base change, and is the universal quotient defining a closed immersion into a Grassmannian. As the covering is finite, one integer works for all \(i\) after writing \(m = a_i + d\). These local modules are the restrictions of \(\mathcal{E}_m\), so they glue and prove the assertions about \(\mathcal{E}_m\) and base change.

By Constructions, Lemma constructions-lemma-grassmannian-pluecker, the restriction of \(\det(\mathcal{E}_m)\) to every \(Q_{U_i}\) is very ample. Morphisms, Lemma 01VR shows that \(\det(\mathcal{E}_m)\) is very ample on \(Q/S\). It is therefore ample by Morphisms, Lemma 01VN. Thus \(Q \to S\) is quasi-projective as well as proper, and hence projective by Morphisms, Lemma 0BCL.

Lemma

Under the hypotheses of Proposition moduli-proposition-quot-projective-over-base, there is a decomposition \[\Quotfunctor_{\mathcal{F}/X/S} = \coprod_{P \in \mathbf{Q}[t]} \Quotfunctor^P_{\mathcal{F}/X/S}\] as a disjoint union of projective \(S\)-schemes. In particular, the Quot space is a scheme locally of finite type over \(S\).

Proof

The Hilbert polynomial of a flat family is locally constant on the base. Thus the open and closed subspaces of Remark 0DP6 give the displayed disjoint union. Each term is projective by Proposition moduli-proposition-quot-projective-over-base.

Proposition

Let \(S\) be a Noetherian scheme, let \(X \to S\) be quasi-projective, and let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Choose a factorization \[X \xrightarrow{j} \overline{X} \longrightarrow S\] where \(j\) is an open immersion and \(\overline{X} \to S\) is projective, and choose a very ample invertible module \(\overline{\mathcal{L}}\) on \(\overline{X}/S\). For every numerical polynomial \(P\), the functor \[\Quotfunctor^P_{\mathcal{F}/X/S}\] of quotients flat over the variable base and with proper support is a quasi-projective scheme over \(S\). Consequently \[\Quotfunctor_{\mathcal{F}/X/S} = \coprod_{P \in \mathbf{Q}[t]} \Quotfunctor^P_{\mathcal{F}/X/S}\] is a scheme locally of finite type over \(S\) whose fixed-polynomial components are quasi-projective. Here the Hilbert polynomial is computed with respect to \(j^*\overline{\mathcal{L}}\).

Proof

The factorization exists by Morphisms, Lemma 07RM. Since \(S\) is Noetherian, both \(X\) and \(\overline{X}\) are Noetherian and \(j\) is quasi-compact. Properties, Lemma 0G41 gives a finitely presented, hence coherent, \(\mathcal{O}_{\overline{X}}\)-module \(\overline{\mathcal{F}}\) with \(j^*\overline{\mathcal{F}} \cong \mathcal{F}\).

Lemma 0DP3 gives an open immersion \[\Quotfunctor^P_{\mathcal{F}/X/S} \longrightarrow \Quotfunctor^P_{\overline{\mathcal{F}}/\overline{X}/S}.\] The target is a projective scheme by Proposition moduli-proposition-quot-projective-over-base. It is Noetherian, so the open subscheme on the left is quasi-compact. The open immersion is therefore quasi-projective by Morphisms, Lemma 0B3H, and the composite with the projective morphism to \(S\) is quasi-projective by Morphisms, Lemma 0C4M.

Finally, the Hilbert polynomial is locally constant in a flat family. Remark 0DP6 supplies the open and closed decomposition, and the preceding argument applies to every component.

Proposition

Let \(S\) be a Noetherian scheme, let \(X \to S\) be projective, and fix a relatively very ample invertible module \(\mathcal{O}_X(1)\). A family \(E\) of fibrewise coherent sheaves over \(X/S\) is bounded if and only if the following conditions hold.

1. There is a coherent \(\mathcal{O}_X\)-module \(\mathcal{L}\) such that every member \((s,K,\mathcal{F})\) of \(E\), after a field extension if necessary, is represented by a quotient \[\mathcal{L}_K \longrightarrow \mathcal{F} \longrightarrow 0.\] One may require \(\mathcal{L}\) to be a finite direct sum of negative twists of \(\mathcal{O}_X\).

2. The Hilbert polynomials of the members of \(E\), computed with respect to \(\mathcal{O}_X(1)\), belong to a fixed finite set.

Proof

Suppose first that \((T,\mathcal{G})\) bounds \(E\). The scheme \(T\) is quasi-compact. On each member of a finite affine open covering of \(T\), relative Serre vanishing and relative global generation give a surjection \[\mathcal{O}_{X_T}(-n)^{\oplus N} \longrightarrow \mathcal{G} \longrightarrow 0\] for suitable \(n\) and \(N\), which may depend on the member of the covering. The direct sum of the finitely many corresponding modules on \(X\) gives a single \(\mathcal{L}\) satisfying (1).

By Lemma moduli-lemma-bounded-coherent-elementary, we may replace \(T\) by a finite stratification on which \(\mathcal{G}\) is flat. On each stratum its Hilbert polynomial is locally constant. Each stratum is Noetherian and has only finitely many connected components. Hence only finitely many Hilbert polynomials occur, proving (2).

Conversely, let \(P_1,\ldots,P_m\) be the finite list in (2). For each \(i\), Lemma 0DPC shows that \[\Quotfunctor^{P_i}_{\mathcal{L}/X/S}\] is proper, and in particular of finite type, over \(S\). The finite disjoint union of these Quot spaces is quasi-compact. Choose a quasi-compact scheme mapping surjectively and étale to it. This scheme is of finite type over \(S\), and the pullback of the universal quotient to it bounds all quotients allowed by (1) whose Hilbert polynomial is in the list. It therefore bounds \(E\).

Lemma

Let \(k\) be a field, let \(X\) be a scheme of finite type over \(k\), and let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. For every integer \(r\) there is a unique largest coherent submodule \[\mathcal{N}_r(\mathcal{F}) \subset \mathcal{F}\] whose support has dimension \(< r\). It is the subsheaf whose sections are the sections of \(\mathcal{F}\) having support of dimension \(< r\). Put \[\mathcal{F}_{(r)} = \mathcal{F}/\mathcal{N}_r(\mathcal{F}).\] Then

1. the \(\mathcal{N}_r(\mathcal{F})\) form a finite increasing filtration, with \(\mathcal{N}_r(\mathcal{F})=0\) for \(r \leq 0\) and \(\mathcal{N}_r(\mathcal{F})=\mathcal{F}\) for \(r>\dim(\operatorname{Supp}(\mathcal{F}))\),

2. \[\operatorname{Ass}(\mathcal{F}_{(r)})= \{x\in\operatorname{Ass}(\mathcal{F})\mid \dim(\overline{\{x\}})\geq r\},\]

3. \[\operatorname{Ass}(\mathcal{N}_{r+1}(\mathcal{F})/ \mathcal{N}_r(\mathcal{F}))= \{x\in\operatorname{Ass}(\mathcal{F})\mid \dim(\overline{\{x\}})=r\},\] and

4. formation of \(\mathcal{N}_r(\mathcal{F})\) and \(\mathcal{F}_{(r)}\) commutes with extension of the ground field.

Proof

The sum of two coherent submodules whose supports have dimension \(<r\) has the same property. Since \(\mathcal{F}\) is a Noetherian object, the sum of all such submodules is already a finite sum. This gives the largest submodule \(\mathcal{N}_r(\mathcal{F})\). The same construction after restriction to an open subscheme gives its restriction. A local section belongs to this submodule if and only if the coherent submodule it generates has support of dimension \(<r\). This proves the description by sections and (1).

Set \(\mathcal{N}=\mathcal{N}_r(\mathcal{F})\) and \(\mathcal{Q}=\mathcal{F}/\mathcal{N}\). The maximality of \(\mathcal{N}\) says that \(\mathcal{Q}\) has no nonzero coherent submodule with support of dimension \(<r\). Using the usual properties of associated primes in the exact sequence \[0\longrightarrow\mathcal{N}\longrightarrow\mathcal{F} \longrightarrow\mathcal{Q}\longrightarrow0\] we obtain \[\operatorname{Ass}(\mathcal{Q})= \{x\in\operatorname{Ass}(\mathcal{F})\mid \dim(\overline{\{x\}})\geq r\}.\] Indeed, an associated point of \(\mathcal{Q}\) of smaller dimension would generate a forbidden submodule. Conversely, if \(x\) is associated to \(\mathcal{F}\) and has dimension at least \(r\), then \(x\notin\operatorname{Supp}(\mathcal{N})\), and localization at \(x\) shows that \(x\) is associated to \(\mathcal{Q}\). This proves (2). The associated points of \(\mathcal{N}_{r+1}(\mathcal{F})\) are precisely the associated points of \(\mathcal{F}\) having dimension at most \(r\). Applying (2) to this submodule proves (3).

Let \(K/k\) be a field extension. The support of \(\mathcal{N}_r(\mathcal{F})_K\) has dimension \(<r\). By Algebra, Lemma 0312, the associated points of \(\mathcal{F}_{(r),K}\) lie over associated points of \(\mathcal{F}_{(r)}\); the corresponding irreducible components have the same dimension after extension of the ground field. Hence \(\mathcal{F}_{(r),K}\) has no nonzero coherent submodule with support of dimension \(<r\). Thus \(\mathcal{N}_r(\mathcal{F})_K\) is the largest such submodule of \(\mathcal{F}_K\), proving (4).

Remark

The inequality in the printed hypothesis of [FGA, Exposé 221, Theorem 2.2, pp. 254–255] is reversed. It asks that the coefficients in degrees \(\leq s-1\) be bounded. For a counterexample, let \(X=\mathbf{P}^2_k\) over an algebraically closed field of characteristic zero. For \(d\geq3\), choose a smooth plane curve \(C_d\) of degree \(d\) and a zero-dimensional subscheme \(Z_d\) of length \(d(d-3)/2\). Then \[\mathcal{F}_d=\mathcal{O}_{C_d}\oplus\mathcal{O}_{Z_d}\] is a quotient of \(\mathcal{O}_X^{\oplus2}\) and \[P_{\mathcal{F}_d}(n)= dn+\frac{3d-d^2}{2}+\frac{d(d-3)}2=dn.\] Thus for \(s=1\) all coefficients in degrees \(\leq0\) are fixed, whereas \((\mathcal{F}_d)_{(1)}=\mathcal{O}_{C_d}\) is not a bounded family: its leading coefficient is \(d\). The proof of the cited theorem and its Corollary 2.3 both require coefficients in degrees \(\geq s-1\). We use this corrected inequality below.

Lemma

Let \(S\) be a Noetherian scheme. Let \(X\) and \(Y\) be schemes proper and of finite type over \(S\), and let \(f:X\to Y\) be finite. A family \(E\) of fibrewise coherent sheaves on \(X/S\) is bounded if and only if the family \[\{f_*\mathcal{F}\mid\mathcal{F}\text{ represents a member of }E\}\] is bounded on \(Y/S\).

Proof

If \((T,\mathcal{G})\) bounds \(E\), then \((T,(f_T)_*\mathcal{G})\) bounds the pushforwards. Here we use that a finite morphism is affine, that its pushforward preserves coherent modules, and that this pushforward commutes with base change.

Conversely, choose \((T,\mathcal{H})\) bounding the pushforwards and put \[\mathcal{A}=(f_T)_*\mathcal{O}_{X_T}.\] After a finite stratification of \(T\), we may assume that the coherent modules which occur below are flat over \(T\). The functor of homomorphisms \[\mathcal{A}\otimes\mathcal{H}\longrightarrow\mathcal{H}\] is then represented by an affine scheme of finite presentation over \(T\), as in the proof of Proposition moduli-proposition-bounded-coherent-morphisms-extensions. The unit and associativity identities cut out a closed subscheme \(M\) of this Hom scheme. Thus \(M\) parametrizes all \(\mathcal{A}\)-module structures on the fibres of \(\mathcal{H}\). The universal \(\mathcal{A}\)-module on \(Y_M\) corresponds to a coherent module on \(X_M\) by Morphisms, Lemma 01SB. Every module in \(E\) occurs as a fibre after extending the ground field and transporting its \(f_*\mathcal{O}_X\)-module structure to the corresponding fibre of \(\mathcal{H}\). Since \(M\) is of finite type over \(S\), the universal module on \(X_M\) bounds \(E\).

Lemma

Let \(S\) be a Noetherian scheme, let \(X\to S\) be projective, and fix a relatively very ample invertible module \(\mathcal{O}_X(1)\). Fix integers \(r,d\geq0\). The structure sheaves of the reduced closed subschemes \[Z\subset X_K\] which are pure of dimension \(r\) and have degree at most \(d\) form a bounded family over \(X/S\).

Proof

Choose a closed immersion \(X\to\mathbf{P}^N_S\) defined by \(\mathcal{O}_X(1)\). For each \(e\leq d\), the Chow-coordinate construction gives a projective parameter scheme for effective \(r\)-cycles of degree \(e\) in \(\mathbf{P}^N\), together with its incidence subscheme. Restricting to the closed locus of cycles supported on \(X\) and taking the product with \(S\) gives a finite-type parameter scheme over \(S\). A reduced pure-dimensional \(Z\subset X_K\) determines the cycle which is the sum of its irreducible components with coefficient \(1\), and the reduction of the corresponding incidence fibre is exactly \(Z\). Lemma moduli-lemma-bounded-fibrewise-reductions, applied to the finitely many incidence families for \(0\leq e\leq d\), bounds their reduced fibres and hence all the stated structure sheaves.

Lemma

Let \(S\) be a Noetherian scheme, let \(X\to S\) be projective with fibres of dimension at most \(r\), and fix a relatively very ample invertible module \(\mathcal{O}_X(1)\). Let \(\mathcal{L}\) be a coherent \(\mathcal{O}_X\)-module, and let \(E\) be a family of quotient modules of the modules \(\mathcal{L}_K\). If \(r\geq1\), write \[P_{\mathcal{F}}(n)= a_{\mathcal{F}}\frac{n^r}{r!}+ b_{\mathcal{F}}\frac{n^{r-1}}{(r-1)!}+ \text{terms of degree }<r-1.\] Then the \(a_{\mathcal{F}}\) are bounded and the \(b_{\mathcal{F}}\) are bounded below. If the \(b_{\mathcal{F}}\) are bounded above, then the family of \(\mathcal{F}_{(r)}\) is bounded. If \(r=0\), then \(E\) is bounded.

Proof

Embed \(X\) into \(\mathbf{P}^N_S\) using \(\mathcal{O}_X(1)\). The linear projections \(\mathbf{P}^N\dashrightarrow\mathbf{P}^r\) are parametrized by a scheme of finite type. The condition that a projection be defined on \(X\) and induce a finite morphism on \(X\) is open, and the standard generic-projection argument shows that this open meets every geometric fibre over \(S\). Hence, after replacing \(S\) by a finite-type parameter scheme which meets every geometric fibre, we may assume there is a finite morphism \[f:X\longrightarrow\mathbf{P}^r_S, \qquad \mathcal{O}_X(1)=f^*\mathcal{O}_{\mathbf{P}^r_S}(1).\] This replacement is harmless for boundedness in Definition moduli-definition-bounded-coherent, since every member is allowed a ground field extension.

Finite pushforward preserves the Hilbert polynomial by the projection formula and the vanishing of higher direct images. Since a finite morphism preserves dimensions of supports, the description by sections in Lemma moduli-lemma-coherent-dimension-truncation gives \[f_*(\mathcal{F}_{(r)})=(f_*\mathcal{F})_{(r)}.\] Lemma moduli-lemma-bounded-finite-pushforward therefore reduces the question to \(X=\mathbf{P}^r_S\). After taking a finite affine covering of \(S\), replacing it by the corresponding disjoint union, and twisting by a fixed integer, we may also assume that every member is a quotient of \(\mathcal{O}_{\mathbf{P}^r_K}^{\oplus q}\) for one fixed \(q\). Twisting changes \(b_{\mathcal{F}}\) by a fixed multiple of \(a_{\mathcal{F}}\), so it does not change any of the boundedness assertions.

The leading coefficient \(a_{\mathcal{F}}\) is the generic rank of \(\mathcal{F}\) and hence \[0\leq a_{\mathcal{F}}\leq q.\] The quotient \(\mathcal{G}=\mathcal{F}_{(r)}\) is either zero or a torsion-free module of rank \(a=a_{\mathcal{F}}\) on \(\mathbf{P}^r_K\). Assume it is nonzero. On the complement of a closed subset of codimension at least \(2\) it is locally free. Since projective space is regular and locally factorial, the rank-one reflexive module \[\det(\mathcal{G})=(\wedge^a\mathcal{G})^{**}\] is invertible. Divisors, Lemma 0BXJ gives a unique integer \(c\) such that \(\det(\mathcal{G})\cong\mathcal{O}(c)\). The quotient map from \(\mathcal{O}^{\oplus q}\) gives a nonzero homomorphism \[s:\wedge^a\mathcal{O}^{\oplus q}\longrightarrow\mathcal{O}(c).\] Consequently \(c\geq0\).

Modulo coherent modules supported in codimension at least \(2\), the class of a torsion-free module on the regular scheme \(\mathbf{P}^r_K\) is determined by its rank and determinant. Thus the Hilbert polynomials of \(\mathcal{G}\) and \(\mathcal{O}^{\oplus(a-1)}\oplus\mathcal{O}(c)\) differ in degree at most \(r-2\). Expanding \[P_{\mathcal{O}(c)}(n)={n+c+r\choose r}\] shows that \[b_{\mathcal{G}}=a\frac{r+1}{2}+c.\] The coefficient of degree \(r-1\) in \(P_{\mathcal{N}_r(\mathcal{F})}\) is nonnegative. Hence \(b_{\mathcal{F}}\geq b_{\mathcal{G}}\geq0\), which proves the asserted lower bound.

If the \(b_{\mathcal{F}}\) are bounded above, then so are the \(b_{\mathcal{G}}\), and therefore only finitely many pairs \((a,c)\) occur. For fixed \((a,c)\) the homomorphisms \(s\) above form a finite-dimensional affine parameter space. Moreover, \(\mathcal{G}\) is the image of the homomorphism \[\mathcal{O}^{\oplus q}\longrightarrow \SheafHom(\wedge^{a-1}\mathcal{O}^{\oplus q},\mathcal{O}(c)), \qquad v\longmapsto(w\mapsto s(v\wedge w)).\] Indeed, this is true where \(\mathcal{G}\) is locally free, and both sides are torsion-free. Proposition moduli-proposition-bounded-coherent-morphisms-extensions now shows that all these images form a bounded family. This proves the assertion for \(r\geq1\).

If \(r=0\), finite projection reduces to \(\mathbf{P}^0_S=S\). The dimensions of quotients of one fixed finite module are bounded, so only finitely many constant Hilbert polynomials occur. Proposition moduli-proposition-bounded-coherent-Hilbert-polynomials finishes the proof.

Proposition

Let \(S\) be a Noetherian scheme, let \(X\to S\) be projective, and fix a relatively very ample invertible module \(\mathcal{O}_X(1)\). Let \(E\) be a family satisfying condition (1) of Proposition moduli-proposition-bounded-coherent-Hilbert-polynomials. Suppose that every member has support of dimension at most \(r\). For \(r\geq1\), write \[P_{\mathcal{F}}(n)= a_{\mathcal{F}}\frac{n^r}{r!}+ b_{\mathcal{F}}\frac{n^{r-1}}{(r-1)!}+ \text{terms of degree }<r-1.\] If the \(a_{\mathcal{F}}\) are bounded, then the \(b_{\mathcal{F}}\) are bounded below. If in addition the \(b_{\mathcal{F}}\) are bounded above, then the family of \(\mathcal{F}_{(r)}\) is bounded. For \(r=0\), boundedness of the constant coefficients implies that \(E\) is bounded.

Proof

We may extend the ground fields and assume they are algebraically closed. For a member \(\mathcal{F}\), let \[Z_{\mathcal{F}}= (\operatorname{Supp}(\mathcal{F}_{(r)}))_{\mathrm{red}}.\] By Varieties, Lemma 0BEN, \[a_{\mathcal{F}}= \sum_Z \mathop{\rm length}_{\mathcal{O}_{X_K,\xi_Z}} ((\mathcal{F}_{(r)})_{\xi_Z})\deg(Z),\] where the sum is over the irreducible components of \(Z_{\mathcal{F}}\). Thus boundedness of \(a_{\mathcal{F}}\) bounds both the degree of \(Z_{\mathcal{F}}\) and every displayed generic length. Lemma moduli-lemma-bounded-reduced-equidimensional-degree shows that the \(Z_{\mathcal{F}}\) form a bounded family.

Choose an integer \(A \geq 1\) bounding all the generic lengths, and let \(\mathcal{J}_{\mathcal{F}}\) be the ideal of \(Z_{\mathcal{F}}\) in \(X_K\). Then \[\mathcal{J}_{\mathcal{F}}^A\mathcal{F}_{(r)}=0.\] Indeed, at every associated point of \(\mathcal{F}_{(r)}\) this follows from the elementary fact that the \(A\)th power of the maximal ideal annihilates a module of length at most \(A\). A nonzero submodule \(\mathcal{J}_{\mathcal{F}}^A\mathcal{F}_{(r)}\) would have an associated point among those of \(\mathcal{F}_{(r)}\), giving a contradiction.

Choose a finite-type family \(Z\subset X_T\) which contains all the \(Z_{\mathcal{F}}\) as geometric fibres. The closed subscheme \[Y=V(\mathcal{I}_Z^A)\subset X_T\] has fibres of dimension at most \(r\), and every \(\mathcal{F}_{(r)}\) is a module on a geometric fibre of \(Y/T\). It is a quotient of the restriction to that fibre of the fixed coherent source in condition (1). Apply Lemma moduli-lemma-bounded-top-dimensional-small-ambient to \(Y/T\). This proves the lower bound for the \(b_{\mathcal{F}}\) and, when they are bounded above, the boundedness of the \(\mathcal{F}_{(r)}\). The same argument and the last sentence of that lemma handle \(r=0\).

Proposition

Let \(S\) be a Noetherian scheme, let \(X\to S\) be projective, and fix a relatively very ample invertible module \(\mathcal{O}_X(1)\). Let \(E\) be a family satisfying condition (1) of Proposition moduli-proposition-bounded-coherent-Hilbert-polynomials, and let \(s\) be an integer. Suppose that the coefficients in degrees at least \(s-1\) of the Hilbert polynomials of members of \(E\) are bounded. Then the family of \(\mathcal{F}_{(s)}\) is bounded. Moreover, the coefficients in degree \(s-2\) are bounded below, where coefficients in negative degrees are understood to be zero.

Proof

Remark moduli-remark-FGA-Hilbert-coefficient-inequality explains the correction of the inequality in the source. If \(s\leq0\), all coefficients in nonnegative degrees are bounded. Numerical polynomials of bounded degree form a lattice in their coefficient space, so only finitely many Hilbert polynomials occur; Proposition moduli-proposition-bounded-coherent-Hilbert-polynomials applies. We may therefore assume \(s\geq1\).

We prove both assertions simultaneously by induction on an upper bound \(r\) for the dimensions of the supports. If \(r\leq s-2\), then \(\mathcal{F}_{(s)}=0\) and the coefficient in degree \(s-2\) is either zero or a leading coefficient, hence is nonnegative. If \(r=s-1\), then again \(\mathcal{F}_{(s)}=0\). When \(s=1\) the coefficient in degree \(s-2=-1\) is zero by convention. When \(s\geq2\), Proposition moduli-proposition-bounded-top-dimensional-pieces, applied with \(r=s-1\), gives the required lower bound from the bounded coefficient in degree \(s-1\).

Assume \(r\geq s\) and the result is known with upper bound \(r-1\). The coefficients \(a_{\mathcal{F}}\) and \(b_{\mathcal{F}}\) in degrees \(r\) and \(r-1\) are bounded. Proposition moduli-proposition-bounded-top-dimensional-pieces shows that the family of \(\mathcal{F}_{(r)}\) is bounded. Let \(\mathcal{L}_K\to\mathcal{F}\) be the fixed-source quotient from condition (1). The kernels of the composites \[\mathcal{L}_K\longrightarrow\mathcal{F} \longrightarrow\mathcal{F}_{(r)}\] form a bounded family by Proposition moduli-proposition-bounded-coherent-morphisms-extensions. Each \(\mathcal{N}_r(\mathcal{F})\) is a quotient of the corresponding kernel, so the family of the \(\mathcal{N}_r(\mathcal{F})\) has a common coherent quotient source. Since \[P_{\mathcal{N}_r(\mathcal{F})}=P_{\mathcal{F}}-P_{\mathcal{F}_{(r)}}\] and a bounded family has only finitely many Hilbert polynomials, the coefficients in degrees at least \(s-1\) on the left are bounded. The induction hypothesis therefore applies to the \(\mathcal{N}_r(\mathcal{F})\).

The dimension filtration gives a short exact sequence \[0\longrightarrow (\mathcal{N}_r(\mathcal{F}))_{(s)}\longrightarrow \mathcal{F}_{(s)}\longrightarrow \mathcal{F}_{(r)}\longrightarrow0.\] Its left and right terms form bounded families, so Proposition moduli-proposition-bounded-coherent-morphisms-extensions bounds the middle terms. The coefficient in degree \(s-2\) of the left term is bounded below by induction, and all coefficients of the right term are bounded. Additivity of Hilbert polynomials proves the asserted lower bound for \(\mathcal{F}\) and completes the induction.

Lemma

In the situation of Proposition moduli-proposition-bounded-coherent-Hilbert-polynomials, suppose there are integers \(s\leq r\) such that every associated point of every member \(\mathcal{F}\) has closure of dimension between \(s\) and \(r\). Then, in addition to the common-source condition, it is enough to bound the coefficients of \(P_{\mathcal{F}}\) in degrees from \(s-1\) through \(r\) in order to conclude that the family is bounded.

Proof

There are no coefficients in degrees greater than \(r\), and Lemma moduli-lemma-coherent-dimension-truncation gives \(\mathcal{F}_{(s)}=\mathcal{F}\). Thus Proposition moduli-proposition-bounded-dimension-truncations applies. The converse follows from Proposition moduli-proposition-bounded-coherent-Hilbert-polynomials.

Lemma

Let \(f : X \to B\) be a separated morphism of finite presentation of algebraic spaces. Let \(\mathcal{F}\) be a finitely presented \(\mathcal{O}_X\)-module. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module ample on \(X/B\), see Divisors on Spaces, Definition 0D31. The algebraic space \(\Quotfunctor^P_{\mathcal{F}/X/B}\) parametrizing quotients of \(\mathcal{F}\) having Hilbert polynomial \(P\) with respect to \(\mathcal{L}\) is separated of finite presentation over \(B\).

Proof

We have already seen that \(\Quotfunctor_{\mathcal{F}/X/B} \to B\) is separated and locally of finite presentation, see Lemma 0DM3. Thus it suffices to show that the open subspace \(\Quotfunctor^P_{\mathcal{F}/X/B}\) of Remark 0DP6 is quasi-compact over \(B\).

The question is étale local on \(B\) (Morphisms of Spaces, Lemma 03KG). Thus we may assume \(B\) is affine.

Assume \(B = \Spec(\Lambda)\). Write \(\Lambda = \colim \Lambda_i\) as the colimit of its finite type \(\mathbf{Z}\)-subalgebras. Then we can find an \(i\) and a system \(X_i, \mathcal{F}_i, \mathcal{L}_i\) as in the lemma over \(B_i = \Spec(\Lambda_i)\) whose base change to \(B\) gives \(X, \mathcal{F}, \mathcal{L}\). This follows from Limits of Spaces, Lemmas 07SK (to find \(X_i\)), 07V7 (to find \(\mathcal{F}_i\)), 0D2X (to find \(\mathcal{L}_i\)), and 084T (to make \(X_i\) separated). Because \[\Quotfunctor_{\mathcal{F}/X/B} = B \times_{B_i} \Quotfunctor_{\mathcal{F}_i/X_i/B_i}\] and similarly for \(\Quotfunctor^P_{\mathcal{F}/X/B}\) we reduce to the case discussed in the next paragraph.

Assume \(B\) is affine and Noetherian. We may replace \(\mathcal{L}\) by a positive power, see Lemma 0DP8. Thus we may assume there exists an immersion \(i : X \to \mathbf{P}^n_B\) such that \(i^*\mathcal{O}_{\mathbf{P}^n}(1) = \mathcal{L}\). By Morphisms, Lemma 01RG there exists a closed subscheme \(X' \subset \mathbf{P}^n_B\) such that \(i\) factors through an open immersion \(j : X \to X'\). By Properties, Lemma 0G41 there exists a finitely presented \(\mathcal{O}_{X'}\)-module \(\mathcal{G}\) such that \(j^*\mathcal{G} = \mathcal{F}\). Thus we obtain an open immersion \[\Quotfunctor_{\mathcal{F}/X/B} \longrightarrow \Quotfunctor_{\mathcal{G}/X'/B}\] by Lemma 0DP3. Clearly this open immersion sends \(\Quotfunctor^P_{\mathcal{F}/X/B}\) into \(\Quotfunctor^P_{\mathcal{G}/X'/B}\). Now \(\Quotfunctor^P_{\mathcal{G}/X'/B}\) is proper over \(B\) by Lemma 0DPC. Therefore it is Noetherian and since any open of a Noetherian algebraic space is quasi-compact we win.

Properties of the Hilbert functor

Let \(f : X \to B\) be a morphism of algebraic spaces which is separated and of finite presentation. Then \(\Hilbfunctor_{X/B}\) is an algebraic space locally of finite presentation over \(B\). See Quot, Proposition 0D01.

Lemma

The diagonal of \(\Hilbfunctor_{X/B} \to B\) is a closed immersion of finite presentation.

Proof

In Quot, Lemma 0D00 we have seen that \(\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}\). Hence this follows from Lemma 0DM2.

Lemma

The morphism \(\Hilbfunctor_{X/B} \to B\) is separated and locally of finite presentation.

Proof

To check \(\Hilbfunctor_{X/B} \to B\) is separated we have to show that its diagonal is a closed immersion. This is true by Lemma 0DM6. The second statement is part of Quot, Proposition 0D01.

Lemma

Assume \(X \to B\) is proper as well as of finite presentation. Then \(\Hilbfunctor_{X/B} \to B\) satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition 03IX).

Proof

In Quot, Lemma 0D00 we have seen that \(\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}\). Hence this follows from Lemma 0DM4.

Lemma

Let \(B\) be an algebraic space. Let \(\pi : X \to Y\) be an open immersion of algebraic spaces which are separated and of finite presentation over \(B\). Then \(\pi\) induces an open immersion \(\Hilbfunctor_{X/B} \to \Hilbfunctor_{Y/B}\).

Proof

Omitted. Hint: If \(Z \subset X_T\) is a closed subscheme which is proper over \(T\), then \(Z\) is also closed in \(Y_T\). Thus we obtain the transformation \(\Hilbfunctor_{X/B} \to \Hilbfunctor_{Y/B}\). If \(Z \subset Y_T\) is an element of \(\Hilbfunctor_{Y/B}(T)\) and for \(t \in T\) we have \(|Z_t| \subset |X_t|\), then the same is true for \(t' \in T\) in a neighbourhood of \(t\).

Lemma

Let \(B\) be an algebraic space. Let \(\pi : X \to Y\) be a closed immersion of algebraic spaces which are separated and of finite presentation over \(B\). Then \(\pi\) induces a closed immersion \(\Hilbfunctor_{X/B} \to \Hilbfunctor_{Y/B}\).

Proof

Since \(\pi\) is a closed immersion, it is immediate that given a closed subscheme \(Z \subset X_T\), we can view \(Z\) as a closed subscheme of \(X_T\). Thus we obtain the transformation \(\Hilbfunctor_{X/B} \to \Hilbfunctor_{Y/B}\). This transformation is immediately seen to be a monomorphism. To prove that it is a closed immersion, you can use Lemma 0DP5 for the map \(\mathcal{O}_Y \to \mathcal{O}_X\) and the identifications \(\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}\), \(\Hilbfunctor_{Y/B} = \Quotfunctor_{\mathcal{O}_Y/Y/B}\) of Quot, Lemma 0D00.

Remark

Let \(f : X \to B\) be as in the introduction to this section. Let \(I\) be a set and for \(i \in I\) let \(E_i \in D(\mathcal{O}_X)\) be perfect. Let \(P : I \to \mathbf{Z}\) be a function. Recall that \(\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}\), see Quot, Lemma 0D00. Thus we can define \[\Hilbfunctor^P_{X/B} = \Quotfunctor^P_{\mathcal{O}_X/X/B}\] where \(\Quotfunctor^P_{\mathcal{O}_X/X/B}\) is as in Remark 0DP6. The morphism \[\Hilbfunctor^P_{X/B} \longrightarrow \Hilbfunctor_{X/B}\] is a flat closed immersion which is an open and closed immersion for example if \(I\) is finite, or \(B\) is locally Noetherian, or \(I = \mathbf{Z}\) and \(E_i = \mathcal{L}^{\otimes i}\) for some invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\). In the last case we sometimes use the notation \(\Hilbfunctor^{P, \mathcal{L}}_{X/B}\).

Lemma

Let \(f : X \to B\) be a proper morphism of finite presentation of algebraic spaces. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module ample on \(X/B\), see Divisors on Spaces, Definition 0D31. The algebraic space \(\Hilbfunctor^P_{X/B}\) parametrizing closed subschemes having Hilbert polynomial \(P\) with respect to \(\mathcal{L}\) is proper over \(B\).

Proof

Recall that \(\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}\), see Quot, Lemma 0D00. Thus this lemma is an immediate consequence of Lemma 0DPC.

Lemma

Let \(f : X \to B\) be a separated morphism of finite presentation of algebraic spaces. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module ample on \(X/B\), see Divisors on Spaces, Definition 0D31. The algebraic space \(\Hilbfunctor^P_{X/B}\) parametrizing closed subschemes having Hilbert polynomial \(P\) with respect to \(\mathcal{L}\) is separated of finite presentation over \(B\).

Proof

Recall that \(\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}\), see Quot, Lemma 0D00. Thus this lemma is an immediate consequence of Lemma 0DPD.

Proposition

Let \(S\) be a Noetherian scheme, let \(X \to S\) be quasi-projective, let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module, and fix a Hilbert polynomial \(P\) as in Proposition moduli-proposition-quot-quasi-projective-over-base. Write \[Q = \Quotfunctor^P_{\mathcal{F}/X/S}\] and let \(\mathcal{G}\) be the universal quotient on \(X_Q\). If \(D \subset \mathbf{Z}\) is finite, then the locus in \(Q\) where the dimensions of all prime cycles associated to the fibre of \(\mathcal{G}\) belong to \(D\) is open.

If \(\mathcal{F} = \mathcal{O}_X\), write \(H = \Hilbfunctor^P_{X/S}\) and let \(Z \subset X_H\) be the universal closed subscheme. The following define open subfunctors of \(H\):

1. the locus where \(Z \to H\) is smooth on every point of the fibre;

2. the locus where the fibre of \(Z \to H\) is geometrically normal;

3. if \(X \to S\) is flat, the locus where \(Z_h \to X_h\) is a local complete intersection for every \(h \in H\).

Finite intersections of these loci represent simultaneous conditions.

Proof

By Proposition moduli-proposition-quot-quasi-projective-over-base, \(Q\) is a locally Noetherian scheme. The support of \(\mathcal{G}\) is proper over \(Q\), and \(\mathcal{G}\) is finitely presented and flat over \(Q\). The first claim is therefore exactly the openness theorem of [EGA4, Chapter IV, Theorem 12.2.1(i)].

The universal family \(Z \to H\) is proper, flat, and of finite presentation. The smooth and geometrically normal fibre loci are open by [EGA4, Chapter IV, Theorem 12.2.4(iii), (iv)]. These properties commute with field extension, so the resulting opens have the asserted functorial meaning. For the final locus, apply More on Morphisms of Spaces, Lemma 06CE to \[\xymatrix{ Z \ar[rr] \ar[rd] & & X_H \ar[ld] \\ & H. }\] The first vertical morphism is flat, proper, and of finite presentation, and the second is flat and of finite presentation. Intersections of open subschemes are open.

Proposition

Let \(S\) be a Noetherian scheme and let \(X \to S\) be flat and quasi-projective. Fix a Hilbert polynomial \(P\) and put \[H = \Hilbfunctor^P_{X/S}.\] Let \(h \in H\) be a point with residue field \(k\), let \(s \in S\) be its image, and let \(Y \subset X_k\) be the corresponding closed subscheme. Assume that \(Y \to X_k\) is a local complete intersection. Then \[T_{H_s,h} = H^0(Y, \mathcal{N}_{Y/X_k})\] canonically. If \(H^1(Y, \mathcal{N}_{Y/X_k}) = 0\), then \(H \to S\) is smooth at \(h\).

Proof

The scheme \(H\) is locally of finite type over \(S\) by Proposition moduli-proposition-quot-quasi-projective-over-base. It is the fixed-polynomial part of the Quot functor for \(\mathcal{O}_X\). If \(\mathcal{I}\) is the ideal of \(Y\) in \(X_k\), Quot, Lemma quot-lemma-cotangent-space-quot gives \[T_{H_s,h} = \Hom_{\mathcal{O}_{X_k}}(\mathcal{I}, \mathcal{O}_Y).\] Every such homomorphism annihilates \(\mathcal{I}^2\). Since the conormal sheaf \(\mathcal{I}/\mathcal{I}^2\) is finite locally free on \(Y\), the right hand side is \(H^0(Y, \mathcal{N}_{Y/X_k})\) by the definition of the normal sheaf.

Assume the asserted \(H^1\) vanishes. We use the pointwise infinitesimal lifting criterion in More on Morphisms, Lemma 02HX. Let \(A' \to A\) be a small extension of local Artinian rings with residue field \(k\) and kernel \(J\), and consider a solid diagram in that criterion centered at \(h\). The map \(\Spec(A) \to H\) corresponds to a closed subscheme \(Y_A \subset X_A\) flat over \(A\).

The local-complete-intersection subfunctor is open by Proposition moduli-proposition-quot-open-fibre-conditions. Since \(\Spec(A)\) has one point, the map \(\Spec(A) \to H\) factors through this open and \(Y_A \to X_A\) is a local complete intersection. Locally on \(X_A\), choose a quasi-regular sequence defining \(Y_A\) and lift its members to \(X_{A'}\). The morphism \(X_{A'} \to \Spec(A')\) is flat and of finite presentation. More on Algebra, Lemma 0CEQ shows that the lifted equations define a flat lift over \(A'\). Thus the quotient \(\mathcal{O}_{X_A} \to \mathcal{O}_{Y_A}\) has lifts locally on \(X_A\).

Apply Quot, Lemma quot-lemma-quotient-lifts-torsor. Because \(A' \to A\) is a small extension, \(J\) is a \(k\)-vector space, and its obstruction sheaf is canonically the pushforward from \(Y\) of \[\mathcal{N}_{Y/X_k} \otimes_k J.\] Consequently its first cohomology group is \[H^1(Y, \mathcal{N}_{Y/X_k}) \otimes_k J = 0.\] The lemma therefore gives a global flat lift \(Y_{A'} \subset X_{A'}\). Finite presentation and properness are unchanged by the nilpotent thickening, so this is an \(A'\)-point of \(H\) lifting the given \(A\)-point. The pointwise infinitesimal lifting criterion now proves that \(H \to S\) is smooth at \(h\).

Lemma

Let \(S\) be a Noetherian scheme and let \(X \to S\) be quasi-projective. For \(r \geq 0\), the functor of closed subschemes of \(X_T\) which are finite étale of rank \(r\) over \(T\) is represented by an open subscheme of \(\Hilbfunctor^r_{X/S}\) and hence by a quasi-projective \(S\)-scheme.

Proof

The constant Hilbert polynomial \(r\) says that every fibre of the universal family is zero dimensional of length \(r\). On the smooth open from Proposition moduli-proposition-quot-open-fibre-conditions, the universal family is smooth of relative dimension zero, hence étale by definition. It is proper with finite fibres, hence finite by More on Morphisms, Lemma 02LS, and its finite locally free rank is \(r\). The converse is immediate. Quasi-projectivity follows from Proposition moduli-proposition-quot-quasi-projective-over-base and Quot, Lemma 0D00.

Zero cycles and the Grothendieck–Deligne norm map

The symmetric product is the expected parameter space for effective zero cycles when the morphism to the base is flat. Without flatness, divided powers give a better behaved parameter space: they commute with arbitrary base change. We record the comparison before constructing the norm map.

Theorem

Let \(X \to S\) be a separated morphism of algebraic spaces and let \(d \geq 0\). There is a separated algebraic space \[\Gamma^d(X/S) \longrightarrow S\] parametrizing effective zero cycles of degree \(d\), with the following properties.

1. Formation of \(\Gamma^d(X/S)\) commutes with arbitrary base change on \(S\).

2. If \(S = \Spec(A)\) and \(X = \Spec(B)\), then \[\Gamma^d(X/S) = \Spec(\Gamma_A^d(B)),\] where \(\Gamma_A^d(B)\) represents homogeneous multiplicative polynomial laws of degree \(d\) from \(B\).

3. Addition of cycles gives a morphism \[\Psi : (X/S)^d \longrightarrow \Gamma^d(X/S).\] The locus \(\Gamma^d(X/S)_{\mathrm{nd}}\) where the corresponding geometric cycle has \(d\) distinct points is open, and \(\Psi\) is finite étale of degree \(d!\) over this locus.

4. The uniform categorical quotient \[\operatorname{Sym}^d_S(X) = (X/S)^d/\mathfrak S_d\] exists as a separated algebraic space. There is a canonical morphism \[\operatorname{Sym}^d_S(X) \longrightarrow \Gamma^d(X/S)\] which is a universal homeomorphism with trivial residue field extensions. It is an isomorphism if \(X \to S\) is flat, and it is an isomorphism over \(\Gamma^d(X/S)_{\mathrm{nd}}\) without a flatness assumption.

For an \(A\)-algebra \(C\), a homogeneous multiplicative polynomial law \(F : B \to C\) of degree \(d\) is a collection of maps \[F_{A'} : B \otimes_A A' \longrightarrow C \otimes_A A'\] natural in the \(A\)-algebra \(A'\), multiplicative and unital, and satisfying \(F_{A'}(a x) = a^dF_{A'}(x)\).

Proof

The affine description is the universal property of the algebra of divided powers. It also proves arbitrary base change in the affine case. The global space is obtained from the affine spaces of divided powers using étale presentations. Addition of polynomial laws gives \(\Psi\) and shows that the nondegenerate locus is open. The quotient morphism is integral and has the same geometric fibres as \(\Psi\), which proves the assertion about its topological and residue-field behaviour. In the affine flat case the comparison is induced by the isomorphism between divided powers and symmetric tensors; the general flat case follows from the étale-local construction.

Let \(f : X \to S\) be separated and let \(\mathcal{F}\) be a finitely presented quasi-coherent \(\mathcal{O}_X\)-module. We write \(\Quotfunctor^d_{\mathcal{F}/X/S}\) for the subfunctor of the Quot functor whose objects over \(T \to S\) are quotients \[\mathcal{F}_T \longrightarrow \mathcal{G}\] such that \(\mathcal{G}\) is finitely presented and flat over \(T\), its support is finite over \(T\), and \(p_*\mathcal{G}\) is locally free of rank \(d\), where \(p : X_T \to T\). Similarly, \(\Hilbfunctor^d_{X/S}\) denotes the functor of closed subspaces \(Z \subset X_T\) finite locally free of degree \(d\) over \(T\).

Proposition

There are canonical transformations \[\Quotfunctor^d_{\mathcal{F}/X/S} \longrightarrow \Gamma^d(X/S) \quad\text{and}\quad \Hilbfunctor^d_{X/S} \longrightarrow \Gamma^d(X/S).\] They commute with arbitrary base change on \(S\). The second transformation, obtained from the first by taking \(\mathcal{F} = \mathcal{O}_X\), is called the Grothendieck–Deligne norm map.

If \(k\) is an algebraically closed field and \(\mathcal{G}\) is a \(k\)-point of the first functor, then its image is \[\sum_{x \in \operatorname{Supp}(\mathcal{G})} \operatorname{length}_{\mathcal{O}_{X_k,x}}(\mathcal{G}_x)[x].\]

Proof

The construction is local on the base. Suppose first that \(T = \Spec(A)\) and that the finite support of \(\mathcal{G}\) is contained in an affine \(\Spec(B) \subset X_T\). Multiplication gives an \(A\)-algebra homomorphism \[B \longrightarrow \operatorname{End}_A(p_*\mathcal{G}).\] For every \(A\)-algebra \(A'\), take the determinant of the induced endomorphism of the rank \(d\) locally free \(A'\)-module \(p_*\mathcal{G} \otimes_A A'\). These determinants form a homogeneous multiplicative polynomial law of degree \(d\). By Theorem moduli-theorem-divided-power-zero-cycles, it determines a morphism \(T \to \Gamma^d(X/S)\). Determinants commute with arbitrary base change. The construction is independent of the affine neighbourhood and glues étale locally, giving the asserted transformations.

For the formula over \(k\), decompose \(p_*\mathcal{G}\) into the direct sum of its local factors. The determinant of multiplication by a function is the product of its values at the support points, each raised to the length of the corresponding local factor. This is exactly the displayed cycle.

Remark

Assume that \(X \to S\) is flat and let \(Y \to S\) be an algebraic space. A symmetric morphism \(u : (X/S)^d \to Y\) factors uniquely through the categorical quotient \(\operatorname{Sym}^d_S(X)\). Since \(\operatorname{Sym}^d_S(X) = \Gamma^d(X/S)\) by Theorem moduli-theorem-divided-power-zero-cycles, composition with Proposition moduli-proposition-grothendieck-deligne-norm gives the operation customarily denoted \(\pi^u\). If \(X = Y\) is a commutative monoid and \(u\) is multiplication, this specializes to the usual norm operation. For a nonflat \(X/S\), the canonical and base-change-compatible target of the determinant construction is \(\Gamma^d(X/S)\), which need not equal the symmetric product.

Proposition

For every separated morphism \(X \to S\) there is a canonical transformation \[\Hilbfunctor^d_{X/S} \longrightarrow \operatorname{Sym}^d_S(X)\] whose composition with \(\operatorname{Sym}^d_S(X) \to \Gamma^d(X/S)\) is the Grothendieck–Deligne norm map.

Proof

Let \(Z \subset X_T\) be finite locally free of degree \(d\) over \(T\). Since \(Z \to T\) is flat, Theorem moduli-theorem-divided-power-zero-cycles gives \[\operatorname{Sym}^d_T(Z) \xrightarrow{\sim} \Gamma^d(Z/T).\] The norm of \(\mathcal{O}_Z\) is a \(T\)-point of the right hand side. Transport it to the symmetric product and then use the morphism \(\operatorname{Sym}^d_T(Z) \to \operatorname{Sym}^d_S(X) \times_S T\) induced by \(Z \to X_T\). This construction commutes with pullback in \(T\) and has the required compatibility with the norm map.

Proposition

Let \(X \to S\) be separated. The Grothendieck–Deligne norm map induces an isomorphism from the subfunctor of \(\Hilbfunctor^d_{X/S}\) parametrizing finite étale closed subspaces onto \(\Gamma^d(X/S)_{\mathrm{nd}}\). Equivalently, the morphism of Proposition moduli-proposition-hilbert-symmetric-product is an isomorphism over the nondegenerate locus of the symmetric product.

Proof

The assertion is étale local on the base. A nondegenerate cycle becomes a sum of \(d\) pairwise distinct sections after an étale base change. Their disjoint union is the unique finite étale closed subspace producing the cycle. This construction descends, and it is inverse to the norm map.

Proposition

Let \(X \to S\) be a separated smooth morphism of schemes of relative dimension \(1\). Then the canonical morphisms \[\Hilbfunctor^d_{X/S} \longrightarrow \operatorname{Sym}^d_S(X) \longrightarrow \Gamma^d(X/S)\] are isomorphisms.

Proof

The second morphism is an isomorphism because \(X \to S\) is flat. A finite locally free closed subspace of a smooth relative curve is a relative effective Cartier divisor; see Picard Schemes of Curves, Lemma 0B9D. Addition of divisors gives an \(\mathfrak S_d\)-invariant morphism \[(X/S)^d \longrightarrow \Hilbfunctor^d_{X/S}.\] It is finite locally free of degree \(d!\) by repeated application of Picard Schemes of Curves, Lemma 0B9G. Hence it is an fpqc covering. It factors through a morphism \[\operatorname{Sym}^d_S(X) \longrightarrow \Hilbfunctor^d_{X/S}.\] Let \(a\) denote this morphism and let \(b\) denote the Hilbert-to-symmetric-product morphism. The equality \(b \circ a = \operatorname{id}\) follows from the categorical quotient property after composition with \((X/S)^d \to \operatorname{Sym}^d_S(X)\). The equality \(a \circ b = \operatorname{id}\) may be checked after the displayed fpqc covering of \(\Hilbfunctor^d_{X/S}\), where it follows directly from the construction by adding the \(d\) sections.

Example

Let \(X\) be smooth of dimension at least \(2\) over an algebraically closed field \(k\), and let \(x \in X(k)\). Put \(R = \mathcal{O}_{X,x}\) and let \(\mathfrak m\) be its maximal ideal. Ideals \[\mathfrak m^2 \subset I \subset \mathfrak m \quad\text{with}\quad \dim_k(\mathfrak m/I) = 1\] give distinct length-two closed subschemes supported at \(x\). They are parametrized by the one-dimensional quotients of \(\mathfrak m/\mathfrak m^2\), while all have norm cycle \(2[x]\). Thus the Hilbert–Chow morphism need not be injective already for \(d = 2\). Adding \(d - 2\) fixed reduced points away from \(x\) gives the same conclusion for every \(d > 1\).

Relative effective Cartier divisors and complete linear systems

Let \(f : X \to S\) be a flat, proper morphism of finite presentation of schemes. For a scheme \(T\) over \(S\), set \[\mathit{Div}_{X/S}(T) = \{D \subset X_T \mid D\text{ is a relative effective Cartier divisor on }X_T/T\}.\] Formation of a relative effective Cartier divisor commutes with arbitrary base change by Divisors, Lemma 056Q. Thus this rule defines a subfunctor of \(\Hilbfunctor_{X/S}\).

Lemma

The transformation \[\mathit{Div}_{X/S} \longrightarrow \Hilbfunctor_{X/S}\] is representable by open immersions. In particular, \(\mathit{Div}_{X/S}\) is a separated algebraic space locally of finite presentation over \(S\).

Proof

A relative effective Cartier divisor on \(X_T/T\) is proper over \(T\) because it is closed in the proper \(T\)-scheme \(X_T\). It is flat and of finite presentation over \(T\), so it gives a point of the Hilbert functor.

Put \(H = \Hilbfunctor_{X/S}\) and let \(Z \subset X_H\) be the universal closed subspace. We construct the required open after replacing \(H\) by a scheme \(U\) étale over it. Then \(Z_U \subset X_U\) is a closed subscheme, \(Z_U \to U\) is flat, proper, and of finite presentation, and \(X_U \to U\) is flat and of finite presentation.

Let \(W \subset Z_U\) be the set of points \(z\) such that the inclusion of the fibre \[(Z_U)_u \subset (X_U)_u\] is an effective Cartier divisor in a neighbourhood of \(z\). This is open. Indeed, at such a point Divisors, Lemma 062Y shows that \(Z_U\) is a relative effective Cartier divisor in a neighbourhood of \(z\). The converse follows by base change. Since \(Z_U \to U\) is proper, the image of the closed subset \(Z_U \setminus W\) is closed in \(U\) by Morphisms, Lemma 01W6. Its complement \(U^0\) has the following functorial description: a morphism \(T \to U\) factors through \(U^0\) if and only if \(Z_T \subset X_T\) is a relative effective Cartier divisor. The opens \(U^0\) are compatible with étale pullback and hence descend to the desired open subspace of \(H\).

The last assertion follows from Quot, Proposition 0D01 and Lemma 0DM7.

Proposition

Let \(S\) be a Noetherian scheme and let \(X \to S\) be projective and flat. Choose an \(f\)-very ample invertible module \(\mathcal{O}_X(1)\). Then \(\mathit{Div}_{X/S}\) is a scheme and there is a decomposition \[\mathit{Div}_{X/S} = \coprod_{P \in \mathbf{Q}[t]} \mathit{Div}^P_{X/S}\] into open and closed subschemes. Every \(\mathit{Div}^P_{X/S}\) is quasi-projective over \(S\) and is the open Cartier-divisor locus in \(\Hilbfunctor^{P, \mathcal{O}_X(1)}_{X/S}\).

Proof

The Hilbert functor has the corresponding open and closed decomposition by Remark 0DPG. Its fixed-polynomial terms are projective schemes by Proposition moduli-proposition-quot-projective-over-base, applied to \(\mathcal{F} = \mathcal{O}_X\). Lemma moduli-lemma-divisors-open-in-hilbert identifies \(\mathit{Div}^P_{X/S}\) with an open subscheme of the corresponding projective Hilbert scheme. It is therefore quasi-projective over \(S\).

If \(D \subset X_T\) is a relative effective Cartier divisor with invertible ideal \(\mathcal{I}_D\), write \[\mathcal{O}_{X_T}(D) = \mathcal{I}_D^{-1}.\] This construction commutes with base change and defines the Abel transformation \[\begin{equation} a : \mathit{Div}_{X/S} \longrightarrow \Picardfunctor_{X/S}, \qquad D \longmapsto [\mathcal{O}_{X_T}(D)]. \end{equation}\]

Lemma

Let \(f : X \to S\) be a proper morphism of finite presentation and let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(S\). Put \(K = Rf_*\mathcal{F}\) and \[\mathcal{Q}_{\mathcal{F}} = H^0\left(R\SheafHom_{\mathcal{O}_S}(K, \mathcal{O}_S)\right).\] Then \(\mathcal{Q}_{\mathcal{F}}\) is of finite presentation, its formation commutes with arbitrary base change, and for every \(g : T \to S\) there is a functorial isomorphism \[\begin{equation} f_{T, *}\mathcal{F}_T \cong \SheafHom_{\mathcal{O}_T}(g^*\mathcal{Q}_{\mathcal{F}}, \mathcal{O}_T). \end{equation}\]

Proof

By Derived Categories of Schemes, Lemma 0B91, \(K\) is perfect, its formation commutes with arbitrary base change, and every derived pullback of \(K\) has vanishing cohomology in negative degrees. Quot, Lemma 08JX shows that locally on \(S\) the complex \(K\) has tor amplitude in \([0, b]\) for some \(b \geq 0\).

On an affine open of \(S\), represent \(K\) by a complex \[K^0 \longrightarrow K^1 \longrightarrow \ldots \longrightarrow K^b\] of finite locally free modules. On this open we have \[\mathcal{Q}_{\mathcal{F}} = \Coker\left((K^1)^\vee \longrightarrow (K^0)^\vee\right).\] This description proves finite presentation and commutation with arbitrary base change. After pulling back to \(T\), it also gives \[\begin{align*} \SheafHom_{\mathcal{O}_T}(g^*\mathcal{Q}_{\mathcal{F}}, \mathcal{O}_T) & = \Ker(g^*K^0 \longrightarrow g^*K^1) \\ & = H^0(Lg^*K) \\ & = f_{T, *}\mathcal{F}_T, \end{align*}\] where the last equality uses the arbitrary base change assertion for \(K\). The construction is canonical, so these local isomorphisms glue.

Theorem

Let \(f : X \to S\) be proper, flat, and of finite presentation with geometrically integral fibres, and let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(\mathcal{Q} = \mathcal{Q}_{\mathcal{L}}\) be the module of Lemma moduli-lemma-universal-module-of-sections. The functor which associates to \(T \to S\) the relative effective Cartier divisors \(D \subset X_T\) for which there exist an invertible \(\mathcal{O}_T\)-module \(\mathcal{N}\) and an isomorphism \[\mathcal{O}_{X_T}(D) \cong \mathcal{L}_T \otimes_{\mathcal{O}_{X_T}} f_T^*\mathcal{N}\] is represented by the projective bundle \(\mathbf{P}(\mathcal{Q})\). Equivalently, this is the fibre of the Abel transformation (moduli-equation-divisor-to-picard) over the class of \(\mathcal{L}\).

Proof

For every \(T \to S\) we have \[\mathcal{O}_T \xrightarrow{\sim} f_{T, *}\mathcal{O}_{X_T}\] by Derived Categories of Schemes, Lemma 0E62. Consequently the final assertion about the fibre follows from Quot, Lemma 0D27: two invertible modules on \(X_T\) define the same element of the Picard functor precisely when they differ by the pullback of an invertible module on \(T\).

By Constructions, Definition 01OB, a \(T\)-point of \(\mathbf{P}(\mathcal{Q})\) is an invertible quotient \[g^*\mathcal{Q} \longrightarrow \mathcal{N}.\] Dualizing and using (moduli-equation-universal-module-of-sections) gives a map \[\mathcal{N}^{-1} \longrightarrow f_{T, *}\mathcal{L}_T,\] or, by adjunction, a section \[s : f_T^*\mathcal{N}^{-1} \longrightarrow \mathcal{L}_T.\] The quotient is surjective if and only if the restriction of \(s\) to every geometric fibre is nonzero. Since the geometric fibres of \(X \to S\) are integral, the zero scheme of every such fibrewise section is an effective Cartier divisor. The zero scheme of \(s\) is locally principal on \(X_T\), so Divisors, Lemma 062Y shows that it is a relative effective Cartier divisor \(D\) on \(X_T/T\). Moreover, \[\mathcal{O}_{X_T}(D) \cong \mathcal{L}_T \otimes f_T^*\mathcal{N}.\]

Conversely, the canonical section of a divisor with such an isomorphism gives a fibrewise nonzero section of \(\mathcal{L}_T \otimes f_T^*\mathcal{N}\) and hence an invertible quotient of \(g^*\mathcal{Q}\). These constructions are inverse. Their independence from the displayed isomorphism follows from \(\mathcal{O}_T^* = f_{T, *}\mathcal{O}_{X_T}^*\), and they commute with base change. Thus they give the asserted isomorphism of functors.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(X \to S\) be projective and flat with geometrically integral fibres. Then the Abel transformation \[\mathit{Div}_{X/S} \longrightarrow \Picardfunctor_{X/S}\] is representable by projective morphisms.

Proof

Let \(T \to \Picardfunctor_{X/S}\) be a morphism from a scheme and set \[Y = T \times_{\Picardfunctor_{X/S}} \mathit{Div}_{X/S}.\] After an fppf covering of \(T\), the given Picard class is represented by an invertible module \(\mathcal{L}\) on \(X_T\). Theorem moduli-theorem-complete-linear-system identifies the corresponding pullback of \(Y\) with \(\mathbf{P}(\mathcal{Q}_{\mathcal{L}})\). It follows that \(Y \to T\) is proper by Descent on Spaces, Lemma 0422.

Choose an \(f\)-very ample invertible module on \(X\). Fppf locally on \(T\), the Hilbert polynomial of a divisor in \(Y\) is \[P_{\mathcal{O}_{X_t}} - P_{\mathcal{L}_t^{-1}}.\] It is locally constant by Derived Categories of Schemes, Lemma 0B9T, and it is unchanged when \(\mathcal{L}\) is tensored by the pullback of an invertible module on the base. Hence it descends to a locally constant function on \(T\). Write \(T_P\) for its open and closed level sets and \(Y_P = Y \times_T T_P\).

The morphism \(Y_P \to T_P\) factors through the open subscheme \(\mathit{Div}^P_{X/S}\) of the projective Hilbert scheme \(\Hilbfunctor^{P, \mathcal{O}_X(1)}_{X/S}\). Since \(Y_P \to T_P\) is proper and \(\Picardfunctor_{X/S} \to S\) is separated by Lemma 0DNJ, \(Y_P\) is a closed subspace of \(\mathit{Div}^P_{X/S} \times_S T_P\) and hence is a scheme. The projective Hilbert scheme is separated over \(T_P\), so the resulting immersion of \(Y_P\) into it is a closed immersion; see Morphisms, Lemma 01W6. Thus \(Y_P\) is a projective \(T_P\)-scheme. The \(T_P\) are pairwise disjoint open and closed subschemes, so these projective embeddings glue. Therefore \(Y \to T\) is projective.

Remark

In Theorem moduli-theorem-complete-linear-system, the coherent module \(\mathcal{Q}\) need not be locally free: the dimension of \(H^0(X_s, \mathcal{L}_s)\) can jump. At a point \(s \in S\), the following are equivalent:

1. \(\mathcal{Q}\) is finite locally free in a neighbourhood of \(s\);

2. the base change map \[(f_*\mathcal{L}) \otimes_{\mathcal{O}_S} \kappa(s) \longrightarrow H^0(X_s, \mathcal{L}_s)\] is surjective.

This follows directly by representing \(Rf_*\mathcal{L}\) near \(s\) by the finite locally free complex used in the proof of Lemma moduli-lemma-universal-module-of-sections. If these conditions hold, then \(\mathbf{P}(\mathcal{Q})\) is a projective space bundle and is smooth over \(S\) near its fibre over \(s\). Conversely, the elementary affine-chart criterion for \(\mathbf{P}(\mathcal{Q})\) shows that flatness at every point over \(s\) forces \(\mathcal{Q}\) to be finite locally free at \(s\). Thus the displayed surjectivity is the exact criterion for smoothness along the whole complete linear system. The vanishing \[H^1(X_s, \mathcal{L}_s) = 0\] is a sufficient condition for this surjectivity, by the same finite-complex calculation and Nakayama’s lemma.

Applied after base change to the Abel transformation, this gives the same criterion for smoothness along one of its complete-linear-system fibres. The word translated here as “smooth” is the historical term “simple” in the printed source.

The positive Picard locus and the classical quotient construction

Let \(S\) be a locally Noetherian scheme and let \(f : X \to S\) be projective and flat with geometrically integral fibres. Fix an \(f\)-very ample invertible module \(\mathcal{O}_X(1)\) and denote its class in the Picard functor by \(\xi\). For \(T \to S\), say that a class \(\alpha \in \Picardfunctor_{X/S}(T)\) is positive if, fppf locally on \(T\), it is represented by an invertible module \(\mathcal{L}\) such that for every geometric point \(\bar t\) of \(T\) we have \[\begin{align*} H^i(X_{\bar t}, \mathcal{L}_{\bar t}(n)) & = 0 && \text{for all }i > 0\text{ and }n \geq 0, \\ H^0(X_{\bar t}, \mathcal{L}_{\bar t}(n)) & \neq 0 && \text{for all }n \geq 0. \end{align*}\] This condition is unchanged if \(\mathcal{L}\) is tensored by the pullback of an invertible module on \(T\). Thus it defines a subfunctor \(\mathit{Pic}^+_{X/S}\) of \(\Picardfunctor_{X/S}\).

Lemma

The functor \(\mathit{Pic}^+_{X/S}\) is represented by an open subspace of \(\Picardfunctor_{X/S}\). Its formation commutes with arbitrary base change, it satisfies \[\mathit{Pic}^+_{X/S} + \xi \subset \mathit{Pic}^+_{X/S},\] and \[\begin{equation} \Picardfunctor_{X/S} = \bigcup_{n \geq 0}(\mathit{Pic}^+_{X/S} - n\xi). \end{equation}\]

Proof

The condition is formulated on geometric fibres, so it commutes with base change and is fppf local on the base. To prove openness, pull back to a locally Noetherian scheme \(T\) over the Picard functor and then pass to an fppf cover on which the class is represented by an invertible module \(\mathcal{L}\) on \(X_T\).

For every \(n\), the complex \[K_n = Rf_{T, *}\mathcal{L}(n)\] is perfect and commutes with arbitrary base change by Derived Categories of Schemes, Lemma 0B91. The dimensions of its fibre cohomology are upper semi-continuous by Lemma 0BDN. At a positive geometric point \(\bar t\), relative Serre vanishing, Cohomology of Schemes, Lemma 02O1, gives an integer \(N\) and a neighbourhood on which the higher fibre cohomology vanishes for every \(n \geq N\). For the finitely many \(0 \leq n < N\), upper semi-continuity gives the same conclusion after shrinking again.

On this neighbourhood the dimension of \(H^0\) equals the Euler characteristic and is therefore locally constant by Lemma 0B9T. Shrink once more so that it is positive for \(0 \leq n \leq N\). If \(n > N\), multiply a nonzero section of \(\mathcal{L}_{\bar t}(N)\) by a nonzero section of \(\mathcal{O}_{X_{\bar t}}(n - N)\). The product is nonzero because \(X_{\bar t}\) is integral. Thus positivity is open. The opens obtained on an fppf cover descend, proving the first assertion.

If \(\mathcal{L}\) is positive, then \(\mathcal{L}(1)\) is positive, which proves the inclusion under translation. Finally, for any invertible module \(\mathcal{M}\) on a geometric fibre, Serre vanishing and global generation show that \(\mathcal{M}(n)\) is positive for all sufficiently large \(n\). Consequently every geometric point of the Picard functor belongs to one of the opens on the right hand side of (moduli-equation-picard-covered-positive-translates), proving the equality.

Lemma

Set \[\mathit{Div}^+_{X/S} = \mathit{Div}_{X/S} \times_{\Picardfunctor_{X/S}} \mathit{Pic}^+_{X/S}.\] Then \(\mathit{Div}^+_{X/S}\) is an open subscheme of \(\mathit{Div}_{X/S}\) and the Abel transformation restricts to a smooth, projective, and surjective morphism \[\begin{equation} q : \mathit{Div}^+_{X/S} \longrightarrow \mathit{Pic}^+_{X/S}. \end{equation}\] There are open and closed decompositions \[\mathit{Div}^+_{X/S} = \coprod_P \mathit{Div}^{+,P}_{X/S}, \qquad \mathit{Pic}^+_{X/S} = \coprod_P \mathit{Pic}^{+,P}_{X/S},\] where \(P\) is the Hilbert polynomial of the divisor and \(\mathit{Pic}^{+,P}_{X/S}\) is the image of \(\mathit{Div}^{+,P}_{X/S}\). Each \(\mathit{Div}^{+,P}_{X/S}\) is quasi-projective over \(S\), and \[R^P = \mathit{Div}^{+,P}_{X/S} \times_{\mathit{Pic}^{+,P}_{X/S}} \mathit{Div}^{+,P}_{X/S}\] is a scheme defining an equivalence relation whose two projections are smooth and projective.

Proof

The first assertion follows from Lemma moduli-lemma-positive-picard-open and Proposition moduli-proposition-divisors-projective-flat. Fppf locally on a scheme mapping to \(\mathit{Pic}^+_{X/S}\), the Picard class is represented by an invertible module \(\mathcal{L}\). Positivity and cohomology and base change show that \(f_*\mathcal{L}\) is finite locally free of positive rank and commutes with base change. Lemma moduli-lemma-universal-module-of-sections identifies \(\mathcal{Q}_{\mathcal{L}}\) with \((f_*\mathcal{L})^\vee\). Theorem moduli-theorem-complete-linear-system therefore identifies the pullback of \(q\) with the projective space bundle \(\mathbf{P}((f_*\mathcal{L})^\vee)\). This proves that \(q\) is smooth, projective, and surjective.

The decomposition of the divisor scheme and its quasi-projectivity were proved in Proposition moduli-proposition-divisors-projective-flat. All divisors above the same Picard class have the same Hilbert polynomial: on a geometric fibre the exact sequence of a divisor gives \[P_{\mathcal{O}_D} = P_{\mathcal{O}_X} - P_{\mathcal{L}^{-1}}.\] Since \(q\) is open, its fixed-polynomial images are open; since they are pairwise disjoint and cover, they are also closed. The projections from \(R^P\) are base changes of \(q\) and hence are smooth and projective. Finally, the separatedness of the Picard functor, Lemma 0DNJ, identifies \(R^P\) with a closed subspace of the scheme \(\mathit{Div}^{+,P}_{X/S} \times_S \mathit{Div}^{+,P}_{X/S}\). Thus \(R^P\) is a scheme.

Remark

Lemma moduli-lemma-positive-divisor-cover gives an fppf presentation of every \(\mathit{Pic}^{+,P}_{X/S}\) by a quasi-projective scheme with a projective flat equivalence relation. Bootstrap, Theorem 04S6, proves from this presentation that the quotient is an algebraic space. The classical proof invokes the stronger effective-quotient theorem for a projective flat equivalence relation on a quasi-projective scheme. That theorem makes \[\mathit{Pic}^{+,P}_{X/S} = \mathit{Div}^{+,P}_{X/S}/R^P\] a quasi-projective scheme, compatibly with base change. The disjoint union over \(P\), followed by the translate covering (moduli-equation-picard-covered-positive-translates), is the additional step which gives the scheme assertion in Remark moduli-remark-classical-projective-picard-theorem. It must not be inferred from algebraic-space bootstrap alone.

The source describes this as Matsusaka’s construction. Its original effective quotient can also be constructed from the Hilbert scheme; older constructions used Chow coordinates.

Remark

The printed remark conjectures that properness, flatness, and the base-change-stable equality \[\mathcal{O}_T \xrightarrow{\sim} f_{T, *}\mathcal{O}_{X_T}\] should suffice for existence of the Picard scheme. The linked correction says explicitly that this conjecture is false; it adds that Mumford’s methods prove a slightly weaker theorem. The same correction inserts the omitted hypothesis “with algebraically closed residue field” in the printed example involving a complete local base and says that Mumford’s counterexample makes that restriction indispensable.

There is no conflict with Quot, Proposition 0D2C: under the displayed universal global-functions hypothesis the current theorem represents the Picard functor by an algebraic space, not necessarily by a scheme. The divisor argument also locates the obstruction. Without geometrically integral fibres, fibrewise nonzero sections cutting out relative effective Cartier divisors form only an open subfunctor of the projective bundle in Theorem moduli-theorem-complete-linear-system. The resulting equivalence relation can be flat without being proper, so the classical scheme-quotient step above no longer applies.

Remark

The construction above does not use a prior construction of Jacobians or the theory of abelian schemes. This order is useful: Picard spaces exist in situations where the answer is not an abelian scheme, as already happens for generalized Jacobians of singular curves, and Picard theory in turn supplies the natural setting for duality of abelian schemes.

For a projective flat family of curves with geometrically integral fibres, the classical theorem applies even when the family is not smooth. The separate smooth-curve construction is discussed in Picard Schemes of Curves, Proposition 0B9Z. Statements in the 1962 remark about then-unknown reducible-fibre cases and compactifications are historical status reports, not current nonexistence assertions; the modern algebraic-space theorem is Quot, Proposition 0D2C.

Reductions for Picard spaces

The results in this section explain how representability of a Picard functor can be transported through a finite Stein factor, a union of closed subschemes, and a nilpotent thickening. We also record two nonflat descent theorems over a field. The word “scheme” in the last two results is essential: algebraic-space representability alone does not contain these statements.

Proposition

Let \(S\) be a locally Noetherian scheme and let \(X \to S' \to S\) be morphisms of schemes such that

1. \(X \to S'\) is projective and flat with geometrically integral fibres, and

2. \(S' \to S\) is finite locally free.

Then there is a canonical isomorphism of fppf sheaves \[\begin{equation} \Picardfunctor_{X/S} \longrightarrow \mathop{\rm Res}\nolimits_{S'/S}(\Picardfunctor_{X/S'}). \end{equation}\] Moreover, both sides are represented by schemes.

Proof

For a scheme \(T\) over \(S\) we have a canonical equality \[X \times_S T = X \times_{S'}(S' \times_S T).\] It identifies the isomorphism-class presheaves of invertible modules which occur on the two sides of (moduli-equation-picard-stein-restriction). The identification is compatible with fppf pullback, and hence gives the displayed isomorphism after sheafification.

By Remark moduli-remark-classical-projective-picard-theorem, the Picard functor over \(S'\) is a scheme covered by open subschemes which are quasi-projective over \(S'\). The classical construction also has the finite-subset property used in the cited Proposition 6.1: every finite set in a fibre is contained in an affine open. Restriction of scalars of any one of these opens is a scheme by Criteria for Representability, Proposition criteria-proposition-restriction-of-scalars-properties. These restrictions of scalars cover the right hand side. Indeed, over a point of \(S\) the finite fibre of \(S' \to S\) has finite image in the Picard scheme. Such an image is contained in one of the quasi-projective opens, and the same containment holds over an open neighbourhood of the point because \(S' \to S\) is finite. Thus the algebraic space supplied by Criteria for Representability, Proposition 05YF is covered by open subspaces which are schemes, and is therefore a scheme.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(X \to S\) be projective and flat with geometrically locally integral fibres. Then \(\Picardfunctor_{X/S}\) is represented by a scheme.

Proof

Let \(X \to S' \to S\) be the Stein factorization. The morphism \(S' \to S\) is finite étale by More on Morphisms of Spaces, Lemma 0E0D. Its geometric fibres index the connected components of the geometric fibres of \(X/S\). A connected locally integral scheme is integral, so \(X \to S'\) has geometrically integral fibres. It is flat, and the result follows from Proposition moduli-proposition-picard-stein-restriction.

Example

If \(X = \coprod_{i = 1}^r X_i\) is a finite disjoint union over \(S\), then Proposition moduli-proposition-picard-stein-restriction specializes to \[\Picardfunctor_{X/S} = \prod_{i = 1}^r \Picardfunctor_{X_i/S}.\]

Proposition

Let \(S\) be locally Noetherian, let \(X \to S\) be proper, and let \(X_1, X_2 \subset X\) be closed subschemes defined by coherent ideals \(\mathcal{I}_1, \mathcal{I}_2\). Assume that

1. \(\mathcal{I}_1 \cap \mathcal{I}_2 = 0\),

2. \(X_1\) and \(X_2\) are flat over \(S\),

3. \(Z = X_1 \cap X_2\) is flat over \(S\), and

4. for every \(s \in S\) the maps \[\kappa(s) \longrightarrow H^0((X_i)_s, \mathcal{O}_{(X_i)_s}), \qquad i = 1, 2,\] are isomorphisms.

Then restriction induces a morphism \[\begin{equation} \Picardfunctor_{X/S} \longrightarrow \Picardfunctor_{X_1/S} \times_S \Picardfunctor_{X_2/S} \end{equation}\] which is representable by affine morphisms. In particular, if the two Picard functors on the right are schemes, then so is the Picard functor on the left.

Proof

The first three assumptions give an exact sequence \[0 \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}_{X_1} \oplus \mathcal{O}_{X_2} \longrightarrow \mathcal{O}_Z \longrightarrow 0.\] In particular \(X\) and \(Z\) are flat and of finite presentation over \(S\). The assertion that a base change of (moduli-equation-picard-closed-union) is affine is fppf local on the base. We may therefore work locally where the two Picard classes are represented by invertible modules \(\mathcal{L}_1\) and \(\mathcal{L}_2\). After separating the open and closed locus on which \(Z\) is empty, we may also make an fppf base change such that \(Z \to S\) has a section \(\sigma\). This section gives sections of \(X_1\) and \(X_2\) and permits the modules to be rigidified; compare Quot, Lemmas 0D28 and 0D29.

An invertible module on the scheme-theoretic union \(X_1 \cup X_2 = X\) is then the same as a triple \[(\mathcal{L}_1, \mathcal{L}_2, u : \mathcal{L}_1|_Z \longrightarrow \mathcal{L}_2|_Z),\] where \(u\) is an isomorphism compatible with the rigidifications. The isomorphism space between the two restrictions is affine over \(Z\). Its functor of sections over the proper flat morphism \(Z \to S\) is affine by Proposition moduli-proposition-sections-proper-flat; compatibility at \(\sigma\) cuts out the fibre over the identity and is again affine. Hence the fibre of (moduli-equation-picard-closed-union) is affine. These affine schemes and their descent data descend along the fppf coverings used above, which proves representability and affineness.

Lemma

Let \(X\) be a proper geometrically reduced scheme over a field \(k\). After a finite separable extension \(k'/k\), write \(X_1, \ldots, X_r\) for the irreducible components of \(X_{k'}\). If the Picard functors of the \(X_i\) are represented by schemes, then the Picard functor of \(X\) is represented by a scheme, and after that extension the canonical morphism \[\Picardfunctor_{X_{k'}/k'} \longrightarrow \prod_{i = 1}^r \Picardfunctor_{X_i/k'}\] is affine.

Proof

The finite separable extension can be chosen so that all irreducible components are geometrically irreducible by Varieties, Lemma 054R. Their reduced scheme-theoretic intersections are flat over the field. Applying Proposition moduli-proposition-picard-closed-union-affine successively proves the assertion after this extension. The construction is equivariant for the finite descent datum; the affine morphism and its affine source descend. This gives the assertion over \(k\).

Proposition

Let \(X\) be a proper scheme over a field \(k\) and let \(X_0 \subset X\) be a closed subscheme with the same underlying topological space. Then \[\begin{equation} \Picardfunctor_{X/k} \longrightarrow \Picardfunctor_{X_0/k} \end{equation}\] is representable by affine morphisms. Consequently, representability of the Picard functor of \(X_0\) by a scheme implies the same for \(X\).

Proof

Filtering the nilpotent ideal by its powers reduces the assertion to a first order thickening with ideal \(\mathcal{I}\). More on Morphisms, Lemma 0C6R gives the deformation sequence \[H^1(X_0, \mathcal{I}) \longrightarrow \Pic(X) \longrightarrow \Pic(X_0) \longrightarrow H^2(X_0, \mathcal{I}).\] The same sequence applies after every base change over \(k\). Since \(X_0\) is proper, the cohomology groups of \(\mathcal{I}\) are finite dimensional and cohomology commutes with extension of the ground field. Thus, after pulling back a class of \(\Picardfunctor_{X_0/k}\) to a test scheme, its obstruction to lifting is a section of the vector bundle associated to \(H^2(X_0, \mathcal{I})\). Its zero scheme is affine over the test scheme. Over that zero scheme the sheaf of lifts is a torsor under the vector group associated to \(H^1(X_0, \mathcal{I})\), and is therefore affine. This description is compatible with fppf descent and proves the claim. Iterating over the filtration by powers of the ideal finishes the proof.

Lemma

For every projective scheme \(X\) over a field \(k\), the fppf Picard functor \(\Picardfunctor_{X/k}\) is represented by a scheme.

Proof

After a finite extension of the ground field, the reduced irreducible components are covered by the classical projective Picard theorem, Remark moduli-remark-classical-projective-picard-theorem. Corollary moduli-corollary-picard-reduced-components glues the components and Proposition moduli-proposition-picard-nilpotent-thickening-affine restores the nilpotent structure. The inseparable and descent steps are the Oort reduction in the cited source.

Theorem

Let \(g : Y \to X\) be a surjective morphism of proper schemes over a field \(k\). Pullback gives an affine morphism \[\begin{equation} g^* : \Picardfunctor_{X/k} \longrightarrow \Picardfunctor_{Y/k}. \end{equation}\]

Proof

This is the affirmative nonflat-descent theorem referenced in the correction to Exposé 232. The Comments on Exposé 236 first give the relative finite-type statement and then state explicitly that over the spectrum of a field the same proof makes the pullback morphism affine. Proposition moduli-proposition-picard-closed-union-affine and Proposition moduli-proposition-picard-nilpotent-thickening-affine are two elementary instances of the same theorem.

Theorem

For every proper scheme \(X\) over a field \(k\), the fppf Picard functor \(\Picardfunctor_{X/k}\) is represented by a scheme.

Proof

The projective case is Corollary moduli-corollary-picard-projective-field. After the component and nilpotent reductions, Chow’s lemma supplies a proper surjection \(Y \to X\) with \(Y\) projective. The Picard functor of \(Y\) is a scheme and Theorem moduli-theorem-picard-pullback-proper-surjective-affine makes the pullback from the Picard functor of \(X\) affine. This is Murre’s affirmative solution of the question printed in Remark 6.6, as recorded by the linked correction.

Lemma

Let \(X\) be a normal scheme of finite type over a field \(k\). The projection \(p : X\times_k\mathbf G_{m,k}\to X\) induces an isomorphism \[p^* : \Pic(X)\longrightarrow \Pic(X\times_k\mathbf G_{m,k}).\]

Proof

The section defined by \(t=1\) shows that \(p^*\) is injective. Since the irreducible components of a normal Noetherian scheme are open and closed, we may assume that \(X\) is integral. Put \(Y=X\times_k\mathbf G_{m,k}\) and let \(K\) be the function field of \(X\).

Represent an invertible module on \(Y\) by a Cartier divisor \(D\). Its restriction to \[Y_K=\Spec(K[t,t^{-1}])\] is principal because \(K[t,t^{-1}]\) is a unique factorization domain. After subtracting the divisor of a rational function, we may therefore assume that no irreducible component of \(D\) dominates \(X\).

Every prime divisor of \(Y\) which does not dominate \(X\) is of the form \(Z\times_k\mathbf G_{m,k}\) for a unique prime divisor \(Z\) of \(X\). Hence \(D=p^*E\) as a Weil divisor for some Weil divisor \(E\) on \(X\). The rank one reflexive module \(\mathcal{O}_X(E)\) pulls back to the invertible module \(\mathcal{O}_Y(D)\). Invertibility is fpqc local, and \(p\) is faithfully flat; thus \(\mathcal{O}_X(E)\) is invertible. Equivalently, \(E\) is Cartier. It follows that the original invertible module is pulled back from \(X\), proving surjectivity.

Theorem

Let \(S\) be a locally Noetherian scheme and let \(f:X\to S\) be proper and smooth. Suppose that the fppf Picard functor is represented by a scheme \[P=\Picardfunctor_{X/S}.\] Then \(P\to S\) is separated. Every closed subscheme \(Z\subset P\) which is of finite type over \(S\) is proper over \(S\).

Proof

The representing scheme \(P\) is locally of finite type over \(S\). By Limits, Lemma 0207, separatedness may be checked using discrete valuation rings. Consider two lifts to \(P\) over a discrete valuation ring which agree over its fraction field. Equality of the lifts is fpqc local on the valuation ring, so we may pass to its strict henselization.

The Stein factorization of the base change of \(X\) is finite étale by More on Morphisms of Spaces, Lemma 0E0D. It therefore splits the base change of \(X\) into a finite disjoint union of proper smooth families with geometrically connected fibres. These fibres are geometrically integral. Example moduli-example-picard-disjoint-union identifies the Picard functor with the product of their Picard functors, and Lemma 0DNJ says that every factor is separated. The two lifts are therefore equal after strict henselization, hence equal over the original valuation ring. Thus \(P\to S\) is separated.

Let \(Z\subset P\) be closed and of finite type over \(S\). Given a valuative diagram for \(Z\) over a discrete valuation ring \(R\), pass to a faithfully flat extension \(R\subset R'\) of valuation rings over which the generic Picard class is represented by an invertible module. Lemma 0DNG extends that module over the proper smooth family and hence gives a point of \(P(R')\). On \(\Spec(R'\otimes_RR')\) its two pullbacks agree over the schematically dense generic fibre. They agree everywhere because \(P\to S\) is separated. Thus the point descends to \(P(R)\); it factors through \(Z\) because its generic point does and \(Z\) is closed. Uniqueness follows from separatedness. Limits, Lemma 0208, now shows that \(Z\to S\) is proper.

The classical proof in the cited source obtains the same extension by closing a Cartier divisor from the generic fibre. In its final sentence the printed phrase “divisor on \(S\)” has to be read as “divisor on \(X\)”: it is the regular local rings of the smooth scheme \(X\) which are factorial.

Lemma

Let \(R\) be a discrete valuation ring with fraction field \(K\). Let \(X\to \Spec(R)\) be proper and flat. Assume every geometric fibre is a local complete intersection and is smooth at all points of codimension at most \(2\). Then every invertible module on \(X_K\) extends to an invertible module on \(X\).

Proof

The fibre hypotheses imply that the geometric fibres are normal: a local complete intersection is Cohen–Macaulay, hence \((S_2)\), and smoothness in codimension at most \(2\) gives \((R_1)\). The total space \(X\) is normal as well. Indeed, at a point of the closed fibre its local ring \(A\) is a complete intersection by Divided Power Algebra, Lemma 09Q7, and is therefore \((S_2)\). At a prime of height at most \(1\), either one is on the generic fibre, where regularity follows from the fibre hypothesis, or one is on the closed fibre and regularity follows from Algebra, Lemma 031E. Thus \(A\) is normal by Serre’s criterion.

Work componentwise. Represent an invertible module on \(X_K\) by a Cartier divisor \(D_K\), and let \(D\) be its closure in \(X\), viewed as a Weil divisor. It remains to prove that \(D\) is Cartier at every point of the closed fibre. Fix such a point and put \(A=\mathcal{O}_{X,x}\). As above, \(A\) is a normal local complete intersection. Let \(\mathfrak p\subset A\) have height at most \(3\). If the uniformizer of \(R\) is not in \(\mathfrak p\), then \(D\) is Cartier at \(\mathfrak p\) because it restricts to \(D_K\). If the uniformizer is in \(\mathfrak p\), then \[\dim(A_\mathfrak p/\pi A_\mathfrak p) =\dim(A_\mathfrak p)-1\leq 2.\] The fibre hypothesis makes this quotient regular, and Algebra, Lemma 031E makes \(A_\mathfrak p\) regular. Hence \(D\) is Cartier at \(\mathfrak p\). Divisors, Theorem divisors-theorem-ci-Cartier-codimension-three now shows that \(D\) is Cartier at \(x\). Thus \(\mathcal{O}_X(D)\) is the required extension.

Theorem

Let \(S\) be a locally Noetherian scheme and let \(f:X\to S\) be proper and flat. Assume every geometric fibre is a local complete intersection and is smooth at all points of codimension at most \(2\). Suppose the fppf Picard functor is represented by a scheme \[P=\Picardfunctor_{X/S}.\] Then \(P\to S\) is separated. Every closed subscheme \(Z\subset P\) which is of finite type over \(S\) is proper over \(S\).

Proof

The scheme \(P\) is locally of finite type over \(S\). The separation argument of Theorem moduli-theorem-picard-smooth-proper-pieces applies without change. Indeed, after strict henselization the finite étale Stein factorization splits \(X\) into proper flat families with geometrically integral fibres, and Lemma 0DNJ applies to every factor.

Let \(Z\subset P\) be closed and of finite type over \(S\), and consider its valuative criterion over a discrete valuation ring \(R\). After a faithfully flat extension which is again a discrete valuation ring, the generic Picard class is represented by an invertible module. Lemma moduli-lemma-pic-flat-lci-codimension-two-extension extends it over the family. Separatedness makes the two pullbacks agree on the overlap, so the resulting point descends to \(P(R)\) and factors through \(Z\). Uniqueness follows from separatedness. Limits, Lemma 0208 proves that \(Z\to S\) is proper.

Example

Let \(k\) be an algebraically closed field of characteristic different from \(2\), let \(R=k[[t]]\), and let \(X\subset\mathbf P^3_R\) be defined by \[x_0x_1-x_2^2+t^2x_3^2=0.\] This is a projective flat family whose generic fibre is a smooth split quadric surface and whose special fibre is a normal quadric cone. Its Picard functor is represented by a scheme, but it has a finite type closed piece which is not proper over \(R\).

Proof

Representability follows from the projective geometrically integral-fibre criterion recorded in Remark moduli-remark-mumford-picard-scheme-boundary.

Over \(K=k((t))\), put \(y=x_2+tx_3\) and \(z=x_2-tx_3\). The generic equation is \(x_0x_1-yz=0\), so \(X_K\cong\mathbf P^1_K\times\mathbf P^1_K\). Let \[\mathcal{L}_K=\mathcal{O}_{\mathbf P^1\times\mathbf P^1}(1,0) \quad\text{and}\quad \mathcal{H}=\mathcal{O}_X(1).\] Then \(\deg(\mathcal{L}_K\cdot\mathcal{H}_K)=1\).

Write \(Q=X_k\). The resolution of the vertex of \(Q\) is the Hirzebruch surface \(\mathbf F_2\). Pullback on Picard groups is injective. If \(C\) is the exceptional section and \(F\) a ruling, then \(C^2=-2\) and the pullback of \(\mathcal{O}_Q(1)\) is \(\mathcal{O}_{\mathbf F_2}(C+2F)\). A line bundle \(\mathcal{O}_{\mathbf F_2}(aC+bF)\) in the image has degree zero on \(C\), namely \(b=2a\). Conversely, every such class is the pullback of \(\mathcal{O}_Q(a)\). Consequently \[\Pic(Q)=\mathbf Z[\mathcal{O}_Q(1)].\] In particular, the intersection degree with \(\mathcal{H}_k\) of every invertible module on \(Q\) is even.

If \(\mathcal{L}_K\) extended after any dominating extension of discrete valuation rings, constancy of the Hilbert polynomial would make its intersection degree with \(\mathcal{H}\) equal to \(1\) on the special fibre. This contradicts the preceding parity calculation. Hence the corresponding \(K\)-point of the Picard scheme has no specialization, even after extending the valuation ring. It is isolated in the generic Picard scheme; its closure therefore gives a finite type closed copy of \(\Spec(K)\) in the relative Picard scheme, and this copy is not proper over \(R\).

The special fibre is a hypersurface with one singular point of codimension \(2\). Thus it is smooth in codimension at most \(1\) and normal, but not smooth in codimension at most \(2\). This proves the sharpness asserted in the cited remark.

Remark

In Theorem moduli-theorem-picard-smooth-proper-pieces, suppose that \[P=\coprod_i P^{(i)}\] and every \(P^{(i)}\) is of finite type over \(S\). Each \(P^{(i)}\) is open and closed in \(P\), so the theorem makes it proper over \(S\). Theorem moduli-theorem-picard-flat-lci-codimension-two gives the same conclusion under its weaker local-complete-intersection fibre hypotheses.

Example moduli-example-picard-quadric-cone-boundary shows that geometrically normal fibres alone do not suffice for this conclusion. The nonproper point in that example is represented on the generic split quadric by \(\mathcal{O}(1,0)\). Its class has infinite order in \[\mathop{\rm NS}(\mathbf P^1\times\mathbf P^1)=\mathbf Z^2,\] so it does not belong to \(P^\tau\). Consequently the example does not answer the narrower question left open in the cited source: whether \(P^\tau\) must be proper for a proper normal family when it is of finite type over the base.

Theorem

Let \(X\) be a proper geometrically normal scheme over a field \(k\). Let \[P=\Picardfunctor_{X/k}\] be its Picard scheme, whose existence is Theorem moduli-theorem-picard-proper-field. Then the identity component \(P^0\) is proper over \(k\).

Proof

Properness may be checked after extending the ground field, and the assertion is componentwise in \(X\). We may therefore assume that \(k\) is algebraically closed and that \(X\) is nonempty and connected. Choose a point \(x\in X(k)\); rigidification along \(x\) identifies maps to \(P\) with rigidified invertible modules.

Let \(H=(P^0)_{red}\). This is a smooth connected subgroup scheme by Groupoids, Lemmas 047R and 047P in positive characteristic, and by Lemma 047N in characteristic zero. Suppose that \(H\) is not proper. Chevalley’s structure theorem for the connected commutative algebraic group \(H\) then gives a positive dimensional connected affine subgroup. Over an algebraically closed field such a group contains a subgroup isomorphic to \(\mathbf G_a\) or \(\mathbf G_m\). In the additive case compose its inclusion with, for example, \(u\mapsto u-1\) on \(\mathbf G_m\); in the multiplicative case use the subgroup itself. In either case there is a nonconstant morphism \[\mathbf G_m\longrightarrow H\longrightarrow P.\]

The corresponding rigidified invertible module on \(X\times_k\mathbf G_m\) is pulled back from \(X\) by Lemma moduli-lemma-pic-normal-laurent-invariance. Hence the displayed morphism is constant, a contradiction. Thus \(H\) is proper. Finally \(P^0\) is of finite type over \(k\) and \(H\subset P^0\) is a thickening. More on Morphisms, Lemma 09ZV shows that \(P^0\) is separated and universally closed because \(H\) is; hence \(P^0\) is proper. Properness descends from the algebraic closure, proving the theorem over \(k\).

Lemma

Let \(X\) be a proper geometrically normal scheme over a field \(k\). Let \[P=\Picardfunctor_{X/k}\] be its Picard scheme. Then \[A=(P^0)_{red}\] is an abelian variety over \(k\) and a closed subgroup scheme of \(P\). Its underlying space is \(|P^0|\). For every field extension \(K/k\) we have \[A_K=((P_K)^0)_{red},\] where \(P_K\) represents \(\Picardfunctor_{X_K/K}\).

Proof

The group scheme \(P^0\) is proper over \(k\) by Theorem moduli-theorem-picard-identity-component-proper-normal; in particular it is of finite type. The group scheme \((P^0)_{\overline{k}}\) contains no closed subgroup scheme isomorphic to \(\mathbf{G}_{a,\overline{k}}\), since a closed subscheme of a proper scheme is proper whereas \(\mathbf{G}_a\) is not proper. Groupoids, Lemma groupoids-lemma-reduced-identity-component-no-additive therefore shows that \(A\) is a smooth connected closed subgroup scheme of \(P^0\). It is proper because it is closed in \(P^0\). Hence \(A\) is an abelian variety by Groupoids, Remark 0H2U.

Formation of the Picard functor commutes with extension of the ground field. Moreover, \((P^0)_K=(P_K)^0\) by Groupoids, Proposition 0B7R: the identity component is geometrically irreducible and hence remains the identity component after a field extension. The smooth scheme \(A_K\) is reduced and has \(|(P_K)^0|\) as its underlying space. It is therefore the reduction of \((P_K)^0\), proving the last assertion.

Theorem

Let \(S\) be a locally Noetherian scheme and let \(f:X\to S\) be proper and flat with geometrically normal fibres. Suppose the fppf Picard functor is represented by a scheme \[P=\Picardfunctor_{X/S}.\] Then \(P\to S\) is separated. The identity-component locus \[P^0=\bigcup_{s\in S}P_s^0\] is closed in \(P\) and, with its reduced induced closed subscheme structure, is proper over \(S\). The torsion-component locus \(P^\tau\) is open and closed, and the residue-characteristic-primary component locus \(P^\rho\) is closed. If all residue fields of \(S\) have the same characteristic exponent, then the prime-to-characteristic component locus \(P^\sigma\) is closed as well.

Proof

The scheme \(P\) is locally of finite type over \(S\). We first prove that it is separated. By the discrete-valuation criterion, fpqc descent, and strict henselization, it is enough to prove uniqueness over a strictly henselian discrete valuation ring. The Stein factorization of \(X\) is finite étale by More on Morphisms of Spaces, Lemma 0E0D, since the geometric fibres are normal and hence reduced. It splits \(X\) into finitely many proper flat families with geometrically connected normal fibres. Those fibres are geometrically integral. Example moduli-example-picard-disjoint-union identifies \(P\) with the product of the Picard functors of the factors, each of which is separated by Lemma 0DNJ. This proves that \(P\to S\) is separated.

For every \(s\in S\), Theorem moduli-theorem-picard-identity-component-proper-normal, applied over \(\kappa(s)\), shows that \(P_s^0\) is proper. More on Morphisms, Lemma more-morphisms-lemma-proper-connected-component-neighbourhood gives, locally on \(S\), closedness of \(P^0\) and properness of its reduced induced structure. Both conclusions are local on the base, so they hold over \(S\).

Finally, More on Morphisms, Lemma more-morphisms-lemma-torsion-component-locus-open makes \(P^\tau\) open. Lemma more-morphisms-lemma-order-in-component-group-constructible, using the closedness of \(P^0\), makes \(P^\tau\) and \(P^\rho\) closed and also makes \(P^\sigma\) closed under the stated equal-characteristic-exponent hypothesis.

Theorem

Let \(B\) be a scheme and let \(X \to B\) be a proper flat morphism of schemes of finite presentation. Suppose that the fppf Picard functor is represented by an algebraic space \[P=\Picardfunctor_{X/B}.\] Then \(P\to B\) is locally of finite presentation. Let \(n \geq 1\). Multiplication by \(n\) \[[n] : P \longrightarrow P\] is étale over the open subscheme of \(B\) on which \(n\) is invertible. In particular, it is étale at every point of \(P\) whose residue characteristic is prime to \(n\).

Proof

First, \(P \to B\) is locally of finite presentation. Indeed, the usual limit argument for invertible modules and fppf descent shows that the fppf Picard functor commutes with directed limits of affine \(B\)-schemes: a finite presentation fppf covering, the invertible modules on its pullback of \(X\), the finite descent data, and all their relations descend to a finite stage. Thus the assertion follows from Limits, Proposition 01ZC. Consequently \([n]\) is locally of finite presentation by Morphisms of Spaces, Lemma 05WT.

We may now replace \(B\) by the open subscheme on which \(n\) is invertible. Let \(T_0 \subset T\) be a first order thickening of affine schemes over \(B\). On the fppf site of \(T\), put \(U_0=U\times_TT_0\) for every \(U/T\) and consider the square of presheaves whose value on \(U\) is \[\xymatrix{ \Pic(X_U) \ar[r]^{[n]} \ar[d] & \Pic(X_U) \ar[d] \\ \Pic(X_{U_0}) \ar[r]^{[n]} & \Pic(X_{U_0}). }\] It is cartesian for every \(U\) by More on Morphisms, Lemma more-morphisms-lemma-pic-roots-first-order-thickening. Sheafification commutes with finite limits by Sites, Lemma 00WJ. After fppf sheafification and evaluation on \(T\), we therefore obtain a cartesian square \[\xymatrix{ P(T) \ar[r]^{[n]} \ar[d] & P(T) \ar[d] \\ P(T_0) \ar[r]^{[n]} & P(T_0). }\] Thus \([n]\) is formally étale. Since it is locally of finite presentation, it is étale by More on Morphisms of Spaces, Lemma 0616.

Lemma

Let \(B\) be a locally Noetherian scheme and let \(X\to B\) be a proper flat morphism of schemes. Suppose that the fppf Picard functor is represented by a scheme \[P=\Picardfunctor_{X/B}.\] Assume that for every geometric point \(\overline{b}\to B\), the identity component \(P_{\overline{b}}^0\) contains no subgroup scheme isomorphic to \(\mathbf G_{a,\overline{b}}\). Then \(P\to B\) is universally open along the prime-to-residue-characteristic component locus \(P^\sigma\), and hence along \(P^0\).

Assume, in addition, that there is a \(p\), equal to \(1\) or to a prime number, which is the characteristic exponent of every residue field of \(B\). If \(P^0\) is closed in \(P\), then \(P^\sigma\), with its reduced induced closed subscheme structure, is universally open over \(B\). In the same equal-characteristic situation, \(P^\rho\) is open in \(P\).

Proof

Let \(x\in P^\sigma\) lie over \(b\in B\). By the definition of \(P^\sigma\), we can choose \(n>1\), invertible in \(\kappa(b)\), such that \([n^h](x)\in P_b^0\) for some \(h\geq 0\). Here, if \(x\in P_b^0\), we may choose any \(n>1\) invertible in \(\kappa(b)\). After replacing \(B\) by a neighbourhood of \(b\), the integer \(n\) is invertible on \(B\).

Theorem moduli-theorem-picard-multiplication-etale shows that \([n]:P\to P\) is étale, hence universally open. More on Morphisms, Lemma more-morphisms-lemma-prime-to-characteristic-torsion-dense verifies the geometric fibrewise-density hypothesis of Lemma more-morphisms-lemma-power-kernel-density-universally-open, even though the residue characteristic may vary, because \(n\) is invertible on the chosen neighbourhood. The latter lemma shows that \(P\to B\) is universally open along \[\bigcup_{r\geq 0}[n^r]^{-1}(P^0),\] which contains \(x\). This proves universal openness along \(P^\sigma\).

Under the additional equal-characteristic and closedness hypotheses, the assertion about \(P^\sigma\) is More on Morphisms, Lemma more-morphisms-lemma-prime-to-characteristic-locus-universally-open. Finally every \([n]\) with \((n,p)=1\) is étale by Theorem moduli-theorem-picard-multiplication-etale. Openness of \(P^\rho\) follows from More on Morphisms, Lemma more-morphisms-lemma-primary-component-locus-open.

Remark

The cited corollary additionally says that, in equal characteristic, \(P^\rho\to B\) is universally open. The two general results invoked there do not prove this: Corollary more-morphisms-corollary-prime-to-characteristic-component-openness gives universal openness along \(P^\sigma\), whereas More on Morphisms, Lemma more-morphisms-lemma-primary-component-locus-open only makes \(P^\rho\) open as a subset of \(P\).

More on Morphisms, Example more-morphisms-example-primary-component-open-not-universally-open shows that this implication is false for general commutative group schemes even when all prime-to-characteristic power maps are automorphisms and all geometric identity components contain no additive subgroup. Thus the stronger Picard-specific assertion requires an additional argument or hypothesis and is not inferred from the cited group-scheme results here.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(X\to S\) be a proper flat morphism of schemes. Suppose that the fppf Picard functor is represented by a scheme \[P=\Picardfunctor_{X/S}.\] Then \[S\longrightarrow\{0,1,2,\ldots\},\qquad s\longmapsto\dim(P_s)\] is upper semicontinuous.

If, for every geometric point \(\overline{s}\to S\), the identity component \(P^0_{\overline{s}}\) contains no subgroup scheme isomorphic to \(\mathbf G_{a,\overline{s}}\), then this function is locally constant.

Proof

Theorem moduli-theorem-picard-multiplication-etale shows that \(P\to S\) is locally of finite presentation. Hence upper semicontinuity follows from More on Morphisms, Lemma more-morphisms-lemma-group-scheme-fibre-dimension-upper-semicontinuous.

Under the additional hypothesis, Lemma moduli-corollary-picard-prime-to-characteristic-openness shows that \(P\to S\) is universally open along \(P^0\). More on Morphisms, Lemma more-morphisms-lemma-identity-component-open-fibre-dimension now gives local constancy.

Remark

Let \(s\) be a specialization of \(s'\) in the base of a represented relative Picard scheme. In the notation of the source, let \[(\alpha,\mu,\lambda) \quad\text{and}\quad (\alpha',\mu',\lambda')\] be the dimensions of the abelian, multiplicative, and additive parts of the geometric identity components at \(s\) and \(s'\), respectively. Thus the two Picard-fibre dimensions are \[\alpha+\mu+\lambda \quad\text{and}\quad \alpha'+\mu'+\lambda'.\] Corollary moduli-corollary-picard-dimension-semicontinuity gives \[\begin{equation} \alpha+\mu+\lambda\geq\alpha'+\mu'+\lambda'. \end{equation}\] If the no-additive-subgroup hypothesis of that corollary holds on a neighbourhood of the specialization, then equality holds.

The cited source also records Serre’s specialization argument using the abelianized fundamental group and Kummer theory. It gives the following inequality when the fibres are geometrically reduced (called “separable” there), or when the relevant torsion Picard functors are separated: \[\begin{equation} 2\alpha+\mu\leq 2\alpha'+\mu'. \end{equation}\] Here the assertion that an \(n\)-torsion Picard functor is étale by Theorem moduli-theorem-picard-multiplication-etale is used only where \(n\) is invertible on the base.

Finally, the source explicitly presents as conjectures both general equality in (moduli-equation-picard-specialization-total-dimension) and the inequalities \[\alpha\leq\alpha',\qquad \lambda\geq\lambda'\] for a locally finite type group scheme of constant fibre dimension. These conjectural clauses are retained here as historical context, not as results of the Stacks Project, and no argument below uses them.

Lemma

Let \(S\) be a reduced locally Noetherian scheme and let \(X\to S\) be a proper flat morphism of schemes. Suppose that the fppf Picard functor is represented by a scheme \[P=\Picardfunctor_{X/S}.\] Assume that every geometric fibre of \(P\to S\) is reduced and that none of its geometric identity components contains a subgroup scheme isomorphic to \(\mathbf G_a\). Then \(P\to S\) is smooth at every point of \(P^\sigma\).

Proof

Theorem moduli-theorem-picard-multiplication-etale shows that \(P\to S\) is locally of finite presentation, and Lemma moduli-corollary-picard-prime-to-characteristic-openness shows that it is universally open along \(P^\sigma\).

Let \(x\in P^\sigma\) lie over \(s\in S\). Every irreducible component of \(P_s\) through \(x\) is contained in the connected component of \(x\), and hence its generic point is still in \(P^\sigma\). The local ring \(\mathcal O_{S,s}\) is reduced and \(P_s\) is geometrically reduced at \(x\). More on Morphisms, Lemma more-morphisms-lemma-universally-open-geometrically-reduced-flat therefore shows that \(P\to S\) is flat at \(x\).

After extending \(\kappa(s)\) to an algebraic closure, the reduced group scheme \(P_{\overline{s}}\) is smooth by Groupoids, Lemmas 047N and 047P. Thus \(P_s\) is smooth at \(x\). Morphisms, Lemma 01V8 now proves that \(P\to S\) is smooth at \(x\).

Remark

Universal openness of a torsion-component locus should not be confused with flatness over a nonreduced base. The source reports Mumford’s infinitesimal deformation of an Igusa surface over an Artinian base for which \(P^\tau\) is not flat; the point exhibiting nonflatness lies in \(P^\rho\). The construction itself is not reproduced in the cited exposé, so this is retained as a historical boundary rather than promoted to a standalone Stacks example.

The source asks whether, for a smooth proper family, \(P^\tau\) might be flat at points of \(P^\sigma\), and expresses doubt even for flatness along \(P^0\) over a discrete valuation ring. It relates the question to fixed loci of finite automorphism groups on abelian schemes. These are questions in the source, not assertions here.

The source also points ahead to its Theorem 3.5 for removal of the reducedness hypothesis in Lemma moduli-lemma-picard-smooth-prime-to-characteristic-reduced-base when the original family is normal. That stronger normal-family result is not inferred from Remark 2.9; it follows instead from the lifting argument in Theorem moduli-theorem-picard-smooth-near-smooth-fibre.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(f:X\to S\) be a proper flat morphism of schemes. Suppose that the fppf Picard functor is represented by a scheme \[P=\Picardfunctor_{X/S}.\] If \(s\in S\) and \[H^2(X_s,\mathcal O_{X_s})=0,\] then there is an open neighbourhood \(U\) of \(s\) such that \(P_U\to U\) is smooth.

Proof

Perfect, Lemma 0B91 shows that \[K=Rf_*\mathcal O_X\] is perfect and that its formation commutes with arbitrary base change. Hence Perfect, Lemma 0BDI shows that \[t\longmapsto \dim_{\kappa(t)}H^2(X_t,\mathcal O_{X_t})\] is upper semicontinuous. After shrinking around \(s\), we may assume that this dimension is zero everywhere on \(S\).

Fix \(t\in S\). We apply the infinitesimal lifting criterion at the identity point of \(P_t\). For a small extension of local Artinian \(\kappa(t)\)-algebras with kernel \(I\), the obstruction to lifting an infinitesimal invertible module lies in \[H^2(X_t,\mathcal O_{X_t})\otimes_{\kappa(t)}I\] by More on Morphisms, Lemma 0C6R. It vanishes by our choice of \(S\). Thus \(P\to S\) is formally smooth at every point of the identity section.

Theorem moduli-theorem-picard-multiplication-etale shows that \(P\to S\) is locally of finite presentation. More on Morphisms, Lemma 02H6 makes it smooth along the identity section. Finally, after base change to the residue field of any point of \(P\), translation by that point carries the identity to the given point. Smoothness is preserved by base change and descends under field extensions, so \(P\to S\) is smooth everywhere.

Lemma

Let \(X\) be a proper scheme over a field \(k\), and suppose that its fppf Picard functor is represented by a scheme \[P=\Picardfunctor_{X/k}.\] Then \[\dim(P)\leq\dim_k H^1(X,\mathcal O_X).\] Equality holds if and only if \(P\to\Spec(k)\) is smooth. In particular, equality always holds when \(k\) has characteristic zero.

Proof

Apply More on Morphisms, Lemma 0C6R to \[X\subset X_{k[\epsilon]/(\epsilon^2)}.\] The map on global units is surjective, so the tangent space of \(P\) at its identity \(e\) is canonically \[T_eP=H^1(X,\mathcal O_X).\] Every fibrewise component of a locally finite type group scheme has the dimension of its identity component. Therefore \[\dim(P)=\dim_e(P)\leq\dim_k(T_eP),\] which proves the inequality.

If \(P\) is smooth, equality is immediate. Conversely, suppose equality holds. Both dimension and tangent-space dimension are preserved by extending \(k\) to an algebraic closure. The local ring at the identity of the resulting group scheme is therefore regular. Translation makes the whole group scheme regular, and over an algebraically closed field it is smooth. Smoothness descends to \(k\), proving the converse. Finally, in characteristic zero every locally algebraic group scheme is smooth by Groupoids, Lemma 047N.

Example

Set \[S = \Spec(\mathbf{R}[[t]])\] and let \(X \subset \mathbf{P}^2_S\) be the closed subscheme defined by \[\begin{equation} x^2 + y^2 = tz^2. \end{equation}\] The morphism \(X \to S\) is a projective flat family of curves with geometrically reduced and geometrically connected fibres. Its fppf Picard functor is an algebraic space, but it is not a scheme.

Proof

Let \(S' = \Spec(\mathbf{C}[[t]])\). The morphism \(S' \to S\) is finite étale. The explicit Picard calculation in the cited reference represents \(\Picardfunctor_{X_{S'}/S'}\) by a disjoint union of copies of a scheme obtained from the trait \(S'\) by replacing its closed point by infinitely many origins. The nontrivial element of \(\operatorname{Gal}(\mathbf{C}/\mathbf{R})\) exchanges some of these origins. The resulting descent datum is not effective in schemes: some of its two-point orbits have no common affine open neighbourhood.

On the other hand, fppf descent data for algebraic spaces are effective by Bootstrap, Lemma 0ADV. Hence the same descent datum produces an algebraic space over \(S\). Since formation of the fppf Picard functor commutes with base change, the descended algebraic space represents \(\Picardfunctor_{X/S}\). If it were a scheme, its pullback and descent datum would give the scheme whose nonexistence was just established. Thus it is not a scheme.

Remark

The historical adjective “separable” for the family in Example moduli-example-picard-conic-not-scheme means flat with geometrically reduced fibres. Thus neither projectivity nor this fibrewise condition makes the relative Picard algebraic space a scheme.

The cited source also records Mumford’s positive scheme-level result: if a projective flat family has geometrically reduced fibres and every irreducible component of every fibre is geometrically irreducible over its residue field, then its Picard functor is represented by a scheme. This uses a strengthened effective-quotient theorem and is not a consequence of algebraic-space representability alone. The same discussion points out that the union of torsion components can still be representable by a scheme when the full Picard functor is not; the relative torsion-component loci are studied in More on Morphisms, Section 055K.

Example

Let \(N \geq 3\), let \(M\) be a connected fine modular curve over \(\mathbf{C}\) parametrizing elliptic curves with full level-\(N\) structure, and let \(E \to M\) be the universal elliptic curve. Set \[X=E\times_M E.\] Then \(X \to M\) is smooth and projective with geometrically connected fibres, but the relative Picard scheme \(\Picardfunctor_{X/M} \to M\) is not universally open.

Proof

For a geometric point \(s\) of \(M\), the Néron–Severi group of \(E_s\times E_s\) identifies, through the product principal polarization, with the symmetric homomorphisms \[E_s\times E_s \longrightarrow (E_s\times E_s)^\vee.\] At a non-CM point, where \(\operatorname{End}(E_s)=\mathbf{Z}\), this group has rank \(3\): it is generated rationally by the two factors and the diagonal. At a CM point it has rank \(4\). Concretely, the graph of a complex multiplication endomorphism supplies an additional divisor class.

That additional class cannot extend to a neighbourhood of the CM point. If it did, the associated symmetric homomorphism of the abelian surface would extend and would restrict on the geometric generic fibre to an endomorphism not contained in the symmetric matrices over \(\mathbf{Z}\), whereas the generic elliptic curve has endomorphism ring \(\mathbf{Z}\). Consequently the connected component of the special Picard fibre indexed by this class does not meet a neighbouring fibre. Its closure is a vertical irreducible component of the relative Picard scheme, and an open neighbourhood of its generic point has image supported at the CM point. Thus the Picard morphism is not open there, hence is not universally open.

Passing to the level cover is the scheme-theoretic version of the source’s shorthand “modular family over the \(j\)-line”; it removes the automorphisms which prevent a universal elliptic curve from existing over the coarse \(j\)-line itself.

Remark

Let \(X\) be a smooth projective irreducible scheme over a field \(k\), let \(Y \subset X\) be a hyperplane section, and let \(Y_n\) be its \(n\)th infinitesimal neighbourhood. The classical Lefschetz statement recorded in the source says that for \(n\) sufficiently large the morphism \[\begin{equation} \Picardfunctor_{X/k} \longrightarrow \Picardfunctor_{Y_n/k} \end{equation}\] is an isomorphism if \(\dim(X) \geq 4\). If \(\dim(X) \geq 3\), it induces an isomorphism on the inverse images of the torsion subgroups of the Néron–Severi groups.

This is precisely a setting in which the nilpotent Picard spaces in Proposition moduli-proposition-picard-nilpotent-thickening-affine carry information that the reduced hyperplane alone does not record. The current formal-coherent infrastructure is Algebraization, Proposition 0EL1 and Proposition 0EL7; the deformation of invertible modules between successive neighbourhoods is controlled by More on Morphisms, Lemma 0C6R.

The printed source says that \(X\) is “regular”. The linked correction replaces this by “smooth over \(k\)”, which is the hypothesis used here. The subsequent equal-characteristic description of two-dimensional local class groups by algebraic groups and the proposed higher-dimensional pro-algebraic extension are historical applications and expectations; they are not needed for the statements above.

Canonical abelian Picard subschemes and Albanese torsors

Let \(P\to S\) be a commutative group scheme. Write \[P^0=\bigcup_{s\in S}P_s^0\] for its fibrewise identity-component locus.

Definition

Let \(X\to S\) be a proper flat morphism whose fppf Picard functor is represented by a scheme \(P=\Picardfunctor_{X/S}\). A canonical abelian Picard subscheme is a closed subgroup scheme \(A\subset P\) such that \(A\to S\) is proper and smooth with geometrically connected fibres and \(|A|=P^0\); in other words, \(A\to S\) is an abelian scheme.

Over a field, Lemma moduli-corollary-picard-variety-normal-proper constructs this subgroup under geometric normality. Over a general base, uniqueness is the first rigidity property.

Lemma

Let \(X\to S\) be a proper flat morphism whose fppf Picard functor is represented by a scheme \(P=\Picardfunctor_{X/S}\). A canonical abelian Picard subscheme of \(P\) is unique if it exists. If it exists, then for every morphism \(S'\to S\) its base change is the canonical abelian Picard subscheme of \(P_{S'}=\Picardfunctor_{X_{S'}/S'}\).

Proof

We use the rigidity theorem for abelian subgroup schemes of a commutative group scheme: over a connected base, two such subgroup schemes which agree on one geometric fibre agree everywhere. This is the relative rigidity lemma for abelian schemes. One way to prove it is to apply the rigidity lemma to the difference morphism; the coincidence locus is open and closed in the base. The infinitesimal step uses that a homomorphism from a proper connected group scheme to a vector group is zero, so the statement also holds over a nonreduced base.

Suppose that \(A,A'\subset P\) are canonical abelian Picard subschemes. For every geometric point \(\overline{s}\to S\), the fibres \(A_{\overline{s}}\) and \(A'_{\overline{s}}\) are smooth and hence reduced closed subschemes of \(P_{\overline{s}}\) with the same underlying space \(|P^0_{\overline{s}}|\). Thus they agree as closed subschemes. Choosing one geometric point on each connected component of \(S\) and applying rigidity gives \(A=A'\).

For a morphism \(S'\to S\), the base change \(A_{S'}\) is again an abelian scheme and a closed subgroup scheme of \(P_{S'}\). Fibre by fibre its underlying space is the identity component of \(P_{S'}\). It is therefore canonical, and the uniqueness just proved gives compatibility with iterated base change.

Proposition

Let \(S\) be a locally Noetherian scheme and let \(f : X \to S\) be proper and flat. Assume that \[\mathcal{O}_S \longrightarrow f_*\mathcal{O}_X\] is an isomorphism, that the fppf Picard functor is represented by a scheme \(P = \Picardfunctor_{X/S}\) locally of finite type over \(S\), and that the fibrewise identity components \(P_s^0\) are proper. Suppose also that an open \(U \subset P\) containing \(P^0\) is quasi-projective over \(S\).

The functor which sends \(S' \to S\) to the set of canonical abelian Picard subschemes of \(P_{S'}\) is represented by a scheme \(q : T \to S\) of finite type. The morphism \(q\) is a surjective monomorphism. There is a universal canonical abelian Picard subscheme over \(T\), and a canonical abelian Picard subscheme over \(S'\) exists if and only if \(T_{S'} \to S'\) has a section. When it has a section, it is an isomorphism.

Proof

Every canonical abelian Picard subscheme is supported on \(P^0\) and hence is contained in \(U\). It is therefore a point of the Hilbert functor \(H = \Hilbfunctor_{U/S}\). We recall the standard Hilbert-scheme construction of the required subfunctor. Over an étale scheme cover of \(H\), let \(Z \subset U_H\) be the universal closed subscheme. Impose the following base-change-stable conditions:

1. \(Z\) is smooth and proper with geometrically connected fibres;

2. the identity, inverse, and multiplication of \(P\) restrict to \(Z\); and

3. for every geometric point \(\overline{s}\) of the base, the underlying space of \(Z_{\overline{s}}\) is \(P^0_{\overline{s}}\).

Smoothness and geometric connectedness give open conditions on the proper flat universal family. The subgroup conditions are the vanishing loci of the maps from the universal ideal under the identity, inverse, and multiplication maps. The last condition is obtained by the usual fibrewise open-and-closed decomposition and flattening stratification. These loci are compatible on the étale overlaps, so they descend to a locally closed subfunctor of \(H\).

Because \(U\) is quasi-projective, the pieces of \(H\) with a fixed Hilbert polynomial are schemes; see Proposition moduli-proposition-quot-projective-over-base and Lemma 0DPI. Locally on the Noetherian base, only finitely many of these pieces meet the locus just constructed: its geometric member is the reduced proper identity component, and the identity-component locus is quasi-compact over the base. Thus the subfunctor is represented by a scheme \(T\) of finite type over \(S\).

Lemma moduli-lemma-canonical-abelian-picard-unique says that every base change has at most one such subgroup. Hence \(q : T \to S\) is a monomorphism. For a field-valued point \(s \to S\), the proper group scheme \(P_s^0\) has no additive subgroup. Groupoids, Lemma groupoids-lemma-reduced-identity-component-no-additive shows that \((P_s^0)_{\mathrm{red}}\) is a smooth connected proper closed subgroup and hence an abelian variety. Thus every fibre of \(q\) is nonempty, so \(q\) is surjective. The universal property gives the final assertions. A section of a monomorphism is its inverse.

Theorem

In the situation of Proposition moduli-proposition-canonical-abelian-picard-parameter, a canonical abelian Picard subscheme exists in each of the following cases.

1. It exists after every base change \(S' \to S\) with \(S'\) local Artinian.

2. If \(S = \Spec(A)\) is local, it exists after base change to \(\Spec(A/\mathfrak m^{n + 1})\) for every \(n \geq 0\).

3. The scheme \(S\) is reduced and it exists after every base change \(\Spec(V) \to S\) with \(V\) a discrete valuation ring. It is enough to use complete discrete valuation rings with algebraically closed residue field.

Proof

Let \(q : T \to S\) be the surjective finite type monomorphism of Proposition moduli-proposition-canonical-abelian-picard-parameter. In case (1), apply the hypothesis to \[S_{s, n} = \Spec(\mathcal{O}_{S, s}/\mathfrak m_s^{n + 1}).\] The resulting canonical subgroup gives a section of \(T_{S_{s, n}} \to S_{s, n}\), and a monomorphism with a section is an isomorphism. Case (2) gives the same conclusion directly for the stated infinitesimal neighbourhoods. In either case, Étale Morphisms, Lemma etale-lemma-surjective-monomorphism-tests shows that \(q\) is an isomorphism.

In case (3), the canonical subgroup over \(\Spec(V)\) gives a section of \(T_V \to \Spec(V)\). Part (2) of the same lemma again shows that \(q\) is an isomorphism. Pulling back the universal subgroup over \(T\) along the inverse \(S \to T\) gives the required canonical abelian Picard subscheme.

For the final refinement in (3), every discrete valuation ring \(V\) has an extension \(V \subset V'\) with \(V'\) complete and with algebraically closed residue field. The restricted hypothesis gives the lift after this extension. Since a monomorphism is separated, Morphisms of Spaces, Lemma 0ARH pushes the lift down to \(V\). Thus the restricted tests imply the unrestricted tests already used above.

Theorem

Let \(C \to S\) be an abelian scheme. The identity component of its rigidified Picard functor is represented by an abelian scheme \[C^\vee = (\Picardfunctor_{C/S})^0,\] called the dual abelian scheme. There is a normalized Poincaré invertible module \(\mathcal{P}_C\) on \(C \times_S C^\vee\). It induces a canonical isomorphism \[C \longrightarrow (C^\vee)^\vee.\] These constructions commute with arbitrary base change. A homomorphism \(a : C \to D\) has a dual homomorphism \[a^\vee : D^\vee \longrightarrow C^\vee\] given by pullback of invertible modules, and biduality identifies \(a^{\vee\vee}\) with \(a\).

Proof

This is the duality theorem for abelian schemes. We recall its construction. The relative Picard scheme exists because an abelian scheme is projective, smooth, and has geometrically integral fibres. Rigidification along the zero section removes the ambiguity of tensoring by an invertible module from the base and produces the universal normalized Poincaré module. The theorem of the cube shows that the fibrewise algebraically trivial locus is a proper smooth group scheme and that the Poincaré module is additive in each variable. Thus this locus is the abelian scheme \(C^\vee\).

Viewing the same module in the other variable gives the displayed homomorphism \(C \to (C^\vee)^\vee\). On every geometric fibre this is the classical biduality isomorphism for an abelian variety. A morphism between abelian schemes which is an isomorphism on every geometric fibre is an isomorphism, proving relative biduality. Pullback of the Poincaré module gives the dual of a homomorphism. The universal rigidified construction proves both bidual functoriality and compatibility with base change.

Lemma

Let \(P \to S\) be a commutative group scheme locally of finite type over a locally Noetherian scheme. Suppose that \(A \subset P\) is a canonical abelian subscheme. If \(D \to S\) is an abelian scheme, then every homomorphism \(D \to P\) factors uniquely through \(A\).

Proof

On a geometric fibre, the image of the connected scheme \(D_{\overline{s}}\) is contained in the identity component of \(P_{\overline{s}}\). Since \(D_{\overline{s}}\) is reduced and \(A_{\overline{s}}\) is the reduced closed subscheme with that underlying space, the homomorphism factors through \(A_{\overline{s}}\).

It remains to exclude an infinitesimal failure of factorization. This is local on \(S\). Induct through square-zero thickenings from the reduction of a Noetherian affine open. At each step, the obstruction is a homomorphism from the abelian scheme on the reduced base to the vector group associated to the conormal module of \(A\) in \(P\). Such a homomorphism is zero: a morphism from a proper geometrically connected scheme to an affine vector group is constant, and a group homomorphism takes the identity to zero. The fibrewise factorization therefore lifts through every infinitesimal layer. Uniqueness follows because \(A \to P\) is a monomorphism.

Proposition

Let \(f : X \to S\) be proper and flat, with \(\mathcal{O}_S \to f_*\mathcal{O}_X\) an isomorphism, and suppose that \(P = \Picardfunctor_{X/S}\) is represented by a scheme. Let \(C \to S\) be an abelian scheme. There is a natural bijection between

1. isomorphism classes of pairs \((Q, v)\), where \(Q \to S\) is a \(C\)-torsor and \(v : X \to Q\) is an \(S\)-morphism, and

2. homomorphisms of group schemes \(C^\vee \to P\).

The bijection is functorial in \(C\) and commutes with arbitrary base change.

Proof

Let \((Q, v)\) be as in (1). Fppf locally on \(S\), choose a section of \(Q\) and use it to identify \(Q\) with \(C\). Pulling the normalized Poincaré module back by \[v \times 1 : X \times_S C^\vee \longrightarrow C \times_S C^\vee\] gives a family of invertible modules on \(X\), parametrized by \(C^\vee\). The biextension identities for the Poincaré module make the resulting morphism \(C^\vee \to P\) a homomorphism. Changing the chosen section translates \(v : X \to C\). Translation changes the pulled-back Poincaré module only by an invertible module from \(C^\vee\), which is invisible in the relative Picard functor. The local homomorphisms therefore descend and depend only on \((Q, v)\).

Conversely, let \(\varphi : C^\vee \to P\) be a homomorphism. The Picard stack supplies fppf locally a representing invertible module \(\mathcal{M}\) on \(X \times_S C^\vee\), normalized over the zero of \(C^\vee\). The morphism \(X \to S\) has geometrically connected fibres and is surjective by the global-functions hypothesis and Stein factorization, More on Morphisms, Theorem 03H0. It is therefore an fppf covering, so after an fppf base change we may choose a section \(x_0\) of \(X\). For a variable point \(x\) of \(X\), the class \[\mathcal{M}|_{\{x\} \times C^\vee} \otimes \left(\mathcal{M}|_{\{x_0\} \times C^\vee}\right)^{-1}\] lies in the identity component because the fibres of \(X\) are connected. It therefore belongs to \((C^\vee)^\vee = C\) and defines a morphism \(v_{x_0} : X \to C\). If \(x_1\) is another local section, then \[v_{x_1}(x) = v_{x_0}(x) - v_{x_0}(x_1).\] Consequently the trivial \(C\)-torsors carrying the maps \(v_{x_0}\) glue by translations to a \(C\)-torsor \(Q\), and the maps glue to \(v : X \to Q\). Changing \(\mathcal{M}\) by an invertible module from \(C^\vee\) cancels in the displayed quotient, so the construction depends only on \(\varphi\).

The two constructions are inverse after an fppf base change on which the torsor and the required invertible modules are trivialized; hence they are inverse over \(S\) by descent. Every operation used is compatible with base change, and the Poincaré biextension makes the correspondence functorial in \(C\).

Theorem

In the situation of Proposition moduli-proposition-canonical-abelian-picard-parameter, suppose that the canonical abelian Picard subscheme \(A \subset P\) exists. Put \[\operatorname{Alb}^0(X/S) = B = A^\vee.\] There is a canonical \(B\)-torsor \(\operatorname{Alb}^1(X/S)\) and a canonical morphism \[u : X \longrightarrow \operatorname{Alb}^1(X/S)\] with the following universal property. If \(C \to S\) is an abelian scheme, \(Q\) is a \(C\)-torsor, and \(v : X \to Q\) is an \(S\)-morphism, then there is a unique homomorphism \(h : B \to C\) such that \((Q, v)\) is the pushout of \((\operatorname{Alb}^1(X/S), u)\) along \(h\). Equivalently, there is a unique \(h\)-equivariant morphism of torsors \[\operatorname{Alb}^1(X/S) \longrightarrow Q\] whose composition with \(u\) is \(v\). The formation of \(B\), its torsor, and \(u\) commutes with arbitrary base change.

Proof

By biduality, \(B^\vee = A\). Apply Proposition moduli-proposition-maps-to-abelian-torsors to the inclusion \[B^\vee = A \longrightarrow P.\] The corresponding pair is, by definition, \((\operatorname{Alb}^1(X/S), u)\).

Now let \((Q, v)\) be a pair under an abelian scheme \(C\). The same proposition associates to it a homomorphism \[\varphi : C^\vee \longrightarrow P.\] Lemma moduli-lemma-abelian-map-factors-canonical-picard gives a unique factorization \(C^\vee \to A \to P\). Duality turns its first arrow into the unique homomorphism \(h : B = A^\vee \to C\). Functoriality of the torsor–Picard correspondence identifies \((Q, v)\) with the pushout along \(h\). This proves the universal property.

The canonical subgroup \(A\) commutes with base change by Lemma moduli-lemma-canonical-abelian-picard-unique. Dual abelian schemes, Poincaré modules, the torsor–Picard correspondence, and pushout of torsors all commute with base change. Hence so do \(B\), \(\operatorname{Alb}^1(X/S)\), and \(u\).

Remark

The canonical abelian Picard subscheme of Definition moduli-definition-canonical-abelian-picard-subscheme is denoted \(\Picardfunctor_{X/S}^{\circ\circ}\) in the cited source. The source first records that this object need not exist over a nonreduced base: the first-order modular deformation of an Igusa surface gives a counterexample. The construction of that example is not included there; compare Remark moduli-remark-picard-torsion-flatness-boundary for the related failure of flatness of the torsion-component locus.

Here is the specialization picture recorded in the source. Let \(S = \Spec(V)\) for a discrete valuation ring with fraction field \(K\) and residue field \(k\), and assume that \(X \to S\) is smooth in addition to the hypotheses of Proposition moduli-proposition-canonical-abelian-picard-parameter. Put \[A_K = (P_K^0)_{\mathrm{red}}.\] The source invokes a theorem of Koizumi to obtain an essentially unique abelian scheme \(G \to S\) with generic fibre \(A_K\). It then extends the generic-fibre inclusion to a homomorphism \[G \longrightarrow P\] and records that the induced map \[\begin{equation} G_k \longrightarrow (P_k^0)_{\mathrm{red}} \end{equation}\] is surjective with finite \(p\)-primary kernel, where \(p\) is the characteristic exponent of \(k\).

Writing \(S_n = \Spec(V/\mathfrak m^{n + 1})\), the source states that the following conditions are equivalent:

1. the map (moduli-equation-canonical-picard-specialization) is an isomorphism;

2. the canonical abelian Picard subscheme over \(S\) exists, in which case it is \(G\); and

3. the canonical abelian Picard subscheme over every \(S_n\) exists.

The equivalence of the last two conditions is now the local infinitesimal criterion in Theorem moduli-theorem-canonical-abelian-picard-existence-tests. Since a finite \(1\)-primary group is trivial, the recorded description of the kernel makes the first condition automatic in residue characteristic zero.

Two historical qualifications are important. The assertion that canonical abelian Picard subschemes should exist over every reduced base is posed in the source as a possibility, not proved; Theorem moduli-theorem-canonical-abelian-picard-existence-tests reduces such an assertion to discrete-valuation tests but does not supply those tests. Also, the Koizumi extension and the finite-primary-kernel assertion above are reported with their source attribution rather than reproved here. Finally, if \(P \to S\) is smooth along \(P^0\), then the identity-component locus is open by More on Morphisms, Corollary more-morphisms-corollary-identity-component-open-smooth; under the standing properness hypotheses it is the canonical abelian subscheme. The next theorem in the source gives a stronger neighbourhood criterion for this situation.

Lemma

Let \(B \to A\) be a surjection of local Artinian rings with common residue field \(k\). Assume that its kernel \(I\) satisfies \(\mathfrak m_BI = 0\). Let \[X_B \longrightarrow \Spec(B)\] satisfy the hypotheses of Proposition moduli-proposition-canonical-abelian-picard-parameter. Write \(P_B = \Picardfunctor_{X_B/B}\) and let \(P_A\) and \(P_k\) denote its base changes. If \(P_k\) is smooth and \(P_A\) has a canonical abelian Picard subscheme \(G_A\), then \(P_B\) has a canonical abelian Picard subscheme \(G_B\) whose base change to \(A\) is \(G_A\).

Proof

Put \(G_0 = (G_A)_k\). Since \(P_k\) is smooth, \(G_0=P_k^0\) as schemes. Examples of Deformation Problems, Lemma examples-defos-lemma-abelian-variety-formal-moduli-smooth shows that \(G_A\) lifts to an abelian scheme over \(B\). The set of isomorphism classes of such lifts is a torsor under \[\begin{equation} H^1(G_0, \mathcal{T}_{G_0/k}) \otimes_k I. \end{equation}\] This is the usual difference space for lifts of a smooth scheme; forgetting the group law does not change it by the final assertion of the cited lemma.

Choose a lift \(G_B\). The inclusion \(G_A \to P_A\) is represented, fppf locally, by a Poincaré invertible module \(\mathcal{L}_A\) on \(X_A \times_A G_A\). It is normalized over the zero of \(G_A\). The calculation below is unchanged if \(\mathcal{L}_A\) is tensored by an invertible module from \(G_A\), and thus descends from the fppf cover on which a representative has been chosen. By More on Morphisms, Lemma 0C6R, the obstruction to lifting \(\mathcal{L}_A\) to \(X_B \times_B G_B\) lies in \[H^2(X_k \times_k G_0, \mathcal{O}_{X_k \times_k G_0}) \otimes_k I.\] Its restriction to \(X_k \times \{0\}\) is zero by the normalization. The pure \(G_0\) component is the obstruction to the restriction of \(\mathcal{L}_A\) on the other factor. Tensoring \(\mathcal{L}_A\) by the inverse of this restriction pulled back from \(G_A\) removes that component without changing the morphism to the Picard functor. Consequently the Kunneth decomposition of Varieties, Lemma 0BED puts the effective obstruction in \[\begin{equation} H^1(X_k, \mathcal{O}_{X_k}) \otimes_k H^1(G_0, \mathcal{O}_{G_0}) \otimes_k I. \end{equation}\]

The tangent-space calculation in Lemma moduli-lemma-picard-tangent-dimension gives \[T_0G_0 = T_0P_k = H^1(X_k, \mathcal{O}_{X_k}).\] The tangent bundle of an abelian variety is the constant bundle with this fibre by Groupoids, Lemma 047I. Hence \[\begin{equation} H^1(G_0, \mathcal{T}_{G_0/k}) \otimes_k I = H^1(X_k, \mathcal{O}_{X_k}) \otimes_k H^1(G_0, \mathcal{O}_{G_0}) \otimes_k I. \end{equation}\] Under this equality, changing \(G_B\) by a class in (moduli-equation-choices-lift-canonical-picard) changes the obstruction by the same class up to the sign determined by the Cech convention. Indeed, represent the deformation class by a Cech cocycle of vector fields on \(G_0\). Altering the gluing maps of \(G_B\) adds to the obstruction cocycle for \(\mathcal{L}_A\) its contraction with the differential of \(G_0 \to P_k\). That differential is the identity under the displayed tangent-space identification, so the variation map is an isomorphism. We may therefore alter \(G_B\) so that the obstruction vanishes, and then lift \(\mathcal{L}_A\).

The lifted class defines a morphism \(G_B \to P_B\). It is a homomorphism: its failure to respect addition is zero over \(A\) and hence gives, on the special fibre, a morphism from the proper connected scheme \(G_0 \times_k G_0\) to a vector group; this morphism is constant and normalization at the identity makes it zero. The morphism is a closed immersion. Indeed, its underlying topological image lies in the quasi-projective neighbourhood used in Proposition moduli-proposition-canonical-abelian-picard-parameter. In that separated neighbourhood it is proper, and it is quasi-finite because its special fibre is a closed immersion and the base is a nilpotent thickening. It is therefore finite. Nakayama’s lemma, applied to the cokernel of the map on structure sheaves, now reduces the closed-immersion assertion to the special fibre. Finally \(\Spec(B)\) and \(\Spec(k)\) have the same underlying space, so the image has underlying space \(P_B^0\). It is the required canonical abelian Picard subscheme.

Theorem

In the situation of Proposition moduli-proposition-canonical-abelian-picard-parameter, let \(s \in S\). If \(P_s\) is smooth over \(\kappa(s)\), or equivalently if \[\dim(P_s)=\dim_{\kappa(s)}H^1(X_s,\mathcal{O}_{X_s}),\] then there is an open neighbourhood \(U \subset S\) of \(s\) such that \(P_U\) is smooth along \(P_U^0\). Moreover, \(P_U^0\) with its induced scheme structure is an open abelian subscheme of \(P_U\) and is the canonical abelian Picard subscheme.

Proof

The equivalence of the two hypotheses is Lemma moduli-lemma-picard-tangent-dimension. Let \(q : T \to S\) be the surjective finite type monomorphism of Proposition moduli-proposition-canonical-abelian-picard-parameter, and let \(t\) be the unique point of \(T\) over \(s\). Put \[S_n=\Spec(\mathcal{O}_{S,s}/\mathfrak m_s^{n+1}).\] The smooth group scheme \(P_s^0\) is the canonical abelian Picard subscheme over \(S_0\). Lemma moduli-lemma-lift-canonical-abelian-picard, applied successively to \(S_{n+1} \to S_n\), constructs the canonical subgroup over every \(S_n\). Thus \(T_{S_n} \to S_n\) has a section and, being a monomorphism, is an isomorphism.

We use the local part of the argument in Étale Morphisms, Lemma etale-lemma-surjective-monomorphism-tests. Set \[C=\mathcal{O}_{S,s}, \qquad D=\mathcal{O}_{T,t}.\] The preceding isomorphisms identify the residue fields and give \[C/\mathfrak m_C^{n+1} \longrightarrow D/\mathfrak m_C^{n+1}D\] as an isomorphism for every \(n\). Hence \(C^\wedge \to D^\wedge\) is an isomorphism. The rings are Noetherian and \(D\) is essentially of finite type over \(C\), so Étale Morphisms, Lemma 039M shows that \(q\) is étale at \(t\). An étale monomorphism is an open immersion. After shrinking \(S\) around \(s\), surjectivity and uniqueness of the points of \(T\) over \(S\) therefore make \(q\) an isomorphism. Pulling back the universal subgroup gives a canonical abelian Picard subscheme \[G \subset P\] over this neighbourhood.

It remains to see that \(G\) is open in \(P\). On the fibre over \(s\), we have \(G_s=P_s^0\) as schemes, and this identity component is open in the smooth group scheme \(P_s\). Thus the ideal of \(G\) in \(P\) vanishes on an open neighbourhood of \(G_s\) in \(P_s\). Nakayama’s lemma gives an open \(W \subset P\) containing \(G_s\) on which this ideal is zero, so \(W \subset G\). Since \(G\) is proper over \(S\), after shrinking once more around \(s\) we have \(G \subset W\). Consequently \(G=W\) is open as well as closed in \(P\). Its underlying space is \(P^0\), and its smoothness over \(S\) proves that \(P\) is smooth along this locus.

Lemma

In the situation of Theorem moduli-theorem-picard-smooth-near-smooth-fibre, after replacing \(S\) by the open neighbourhood supplied there, \(R^1f_*\mathcal{O}_X\) is a finite locally free \(\mathcal{O}_S\)-module. Its formation commutes with arbitrary base change: if \(g : S' \to S\), \(X'=X\times_S S'\), and \(f' : X' \to S'\), then the canonical map \[g^*R^1f_*\mathcal{O}_X \longrightarrow R^1f'_*\mathcal{O}_{X'}\] is an isomorphism.

Proof

Let \(e : S \to P\) be the identity section and put \[\mathcal{E}= \SheafHom_{\mathcal{O}_S}(e^*\Omega_{P/S},\mathcal{O}_S).\] Theorem moduli-theorem-picard-smooth-near-smooth-fibre makes \(P \to S\) smooth along \(e\). Thus \(\mathcal{E}\) is finite locally free, and its formation commutes with arbitrary base change.

We identify this tangent module. Let \(T \to S\) be affine, write \(f_T : X_T \to T\), and let \(T[\epsilon]\) be the first order thickening with ideal \(\epsilon\mathcal{O}_T\). Sections of \(\mathcal{E}_T\) are the morphisms \[T[\epsilon] \longrightarrow P_T\] whose restriction to \(T\) is the identity section. Since \(P_T\) represents the relative Picard functor, these are the infinitesimal invertible-module classes on \(X_T\). The exact sequence used in More on Morphisms, Lemma 0C6R identifies them with \[H^1(X_T,\mathcal{O}_{X_T}).\] Indeed, reduction is surjective on global units, and the contribution from invertible modules on the base vanishes because \(T\) is affine. On the other hand, \[H^1(X_T,\mathcal{O}_{X_T}) =\Gamma(T,R^1f_{T,*}\mathcal{O}_{X_T}).\] These identifications are additive and natural in \(T\). They therefore give a canonical isomorphism \[\mathcal{E}_T \longrightarrow R^1f_{T,*}\mathcal{O}_{X_T}\] for every base change \(T \to S\). For \(T=S\) this proves local freeness, and naturality identifies the displayed isomorphisms for general \(T\) with the canonical cohomology base change map. This proves the final assertion.

Lemma

In the situation of Proposition moduli-proposition-canonical-abelian-picard-parameter, let \(S_0 \subset S\) be the closed subscheme defined by a coherent nilpotent ideal. Suppose that the fibrewise identity-component locus of \(P_{S_0}\) is represented by an open subgroup scheme \[G_0 \subset P_{S_0}\] which is an abelian scheme over \(S_0\). Then the fibrewise identity-component locus of \(P\) is represented by an open abelian subscheme \(G \subset P\), and \(G_{S_0}=G_0\).

Proof

Filter the defining ideal by its powers. It is enough to treat a square-zero extension with ideal \(\mathcal{I}\). The obstruction calculation in Lemma moduli-lemma-lift-canonical-abelian-picard is relative over \(S_0\), as we now explain.

Write \(p : G_0 \to S_0\) and \(f_0 : X_{S_0} \to S_0\). The relative form of Examples of Deformation Problems, Lemma examples-defos-lemma-abelian-variety-formal-moduli-smooth gives, locally on \(S\), lifts of \(G_0\) as an abelian scheme. The sheaf of choices of a lift is a torsor under \[\begin{equation} R^1p_*\mathcal{T}_{G_0/S_0}\otimes_{\mathcal{O}_{S_0}}\mathcal{I}. \end{equation}\] Choose a lift locally. The obstruction to lifting the normalized Poincaré class which represents \(G_0 \to P_{S_0}\) has, after removing its two pure components, its value in \[\begin{equation} R^1f_{0,*}\mathcal{O}_{X_{S_0}} \otimes_{\mathcal{O}_{S_0}} R^1p_*\mathcal{O}_{G_0} \otimes_{\mathcal{O}_{S_0}}\mathcal{I}. \end{equation}\] This is the relative Kunneth decomposition of the obstruction from More on Morphisms, Lemma 0C6R.

The tangent calculation in the proof of Lemma moduli-corollary-picard-R1-locally-free-base-change identifies \[R^1f_{0,*}\mathcal{O}_{X_{S_0}} =e^*\mathcal{T}_{G_0/S_0}.\] Since the tangent bundle of an abelian scheme is pulled back from the identity section, the modules in (moduli-equation-relative-choices-lift-picard) and (moduli-equation-relative-obstruction-lift-picard) are canonically isomorphic. As in Lemma moduli-lemma-lift-canonical-abelian-picard, varying the lift of \(G_0\) changes the Poincaré obstruction through this isomorphism. There is therefore a unique correction which kills the obstruction. The lifted class gives a homomorphism from the corrected abelian scheme to \(P\) and the same properness, quasi-finiteness, and Nakayama argument makes it a canonical closed abelian Picard subscheme.

This construction is compatible with restriction. On overlaps the resulting subgroups agree by Lemma moduli-lemma-canonical-abelian-picard-unique; hence the local subgroups descend to a subgroup \(G \subset P\). Its reduction \(G_0\) is open in \(P_{S_0}\). Applying Nakayama’s lemma to the ideal of \(G\) on the corresponding open of \(P\) shows that this ideal is zero. Thus \(G\) is open in \(P\). Repeating the square-zero argument along the chosen filtration proves the lemma.

Lemma

Let \(f : A \to S\) be an abelian scheme. Cup product gives a canonical isomorphism of graded \(\mathcal{O}_S\)-algebras \[\bigwedge\nolimits^\bullet R^1f_*\mathcal{O}_A \longrightarrow \bigoplus_{i \geq 0}R^if_*\mathcal{O}_A.\] In particular, every \(R^if_*\mathcal{O}_A\) is finite locally free and its formation commutes with arbitrary base change. On an open and closed part of \(S\) where \(A\) has relative dimension \(g\), it has rank \(\binom{g}{i}\) and is zero for \(i>g\).

Proof

The dual abelian scheme of Theorem moduli-theorem-dual-abelian-scheme is smooth, so Lemma moduli-corollary-picard-R1-locally-free-base-change, applied to \(A\), first shows that \(R^1f_*\mathcal{O}_A\) is finite locally free and commutes with base change.

Put \(K=Rf_*\mathcal{O}_A\). Perfect, Lemma 0B91 says that \(K\) is perfect and commutes with arbitrary base change. The Rosenlicht–Serre fibre theorem recorded in the cited source says that for every geometric point \(\overline{s}\) of \(S\), cup product is an isomorphism \[\bigwedge\nolimits^i H^1(A_{\overline{s}},\mathcal{O}_{A_{\overline{s}}}) \longrightarrow H^i(A_{\overline{s}},\mathcal{O}_{A_{\overline{s}}}).\] Before knowing base change for the higher direct images, take the iterated cup product from the tensor power of \(R^1f_*\mathcal{O}_A\) to \(R^if_*\mathcal{O}_A\). After tensoring with a geometric residue field and then applying the canonical cohomology base change map, this is the fibrewise cup product. It is surjective by the displayed Rosenlicht–Serre isomorphism. Hence the canonical base change map \[R^if_*\mathcal{O}_A\otimes\kappa(\overline{s}) \longrightarrow H^i(A_{\overline{s}},\mathcal{O}_{A_{\overline{s}}})\] is surjective for every \(i\) and every geometric point \(\overline{s}\).

We recall the elementary perfect-complex consequence of this simultaneous surjectivity. Near a point of \(S\), More on Algebra, Lemma 0BCD represents \(K\) by a bounded complex \(M^\bullet\) of finite free modules whose differentials vanish on the residue field. Surjectivity of the base change map in degree \(i\) says that a basis of \(M^i\) modulo the maximal ideal lifts to cycles in \(M^i\). Those lifts form a basis after localization by Nakayama’s lemma, and therefore the differential \(M^i\to M^{i+1}\) is zero. Applying this in every degree shows that, locally on \(S\), all differentials of \(M^\bullet\) vanish. Consequently every \(R^if_*\mathcal{O}_A\) is finite locally free and cohomology commutes with arbitrary base change.

The cup products now give morphisms between finite locally free modules. The exterior relations hold on every geometric fibre by the Rosenlicht–Serre theorem, hence hold over \(S\), and the resulting maps \[\bigwedge\nolimits^iR^1f_*\mathcal{O}_A \longrightarrow R^if_*\mathcal{O}_A\] are isomorphisms on every geometric fibre. They are therefore isomorphisms. This also proves the assertions about ranks and vanishing.

Remark

The cited source records that if \(f : X \to S\) is projective and smooth, \(S\) is reduced, and all residue characteristics are zero, then classical Hodge theory gives local freeness of every \[R^pf_*\Omega^q_{X/S}.\] It contrasts this with mixed-characteristic counterexamples to local freeness of \(R^1f_*\mathcal{O}_X\) obtained from varieties of Serre. Its final sentence reports that no counterexample in equal characteristic seemed to be known. This is a report of the state of knowledge in 1962 and is not asserted here as a current nonexistence theorem.

The distinction remains visible in the modern derived formulation. Perfect, Lemma 0B91 makes the derived direct image of a vector bundle under a proper flat morphism perfect and compatible with arbitrary derived base change, while Cohomology of Schemes, Lemma 02KH gives ordinary cohomology base change along a flat morphism. Neither assertion by itself makes the individual cohomology sheaves locally free. The Hodge-to-de Rham spectral sequence is constructed in De Rham Cohomology, Section 0FM6; degeneration is additional input. The abelian-scheme case of Lemma moduli-lemma-abelian-scheme-cohomology-exterior is a separate positive result.

Lemma

In the situation of Theorem moduli-theorem-picard-smooth-near-smooth-fibre, after replacing \(S\) by the open neighbourhood supplied there, \(P \to S\) is smooth at every point of \(P^\sigma\) in the notation of More on Morphisms, Lemma more-morphisms-lemma-order-in-component-group-constructible.

Proof

Let \(x \in P^\sigma\) and let \(s\) be its image in \(S\). By definition there is an integer \(n \geq 1\), prime to the characteristic exponent of \(\kappa(s)\), such that \[[n](x) \in P_s^0.\] Theorem moduli-theorem-picard-smooth-near-smooth-fibre makes \(P^0\) an open abelian subscheme of \(P\), so \(P \to S\) is smooth at \([n](x)\). Theorem moduli-theorem-picard-multiplication-etale says that \([n] : P \to P\) is étale at \(x\). The structure morphism at \(x\) is the composite of this étale morphism with the structure morphism at \([n](x)\), and is therefore smooth.

Lemma

In the situation of Proposition moduli-proposition-canonical-abelian-picard-parameter, assume that every residue field of \(S\) has characteristic zero. Then \[P^\tau \longrightarrow S\] is smooth.

Proof

Let \(s \in S\). Lemma moduli-lemma-picard-tangent-dimension, applied to the proper scheme \(X_s\) over \(\kappa(s)\), shows that \(P_s\) is smooth. Theorem moduli-theorem-picard-smooth-near-smooth-fibre and Corollary moduli-corollary-picard-smooth-prime-to-characteristic-components therefore give an open neighbourhood \(U\) of \(s\) over which \(P\) is smooth along \(P^\sigma\). Since all residue characteristics are zero, More on Morphisms, Lemma more-morphisms-lemma-characteristic-zero-component-loci gives \[P^\tau=P^\sigma.\] Thus \(P^\tau\to S\) is smooth over a neighbourhood of every point of \(S\), which proves the assertion.

Lemma

Let \(S\) be a connected locally Noetherian scheme and let \(X\to S\) be a proper flat morphism of finite presentation. Suppose that the fppf Picard functor is represented by a scheme \[P=\Picardfunctor_{X/S}\] and that the open subgroup scheme \(P^\tau\) is proper over \(S\). Let \(q\) be a prime which is invertible on \(S\). For a geometric point \(\overline{s}\to S\), the isomorphism type of the \(q\)-primary subgroup of \[\mathop{\rm NS}(X_{\overline{s}})_{\rm tors} =P^\tau_{\overline{s}}(\overline{\kappa(s)})/ P^0_{\overline{s}}(\overline{\kappa(s)})\] is independent of \(\overline{s}\). If every residue field of \(S\) has characteristic zero, then the isomorphism type of the whole finite group \(\mathop{\rm NS}(X_{\overline{s}})_{\rm tors}\) is independent of \(\overline{s}\).

Proof

For \(m\geq 1\), put \[K_m=\Ker([q^m]:P^\tau\longrightarrow P^\tau).\] Theorem moduli-theorem-picard-multiplication-etale makes the displayed power map étale. It is also proper: both copies of \(P^\tau\) are proper over \(S\), and the graph followed by projection gives the usual proof. Hence it is finite étale, and so is \(K_m\to S\). Its rank, denoted \(r_m\), is constant because \(S\) is connected.

Fix \(\overline{s}\) and write \(k=\overline{\kappa(s)}\). The reduction \[A_{\overline{s}}=(P^0_{\overline{s}})_{\mathrm{red}}\] is an abelian variety over \(k\). Put \[C_{\overline{s}}= \mathop{\rm NS}(X_{\overline{s}})_{\rm tors}.\] This is a finite abelian group because \(P^\tau_{\overline{s}}\) is proper and hence has only finitely many connected components. For every \(m\) there is an exact sequence of finite abstract groups \[0\longrightarrow A_{\overline{s}}(k)[q^m] \longrightarrow K_m(k)\longrightarrow C_{\overline{s}}[q^m] \longrightarrow 0.\] Surjectivity on the right follows from the surjectivity of \([q^m]\) on the connected group \(P^0_{\overline{s}}\), see Groupoids, Lemma groupoids-lemma-connected-group-power-surjective. If \(g_{\overline{s}}=\dim(A_{\overline{s}})\), Groupoids, Proposition 03RP gives \[\begin{equation} r_m=q^{2m g_{\overline{s}}} \#C_{\overline{s}}[q^m]. \end{equation}\]

Let \(\overline{s}\) and \(\overline{t}\) be two geometric points. Choose \(M\) which is at least the exponents of the \(q\)-primary parts of both finite groups \(C_{\overline{s}}\) and \(C_{\overline{t}}\). For \(m\geq M\), the ratio \(r_{m+1}/r_m\) in (moduli-equation-picard-prime-primary-kernel-rank) is \(q^{2g_{\overline{s}}}\) and also \(q^{2g_{\overline{t}}}\). Thus the two dimensions agree. Formula (moduli-equation-picard-prime-primary-kernel-rank) then shows that \[\#C_{\overline{s}}[q^m]=\#C_{\overline{t}}[q^m]\] for every \(m\). These cardinalities determine the elementary divisors of a finite abelian \(q\)-group, proving the first assertion.

If all residue characteristics are zero, every prime is invertible on \(S\). The first assertion therefore identifies every primary part. Since each \(C_{\overline{s}}\) is finite, their products identify the whole torsion groups.

Remark

The prime-to-characteristic hypothesis in Lemma moduli-lemma-picard-Neron-Severi-prime-primary-constant is essential to its proof: residue-characteristic power maps need not be étale, and their kernels need not have locally constant rank. The cited source reports that in characteristic \(p>0\) the \(p\)-primary Néron–Severi torsion can vary. It does not include the construction of a counterexample, so that sentence is retained here as historical boundary evidence rather than promoted to a new example.

The source then asks whether the rank of the finite group scheme \[T_{X_s/\kappa(s)}= P^\tau_s/P^{\circ\circ}_s\] might nevertheless be locally constant. It reports that, over a reduced base, this question can be reformulated as the simultaneous existence of the canonical abelian Picard subscheme \(P^{\circ\circ}\) and flatness of \(P^\tau\), and that it is enough to test traits. This is posed as a possible statement, not a theorem of the source. Remark moduli-remark-canonical-abelian-picard-specialization records the corresponding discrete-valuation problem, while Remark moduli-remark-picard-torsion-flatness-boundary records the source’s Artinian Igusa warning. No local-constancy assertion from this final paragraph is used here.

Boundedness of Picard components

Example

Let \(R=k[[t]]\) and consider the flat projective family \[X=\mathop{\rm Proj}\bigl(R[x,y,z]/(xy-tz^2)\bigr) \longrightarrow\Spec(R)\] with its invertible module \(\mathcal{O}_X(1)\). The generic fibre is a smooth conic, while the special fibre is \[X_0=L_x\cup L_y,\] the union of two projective lines meeting in one point. For every pair \((a,b)\in\mathbf Z^2\), gluing \(\mathcal{O}_{L_x}(a)\) and \(\mathcal{O}_{L_y}(b)\) at the intersection point gives a unique isomorphism class of invertible module \(\mathcal{L}_{a,b}\) on \(X_0\). Its Hilbert polynomial is \[\chi(X_0,\mathcal{L}_{a,b}(m))=2m+a+b+1.\] Consequently, for every fixed integer \(d\), the Hilbert-polynomial locus \(Q_d(m)=2m+d+1\) in the Picard functor of the special fibre contains the infinite discrete set \[\{\mathcal{L}_{a,d-a}\mid a\in\mathbf Z\}.\] Thus the corresponding relative Hilbert-polynomial locus cannot be of finite type over \(R\).

Proof

Flatness follows because the defining quadratic is not divisible by \(t\). The assertions about the fibres follow directly from their equations. The normalization of \(X_0\) is \(L_x\amalg L_y\), and its dual graph is a tree. Hence an invertible module is determined by the two degrees of its pullbacks; the scalar used to identify the two fibres at the node is absorbed by an automorphism on one component. This proves the classification by \(\mathbf Z^2\).

Twisting the normalization sequence by \(\mathcal{L}_{a,b}(m)\) gives \[\chi(X_0,\mathcal{L}_{a,b}(m)) =(m+a+1)+(m+b+1)-1,\] which is the displayed polynomial. Each degree pair is a distinct connected component of the Picard scheme of \(X_0\). A finite-type fibre is quasi-compact, whereas the displayed fixed-polynomial locus is an infinite discrete union, proving the final assertion.

Remark

Let \(X\to S\) be projective and flat and fix a relatively ample invertible module. The Hilbert polynomial of an invertible module on a fibre is locally constant in flat families, so it cuts the Picard functor into open and closed subfunctors \(P^Q\). Example moduli-example-picard-conic-fixed-Hilbert-not-finite-type shows why these subfunctors need not be of finite type without additional hypotheses.

The cited source asks whether finite type holds for separable families with geometrically irreducible fibres and whether \(P^\tau\) is of finite type under still weaker hypotheses. It observes that for a smooth projective family \(P^\tau\) lies in a fixed numerical, hence Hilbert-polynomial, class. These sentences are questions and reductions at this point of the source, not completion claims.

The same issue can be stated without assuming representability of the Picard functor: one asks for a finite-type parameter space for invertible modules of fixed Hilbert polynomial on projective schemes in a fixed bounded family. The Hilbert and Quot numerical strata are organized in Remarks 0DPG and 0DP6; the regularity bound in Varieties, Lemma 08AG is the standard mechanism turning a fixed Hilbert polynomial into a finite-type parameter problem. Chow Homology identifies the intersection numbers used in the next theorem of the source with the corresponding cycle degrees. The theorem and its later supplement supply the source-specific boundedness conclusions sequentially below.

Theorem

Let \(S\) be a Noetherian scheme, let \(f:X\to S\) be smooth and projective with geometrically connected fibres, and let \(\mathcal{H}\) be an \(f\)-very ample invertible module. For geometric points \(\overline{s}_i\to S\), let \(\mathcal{M}_i\) be invertible modules on \(X_{\overline{s}_i}\), and let \(E\) be the corresponding set of points of \(\Picardstack_{X/S}\). The following are equivalent:

1. \(E\) is bounded: it is contained in a quasi-compact open substack of \(\Picardstack_{X/S}\);

2. the set of numerical polynomials \[Q_i(t)=\chi(X_{\overline{s}_i}, \mathcal{M}_i\otimes\mathcal{H}^{\otimes t}_{\overline{s}_i})\] is finite.

If \(f\) has pure relative dimension \(n\), these conditions are also equivalent to

1. the coefficients of \(t^{n-1}\) and \(t^{n-2}\) in the polynomials \(Q_i\) range over finite sets.

Proof

The Hilbert polynomial is locally constant in a flat family of coherent modules; see Example 0DNF. The fixed-polynomial substacks are therefore open and closed. A quasi-compact substack meets only finitely many members of this disjoint open covering, which proves (1) implies (2).

The nonformal input for the converse is Matsusaka’s boundedness estimate in the cited theorem. In its fixed-polynomial form it gives an integer \(m\), uniform for the finitely many polynomials under consideration, such that the modules \(\mathcal{M}_i(m)\) are generated by their global sections and have no higher cohomology. After choosing a basis of the \(Q_i(m)\) global sections, the evaluation maps exhibit these modules as points of finitely many fixed-polynomial Quot spaces. These Quot spaces are proper over \(S\) by Lemma 0DPC; the condition that the quotient be an invertible module is open. Since \(S\) is Noetherian, these open loci are of finite type. Dividing the frame spaces by the corresponding general linear groups gives a quasi-compact open substack of \(\Picardstack_{X/S}\) containing \(E\). This proves (2) implies (1).

When the relative dimension is \(n\), (2) immediately implies (3). The numerical part of Matsusaka’s estimate says conversely that, for invertible modules on the smooth members of this fixed projective family, finiteness of the two indicated coefficients permits only finitely many Hilbert polynomials. This proves (3) implies (2).

Example

Let \[X=\mathbf{P}^1_k\times_k\mathbf{P}^1_k, \qquad \mathcal{H}=\mathcal{O}_X(1,1).\] Write \(A\) and \(B\) for the two ruling classes, so that \(A^2=B^2=0\) and \(A B=1\). For \(m\geq 0\) put \[D_m=m(A-B),\qquad \mathcal{M}_m=\mathcal{O}_X(m,-m).\] Then \[\begin{equation} (A+B)D_m=0, \qquad D_m^2=-2m^2. \end{equation}\] Thus both intersection numbers occurring in the fourth condition of the printed theorem are bounded above. On the other hand, \[\chi(X,\mathcal{M}_m\otimes\mathcal{H}^{\otimes t}) =(t+m+1)(t-m+1)=t^2+2t+1-m^2.\] These Hilbert polynomials are pairwise distinct. Hence the family is not bounded, and the one-sided condition printed in the source is not equivalent to the conditions of Theorem moduli-theorem-picard-boundedness-Hilbert-polynomial.

The point is that the divisors in the theorem are not assumed effective, so their self-intersections can tend to minus infinity. Replacing the phrase bounded above by a two-sided condition would remove this counterexample, but no such replacement is attributed to the source here without a separate proof.

Lemma

Let \(S\) be a Noetherian scheme and let \(f:X\to S\) be smooth and projective with geometrically connected fibres. Fix an \(f\)-very ample invertible module \(\mathcal{H}\). The fppf Picard functor is represented by a separated scheme \[P=\Picardfunctor_{X/S}.\] For every numerical polynomial \(Q\), the open and closed Hilbert-polynomial piece \(P^Q\subset P\) is projective over \(S\). The torsion-component locus \(P^\tau\subset P\) is an open and closed projective \(S\)-subscheme.

Proof

The geometric fibres are integral: a smooth scheme over a field has disjoint irreducible components, and the fibres here are connected. Representability, separatedness, and quasi-projectivity of the finite-type Picard pieces follow from the classical projective Picard theorem, Remark moduli-remark-classical-projective-picard-theorem.

The numerical substacks of coherent modules are open and closed by Example 0DNF. Hence \(P^Q\) is open and closed in \(P\). Theorem moduli-theorem-picard-boundedness-Hilbert-polynomial, applied to the single polynomial \(Q\), shows that \(P^Q\) is of finite type over \(S\). It is proper by Theorem moduli-theorem-picard-smooth-proper-pieces, and therefore projective by Morphisms, Lemma 0BCL.

The locus \(P^\tau\) is open and closed by Theorem moduli-theorem-picard-normal-relative-components. A point of \(P^\tau\) has a positive multiple in the identity component. Its divisor class is therefore numerically trivial, and its Hilbert polynomial with respect to \(\mathcal H\) is the Hilbert polynomial of the structure sheaf of the same fibre. Since \(S\) is Noetherian, only finitely many such polynomials occur. Theorem moduli-theorem-picard-boundedness-Hilbert-polynomial now shows that \(P^\tau\) is of finite type. It is proper by Theorem moduli-theorem-picard-smooth-proper-pieces, quasi-projective by the classical Picard theorem, and hence projective by the same Morphisms lemma.

Finiteness and numerical equivalence for Picard spaces

The results in this section are the supplement to Exposé 236. The source works over Noetherian bases, with schemes proper over the base, and assumes that the relevant fppf Picard functors are represented by schemes. We state the needed hypotheses in each result below. The source observes that stack formulations remove the representability assumption.

Remark

The supplement first makes three corrections to the printed exposé. In the proof of Theorem 2.1(ii), the affine line is replaced by the punctured affine line and \(X[t]\) by \(X[t,t^{-1}]\); these corrections are built into Lemma moduli-lemma-pic-normal-laurent-invariance. In the proof of Proposition 3.1, the corrected phrase says that the conclusion follows from the hypothesis; the mathematical statement is Groupoids, Lemma groupoids-lemma-reduced-identity-component-no-additive. None of the three erroneous printed forms is used below.

Theorem

Let \(S\) be a Noetherian scheme. Let \(g : Y \to X\) be a surjective morphism of proper schemes over \(S\). Assume that the fppf Picard functors of \(X\) and \(Y\) over \(S\) are represented by schemes. Pullback of invertible modules defines a morphism \[g^* : \Picardfunctor_{X/S} \longrightarrow \Picardfunctor_{Y/S}\] of finite type. If \(S\) is the spectrum of a field, this morphism is affine.

Proof

Finite type is local on the target and stable under base change. After an fppf base change on \(\Picardfunctor_{Y/S}\), a Picard class is represented by an invertible module on \(Y\). The nonflat-descent construction in the cited source parametrizes its possible antecedents on \(X\) by the finite-presentation space of descent data for the two projections from \(Y \times_X Y\) to \(Y\). Only a finiteness assertion is needed: the construction imposes the cocycle equalities as finite-presentation conditions and does not require every descent datum to be effective. Its resulting parameter spaces give finite-type covers of all base changes of the displayed fibre, proving that \(g^*\) is of finite type. Over a field the same descent-data spaces and their equalizers are affine. This is also recorded separately in Theorem moduli-theorem-picard-pullback-proper-surjective-affine.

Lemma

Let \(S\) be a Noetherian scheme. Let \(g : Y \to X\) be a surjective morphism of proper schemes over \(S\). Assume that the fppf Picard functors of \(X\) and \(Y\) over \(S\) are represented by schemes. Let \(M\) be a set of points of \(\Picardfunctor_{X/S}\). Then \(M\) is contained in a quasi-compact open subscheme if and only if \(g^*(M)\) is. Moreover, \[\Picardfunctor_{X/S}^{\tau} = (g^*)^{-1}(\Picardfunctor_{Y/S}^{\tau}).\]

Proof

A morphism of finite type between locally Noetherian schemes is quasi-compact. It therefore sends a set contained in a quasi-compact open to a set contained in a quasi-compact open, and the inverse image of a quasi-compact open is quasi-compact. This proves the first assertion.

An invertible module is \(\tau\)-equivalent to zero precisely when the set of all its integral tensor powers is bounded. Pullback commutes with tensor powers, so the first assertion applied to that cyclic set proves the formula.

Lemma

Let \(S\) be a Noetherian scheme. Let \(X \to S\) be projective and flat with geometrically integral fibres satisfying Serre’s condition \((S_2)\). Let \(\mathcal{H}\) be relatively ample. A collection of invertible modules on geometric fibres of \(X \to S\) is bounded if its Hilbert polynomials with respect to \(\mathcal{H}\) range over a finite set.

Proof

Mumford’s argument, inspired by the relative Nakai criterion, gives a uniform regularity bound for invertible modules with a fixed Hilbert polynomial. The \((S_2)\) hypothesis is the depth input needed in that argument. There are only finitely many polynomials in the statement, so the resulting regularity bounds place the modules in finitely many fixed-polynomial Quot spaces. This is precisely the restricted criterion isolated in the proof comments of the cited source.

Theorem

Let \(S\) be a Noetherian scheme. Let \(Y \to S\) be projective. Let \(\mathcal{H}\) be relatively ample. Let \(X \subset Y\) be the zero scheme of a section of \(\mathcal{H}\). Assume that the fppf Picard functors of \(X\) and \(Y\) over \(S\) are represented by schemes. Suppose every irreducible component of every fibre of \(Y \to S\) has dimension at least \(3\). Then restriction \[\Picardfunctor_{Y/S} \longrightarrow \Picardfunctor_{X/S}\] is of finite type. If \(S\) is the spectrum of a field, it is affine.

Proof

The finite-type pullback theorem permits the proper modifications used in the source to reduce to the case where \(Y/S\) is flat with geometrically integral normal fibres. Lemma moduli-lemma-picard-boundedness-s2-full-polynomial makes the further reduction to the case where \(X/S\) has the same properties. Normality supplies \((S_2)\). The dimension hypothesis then gives depth at least \(2\) at the closed points of the geometric fibres of both \(Y\) and \(X\). The equivalence criterion for an ample divisor identifies the formal restriction data in this depth range, and the restricted boundedness criterion makes the remaining parameter spaces finite type. This is the proof architecture given on page 306 of the cited source; the corresponding formal-neighbourhood setting is described in Remark moduli-remark-picard-formal-neighbourhoods-lefschetz. Over a field, the parameter spaces in this construction and their equalizers are affine, which proves the final assertion.

Theorem

Let \(S\) be a Noetherian scheme. Let \(X \to S\) be projective with integral geometric fibres of dimension \(n\). Let \(\mathcal{H}\) be relatively ample. Let \(\overline{s}_i \to S\) be geometric points. Let \(\mathcal{M}_i\) be invertible modules on \(X_{\overline{s}_i}\). Write \[\chi(X_{\overline{s}_i}, \mathcal{M}_i \otimes \mathcal{H}_{\overline{s}_i}^{\otimes t}) = a_{0, i}t^n + a_{1, i}t^{n-1} + a_{2, i}t^{n-2} + \cdots.\] The collection of the \(\mathcal{M}_i\) is bounded if and only if the numbers \(a_{1, i}\) and \(a_{2, i}\) are bounded. A coefficient whose degree would be negative is omitted; thus only \(a_{1, i}\) occurs when \(n = 1\).

Proof

In a bounded family the coefficients are locally constant and hence take only finitely many values. Conversely, the restricted Mumford criterion of Lemma moduli-lemma-picard-boundedness-s2-full-polynomial proves the assertion under the \((S_2)\) hypothesis once all coefficients have been controlled. The intersection estimates in the cited source show that bounds on the first two nonleading coefficients provide that control. The finite-type pullback theorem and Theorem moduli-theorem-picard-ample-divisor-restriction-finite-type then remove the normality and \((S_2)\) hypotheses by the reduction described on page 306 of the source. Uniform regularity places the resulting invertible modules in finitely many fixed-polynomial Quot spaces. Since the coefficients of a numerical polynomial of degree \(n\) have bounded denominators depending only on \(n\), boundedness of the displayed coefficients is equivalently finiteness of their sets of values.

Theorem

Let \(S\) be a Noetherian scheme. Let \(X \to S\) be proper. Assume that the fppf Picard functor of \(X\) over \(S\) is represented by a scheme. For every nonzero integer \(r\), multiplication by \(r\) on the Picard scheme, \[[r] : \Picardfunctor_{X/S} \longrightarrow \Picardfunctor_{X/S},\] is a morphism of finite type.

Proof

Changing \(r\) to \(-r\) only composes with inversion. On the locus where \(r\) is invertible, the power map is étale by Theorem moduli-theorem-picard-multiplication-etale; the finite-type assertion is the prime-to-characteristic part of the cited argument. For a prime \(p\) dividing \(r\) in the residue characteristic, the \(p\)th-power map on invertible modules is pullback by the relative Frobenius, a finite surjective radicial morphism. Theorem moduli-theorem-picard-pullback-proper-surjective-finite-type applies to that pullback. Factoring \(r\) into prime powers and working on the finitely many corresponding Noetherian strata proves the result.

Theorem

Let \(S\) be a Noetherian scheme. Let \(X \to S\) be proper. Assume that the fppf Picard functor of \(X\) over \(S\) is represented by a scheme. Then \[\Picardfunctor_{X/S}^{\tau} \longrightarrow S\] is of finite type.

Proof

Chow’s lemma gives a proper surjection \(X' \to X\) with \(X'\) projective. Theorem moduli-theorem-picard-pullback-proper-surjective-finite-type and Lemma moduli-lemma-picard-proper-pullback-detects-boundedness reduce the assertion to \(X'\). Finite proper modifications and ample-divisor sections reduce the projective case to geometrically integral fibres. On those fibres a \(\tau\)-equivalent class is numerically trivial, so its first two nonleading Hilbert coefficients equal those of the structure sheaf. Theorem moduli-theorem-picard-boundedness-integral-fibres makes the locus bounded and hence of finite type. Reversing the finite-type reductions proves the theorem for \(X\).

Theorem

Let \(S\) be a Noetherian scheme. Let \(X \to S\) be a proper morphism. For a geometric point \(\overline{s} \to S\), let \(P_{\overline{s}}\) be the Picard scheme of \(X_{\overline{s}}\) and set \[\mathop{\rm NS}(X_{\overline{s}}) = P_{\overline{s}}(\overline{\kappa(s)})/ P^0_{\overline{s}}(\overline{\kappa(s)}).\] These groups are finitely generated. Moreover, there is an integer \(N\) such that \[\mathop{\rm rank} \mathop{\rm NS}(X_{\overline{s}}) \leq N, \qquad \#\mathop{\rm NS}(X_{\overline{s}})_{\rm tors} \leq N\] for every geometric point \(\overline{s} \to S\).

Proof

We recall the reduction in the cited source. Noetherian induction and flattening stratification reduce to finitely many families on which the Picard spaces and the required modifications are defined. Theorem moduli-theorem-picard-pullback-proper-surjective-finite-type, applied first to finite morphisms, and Theorem moduli-theorem-picard-ample-divisor-restriction-finite-type show that the kernels introduced by proper modification and ample-divisor restriction have uniformly bounded component groups. Chow’s lemma and successive ample sections therefore reduce finite generation and uniform boundedness to geometrically integral normal fibres of dimension at most \(2\).

For surfaces, resolution of singularities and another application of the proper-pullback theorem reduce to smooth projective surfaces. The Néron–Severi theorem and nondegeneracy of the intersection pairing give finite generation there, while the Picard–Igusa inequality gives the rank bound. The torsion subgroup is the component group of the finite-type locus \(P^\tau\) from Theorem moduli-theorem-picard-torsion-components-finite-type; hence its order is bounded on each of the finitely many strata. Reversing the finite-type reductions and taking the maximum over those strata proves both uniform bounds.

Lemma

Let \(X\) be a proper scheme over an algebraically closed field \(k\). Put \(P = \Picardfunctor_{X/k}\). Put \[r = \mathop{\rm rank} \mathop{\rm NS}(X).\] There are closed integral curves \(C_1,\ldots,C_r\) in \(X\) such that, for a set \(M\) of isomorphism classes of invertible modules on \(X\), the following are equivalent:

1. \(M\) is contained in a quasi-compact open subscheme of \(P\);

2. for every \(i\), the integers \[\deg_{\widetilde C_i}(\mathcal{L}|_{\widetilde C_i}), \qquad \mathcal{L} \in M,\] are bounded in absolute value, where \(\widetilde C_i \to C_i\) is the normalization.

If \(r = 0\), the second condition is empty.

Proof

Theorem moduli-theorem-picard-neron-severi-uniform-bounds, over \(\Spec(k)\), makes \(\mathop{\rm NS}(X)\) a finitely generated abelian group. The intersection theorem used in its surface reduction says that the degree homomorphisms defined by integral curves vanish simultaneously on precisely the torsion subgroup. We may therefore choose \(r\) curves such that \[\mathop{\rm NS}(X)/\mathop{\rm NS}(X)_{\rm tors} \longrightarrow \mathbf{Z}^r,\qquad [\mathcal{L}] \longmapsto (\deg_{\widetilde C_i}(\mathcal{L}|_{\widetilde C_i}))_{i = 1}^r\] is injective. Indeed, after tensoring with \(\mathbf{Q}\), choose a basis from the curve-degree linear forms on the \(r\)-dimensional vector space.

Every connected component of \(P\) is a translate of the finite-type identity component. Consequently \(M\) is contained in a quasi-compact open if and only if its image in \(\mathop{\rm NS}(X)\) is finite. The displayed injection and finiteness of the torsion subgroup show that this happens if and only if all the displayed degree coordinates are bounded.

Theorem

Let \(X\) be a proper scheme over a field \(k\). Let \(\mathcal{L}\) be an invertible module on \(X\). The following are equivalent:

1. The class of \(\mathcal{L}\) is in \(\Picardfunctor_{X/k}^{\tau}\).

2. For every coherent module \(\mathcal{F}\) on \(X\), \[\chi(X,\mathcal{F} \otimes \mathcal{L}) = \chi(X,\mathcal{F}).\]

3. The equality in (b) holds for \(\mathcal{F} = \mathcal{O}_Y\), for every closed integral subscheme \(Y \subset X\) of dimension \(1\).

4. For every such \(Y\), if \(\widetilde Y \to Y\) is the normalization, then \[\deg_{\widetilde Y}(\mathcal{L}|_{\widetilde Y}) = 0.\]

If \(X\) is projective and \(\mathcal{H} = \mathcal{O}_X(1)\) is ample, these conditions are also equivalent to

1. \(\mathcal{L}^{\otimes m} \otimes \mathcal{H}\) is ample for every \(m \in \mathbf{Z}\).

If, in addition, \(X\) is integral, they are also equivalent to

1. For every \(m, n \in \mathbf{Z}\), \[\chi(X,\mathcal{L}^{\otimes m} \otimes \mathcal{H}^{\otimes n}) = \chi(X,\mathcal{H}^{\otimes n}).\]

Proof

Condition (a) says exactly that the set of all integral tensor powers of \(\mathcal{L}\) is bounded. For a coherent module \(\mathcal{F}\), Varieties, Lemma 0BEM says that \[m \longmapsto \chi(X,\mathcal{F} \otimes \mathcal{L}^{\otimes m})\] is a numerical polynomial. Under (a), its values range over a finite set: Euler characteristic is locally constant in a proper flat family and the powers lie in a quasi-compact part of the Picard scheme. The polynomial is therefore constant. Its values at \(m = 0\) and \(m = 1\) give (b), and (b) plainly implies (b\('\)).

For an integral proper curve \(Y\), Riemann–Roch and normalization give \[\chi(Y,\mathcal{L}|_Y) - \chi(Y,\mathcal{O}_Y) = \deg_{\widetilde Y}(\mathcal{L}|_{\widetilde Y}).\] Thus (b\('\)) and (c) are equivalent. To prove that (c) implies (a), we may extend the ground field to an algebraic closure: both degree zero and \(\tau\)-equivalence are insensitive to this extension. Lemma moduli-lemma-picard-boundedness-finite-curves, applied to the powers of \(\mathcal{L}\), then makes that cyclic set bounded.

Suppose now that \(X\) is projective. If (d) holds, restriction to the normalization of an integral curve gives \[\deg(\mathcal{H}|_{\widetilde Y}) + m\deg(\mathcal{L}|_{\widetilde Y}) > 0\] for every \(m \in \mathbf{Z}\), and hence (c). Conversely, under (c) every \(\mathcal{L}^{\otimes m} \otimes \mathcal{H}\) has the same numerical class as \(\mathcal{H}\); the numerical criterion for ampleness makes it ample. This proves the equivalence with (d).

Condition (b), applied successively with \(\mathcal{F} = \mathcal{L}^{\otimes m} \otimes \mathcal{H}^{\otimes n}\), gives (e). Conversely, (e) says that all powers of \(\mathcal{L}\) have the same Hilbert polynomial. The integral field case of the boundedness criterion in Theorem moduli-theorem-picard-boundedness-integral-fibres, with the proper-pullback reduction used there after extending the ground field, makes the set of powers bounded. This is (a).

Theorem

Let \(S\) be a Noetherian scheme. Let \(X \to S\) be projective and flat with integral geometric fibres. Assume that the fppf Picard functor is represented by a scheme \[P = \Picardfunctor_{X/S}.\] Then \(P^\tau\) is open and closed in \(P\).

Proof

Choose a relatively ample invertible module \(\mathcal{H}\) on \(X\). Fppf locally on \(P\), a Picard class is represented by an invertible module \(\mathcal{L}\) on \(X \times_S P\). For integers \(m, n\), the function on geometric points of this cover given by \[\chi(X_s,\mathcal{L}_s^{\otimes m} \otimes \mathcal{H}_s^{\otimes n}) - \chi(X_s,\mathcal{H}_s^{\otimes n})\] is locally constant. Hence its zero locus is open and closed.

We may work on an open and closed component of \(S\) on which the fibre dimension is a fixed integer \(d\). By Varieties, Lemma 0BEM, the displayed difference is a numerical polynomial in \((m, n)\) of total degree at most \(d\). It vanishes for all pairs of integers if and only if it vanishes on the finite grid \[0 \leq m, n \leq d.\] Theorem moduli-theorem-picard-tau-numerical-criteria, condition (e), therefore identifies \(P^\tau\) on the fppf cover with a finite intersection of open and closed loci. This locus descends to an open and closed subscheme of \(P\).

Remark

The first three results of the supplement form a dependency chain. Nonflat descent proves finite-type proper pullback; Mumford’s fixed-polynomial criterion under \((S_2)\) then proves finite-type restriction to an ample divisor; those two results remove the normality hypothesis and yield the two-coefficient boundedness theorem. Power maps in residue characteristic are handled separately by applying proper pullback to Frobenius. The Néron–Severi theorem and its curve test then turn that boundedness theory into the numerical criteria and the relative open-and-closed statement. Over a field, the proper-pullback and ample-restriction morphisms are affine, as recorded in Theorem moduli-theorem-picard-pullback-proper-surjective-finite-type and the cited supplement.

Properties of the Picard stack

Let \(f : X \to B\) be a morphism of algebraic spaces which is flat, proper, and of finite presentation. Then the stack \(\Picardstack_{X/B}\) parametrizing invertible sheaves on \(X/B\) is algebraic, see Quot, Proposition 0D04.

Lemma

The diagonal of \(\Picardstack_{X/B}\) over \(B\) is affine and of finite presentation.

Proof

In Quot, Lemma 0D03 we have seen that \(\Picardstack_{X/B}\) is an open substack of \(\Cohstack_{X/B}\). Hence this follows from Lemma 0DLY.

Lemma

The morphism \(\Picardstack_{X/B} \to B\) is quasi-separated and locally of finite presentation.

Proof

In Quot, Lemma 0D03 we have seen that \(\Picardstack_{X/B}\) is an open substack of \(\Cohstack_{X/B}\). Hence this follows from Lemma 0DLZ.

Lemma

Assume \(X \to B\) is smooth in addition to being proper. Then \(\Picardstack_{X/B} \to B\) satisfies the existence part of the valuative criterion (Morphisms of Stacks, Definition 0CLK).

Proof

Taking base change, this immediately reduces to the following problem: given a valuation ring \(R\) with fraction field \(K\) and an algebraic space \(X\) proper and smooth over \(R\) and an invertible \(\mathcal{O}_{X_K}\)-module \(\mathcal{L}_K\), show there exists an invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) whose generic fibre is \(\mathcal{L}_K\). Observe that \(X_K\) is Noetherian, separated, and regular (use Morphisms of Spaces, Lemma 04ZL and Spaces over Fields, Lemma 06M1). Thus we can write \(\mathcal{L}_K\) as the difference in the Picard group of \(\mathcal{O}_{X_K}(D_K)\) and \(\mathcal{O}_{X_K}(D'_K)\) for two effective Cartier divisors \(D_K, D'_K\) in \(X_K\), see Divisors on Spaces, Lemma 0DMM. Finally, we know that \(D_K\) and \(D'_K\) are restrictions of effective Cartier divisors \(D, D' \subset X\), see Divisors on Spaces, Lemma 0DMC.

Lemma

Assume \(f_{T, *}\mathcal{O}_{X_T} \cong \mathcal{O}_T\) for all schemes \(T\) over \(B\). Then the inertia stack of \(\Picardstack_{X/B}\) is equal to \(\mathbf{G}_m \times \Picardstack_{X/B}\).

Proof

This is explained in Examples of Stacks, Example 0375.

Lemma

Assume \(f : X \to B\) has relative dimension \(\leq 1\) in addition to the other assumptions in this section. Then \(\Picardstack_{X/B} \to B\) is smooth.

Proof

We already know that \(\Picardstack_{X/B} \to B\) is locally of finite presentation, see Lemma 0DMB. Thus it suffices to show that \(\Picardstack_{X/B} \to B\) is formally smooth, see More on Morphisms of Stacks, Lemma 0DP0. Taking base change, this immediately reduces to the following problem: given a first order thickening \(T \subset T'\) of affine schemes, given \(X' \to T'\) proper, flat, of finite presentation and of relative dimension \(\leq 1\), and for \(X = T \times_{T'} X'\) given an invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\), prove that there exists an invertible \(\mathcal{O}_{X'}\)-module \(\mathcal{L}'\) whose restriction to \(X\) is \(\mathcal{L}\). Since \(T \subset T'\) is a first order thickening, the same is true for \(X \subset X'\), see More on Morphisms of Spaces, Lemma 09ZX. By More on Morphisms of Spaces, Lemma 0DNM we see that it suffices to show \(H^2(X, \mathcal{I}) = 0\) where \(\mathcal{I}\) is the quasi-coherent ideal cutting out \(X\) in \(X'\). Denote \(f : X \to T\) the structure morphism. By Cohomology of Spaces, Lemma 0A4T we see that \(R^pf_*\mathcal{I} = 0\) for \(p > 1\). Hence we get the desired vanishing by Cohomology of Spaces, Lemma 08EX (here we finally use that \(T\) is affine).

Properties of the Picard functor

Let \(f : X \to B\) be a morphism of algebraic spaces which is flat, proper, and of finite presentation such that moreover for every \(T/B\) the canonical map \[\mathcal{O}_T \longrightarrow f_{T, *}\mathcal{O}_{X_T}\] is an isomorphism. Then the Picard functor \(\Picardfunctor_{X/B}\) is an algebraic space, see Quot, Proposition 0D2C. There is a closed relationship with the Picard stack.

Lemma

The morphism \(\Picardstack_{X/B} \to \Picardfunctor_{X/B}\) turns the Picard stack into a gerbe over the Picard functor.

Proof

The definition of \(\Picardstack_{X/B} \to \Picardfunctor_{X/B}\) being a gerbe is given in Morphisms of Stacks, Definition 06QC, which in turn refers to Stacks, Definition 06P2. To prove it, we will check conditions (2)(a) and (2)(b) of Stacks, Lemma 06P1. This follows immediately from Quot, Lemma 0D26; here is a detailed explanation.

Condition (2)(a). Suppose that \(\xi \in \Picardfunctor_{X/B}(U)\) for some scheme \(U\) over \(B\). Since \(\Picardfunctor_{X/B}\) is the fppf sheafification of the rule \(T \mapsto \Pic(X_T)\) on schemes over \(B\) (Quot, Situation 0D25), we see that there exists an fppf covering \(\{U_i \to U\}\) such that \(\xi|_{U_i}\) corresponds to some invertible module \(\mathcal{L}_i\) on \(X_{U_i}\). Then \((U_i \to B, \mathcal{L}_i)\) is an object of \(\Picardstack_{X/B}\) over \(U_i\) mapping to \(\xi|_{U_i}\).

Condition (2)(b). Suppose that \(U\) is a scheme over \(B\) and \(\mathcal{L}, \mathcal{N}\) are invertible modules on \(X_U\) which map to the same element of \(\Picardfunctor_{X/B}(U)\). Then there exists an fppf covering \(\{U_i \to U\}\) such that \(\mathcal{L}|_{X_{U_i}}\) is isomorphic to \(\mathcal{N}|_{X_{U_i}}\). Thus we find isomorphisms between \((U \to B, \mathcal{L})|_{U_i} \to (U \to B, \mathcal{N})|_{U_i}\) as desired.

Lemma

The diagonal of \(\Picardfunctor_{X/B}\) over \(B\) is a quasi-compact immersion.

Proof

The diagonal is an immersion by Quot, Lemma 0D2D. To finish we show that the diagonal is quasi-compact. The diagonal of \(\Picardstack_{X/B}\) is quasi-compact by Lemma 0DMA and \(\Picardstack_{X/B}\) is a gerbe over \(\Picardfunctor_{X/B}\) by Lemma 0DME. We conclude by Morphisms of Stacks, Lemma 0DQL.

Lemma

The morphism \(\Picardfunctor_{X/B} \to B\) is quasi-separated and locally of finite presentation.

Proof

To check \(\Picardfunctor_{X/B} \to B\) is quasi-separated we have to show that its diagonal is quasi-compact. This is immediate from Lemma 0DMF. Since the morphism \(\Picardstack_{X/B} \to \Picardfunctor_{X/B}\) is surjective, flat, and locally of finite presentation (by Lemma 0DME and Morphisms of Stacks, Lemma 06QI) it suffices to prove that \(\Picardstack_{X/B} \to B\) is locally of finite presentation, see Morphisms of Stacks, Lemma 06Q9. This follows from Lemma 0DMB.

Lemma

Assume \(X\) is a scheme in addition to the other assumptions in this section. Let \(n\) be an integer which is invertible on \(B\). Then multiplication by \(n\) \[[n] : \Picardfunctor_{X/B} \longrightarrow \Picardfunctor_{X/B}\] is an étale morphism.

Proof

The morphism \([n]\) is locally of finite presentation by Lemma 0DNI and Morphisms of Spaces, Lemma 05WT. We prove that it is formally étale. Let \(T_0 \subset T\) be a first order thickening of affine schemes over \(B\). Suppose we have \[\xi_0 \in \Picardfunctor_{X/B}(T_0), \qquad \zeta \in \Picardfunctor_{X/B}(T)\] such that \(n\xi_0 = \zeta|_{T_0}\).

By Lemma 0DME and Morphisms of Stacks, Lemma 0DN8, elements of the Picard functor are represented by invertible modules étale locally on the base; here we also use More on Morphisms, Lemma 055U. Moreover, an étale covering of \(T_0\) lifts to an étale covering of \(T\) by Étale Morphisms, Theorem 039R. Consequently, after replacing \(T\) by an étale covering, we may choose \[\mathcal{L} \in \Pic(X_T), \qquad \mathcal{M}_0 \in \Pic(X_{T_0})\] representing \(\zeta\) and \(\xi_0\). By Quot, Lemma 0D27, and after a further Zariski refinement, we may arrange that \[n[\mathcal{M}_0] = [\mathcal{L}|_{X_{T_0}}] \quad\text{in}\quad \Pic(X_{T_0}).\]

The closed immersion \(X_{T_0} \subset X_T\) is a first order thickening. More on Morphisms, Lemma more-morphisms-lemma-pic-roots-first-order-thickening gives a unique \(\mathcal{M} \in \Pic(X_T)\) restricting to \(\mathcal{M}_0\) and satisfying \(n[\mathcal{M}] = [\mathcal{L}]\).

We claim that any two lifts in the original lifting problem are equal. Their difference is an element \(\delta \in \Picardfunctor_{X/B}(T)\) such that \(\delta|_{T_0} = 0\) and \(n\delta = 0\). After étale localization it is represented by some \(\mathcal{D} \in \Pic(X_T)\). Using Quot, Lemma 0D27, and further Zariski localization, we may arrange that \(\mathcal{D}|_{X_{T_0}}\) and \(\mathcal{D}^{\otimes n}\) are trivial. More on Morphisms, Lemma 0C6S then shows that \(\mathcal{D}\) is trivial. Thus \(\delta\) is zero. It follows that the local lifts constructed above agree on overlaps. They descend because \(\Picardfunctor_{X/B}\) is an fppf sheaf. Hence \([n]\) is formally étale.

We conclude that \([n]\) is étale by More on Morphisms of Spaces, Lemma 0616.

Lemma

Assume the geometric fibres of \(X \to B\) are integral in addition to the other assumptions in this section. Then \(\Picardfunctor_{X/B} \to B\) is separated.

Proof

Since \(\Picardfunctor_{X/B} \to B\) is quasi-separated, it suffices to check the uniqueness part of the valuative criterion, see Morphisms of Spaces, Lemma 03KV. This immediately reduces to the following problem: given

1. a valuation ring \(R\) with fraction field \(K\),

2. an algebraic space \(X\) proper and flat over \(R\) with integral geometric fibre,

3. an element \(a \in \Picardfunctor_{X/R}(R)\) with \(a|_{\Spec(K)} = 0\),

then we have to prove \(a = 0\). Applying Morphisms of Stacks, Lemma 0DN5 to the surjective flat morphism \(\Picardstack_{X/R} \to \Picardfunctor_{X/R}\) (surjective and flat by Lemma 0DME and Morphisms of Stacks, Lemma 06QI) after replacing \(R\) by an extension we may assume \(a\) is given by an invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\). Since \(a|_{\Spec(K)} = 0\) we find \(\mathcal{L}_K \cong \mathcal{O}_{X_K}\) by Quot, Lemma 0D27.

Denote \(f : X \to \Spec(R)\) the structure morphism. Let \(\eta, 0 \in \Spec(R)\) be the generic and closed point. Consider the perfect complexes \(K = Rf_*\mathcal{L}\) and \(M = Rf_*(\mathcal{L}^{\otimes -1})\) on \(\Spec(R)\), see Derived Categories of Spaces, Lemma 0CTM. Consider the functions \(\beta_{K, i}, \beta_{M, i} : \Spec(R) \to \mathbf{Z}\) of Derived Categories of Spaces, Lemma 0D1Y associated to \(K\) and \(M\). Since the formation of \(K\) and \(M\) commutes with base change (see lemma cited above) we find \(\beta_{K, 0}(\eta) = \beta_{M, 0}(\eta) = 1\) by Spaces over Fields, Lemma 0DMY and our assumption on the fibres of \(f\). By upper semi-continuity we find \(\beta_{K, 0}(0) \geq 1\) and \(\beta_{M, 0}(0) \geq 1\). By Spaces over Fields, Lemma 0DMZ we conclude that the restriction of \(\mathcal{L}\) to the special fibre \(X_0\) is trivial. In turn this gives \(\beta_{K, 0}(0) = \beta_{M, 0}(0) = 1\) as above. Then by More on Algebra, Lemma 0BCC we can represent \(K\) by a complex of the form \[\ldots \to 0 \to R \to R^{\oplus \beta_{K, 1}(0)} \to R^{\oplus \beta_{K, 2}(0)} \to \ldots\] Now \(R \to R^{\oplus \beta_{K, 1}(0)}\) is zero because \(\beta_{K, 0}(\eta) = 1\). In other words \(K = R \oplus \tau_{\geq 1}(K)\) in \(D(R)\) where \(\tau_{\geq 1}(K)\) has tor amplitude in \([1, b]\) for some \(b \in \mathbf{Z}\). Hence there is a global section \(s \in H^0(X, \mathcal{L})\) whose restriction \(s_0\) to \(X_0\) is nonvanishing (again because formation of \(K\) commutes with base change). Then \(s : \mathcal{O}_X \to \mathcal{L}\) is a map of invertible sheaves whose restriction to \(X_0\) is an isomorphism and hence is an isomorphism as desired.

Remark

Let \(X\) and \(B\) be locally Noetherian schemes and suppose that \(X \to B\) is projective and flat with integral geometric fibres. The classical projective Picard theorem strengthens Quot, Proposition 0D2C, Lemma 0DNI, and Lemma 0DNJ: in this situation \(\Picardfunctor_{X/B}\) is represented by a separated scheme. If \(\mathcal{O}_X(1)\) is relatively very ample and \(\xi\) is its class, the classical construction gives an open subspace \(U\) which is a disjoint union of open subschemes quasi-projective over \(B\) and such that \[U + \xi \subset U \quad\text{and}\quad \Picardfunctor_{X/B} = \bigcup_{n \geq 0}(U - n\xi).\] Translation identifies every \(U - n\xi\) with \(U\). The scheme and quasi-projectivity assertions use the divisor and Hilbert scheme construction; they do not follow from algebraic-space representability and local finite presentation alone.

The printed source also asks whether the Picard scheme is a disjoint union of open subschemes of finite type, and hence quasi-projective in this setting, over \(B\). The correction appended to the collected exposés says that Mumford answered this question affirmatively. Thus the question in the printed Remark 3.3 is not open.

Remark

Fibrewise constancy of global functions is not a substitute for the integrality hypothesis in Lemma 0DNJ. More precisely, the condition \[\kappa(b) \longrightarrow H^0(X_b, \mathcal{O}_{X_b})\] being an isomorphism for every \(b \in B\) does not by itself force the relative Picard functor to be separated. The cited source records examples over discrete valuation rings in which an integral generic curve specializes either to two intersecting irreducible components, as for a degeneration of a conic to two lines, or to a nonreduced multiple fibre with irreducible support, as for a double elliptic curve. Both kinds of example occur in every characteristic.

Remark

Replacing projectivity by projectivity locally on the base preserves the local representability problem, but it does not force connected components of the Picard scheme to be of finite type. Here is the construction recorded in the cited source. Let \(X_0\) be a smooth projective variety over an algebraically closed field \(k\). Choose an automorphism \(u\) and a class \(\xi\) in its Néron–Severi group such that the classes \(u^n(\xi)\) are pairwise distinct. One example is \(X_0 = E \times_k E\) for an elliptic curve \(E\), with \[u(x, y) = (x, y + x).\] Let \(B\) be the union of two smooth irreducible curves meeting in two points and let \(P \to B\) be a connected principal \(\mathbf{Z}\)-cover. The family associated to the action of \(\mathbf{Z}\) on \(X_0\) is locally projective over \(B\); for the displayed example it is an abelian scheme. Its Picard scheme has a connected component isomorphic to \[P \times_k \underline{\Picardfunctor}^{0}_{X_0/k}.\] This component is not of finite type over \(B\). In particular, local finite presentation of the entire Picard space does not imply that each of its connected components is of finite type.

Lemma

Assume \(f : X \to B\) has relative dimension \(\leq 1\) in addition to the other assumptions in this section. Then \(\Picardfunctor_{X/B} \to B\) is smooth.

Proof

By Lemma 0DPJ we know that \(\Picardstack_{X/B} \to B\) is smooth. The morphism \(\Picardstack_{X/B} \to \Picardfunctor_{X/B}\) is surjective and smooth by combining Lemma 0DME with Morphisms of Stacks, Lemma 0DN8. Thus if \(U\) is a scheme and \(U \to \Picardstack_{X/B}\) is surjective and smooth, then \(U \to \Picardfunctor_{X/B}\) is surjective and smooth and \(U \to B\) is surjective and smooth (because these properties are preserved by composition). Thus \(\Picardfunctor_{X/B} \to B\) is smooth for example by Descent on Spaces, Lemma 0AHD.

Properties of relative morphisms

Let \(B\) be an algebraic space. Let \(X\) and \(Y\) be algebraic spaces over \(B\) such that \(Y \to B\) is flat, proper, and of finite presentation and \(X \to B\) is separated and of finite presentation. Then the functor \(\mathit{Mor}_B(Y, X)\) of relative morphisms is an algebraic space locally of finite presentation over \(B\). See Quot, Proposition 0D1C.

Lemma

The diagonal of \(\mathit{Mor}_B(Y, X) \to B\) is a closed immersion of finite presentation.

Proof

There is an open immersion \(\mathit{Mor}_B(Y, X) \to \Hilbfunctor_{Y \times_B X/B}\), see Quot, Lemma 0D1B. Thus the lemma follows from Lemma 0DM6.

Lemma

The morphism \(\mathit{Mor}_B(Y, X) \to B\) is separated and locally of finite presentation.

Proof

To check \(\mathit{Mor}_B(Y, X) \to B\) is separated we have to show that its diagonal is a closed immersion. This is true by Lemma 0DPM. The second statement is part of Quot, Proposition 0D1C.

Lemma

With \(B, X, Y\) as in the introduction of this section, in addition assume \(X \to B\) is proper. Then the subfunctor \(\mathit{Isom}_B(Y, X) \subset \mathit{Mor}_B(Y, X)\) of isomorphisms is an open subspace.

Proof

Follows immediately from More on Morphisms of Spaces, Lemma 05XD.

Remark

Let \(B, X, Y\) be as in the introduction to this section. Let \(I\) be a set and for \(i \in I\) let \(E_i \in D(\mathcal{O}_{Y \times_B X})\) be perfect. Let \(P : I \to \mathbf{Z}\) be a function. Recall that \[\mathit{Mor}_B(Y, X) \subset \Hilbfunctor_{Y \times_B X/B}\] is an open subspace, see Quot, Lemma 0D1B. Thus we can define \[\mathit{Mor}^P_B(Y, X) = \mathit{Mor}_B(Y, X) \cap \Hilbfunctor^P_{Y \times_B X/B}\] where \(\Hilbfunctor^P_{Y \times_B X/B}\) is as in Remark 0DPG. The morphism \[\mathit{Mor}^P_B(Y, X) \longrightarrow \mathit{Mor}_B(Y, X)\] is a flat closed immersion which is an open and closed immersion for example if \(I\) is finite, or \(B\) is locally Noetherian, or \(I = \mathbf{Z}\), \(E_i = \mathcal{L}^{\otimes i}\) for some invertible \(\mathcal{O}_{Y \times_B X}\)-module \(\mathcal{L}\). In the last case we sometimes use the notation \(\mathit{Mor}^{P, \mathcal{L}}_B(Y, X)\).

Lemma

With \(B, X, Y\) as in the introduction of this section, let \(\mathcal{L}\) be ample on \(X/B\) and let \(\mathcal{N}\) be ample on \(Y/B\). See Divisors on Spaces, Definition 0D31. Let \(P\) be a numerical polynomial. Then \[\mathit{Mor}^{P, \mathcal{M}}_B(Y, X) \longrightarrow B\] is separated and of finite presentation where \(\mathcal{M} = \text{pr}_1^*\mathcal{N} \otimes_{\mathcal{O}_{Y \times_B X}} \text{pr}_2^*\mathcal{L}\).

Proof

By Lemma 0DPN the morphism \(\mathit{Mor}_B(Y, X) \to B\) is separated and locally of finite presentation. Thus it suffices to show that the open and closed subspace \(\mathit{Mor}^{P, \mathcal{M}}_B(Y, X)\) of Remark 0DPQ is quasi-compact over \(B\).

The question is étale local on \(B\) (Morphisms of Spaces, Lemma 03KG). Thus we may assume \(B\) is affine.

Assume \(B = \Spec(\Lambda)\). Note that \(X\) and \(Y\) are schemes and that \(\mathcal{L}\) and \(\mathcal{N}\) are ample invertible sheaves on \(X\) and \(Y\) (this follows immediately from the definitions). Write \(\Lambda = \colim \Lambda_i\) as the colimit of its finite type \(\mathbf{Z}\)-subalgebras. Then we can find an \(i\) and a system \(X_i, Y_i, \mathcal{L}_i, \mathcal{N}_i\) as in the lemma over \(B_i = \Spec(\Lambda_i)\) whose base change to \(B\) gives \(X, Y, \mathcal{L}, \mathcal{N}\). This follows from Limits, Lemmas 01ZM (to find \(X_i\), \(Y_i\)), 0B8W (to find \(\mathcal{L}_i\), \(\mathcal{N}_i\)), 01ZQ (to make \(X_i \to B_i\) separated), 081F (to make \(Y_i \to B_i\) proper), and 09MT (to make \(\mathcal{L}_i\), \(\mathcal{N}_i\) ample). Because \[\mathit{Mor}_B(Y, X) = B \times_{B_i} \mathit{Mor}_{B_i}(Y_i, X_i)\] and similarly for \(\mathit{Mor}^P_B(Y, X)\) we reduce to the case discussed in the next paragraph.

Assume \(B\) is a Noetherian affine scheme. By Properties, Lemma 0DNK we see that \(\mathcal{M}\) is ample. By Lemma 0DPI we see that \(\Hilbfunctor^{P, \mathcal{M}}_{Y \times_B X/B}\) is of finite presentation over \(B\) and hence Noetherian. By construction \[\mathit{Mor}^{P, \mathcal{M}}_B(Y, X) = \mathit{Mor}_B(Y, X) \cap \Hilbfunctor^{P, \mathcal{M}}_{Y \times_B X/B}\] is an open subspace of \(\Hilbfunctor^{P, \mathcal{M}}_{Y \times_B X/B}\) and hence quasi-compact (as an open of a Noetherian algebraic space is quasi-compact).

Proposition

Let \(S\) be a Noetherian scheme.

1. Let \(Y \to S\) be flat and proper and let \(Z \to Y\) be an \(S\)-morphism such that \(Z \to S\) is quasi-projective. The functor \(\mathit{Sec}_{Y/S}(Z/Y)\) of sections is a scheme locally of finite type over \(S\). It is an open subscheme of \(\Hilbfunctor_{Z/S}\), and each fixed-Hilbert-polynomial component is quasi-projective over \(S\).

2. Let \(Y \to S\) be flat and projective and let \(X \to S\) be quasi-projective. Then \(\mathit{Mor}_S(Y, X)\) is a scheme locally of finite type over \(S\). The graph construction identifies it with an open subscheme of \(\Hilbfunctor_{Y \times_S X/S}\), and every fixed-polynomial subfunctor \(\mathit{Mor}^{P,\mathcal{M}}_S(Y, X)\) is quasi-projective over \(S\), where \[\mathcal{M} = \operatorname{pr}_1^*\mathcal{N} \otimes \operatorname{pr}_2^*\mathcal{L}\] for relatively very ample invertible modules \(\mathcal{N}\) on \(Y/S\) and \(\mathcal{L}\) on \(X/S\). The subfunctor of morphisms \(Y_T \to X_T\) which are immersions is open.

3. If \(X \to S\) is projective as well, then \(\mathit{Isom}_S(Y, X)\) is an open subscheme of \(\mathit{Mor}_S(Y, X)\).

Proof

A section of \(Z_T \to Y_T\) has an image which is a closed subscheme of \(Z_T\), and a closed subscheme \(\Gamma \subset Z_T\) is the image of a section exactly when the projection \(\Gamma \to Y_T\) is an isomorphism. On the universal family over \(\Hilbfunctor_{Z/S}\) this is an open condition by More on Morphisms of Spaces, Lemma 05XD. Thus the section functor is an open subspace of \(\Hilbfunctor_{Z/S}\). Proposition moduli-proposition-quot-quasi-projective-over-base and Quot, Lemma 0D00 show that the fixed-polynomial Hilbert spaces are quasi-projective schemes. Their open section loci are quasi-compact, because these Hilbert spaces are Noetherian, and hence are quasi-projective. Taking their disjoint union proves (1).

For (2), use \[\mathit{Mor}_S(Y, X) = \mathit{Sec}_{Y/S}((Y \times_S X)/Y).\] The product \(Y \times_S X\) is quasi-projective over \(S\): its projection to \(X\) is a base change of the projective morphism \(Y \to S\), and one applies Morphisms, Lemmas 0B3G and 0C4M. The statement about graphs also follows directly from Quot, Lemma 0D1B. The displayed invertible module is relatively very ample: locally on \(S\), take the projective embeddings defined by \(\mathcal{N}\) and \(\mathcal{L}\) and compose their product with the Segre embedding of Constructions, Lemma 01WD. The fixed-polynomial assertion follows from (1) or from Remark 0DPQ. Finally, every immersion \(Y_T \to X_T\) is closed: it is proper because \(Y_T \to T\) is proper and \(X_T \to T\) is separated. The closed-immersion locus of the universal morphism is open by More on Morphisms of Spaces, Lemma 05XA.

If \(X\) is projective, the isomorphism locus is open by Lemma 0DPP. This proves (3).

Remark

For a projective scheme over a field, the automorphism functor is represented by the open subscheme \(\mathit{Isom}(X, X)\) above. Its scheme structure is essential: the reduction seen by classical constructions over an algebraically closed field can discard infinitesimal automorphisms. Over an imperfect field, the nilradical appearing after extending the ground field need not descend, so replacing the automorphism scheme by that reduction is not a base-field construction.

Proposition

Let \(Y \to S\) be a flat proper morphism of schemes of finite presentation. For a morphism \(Z \to Y\), let \(\mathit{Sec}_{Y/S}(Z/Y)\) be the functor which assigns to \(T \to S\) the set of sections of \(Z_T \to Y_T\).

1. If \(Z \to Y\) is separated and of finite presentation, then \(\mathit{Sec}_{Y/S}(Z/Y)\) is a separated algebraic space locally of finite presentation over \(S\).

2. If \(Z \to Y\) is a closed immersion, then \(\mathit{Sec}_{Y/S}(Z/Y) \to S\) is represented by a closed immersion. It is of finite presentation if \(Z \to S\) is of finite presentation.

3. If \(Z \to Y\) is affine, then \(\mathit{Sec}_{Y/S}(Z/Y)\) is an affine scheme over \(S\). If in addition \(Z \to Y\) is of finite presentation, then this affine scheme is of finite presentation over \(S\).

4. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_Y\)-module. The functor of sections of the vector bundle \(\mathbf{V}(\mathcal{E}) \to Y\) is a vector bundle over \(S\) in the sense of Constructions, Definition 01M2. Its associated quasi-coherent module is of finite presentation.

Proof

For (1), composition with \(Z \to Y\) and the identity of \(Y\) give a cartesian diagram \[\xymatrix{ \mathit{Sec}_{Y/S}(Z/Y) \ar[r] \ar[d] & \mathit{Mor}_S(Y, Z) \ar[d] \\ S \ar[r]^-{\operatorname{id}_Y} & \mathit{Mor}_S(Y, Y). }\] Both relative morphism functors in the right column are algebraic spaces locally of finite presentation by Quot, Proposition 0D1C. The first is separated over \(S\) by Lemma 0DPN. The assertions follow by base change.

For (2), a proper morphism is universally pure by More on Flatness, Lemma 05K3. The result is therefore exactly More on Flatness, Lemma 07AI.

For (3), write \[Z = \underline{\Spec}_Y(\mathcal{A}).\] Choose a coequalizer presentation of the quasi-coherent algebra \(\mathcal{A}\) by free commutative algebras \[\xymatrix{ \operatorname{Sym}(\mathcal{F}_1) \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \operatorname{Sym}(\mathcal{F}_0) \ar[r] & \mathcal{A}. }\] After arbitrary base change on \(S\), algebra maps from \(\mathcal{A}\) to the structure sheaf are the equalizer of the corresponding pair of module-map functors \[\xymatrix{ \mathit{Hom}(\mathcal{F}_0, \mathcal{O}_Y) \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathit{Hom}(\mathcal{F}_1, \mathcal{O}_Y). }\] Each of these functors is affine over \(S\) by Quot, Proposition 08K6: the module \(\mathcal{O}_Y\) is flat over \(S\) and has proper support. An equalizer of morphisms of affine \(S\)-schemes is affine. Under the additional finite-presentation hypothesis, (1) shows that the result is of finite presentation.

For (4), a section of \(\mathbf{V}(\mathcal{E})_T \to Y_T\) is an \(\mathcal{O}_{Y_T}\)-linear map \(\mathcal{E}_T \to \mathcal{O}_{Y_T}\). Apply Quot, Proposition 08K6. The explicit construction in Quot, Lemma 08JX presents this affine functor, locally on \(S\), by homogeneous linear equations in a finite affine space. It is consequently \(\mathbf{V}(\mathcal{Q})\) for a finitely presented quasi-coherent \(\mathcal{O}_S\)-module \(\mathcal{Q}\).

Remark

Let \(S = \Spec(k)\), let \(Y\) be a proper \(k\)-scheme, and let \(Z \to Y\) be quasi-projective. The argument recorded in [FGA, Exposé 221, Section 7] uses Chow’s lemma and factorization of a finite morphism to show that \(\mathit{Sec}_{Y/k}(Z/Y)\) is a scheme which is a disjoint union of quasi-projective \(k\)-schemes. The projective cover in Chow’s lemma is automatically flat over the field. Proposition moduli-proposition-sections-proper-flat gives the base-independent algebraic-space statement; representability by a scheme and quasi-projectivity are additional conclusions, not part of the general algebraic-space theorem.

Remark

There are a quadratic field extension \(k'/k\) and a smooth proper nonprojective threefold \(X\) over \(k'\) for which the restriction-of-scalars functor \(\operatorname{Res}_{k'/k}(X)\) is not representable by a scheme. The corresponding finite étale locus in the degree-two Hilbert functor and the symmetric square likewise need not be schemes.

In the terminology of the cited source, these objects “do not exist” because only schemes are admitted as representing objects. As fppf sheaves they do exist as algebraic spaces: use Criteria for Representability, Proposition 05YF, Quot, Proposition 0D01, and Theorem moduli-theorem-divided-power-zero-cycles. Thus the example is an obstruction to scheme representability, not to modern algebraic-space representability.

Remark

The addendum to [FGA, Exposé 221] retracts the quotient conjecture in Exposé 212, Section 8. Even for a smooth variety in characteristic zero, having a closed action graph does not imply existence or quasi-projectivity of a scheme quotient. A free action has an fppf quotient algebraic space by Bootstrap, Lemma 06PH; a scheme or a quasi-projective quotient requires additional hypotheses. This distinction is not removed by the Keel–Mori theorem of Morphisms of Algebraic Stacks, Theorem 0DUT.

Properties of the stack of polarized proper schemes

In this section we discuss properties of the moduli stack \[\Polarizedstack \longrightarrow \Spec(\mathbf{Z})\] whose category of sections over a scheme \(S\) is the category of proper, flat, finitely presented scheme over \(S\) endowed with a relatively ample invertible sheaf. This is an algebraic stack by Quot, Theorem 0D4X.

Lemma

The diagonal of \(\Polarizedstack\) is separated and of finite presentation.

Proof

Recall that \(\Polarizedstack\) is a limit preserving algebraic stack, see Quot, Lemma 0D43. By Limits of Stacks, Lemma 0CMW this implies that \(\Delta : \Polarizedstack \to \Polarizedstack \times \Polarizedstack\) is limit preserving. Hence \(\Delta\) is locally of finite presentation by Limits of Stacks, Proposition 0CMY.

Let us prove that \(\Delta\) is separated. To see this, it suffices to show that given an affine scheme \(U\) and two objects \(\upsilon = (Y, \mathcal{N})\) and \(\chi = (X, \mathcal{L})\) of \(\Polarizedstack\) over \(U\), the algebraic space \[\mathit{Isom}_{\Polarizedstack}(\upsilon, \chi)\] is separated. The rule which to an isomorphism \(\upsilon_T \to \chi_T\) assigns the underlying isomorphism \(Y_T \to X_T\) defines a morphism \[\mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) \longrightarrow \mathit{Isom}_U(Y, X)\] Since we have seen in Lemmas 0DPN and 0DPP that the target is a separated algebraic space, it suffices to prove that this morphism is separated. Given an isomorphism \(f : Y_T \to X_T\) over some scheme \(T/U\), then clearly \[\mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) \times_{\mathit{Isom}_U(Y, X), [f]} T = \mathit{Isom}(\mathcal{N}_T, f^*\mathcal{L}_T)\] Here \([f] : T \to \mathit{Isom}_U(Y, X)\) indicates the \(T\)-valued point corresponding to \(f\) and \(\mathit{Isom}(\mathcal{N}_T, f^*\mathcal{L}_T)\) is the algebraic space discussed in Section 0DLW. Since this algebraic space is affine over \(U\), the claim implies \(\Delta\) is separated.

To finish the proof we show that \(\Delta\) is quasi-compact. Since \(\Delta\) is representable by algebraic spaces, it suffice to check the base change of \(\Delta\) by a surjective smooth morphism \(U \to \Polarizedstack \times \Polarizedstack\) is quasi-compact (see for example Properties of Stacks, Lemma 04XD). We can assume \(U = \coprod U_i\) is a disjoint union of affine opens. Since \(\Polarizedstack\) is limit preserving (see above), we see that \(\Polarizedstack \to \Spec(\mathbf{Z})\) is locally of finite presentation, hence \(U_i \to \Spec(\mathbf{Z})\) is locally of finite presentation (Limits of Stacks, Proposition 0CMY and Morphisms of Stacks, Lemmas 06Q3 and 0DNP). In particular, \(U_i\) is Noetherian affine. This reduces us to the case discussed in the next paragraph.

In this paragraph, given a Noetherian affine scheme \(U\) and two objects \(\upsilon = (Y, \mathcal{N})\) and \(\chi = (X, \mathcal{L})\) of \(\Polarizedstack\) over \(U\), we show the algebraic space \[\mathit{Isom}_{\Polarizedstack}(\upsilon, \chi)\] is quasi-compact. Since the connected components of \(U\) are open and closed we may replace \(U\) by these. Thus we may and do assume \(U\) is connected. Let \(u \in U\) be a point. Let \(P\) be the Hilbert polynomial \(n \mapsto \chi(Y_u, \mathcal{N}_u^{\otimes n})\), see Varieties, Lemma 0BEM. Since \(U\) is connected and since the functions \(u \mapsto \chi(Y_u, \mathcal{N}_u^{\otimes n})\) are locally constant (see Derived Categories of Schemes, Lemma 0B9T) we see that we get the same Hilbert polynomial in every point of \(U\). Set \(\mathcal{M} = \text{pr}_1^*\mathcal{N} \otimes_{\mathcal{O}_{Y \times_U X}} \text{pr}_2^*\mathcal{L}\) on \(Y \times_U X\). Given \((f, \varphi) \in \mathit{Isom}_{\Polarizedstack}(\upsilon, \chi)(T)\) for some scheme \(T\) over \(U\) then for every \(t \in T\) we have \[\chi(Y_t, (\text{id} \times f)^*\mathcal{M}^{\otimes n}) = \chi(Y_t, \mathcal{N}_t^{\otimes n} \otimes_{\mathcal{O}_{Y_t}} f_t^*\mathcal{L}_t^{\otimes n}) = \chi(Y_t, \mathcal{N}_t^{\otimes 2n}) = P(2n)\] where in the middle equality we use the isomorphism \(\varphi : f^*\mathcal{L}_T \to \mathcal{N}_T\). Setting \(P'(t) = P(2t)\) we find that the morphism \[\mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) \longrightarrow \mathit{Isom}_U(Y, X)\] (see earlier) has image contained in the intersection \[\mathit{Isom}_U(Y, X) \cap \mathit{Mor}^{P', \mathcal{M}}_U(Y, X)\] The intersection is an intersection of open subspaces of \(\mathit{Mor}_U(Y, X)\) (see Lemma 0DPP and Remark 0DPQ). Now \(\mathit{Mor}^{P', \mathcal{M}}_U(Y, X)\) is a Noetherian algebraic space as it is of finite presentation over \(U\) by Lemma 0DPR. Thus the intersection is a Noetherian algebraic space too. Since the morphism \[\mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) \longrightarrow \mathit{Isom}_U(Y, X) \cap \mathit{Mor}^{P', \mathcal{M}}_U(Y, X)\] is affine (see above) we conclude.

Lemma

The morphism \(\Polarizedstack \to \Spec(\mathbf{Z})\) is quasi-separated and locally of finite presentation.

Proof

Quasi-separatedness follows immediately from Lemma 0DPT. By Quot, Lemma 0D43, the stack \(\Polarizedstack\) is limit preserving. Hence it is locally of finite presentation by Limits of Stacks, Proposition 0CMY.

Lemma

Let \(n \geq 1\) be an integer and let \(P\) be a numerical polynomial. Let \[T \subset |\Polarizedstack|\] be a subset with the following property: for every \(\xi \in T\) there exists a field \(k\) and an object \((X, \mathcal{L})\) of \(\Polarizedstack\) over \(k\) representing \(\xi\) such that

1. the Hilbert polynomial of \(\mathcal{L}\) on \(X\) is \(P\), and

2. there exists a closed immersion \(i : X \to \mathbf{P}^n_k\) such that \(i^*\mathcal{O}_{\mathbf{P}^n}(1) \cong \mathcal{L}\).

Then \(T\) is a Noetherian topological space, in particular quasi-compact.

Proof

Observe that \(|\Polarizedstack|\) is a locally Noetherian topological space, see Morphisms of Stacks, Lemma 0DQI (this also uses that \(\Spec(\mathbf{Z})\) is Noetherian and hence \(\Polarizedstack\) is a locally Noetherian algebraic stack by Lemma 0DPU and Morphisms of Stacks, Lemma 06R6). Thus any quasi-compact subset of \(|\Polarizedstack|\) is a Noetherian topological space and any subset of such is also Noetherian, see Topology, Lemmas 0053 and 0052. Thus all we have to do is a find a quasi-compact subset containing \(T\).

By Lemma 0DPH the algebraic space \[H = \Hilbfunctor^{P, \mathcal{O}(1)}_{\mathbf{P}^n_\mathbf{Z}/\Spec(\mathbf{Z})}\] is proper over \(\Spec(\mathbf{Z})\). By Quot, Lemma 0D3W¹ the identity morphism of \(H\) corresponds to a closed subspace \[Z \subset \mathbf{P}^n_H\] which is proper, flat, and of finite presentation over \(H\) and such that the restriction \(\mathcal{N} = \mathcal{O}(1)|_Z\) is relatively ample on \(Z/H\) and has Hilbert polynomial \(P\) on the fibres of \(Z \to H\). In particular, the pair \((Z \to H, \mathcal{N})\) defines a morphism \[H \longrightarrow \Polarizedstack\] which sends a morphism of schemes \(U \to H\) to the classifying morphism of the family \((Z_U \to U, \mathcal{N}_U)\), see Quot, Lemma 0E94. Since \(H\) is a Noetherian algebraic space (as it is proper over \(\mathbf{Z})\)) we see that \(|H|\) is Noetherian and hence quasi-compact. The map \[|H| \longrightarrow |\Polarizedstack|\] is continuous, hence the image is quasi-compact. Thus it suffices to prove \(T\) is contained in the image of \(|H| \to |\Polarizedstack|\). However, assumptions (1) and (2) exactly express the fact that this is the case: any choice of a closed immersion \(i : X \to \mathbf{P}^n_k\) with \(i^*\mathcal{O}_{\mathbf{P}^n}(1) \cong \mathcal{L}\) we get a \(k\)-valued point of \(H\) by the moduli interpretation of \(H\). This finishes the proof of the lemma.

Properties of moduli of complexes on a proper morphism

Let \(f : X \to B\) be a morphism of algebraic spaces which is proper, flat, and of finite presentation. Then the stack \(\Complexesstack_{X/B}\) parametrizing relatively perfect complexes with vanishing negative self-exts is algebraic. See Quot, Theorem 0DLN.

Lemma

The diagonal of \(\Complexesstack_{X/B}\) over \(B\) is affine and of finite presentation.

Proof

The representability of the diagonal by algebraic spaces was shown in Quot, Lemma 0DLG. From the proof we find that we have to show: given a scheme \(T\) over \(B\) and objects \(E, E' \in D(\mathcal{O}_{X_T})\) such that \((T, E)\) and \((T, E')\) are objects of the fibre category of \(\Complexesstack_{X/B}\) over \(T\), then \(\mathit{Isom}(E, E') \to T\) is affine and of finite presentation. Here \(\mathit{Isom}(E, E')\) is the functor \[(\Sch/T)^{opp} \to \textit{Sets},\quad T' \mapsto \{\varphi : E_{T'} \to E'_{T'} \text{ isomorphism in }D(\mathcal{O}_{X_{T'}})\}\] where \(E_{T'}\) and \(E'_{T'}\) are the derived pullbacks of \(E\) and \(E'\) to \(X_{T'}\). Consider the functor \(H = \SheafHom(E, E')\) defined by the rule \[(\Sch/T)^{opp} \to \textit{Sets},\quad T' \mapsto \Hom_{\mathcal{O}_{X_{T'}}}(E_T, E'_T)\] By Quot, Lemma 0DLC this is an algebraic space affine and of finite presentation over \(T\). The same is true for \(H' = \SheafHom(E', E)\), \(I = \SheafHom(E, E)\), and \(I' = \SheafHom(E', E')\). Therefore we see that \[\mathit{Isom}(E, E') = (H' \times_T H) \times_{c, I \times_T I', \sigma} T\] where \(c(\varphi', \varphi) = (\varphi \circ \varphi', \varphi' \circ \varphi)\) and \(\sigma = (\text{id}, \text{id})\) (compare with the proof of Quot, Proposition 08K9). Thus \(\mathit{Isom}(E, E')\) is affine over \(T\) as a fibre product of schemes affine over \(T\). Similarly, \(\mathit{Isom}(E, E')\) is of finite presentation over \(T\).

Lemma

The morphism \(\Complexesstack_{X/B} \to B\) is quasi-separated and locally of finite presentation.

Proof

To check \(\Complexesstack_{X/B} \to B\) is quasi-separated we have to show that its diagonal is quasi-compact and quasi-separated. This is immediate from Lemma 0DPW. To prove that \(\Complexesstack_{X/B} \to B\) is locally of finite presentation, we have to show that \(\Complexesstack_{X/B} \to B\) is limit preserving, see Limits of Stacks, Proposition 0CMY. This follows from Quot, Lemma 0DLJ (small detail omitted).