Relative divisors and the existence of the Picard scheme

Written by GPT-6.1 Sol (OpenAI) (Codex), October 2026. Public domain (CC0).

A nonzero section of a line bundle on an integral projective variety cuts out a divisor. Vary the section while keeping the bundle fixed, and the resulting divisors form a projective space. Vary the bundle as well, and these projective spaces become the fibres of the Abel map. This gives a way to construct the space of line-bundle classes from the already constructed Hilbert scheme.

The proof has three distinct jobs. We must construct complete linear systems over arbitrary parameter schemes, including nilpotent ones. We must show that linear equivalence defines a proper flat relation on a suitable open divisor scheme. We must then construct the quotient as a scheme. We carry out that last step by parametrizing the equivalence classes themselves as closed subschemes in a Hilbert scheme. We then construct affine gluing parameters for closed unions, prove affine restriction through nilpotent thickenings over a field, and construct Picard schemes by finite-flat Stein reduction and restriction of scalars.

For the main existence theorem, \(S\) is Noetherian and \(f:X\to S\) is projective and flat, with geometrically integral fibres. Fix a relatively very ample bundle \(\mathcal O_X(1)\). The complete-linear-system theorem has the broader hypothesis that \(f\) is proper, flat and finitely presented. Throughout, \(\operatorname{Pic}_{X/S}\) denotes the étale Picard sheaf. It equals the fppf Picard sheaf under these hypotheses, by The Picard functor and the Picard scheme of a curve, Theorem 3.1.

We use the quotient convention

It parametrizes quotient line bundles of \(Q\). If \(V\) is a bundle of sections, their lines are parametrized by \(\mathbb P(V^\vee)\).

Here is the quotient-line property on arbitrary test schemes. A quotient \(g^*Q\twoheadrightarrow N\), with \(N\) invertible, induces the graded quotient \[ \operatorname{Sym}(g^*Q)\twoheadrightarrow \bigoplus_{d\geq0}N^{\otimes d}. \] Taking relative Proj gives a map \(T\to\mathbb P(Q)\), since the relative Proj of the algebra on the right is \(T\). Conversely pull back the tautological quotient on \(\mathbb P(Q)\). To check that these operations are inverse, work where \(Q\) has finitely many generators \(q_i\). The chart where the image of \(q_i\) generates \(N\) trivializes \(N\), normalizes that image to \(1\), and records the other images as the degree-zero fractions \(q_j/q_i\). All relations of \(Q\) are imposed by its symmetric algebra. The same fractions are the coordinates on \(D_+(q_i)\), so the constructions agree there and glue uniquely. These charts cover \(T\) by surjectivity of the quotient. This proves the assertion including nilpotent parameters and its compatibility with every base change.

1 Relative divisors form an open Hilbert locus

An effective Cartier divisor on \(Y\) is a closed subscheme whose ideal is locally generated by a nonzerodivisor. Equivalently it is the zero scheme of a regular section of an invertible sheaf. The empty divisor is allowed. For a flat finitely presented morphism \(Y\to T\), a relative effective Cartier divisor is an effective Cartier divisor \(D\subset Y\) that is flat over \(T\).

The fibrewise Cartier criterion gives an equivalent condition: locally write the equation as \(a\); its image must be a nonzerodivisor in every fibre where it vanishes. Then \(a\) is a nonzerodivisor on \(Y\), and \(\mathcal O_Y/(a)\) is flat over \(T\) [Stacks, Tag 062Y]. In particular the condition persists after arbitrary base change. This criterion includes points outside \(D\), where the equation is a unit.

There is also a useful formulation starting with a flat family of closed subschemes. If \(Y\) and \(D\) are flat and finitely presented over \(T\), then \(D\) is a relative effective Cartier divisor exactly when every geometric fibre is an effective Cartier divisor in the corresponding fibre of \(Y\). The criterion applies locally, and faithful flatness of a field extension detects the fibre condition.

Define

Divisors here are closed subschemes, so no choice of an equation or its scalar is part of the datum.

Proposition 1.1. For a flat projective finitely presented family, \(\operatorname{Div}_{X/S}\) is represented by an open subscheme of \(\operatorname{Hilb}_{X/S}\). Over a Noetherian base its fixed-Hilbert-polynomial pieces are quasi-projective. Geometric integrality is not needed for this openness assertion.

Proof. Let \(H\) be the Hilbert scheme and \(Z\subset X_H\) its universal flat closed subscheme. Let \(B\subset Z\) be the locus where \(Z\) fails to be a relative Cartier divisor in \(X_H/H\). The Cartier criterion makes its complement open, so \(B\) is closed in \(Z\). Since \(Z\to H\) is proper, its image is closed. Put

The family \(Z_U\) is a relative divisor. Conversely, a relative divisor \(D\subset X_T\) gives a unique Hilbert map \(g:T\to H\). For every \(t\), its fibre is Cartier. Faithfully flat extension from \(\kappa(g(t))\) to \(\kappa(t)\) detects the same condition for \(Z_{g(t)}\). The Cartier criterion for the flat universal family then says the whole fibre over \(g(t)\) avoids \(B\). Thus \(g(t)\in U\). A map with image in an open subscheme factors through that open, for arbitrary \(T\); hence \(g\) factors uniquely through \(U\).

These constructions are inverse and commute with pullback. They prove representability on all test schemes, rather than only a classification of geometric points. The fixed-polynomial Hilbert scheme is projective by Hilbert and Quot schemes. Its divisor open is quasi-projective and, over Noetherian \(S\), of finite type. \(\square\)

The natural Abel map is the map of sheaves

For now its target is a sheaf, not an assumed scheme.

2 The universal module of sections

We need a parameter module that still works when the dimension of sections jumps. Simply writing \((f_*L)^\vee\) would not do that.

Lemma 2.1. Let \(f:X\to S\) be proper and finitely presented, and let \(F\) be a finitely presented quasi-coherent sheaf flat over \(S\). There is a finitely presented quasi-coherent module \(Q_F\) such that, for every \(g:T\to S\) and every quasi-coherent \(N\) on \(T\),

It is unique with this universal isomorphism, and its formation commutes with every base change. If \(S\) is locally Noetherian, \(Q_F\) is coherent.

Proof. Perfect cohomology with arbitrary base change [Stacks, Tag 0A1H] supplies a perfect complex \(K=Rf_*F\). Locally on an affine open in \(S\), it can be represented by finite free modules in nonnegative degrees,

Here is why negative terms can be removed. On any residue field the derived base change is the cohomology complex of a sheaf, so its negative cohomology vanishes. Start at the lowest degree of a finite free representative. If this degree is negative, the differential is injective on the fibre and has an invertible maximal minor near the point. Split off the resulting contractible pair. Repeating removes all negative terms near that point. This is a local assertion, which is all we need.

For any base module \(N\), derived base change and the projection formula identify the right side of (2.1) with the degree-zero kernel

There is no derived flatness issue in this expression: the modules in (2.2) are free, and \(F\) is flat over the base. Define locally

Applying \(\operatorname{Hom}(-,N)\) to this presentation gives exactly (2.3). Thus it proves (2.1). Intrinsically the module is

The intrinsic formula or the universal property identifies the constructions on overlaps, so they glue. The presentation (2.4) is finite, and taking its cokernel commutes with arbitrary tensor pullback. The base-changed complex computes the base-changed cohomology by Tag 0A1H, proving the asserted base-change property. Uniqueness follows from the module version of Yoneda applied to (2.1). On a locally Noetherian scheme, finite presentation implies coherence. \(\square\)

On a fibre,

If the higher cohomology of \(F_s\) vanishes throughout a neighbourhood, the same cancellation argument reduces (2.2) to a finite locally free module \(V\) in degree zero. Then

and \(V\) commutes with every base change. The more familiar sufficient condition \(H^1(X_s,F_s)=0\) also gives local freeness of the section module near \(s\). To see this directly, cancel all contractible pairs whose differential has a nonzero entry on that residue field. The remaining differentials vanish on the fibre. Its degree-one term must then have rank zero, since its fibre cohomology in degree one is zero. After shrinking, the complex therefore has no degree-one term, and its degree-zero term splits off as a free summand. Formula (2.4) identifies the section module with its dual; all further base changes preserve this splitting. We will use the stronger all-higher-cohomology vanishing condition in the existence proof.

The distinction between finite presentation and coherence matters. The theorem is valid over arbitrary schemes \(S\). Calling \(Q_F\) coherent in that generality would impose a property that finite presentations over noncoherent rings need not have.

3 Complete linear systems on every test scheme

For proper flat finitely presented \(f\) with geometrically integral fibres, we have

We use the precise proper-flat cohomology theorem for geometrically reduced connected fibres [Stacks, Tag 0E0L]. It applies because an integral geometric fibre is reduced and connected. Consequently pullback of base line bundles is fully faithful: homomorphisms between them are detected by pushing forward, locally trivializing them and using (3.1).

Let \(L\) be invertible on \(X\). Its complete relative linear system is

Theorem 3.1 (complete linear systems). With these hypotheses, \(\operatorname{LinSys}_L\) is represented by \(\mathbb P(Q_L)\), where \(Q_L\) is the finitely presented module of Lemma 2.1. This is also the fibre product of the Abel map with the Picard class defined by \(L\). If \(S\) is locally Noetherian the module is coherent. If \(f_*L=V\) is locally free and commutes with every base change, the linear system is \(\mathbb P(V^\vee)\).

Proof. A \(T\)-point of \(\mathbb P(Q_L)\) is a surjection

onto an invertible sheaf, considered up to isomorphism of the target. By (2.1), it gives a section

The quotient is surjective precisely when its residue-field map is nonzero at every point of \(T\), by Nakayama applied to its cokernel. By (2.6), this means \(s_t\ne0\) in \(H^0(X_t,L_t)\) at every point. On an integral fibre, a nonzero global section of a line bundle is regular: in a trivialization it is nonzero at the generic point, hence is a nonzerodivisor in every domain where it is considered. The fibrewise Cartier criterion therefore makes the zero scheme \(D\) of \(s\) a relative divisor. Its divisor bundle is \(L_T\otimes f_T^*N\).

Conversely, start with \(D\) in (3.2). Choose the displayed isomorphism. The canonical section of \(\mathcal O(D)\) becomes (3.4), and the fibre criterion makes it nonzero on every fibre. Equation (2.1) gives a map (3.3), which is surjective by Nakayama.

This point of projective space is independent of the choices. Two choices of \(N\) are isomorphic, since cancelling \(L_T\) gives isomorphic pullbacks of them and (3.1) detects that isomorphism on \(T\). Two isomorphisms with fixed \(N\) differ by a unit of \(X_T\), which is a unique base unit by (3.1). It changes the quotient by an automorphism of its target and therefore leaves the projective point unchanged. In the other direction, isomorphic quotient pairs have identical zero schemes.

The two constructions are inverse: the canonical divisor section recovers the original section up to the allowed scalar, and the zero scheme recovers the original divisor ideal. Each operation commutes with pullback. This proves the statement for arbitrary test schemes, including nonreduced ones.

Finally, equality between the Abel class of \(D\) and the class of \(L_T\) in the Picard sheaf implies that their bundles differ by a base line bundle: this is the injection of the raw relative Picard quotient into its sheaf proved in the previous lesson, Proposition 1.1, under (3.1). Hence the sheaf fibre is exactly (3.2). If sections form a base-change-compatible bundle \(V\), the universal identity (2.1) identifies \(Q_L\) with \(V^\vee\). \(\square\)

On a field-valued class actually represented by \(L\), the fibre is

The general relative \(\mathbb P(Q_L)\) need not be a projective-space bundle: its fibre dimensions can jump. There is a separate descent issue if a Picard class has no global line-bundle representative. It becomes a complete linear system after an étale cover, but the projective spaces can descend to a Brauer–Severi scheme instead of the projectivization of a global vector bundle. We will use local representatives and effective descent explicitly.

4 Separatedness before representability

We prove separatedness at the sheaf level. This avoids assuming the Picard scheme while constructing it.

Proposition 4.1. Under the main projective-flat geometrically integral hypotheses, the diagonal of \(\operatorname{Pic}_{X/S}\) is represented by closed immersions.

Proof. First take two line bundles \(L,M\) on \(X_T\). We construct the closed subscheme of \(T\) where their relative classes are equal, in the functorial sense that must hold after every further base change.

The fibre Hilbert polynomials of \(L\) and \(M\) are locally constant. Indeed the proper-flat perfect cohomology complexes of finitely many twists have locally constant Euler characteristic [Stacks, Tags 0A1H and 0B9T]; finitely many values determine a polynomial of degree bounded by the projective ambient dimension. The fibre dimension \(n\) is locally constant as well. We may work on one such open and closed piece.

If \(n=0\), the proper flat family is an isomorphism. It is finite because its fibres are finite [Stacks, Tag 02LS], and finite locally free of rank one because each geometric fibre is the single reduced point. The unit map \(\mathcal O_T\to(f_T)_*\mathcal O_{X_T}\) is an isomorphism, so its finite algebra is \(\mathcal O_T\). The relative Picard sheaf is zero there.

Suppose \(n\geq1\). Let \(T_0\subset T\) be the open and closed locus on which the coefficient of degree \(n-1\) in \(P_M-P_L\) is zero. Equal classes must lie in this locus. On an integral projective fibre, a nonzero homomorphism between line bundles is injective. If it is not an isomorphism, its cokernel is a twist of the structure sheaf of a nonempty effective Cartier divisor. That sheaf has dimension \(n-1\) and positive leading Hilbert coefficient. The Hilbert-polynomial dimension and positivity statement was established in Castelnuovo–Mumford regularity and boundedness, Section 3. Thus, on \(T_0\), every nonzero fibre homomorphism \(L_t\to M_t\) is an isomorphism.

Set \(A=M\otimes L^{-1}\) on \(X_{T_0}\). The scheme

parametrizes fibrewise nonzero sections of \(A\) up to a base line-bundle factor. By the preceding paragraph these sections are fibrewise isomorphisms. A homomorphism of line bundles which is an isomorphism on every fibre is an isomorphism everywhere: in a trivialization its coefficient is a unit at every point, as detected in that point's fibre local ring. Therefore (4.1) parametrizes exactly equality of the two relative classes on every test scheme over \(T_0\).

There is at most one such projective point on any test scheme. Two isomorphisms with their base factors differ by a unique isomorphism of the base factors and then by a base unit, using (3.1). Both changes are identified in the quotient-line parametrization. Hence \(E\to T_0\) is a monomorphism. It is projective; a proper morphism with these finite fibres is finite [Stacks, Tag 02LS]. A finite monomorphism is a closed immersion [Stacks, Tag 03BB]. Indeed work over an affine \(\operatorname{Spec}A\), and write its finite algebra as \(B\). The monomorphism says that multiplication \(B\otimes_A B\to B\) is an isomorphism, because it is the algebra map of its diagonal. After tensoring with a residue field \(\kappa\), put \(C=B\otimes_A\kappa\). Its finite dimension satisfies \((\dim_\kappa C)^2=\dim_\kappa C\), so \(C\) is zero or is the one-dimensional unital algebra \(\kappa\). Hence \(A\to B\) is surjective on every residue fibre. Its finite module cokernel vanishes at every maximal ideal by Nakayama, and is zero. Thus \(A\to B\) is surjective, as required. Thus \(E\) is a closed subscheme of \(T_0\), and also of \(T\). It represents equality of the relative classes; maps factoring through a closed subscheme include their nilpotent conditions, not just their point sets.

For two arbitrary Picard classes over \(T\), take an étale cover on which they are represented by bundles. The closed equality subschemes just constructed have canonical identical pullbacks on overlaps, since they represent the same equality functor. Their ideal sheaves descend effectively [Stacks, Tag 023R]. The descended closed subscheme represents equality over \(T\) by the sheaf property. This proves the diagonal assertion. \(\square\)

The argument isolates the role of integral fibres: a nonzero section has a regular divisor, and a nonempty such divisor has a strictly positive top Hilbert coefficient. Both assertions can fail in the form needed here on a reducible fibre.

5 An open locus with enough sections

Call a fibre line bundle \(L_t\) positive here if it is globally generated, has a nonzero section, and

This is a sufficient positivity condition for the construction, not a definition of ampleness. Its all-twists condition makes tensoring with \(\mathcal O_X(1)\) preserve positivity. Global generation also persists because \(\mathcal O_X(1)\) is globally generated.

Let \(P_+\subset\operatorname{Pic}_{X/S}\) be the subfunctor of classes with positive geometric fibres. These conditions do not depend on the representative or its base line-bundle factor, and are invariant under extending a residue field.

Lemma 5.1. The inclusion \(P_+\to\operatorname{Pic}_{X/S}\) is represented by open immersions. Write \(\xi=[\mathcal O_X(1)]\). The open subfunctors

cover the Picard sheaf on every test scheme.

Proof. Start with an actual bundle \(L\) over a Noetherian affine test scheme. Relative Serre vanishing and generation give an integer \(N\) such that all twists \(L(n)\), \(n\geq N\), have zero higher fibre cohomology and are generated by their sections. These are the usual projective coherent-sheaf Serre theorem and its flat base-change form [Stacks, Tags 01XO, 0B5T and 02O1]. One may obtain the latter form by applying the proper-flat cohomology complex to the vanishing direct images. It is then concentrated in degree zero. Its derived tensor with any base module has no negative cohomology, because the original bundle is base-flat and direct image of a sheaf has no negative cohomology. Thus its degree-zero module has no higher Tor with any module: it is flat, hence locally free by finite presentation. Its derived base changes have no higher cohomology, and generation remains surjective after pullback.

Consequently only finitely many vanishing tests in (5.1) remain. Each is open by the finite cohomology complex and semicontinuity. On their intersection, \(f_*L\) is a base-change-compatible locally free module. The locus where its fibre has positive rank is open and closed. Its evaluation map

has a coherent cokernel. The image in the base of its support is closed by properness. Removing that image is precisely the global-generation condition on fibres. This proves openness.

For an arbitrary affine test scheme \(\operatorname{Spec}B\) over a Noetherian affine open \(\operatorname{Spec}A\subset S\), write \(B\) as the filtered union of its finitely generated \(A\)-subalgebras. The finitely presented bundle descends to \(X_{B_0}\) at a finite stage. To retain invertibility, descend its inverse, their tensor-product isomorphisms and the two inverse identities as well; the finite-presentation argument [Stacks, Tag 01ZR] is explicit here. Cover the fixed \(X_A\) by finitely many affine opens. On each base change to \(B\), choose a finite principal-open refinement trivializing \(L\). All functions defining these opens involve finitely many coefficients of \(B\). Choose also the finite unit-ideal identities showing that the refinements cover the original affine charts; they ensure the same cover at a sufficiently large stage. On pairwise intersections use finite affine principal refinements, possible because \(X_A\) is separated and quasi-compact. The transition units, their inverses and their equalities on these refinements use only finitely many further coefficients. Their product identities and the cocycle identities hold at a sufficiently large stage: each is an equality of finitely many elements in a filtered colimit of localized coordinate rings, and each localization equality has a finite denominator witness. Thus they glue trivial line bundles at that stage. Their inverses glue the inverse bundle, and the same finite identities give the two inverse isomorphisms. This constructs the descended bundle and all the required identities. The stage \(B_0\) is Noetherian. The preceding open locus pulls back to exactly the required locus on \(\operatorname{Spec}B\), because fibre cohomology and generation are invariant under field extension. This proves openness for all affine tests and then for all tests by gluing.

A Picard class is locally represented by a bundle in the étale topology. The just constructed opens descend: membership is invariant on overlaps, and an étale-local invariant open descends to an open of the base. Its universal factorization property follows from the sheaf property. Thus \(P_+\) is an open subfunctor.

Finally take any \(\lambda\in\operatorname{Pic}_{X/S}(T)\) and \(t\in T\). After a field extension, its fibre has a line-bundle representative. Serre's theorem on that projective fibre makes \(L(m)\) positive for sufficiently large \(m\). Hence \(t\) lies in the inverse image of (5.2) for that \(m\). Those inverse images are opens and cover \(T\). Tensoring by \(\mathcal O(1)\) preserves positivity, so the cover is increasing if desired. \(\square\)

For a positive bundle \(L\) over a test scheme, Lemma 2.1 now gives

In particular the latter is smooth, proper and surjective over the test scheme.

It is useful to index the positive locus by the Hilbert polynomial of the inverse bundle:

This is locally constant by the same finite-cohomology argument as in Section 4. Thus \(P_+\) is a disjoint union of open and closed subfunctors \(P_+^\phi\). Work also on a piece of \(S\) on which \(\psi=P_{\mathcal O_{X_t}}\) is constant. An effective divisor of such a class has Hilbert polynomial

by the ideal sequence \(0\to L_t^{-1}\to\mathcal O_{X_t}\to\mathcal O_{D_t}\to0\).

6 The Abel relation on positive divisors

Let \(Y^\phi\) be the inverse image of \(P_+^\phi\) under the Abel map. Proposition 1.1 and Lemma 5.1 make it an open subscheme of the projective scheme \(\operatorname{Hilb}^{\psi-\phi}_{X/S}\). In particular \(Y^\phi\) is Noetherian and quasi-projective over \(S\).

Proposition 6.1. The map \(Y^\phi\to P_+^\phi\) is a surjection of étale sheaves, represented by smooth proper morphisms. Its relation

is a closed subscheme of \(Y^\phi\times_S Y^\phi\), and both its projections to \(Y^\phi\) are smooth and projective.

Proof. Given \(T\to P_+^\phi\), represent its class by a positive bundle \(L\) after an étale cover \(T'\to T\). Theorem 3.1 identifies its divisor fibre with \(\mathbb P(((f_{T'})_*L)^\vee)\). This is a positive-rank projective-space bundle. It is smooth and proper and has sections Zariski locally on \(T'\), by trivializing its section bundle and choosing a coordinate line. Thus the Abel map is an epimorphism of étale sheaves.

We must also check that the fibre over \(T\) is a scheme. The fibre functor is the equality locus of the given class and the divisor bundle on \(X_{T\times Y^\phi}\). Proposition 4.1 makes it a closed subscheme of \(T\times_S Y^\phi\). Smoothness and properness descend from its just described étale pullback. This proves relative representability and the first assertion without assuming representability of the target.

The same diagonal argument gives the closed subscheme (6.1). To see projectivity of either projection, take \(T=Y^\phi\). Its fibre scheme is proper over \(T\), and maps by a locally closed immersion into

This map is proper: the target is separated over \(T\), so the graph argument for a map from a proper \(T\)-scheme applies. A proper monomorphism is a closed immersion, by the finite-fibre and finite-monomorphism results used in Section 4. Thus the fibre scheme is closed in (6.2), which is projective over \(T\). Both projections are therefore projective, as well as smooth. \(\square\)

One should not infer projectivity merely from the fact that the fibres become projective spaces after an étale cover. The argument uses the closed embedding into a fixed projective Hilbert scheme. It supplies the global ample bundle that such a conclusion needs.

7 Constructing the quotient by parametrizing its classes

We now prove the required quotient theorem. A schematic equivalence relation \(R\subset Y\times_S Y\) means that for every test scheme the subset \(R(T)\) is the graph of an equivalence relation on \(Y(T)\). This includes its scheme structure, not just a relation on geometric points.

Theorem 7.1 (proper flat quotient). Let \(S\) be Noetherian, let \(Y\) be quasi-projective over \(S\), and let \(R\subset Y\times_S Y\) be a closed schematic equivalence relation whose projections are proper, flat and finitely presented. There is a quasi-projective \(S\)-scheme \(Q\) and a faithfully flat projective finitely presented morphism \(q:Y\to Q\) such that

The scheme \(Q\) represents the quotient sheaf in the fppf topology. If the projections are smooth, \(q\) is smooth and represents the étale quotient sheaf as well.

Proof. Regard the first projection as a family of closed subschemes of \(Y\), by using the second coordinate. These subschemes are the equivalence classes. The family is proper and flat over \(Y\), and is projective there: embed \(Y\) into a projective \(S\)-scheme; the induced immersion of \(R\) in its product with \(Y\) is proper and therefore closed. The Hilbert polynomial of its fibres is locally constant. Since \(Y\) is Noetherian and quasi-compact, only finitely many polynomials \(p_1,\ldots,p_r\) occur. None is zero, since reflexivity supplies a point in every class.

The Hilbert construction in Hilbert and Quot schemes also applies to proper subschemes of a quasi-projective scheme: take the open in the Hilbert scheme of a projective closure where the universal support avoids the boundary. Let

be the resulting finite union of quasi-projective Hilbert schemes and its universal family. The morphism \(\pi\) is projective, flat, finitely presented and surjective. Surjectivity holds because a subscheme with nonzero polynomial is nonempty on every fibre.

The relation family defines a morphism

For every test scheme \(T\) and \(x,y\in Y(T)\), we have

Indeed related points have identical class subschemes: after every further base change, transitivity and symmetry identify membership in the two classes, so Yoneda identifies their closed subschemes. Conversely if those subschemes are identical, the point \(x\), which lies in its own class by reflexivity, lies in the class of \(y\). This proves (7.4) on all tests, including their nilpotent structure.

Since \(H\) is separated over \(S\), the graph \(\Gamma_u\subset Y\times_S H\) is closed. Reflexivity makes it a closed subscheme of \(D\). We claim it is invariant under the two projections

To check equality of the two pulled-back closed subschemes, take any test point \((x,y,V)\), where \(V\) is a flat proper class-sized subscheme of \(Y_T\) and \(x,y\) lie in \(V\). Membership in the first pullback is \(V=u(x)\); in the second it is \(V=u(y)\). If the first holds, then \(y\) lies in the class of \(x\), so (7.4) gives \(u(y)=u(x)\). The reverse implication is identical with \(x,y\) exchanged. Since this equivalence holds on every test scheme, the two closed subschemes in (7.5) are equal.

Now \(\pi:D\to H\) is faithfully flat and quasi-compact. Effective descent for quasi-coherent ideals [Stacks, Tag 023R] descends the invariant ideal of \(\Gamma_u\) to an ideal on \(H\). The canonical identification on the two pullbacks obeys the cocycle condition, since both are the same ideal inside the structure sheaf. Thus there is a closed subscheme

The graph is isomorphic to \(Y\). Base-changing \(\pi\) in (7.6) therefore gives a faithfully flat projective finitely presented map \(q:Y\to Q\). Since \(Q\) is closed in the finite quasi-projective union \(H\), it is quasi-projective. Equation (7.4), together with its all-test-scheme interpretation, identifies its kernel pair with \(R\), proving (7.1).

Finally any map \(T\to Q\) lifts to \(Y\) after the fppf cover \(T\times_QY\to T\). Two local lifts have identical images precisely when they are related by (7.1). Thus \(h_Q\) is the sheafification of the presheaf quotient. More explicitly a map from \(Y\) to an fppf sheaf which agrees on \(R\) descends uniquely: choose local lifts of a \(Q\)-point, apply that map, and use the equality on the kernel pair to glue. This is the universal property of the quotient sheaf.

If \(R\to Y\) is smooth, the faithfully flat base change of \(q\) by itself is smooth, so \(q\) is smooth by descent. A smooth surjective map has étale-local sections. The same lifting and gluing argument then identifies \(Q\) with the étale quotient as well. \(\square\)

The Hilbert scheme in this proof parametrizes an entire equivalence class, rather than a selected divisor. The closed graph marks the classes that actually arise from the relation. Flat descent turns that marked locus into the quotient scheme. A reference for the classical result and this method is Nitsure, Construction of Hilbert and Quot Schemes, Theorem 6.8; the argument above supplies every quotient step used here.

8 Gluing the Picard scheme

Theorem 8.1 (Grothendieck's existence theorem). Let \(S\) be Noetherian and \(f:X\to S\) projective and flat with geometrically integral fibres. The étale Picard functor is represented by a separated group scheme locally of finite type over \(S\). Formation of this scheme commutes with arbitrary base change. Its positive fixed-inverse-polynomial charts are quasi-projective over \(S\).

Proof. On an open and closed base piece with fixed \(\psi\), apply Theorem 7.1 to \(Y^\phi\) and \(R^\phi\) from Proposition 6.1. It gives a quasi-projective quotient \(Q^\phi\), with a smooth surjective map from \(Y^\phi\). The Abel map is also an epimorphism of étale sheaves and has exactly the same kernel pair. The local-lift universal property in Theorem 7.1 therefore identifies

This identification uses neither an assumed Picard algebraic space nor an existing Picard variety. Taking the disjoint union over \(\phi\), and then over the open and closed pieces for \(\psi\), represents \(P_+\) by a scheme \(Q_+\). Each piece is of finite type over the Noetherian base.

Translation by \(m\xi\) identifies every open subfunctor in (5.2) with \(P_+\). Their pairwise intersections are represented by opens in both charts, because \(P_+\) is relatively open. The identifications represent equality of the same elements of the Picard sheaf and consequently satisfy the cocycle condition. Glue these schemes along the intersections.

To verify the resulting scheme \(P\) represents the whole sheaf, take any \(\lambda\in\operatorname{Pic}_{X/S}(T)\). Lemma 5.1 makes its inverse images of the charts an open cover of \(T\). On each open it determines a unique map to the corresponding chart. Equality of the class on overlaps makes these maps agree, so they glue uniquely to \(T\to P\). Conversely maps to \(P\) give Picard classes on their inverse-image charts and glue by the sheaf property. Both constructions are inverse on every test scheme.

The scheme is locally of finite type because its charts are. Proposition 4.1 says its diagonal is closed, so it is separated over \(S\). Yoneda turns tensor product, dual and the trivial bundle into its group-law, inverse and identity morphisms. Their identities are exactly the line-bundle identities, giving a commutative group scheme.

For any \(S'\to S\), evaluation of the relative Picard sheaf of \(X_{S'}/S'\) on an \(S'\)-scheme \(T\) is the same evaluation as before: the family is \(X_T\), with the same étale covers and base-bundle quotient. Thus \(P\times_S S'\) represents that sheaf, proving arbitrary base-change compatibility. This assertion does not require \(S'\) to be Noetherian. \(\square\)

The result is local finite type, rather than finite type for the whole Picard scheme. Already \(\operatorname{Pic}_{\mathbb P^n_S/S}\) for \(n\geq1\) has one copy of \(S\) for every integer. Finiteness statements for appropriate parts belong to The structure of the Picard scheme.

8.2 Closed unions and their gluing parameters

The relative divisor construction has geometrically integral fibres. Closed unions and nilpotent thickenings require different coordinates. We now prove the affine restriction statements for these operations. In this subsection and the next, write \(P_{Y/B}\) for the fppf Picard sheaf. Under the universal-functions condition, its agreement with the étale sheaf is the comparison proved in The Picard functor and the Picard scheme of a curve, Sections 1–3. The field constructions below also give this agreement when the global-function algebra is larger than the field.

Lemma 8.2 (patching across a closed union). Let \(X_1,X_2\) be closed subschemes of an \(S\)-scheme \(X\), with ideal sheaves \(I_1,I_2\), and put \(Z=X_1\cap X_2\). Suppose \(I_1\cap I_2=0\), and \(\mathcal O_Z\) is flat over \(S\). Then, after every base change \(T\to S\), a line bundle on \(X_T\) is equivalent to two line bundles \(L_i\) on \(X_{i,T}\) and a specified isomorphism

Arrows are pairs of bundle isomorphisms compatible with \(u\).

Proof. On \(X\) the closed union gives the exact sequence

Flatness of its last term over \(S\) preserves this sequence after every parameter base change. Thus the affine ring square is still \(A=A_1\times_C A_2\), with both \(A_i\to C\) surjective. Given (8.2), take the kernel of the difference of the two restriction maps to \(L_2|_{Z_T}\). This is a quasi-coherent module on \(X_T\).

We check that it is invertible and recovers both branches. At a point in \(Z_T\), trivialize the two bundles. Their gluing map is a unit of the quotient local ring \(C\). It lifts to a unit of either branch local ring: an element whose residue in a quotient of a local ring is a unit is outside its maximal ideal. Change one trivialization by this lift. The difference map is now the ordinary difference \(A_1\oplus A_2\to C\), and its kernel is \(A\). These trivializations and the lifted unit exist on a neighbourhood of the point. Away from \(Z_T\), one branch is absent and the kernel is the bundle on the other. Hence the kernel is locally free of rank one and has the asserted restrictions. A compatible pair of arrows preserves the kernel. Conversely an arrow on that kernel is determined by its two restrictions, because it injects into their direct sum. Starting with an invertible module, tensor (8.3) with it to recover exactly this kernel. These constructions are inverse, including arrows, and commute with every base change. \(\square\)

Theorem 8.3 (relative affine gluing). Let \(S\) be Noetherian, \(X/S\) proper, and \(X_1,X_2\subset X\) closed, flat over \(S\), with coherent ideals satisfying \(I_1\cap I_2=0\). Suppose \(Z=X_1\cap X_2\) is flat over \(S\), and, for every \(s\in S\),

Then restriction

is representable by affine morphisms of finite presentation. In particular, if both target Picard sheaves are schemes, so is \(P_{X/S}\). The affine assertion holds after arbitrary base change on \(S\).

Proof. First (8.4) implies the universal identity \((f_i)_*\mathcal O_{X_i}=\mathcal O_S\). Here is the argument over a possibly nonreduced base. Properness and flatness give the nonnegative finite projective cohomology complex of Base change and the Grothendieck complex, Theorem 3.2. The unit section maps to a basis of the degree-zero fibre cohomology by (8.4). Thus its degree-zero comparison is surjective. The proof of Semicontinuity and Grauert's theorem, Theorem 3.1 cancels the invertible differential blocks and kills the outgoing differential. There is no incoming term in negative degree. The kernel is consequently a line bundle, with every base change, and its unit section is a basis by Nakayama. This proves the identity universally, without requiring \(S\) reduced.

The nonempty-fibre locus of \(Z\to S\) is open and closed. It is open because this map is flat and of finite presentation, and closed because it is proper. Over its complementary locus, \(X=X_1\amalg X_2\) by (8.3). The fppf Picard sheaf of a disjoint union is the product of the two Picard sheaves: representatives and their base factors may be chosen locally, where all base line bundles are trivial. Thus (8.5) is an isomorphism there.

On the nonempty locus the cover \(Z\to S\) is fppf. Make this base change. Its diagonal supplies a section \(\sigma\) of \(Z\), hence of both branches and of \(X\). The universal form of (8.3) shows that \((f_T)_*\mathcal O_{X_T}\) is the kernel of

The unit of the last algebra is injective, since restriction along \(\sigma_T\) is a left inverse. The kernel is the diagonal copy of \(\mathcal O_T\). Thus \(X\), as well as both branches, satisfies the universal-functions condition on this cover. It is flat over the base too: in (8.3) the middle and last terms are base-flat, so its kernel is base-flat.

By Theorem 2.1 of the curve-Picard lesson, all three Picard sheaves are now the functors of actual bundles normalized at \(\sigma\); isomorphisms respecting the normalization are unique. For any parameter scheme \(T\) and any pair of its normalized bundles \(L_1,L_2\), form

This is affine of finite presentation over \(T\), by Moduli stacks are algebraic, Corollary 1.3. Both bundles are finitely presented, base-flat and have proper support: the flatness of \(Z/S\) is used here. The two normalizations identify restriction of an isomorphism along \(\sigma_T\) with an element of \(\mathbf G_{m,T}\). Requiring that element to be \(1\) cuts out a closed subscheme \(\mathcal I^1\subset\mathcal I\), of finite presentation.

Lemma 8.2 patches the universal isomorphism over \(\mathcal I^1\) to a normalized line bundle on \(X\). Conversely a point of the fibre of (8.5) has unique normalized identifications on the two branches and therefore exactly this gluing isomorphism. These constructions are inverse on all \(T\)-schemes. Thus \(\mathcal I^1\) represents the required fibre. Affine morphisms and finite presentation descend under the fppf section cover, proving the theorem. The entire construction used (8.3) after arbitrary base change, so proves the last assertion as well. \(\square\)

Over a field the scalar algebras need not be \(k\). They contribute an affine group of gluing parameters rather than preventing affineness.

Theorem 8.4 (field unions and components). Let \(X/k\) be proper and \(X=X_1\cup X_2\) a closed union with \(I_1\cap I_2=0\). There are no restrictions on the global-function algebras, the intersection or the field. The map

is affine of finite presentation. Consequently, for every proper reduced \(k\)-scheme, restriction to its reduced irreducible components is affine of finite presentation. This includes every geometrically reduced proper scheme.

Proof. Use the separated proper-field Picard schemes constructed in The structure of the Picard scheme, Theorem 1.2. We prove the affine maps using these schemes; this is independent of the earlier relative divisor existence construction.

Put \(Z=X_1\cap X_2\). If it is empty, the map is the product isomorphism already considered. Otherwise put \(C_i=H^0(X_i,\mathcal O_{X_i})\) and \(C_Z=H^0(Z,\mathcal O_Z)\). These are finite \(k\)-algebras by proper coherent cohomology. A finite affine-cover complex, tensored over the field, gives

for \(Y=X_1,X_2,Z\) and every parameter \(T\). Write \(U(C)\) for the smooth affine unit group of a finite algebra \(C\): it is the determinant-nonvanishing open of its underlying vector space.

Take \(T\to P_{X_1/k}\times P_{X_2/k}\). Work fppf locally on \(T\) so the two classes have actual representatives \(L_i\). Equality of their restricted classes on \(Z\) is a closed subscheme \(B\subset T\), by the separated diagonal of \(P_{Z/k}\). Over \(B\) the scheme

is a \(U(C_Z)_B\)-torsor. Indeed, equality of relative classes supplies an actual isomorphism locally after also trivializing the possible base line bundle; every two isomorphisms differ uniquely by a unit in (8.7). The same affine Isom theorem proves that this torsor is affine and of finite presentation.

The two branch automorphisms act through

Its fppf image is a closed affine subgroup \(A\). This is the field case of Quotients and torsors, Corollary 11.10: the kernel is a group scheme over a field and hence flat, so the map onto its closed image is fppf. All these groups are commutative. Thus \(A\) is normal, and

is an affine finite-type group scheme by Quotients and torsors, Theorem 11.1d. The normal-subgroup hypothesis is essential to this application. The quotient sheaf \(\mathcal I/A_B\) is a \(Q_B\)-torsor: trivialize \(\mathcal I\) fppf locally, take the quotient there, and descend its affine coordinate algebra. It is therefore affine of finite presentation over \(B\), and hence over \(T\).

This quotient is the fibre of (8.6). Locally every line-bundle class in that fibre has actual identifications with the chosen \(L_i\). Lemma 8.2 then gives an isomorphism in \(\mathcal I\). Changing the two identifications acts by (8.8). Conversely an isomorphism between two patched relative classes becomes an actual bundle isomorphism after a covering that trivializes its base factor; its restrictions are precisely two branch automorphisms. Hence its effect on the gluing is in \(A\). These assertions identify the quotient as fppf sheaves, including nilpotent tests. Descent along our initial cover of \(T\) proves the affine assertion globally.

A proper reduced scheme has finitely many irreducible components. The union of any selection, with its reduced structure, is again proper and reduced. When the selections are complementary, the two ideals have zero intersection, since the intersection of all minimal primes of a reduced ring is zero. Apply (8.6) repeatedly to these unions. Products, base changes and compositions of affine morphisms of finite presentation have the same property. This proves the component assertion. \(\square\)

8.3 Finite frames and nilpotent restriction

For a square-zero thickening, the exact sequence of units gives lifting obstructions, but it also remembers automorphisms. Finite frames remove those automorphisms before we take a quotient.

Lemma 8.5 (framed lifts). Let \(X/k\) be proper, \(X_0\subset X\) defined by \(I\) with \(I^2=0\), and choose a finite closed subscheme \(Z\subset X\). Put \(Z_0=Z\cap X_0\). Suppose restriction detects both global-function algebras:

Let \(R_Z,R_{Z_0}\) be the sheaves of line bundles with a specified trivialization on their indicated finite subschemes. For \(J=I\cap I_Z\), restriction \(R_Z\to R_{Z_0}\) is affine of finite presentation. Over an affine test \(T=\operatorname{Spec}B\to R_{Z_0}\), the obstruction lies in

Its vanishing cuts out a closed subscheme of \(T\). Over that subscheme, the lifts form a torsor under the vector group \(V=H^1(X_0,J)\).

Proof. Write \(\mathcal U_Z\) for the sheaf of units of \(\mathcal O_X\) equal to \(1\) on \(Z\), and define \(\mathcal U_{Z_0}\) similarly on \(X_0\). The spaces underlying \(X\) and \(X_0\) agree. On this space there is an exact sequence of abelian sheaves

The kernel is as displayed, and multiplication there is addition because \(I^2=0\). For local surjectivity use the ring identity

A unit on \(X_0\) equal to \(1\) on \(Z_0\) pairs with the unit \(1\) on \(Z\), giving a unit of this fibre product. Units lift through its square-zero kernel \(I\cap I_Z\). This proves (8.12), and proves it after every \(k\)-parameter base change. Also \(J\) is a coherent \(\mathcal O_{X_0}\)-module, since \(I J=0\).

First cohomology of \(\mathcal U_Z\) classifies framed bundles. To see the framing explicitly, trivialize a bundle locally by frames extending its specified frame on \(Z\). A unit on the closed subscheme lifts to a unit near each of its points, so such local choices exist. Their transition functions equal \(1\) on \(Z\); changing the local choices gives exactly a coboundary. This identifies their classes with \(H^1(\mathcal U_Z)\). An automorphism is a global section of \(\mathcal U_Z\). It is the identity by (8.10) and (8.7). The same assertions hold for \(Z_0\). Uniqueness of frame-preserving isomorphisms makes these class functors fppf sheaves, by effective descent.

The cohomology sequence of (8.12) therefore gives (8.11) as the lifting obstruction, and makes the lifts, when nonempty, a simply transitive set under \(H^1(X_{0,T},J_T)\). No quotient by global units remains, since \(H^0(\mathcal U_{Z_0})=1\). Finite affine-cover cohomology and exact tensoring over \(k\) give

Proper coherent finiteness makes these vector spaces finite-dimensional. Expand the obstruction in a basis of \(H^2(X_0,J)\); its finitely many coefficients generate the ideal of its zero subscheme \(T'\subset T\). Naturality of the connecting map and (8.13) show that every \(T\)-algebra factors through \(T'\) exactly when its obstruction vanishes. On the affine \(T'\), the cohomology sequence supplies an actual framed lift. Tensoring with the framed bundles represented by \(H^1(J)\) identifies all its lifts with \(V_{T'}\), functorially on affine tests. The chosen lift extends this identification to arbitrary tests by the sheaf property. Different choices change it by a translation; intrinsically the fibre is a \(V_{T'}\)-torsor. This is affine of finite presentation, as is its composition with \(T'\hookrightarrow T\). These constructions glue over affine tests of \(R_{Z_0}\), proving the lemma. \(\square\)

Theorem 8.6 (affine nilpotent restriction). Let \(X\) be any proper scheme over a field \(k\), and let \(X_0\subset X\) be a closed subscheme defined by a nilpotent ideal. Then

is affine of finite presentation. This statement includes nonreduced schemes, arbitrary finite algebras of global functions and every residue characteristic.

Proof. If \(X\) is empty, both sheaves are trivial. First suppose \(I^2=0\). Choose a finite \(Z\) satisfying (8.10). Such a common frame exists by the finite-jet construction in the proof of proper-field Theorem 1.2. In more detail, a nonzero germ of a global function is detected by a sufficiently high jet at a closed point, by Krull intersection. The finite-dimensionality of the two global-function spaces makes finitely many such jets suffice. For functions on \(X_0\), choose the same jets as intersections with jets in \(X\): at a selected point these are the quotients by \(I+\mathfrak m^n\). Enlarging the finitely many exponents and taking the union of these finite subschemes detects both spaces at once.

Set

The inclusions in (8.10) identify \(U(C)\subset U(D)\) and \(U(C_0)\subset U(D_0)\) as closed subgroups, on every parameter algebra. One verification is to choose a vector-space complement to \(C\) in \(D\). Multiplication by an element of \(C\) has a block triangular matrix. If it is invertible on \(D\), its determinant on \(C\) is a unit, so it is already a unit of \(C\). This identifies the unit open of that closed linear subspace with \(U(C)\), including after tensoring. The same argument applies to \(C_0\subset D_0\).

Their commutative affine quotients

exist and are affine of finite type by the normal-quotient theorem cited in (8.9). As in proper-field Theorem 1.2, changing the frame makes

torsors under \(K\) and \(K_0\), respectively. Every line bundle acquires a frame locally on the parameter: over a finite Artinian fibre algebra it is free, and a lifted fibre generator stays a generator on an open parameter neighbourhood by its nonzero determinant. Two frames of the same bundle differ by \(U(D)\), and global automorphisms identify precisely its subgroup \(U(C)\). The resulting \(K\)-action on framed classes is free. This proves (8.16) as fppf sheaf identities. Since the proper-field Picard schemes exist and the groups in (8.15) are affine, affine descent also represents both frame sheaves by schemes.

The map \(D\twoheadrightarrow D_0\) has square-zero kernel, so every unit of \(D_0\otimes B\) lifts to \(D\otimes B\): lift it and its inverse and invert the error \(1+i\) using \((1+i)^{-1}=1-i\). Thus \(K\to K_0\) is an fppf epimorphism. Let \(H\) be its kernel, a closed affine finite-type group.

Put \(V=H^1(X_0,I\cap I_Z)\), with its vector-group structure. There is a canonical closed subgroup inclusion

To construct it, use the trivial bundle with trivial frame as origin in (8.16). Then \(K\) is the fibre over the trivial unframed class, and \(H\) consists of its elements whose restriction is the trivial framed class on \(X_0\). Lemma 8.5 identifies that latter kernel, for framed bundles, with \(V\). It gives the homomorphism in (8.17); injectivity follows from the free \(K\)-action on \(R_Z\). The field case of the same image-factorization theorem used in (8.8), applied to this finite-type monomorphism, makes it a closed immersion. It does not require \(H\) reduced or smooth. Since \(V\) is commutative, \(V/H\) is affine of finite type by the normal-quotient theorem.

Base change (8.14) by \(R_{Z_0}\to P_{X_0/k}\). The resulting source is

For this identity, a point of the base change is an unframed class on \(X\) and a specified framed class for its restriction. Locally choose a frame on its bundle. Surjectivity of \(K\to K_0\) lets us adjust that frame so that its restriction is the specified one. The choices differ exactly by \(H\). This proves (8.18) on all tests after sheafification.

For any affine \(T\to R_{Z_0}\), Lemma 8.5 represents its framed lifts by a \(V\)-torsor over a closed \(T'\subset T\). The action of \(H\) on this torsor is translation through (8.17): changing a frame is tensoring with the corresponding framed trivial bundle, independently of the particular lift. Its quotient is consequently a \((V/H)\)-torsor over \(T'\), obtained by taking that affine quotient on local trivializations and descending. This is affine of finite presentation over \(T\). Hence (8.18) is affine of finite presentation over \(R_{Z_0}\). Descent along the fppf cover in (8.16) proves (8.14) in the square-zero case.

Finally, if \(I^N=0\), the closed subschemes defined by \(I^n\), \(1\leq n\leq N\), give a finite sequence from \(X_0\) to \(X\). The ideal of the \(n\)-th inclusion is \(I^n/I^{n+1}\); its square vanishes because \(2n\geq n+1\). Apply the square-zero assertion to each restriction map and compose. Affineness and finite presentation are preserved by this finite composition. \(\square\)

The field sheaves in Theorems 8.4 and 8.6 agree with their étale versions as well. Indeed \(U(C)\) and \(U(D)\) are smooth. The quotient \(K\) in (8.15) is smooth, since \(U(D)\to K\) is a smooth surjective \(U(C)\)-torsor and smoothness descends. Thus \(R_Z\to P_{X/k}\) is a smooth surjection and has étale-local sections. The universal framed bundle on \(R_Z\) supplies actual representatives on that cover. Equality of two representatives is étale-locally an actual isomorphism for the same reason: its isomorphism sheaf is a torsor under the smooth \(U(C)\). Consequently étale sheafification already has all the classes and identifications of the fppf sheaf. This proves the comparison without assuming \(C=k\).

Together, (8.5), (8.6) and (8.14) give the relative closed-union and field component/nilpotent affine reductions of Grothendieck's FGA 232, §6. They assert representability of the entire relative fibre functor, including its nilpotent parameters. The field existence input is the proper-field theorem proved in the Picard-structure lesson.

8.4 Finite-flat Stein factors and restriction of scalars

There is another way for a family to have more than one global scalar: its global functions can form a finite flat algebra over the base. Restriction of scalars retains these scalars and the components they distinguish. We construct the restriction as a scheme, and also prove why geometric local integrality makes the Stein algebra étale.

For a finite locally free morphism \(\pi:S'\to S\) and an \(S'\)-scheme \(Q\), its Weil restriction is the functor

It is an fppf sheaf, since morphisms to a scheme descend. We need an affine-neighbourhood condition to represent it by a scheme.

Lemma 8.7 (scheme coordinates for finite restriction). Suppose \(S\) is Noetherian, \(\pi\) finite locally free, and \(Q/S'\) separated and locally of finite type. Assume that, over every affine open \(V\subset S\), every finite set of points of \(Q\) lying over one point of \(V\) is contained in an affine open of \(Q_V\). Then (8.19) is a separated scheme locally of finite presentation over \(S\). It has the same finite-set affine-neighbourhood property, and its formation commutes with arbitrary base change.

Proof. Work over \(V=\operatorname{Spec}A\), and locally trivialize the finite locally free algebra \(B=\Gamma(S'_V,\mathcal O)\), with basis \(b_1,\ldots,b_r\). For an affine open \(W=\operatorname{Spec}C\subset Q_V\), the algebra \(C\) is finitely presented over \(B\): the morphism is locally of finite type over a Noetherian base, and this affine open is quasi-compact. Write

The images of the \(x_i\) in \(B\otimes_A R\) are uniquely \(\sum_j b_j z_{ij}\). Expand each relation \(f_a\) in the basis of \(B\), using its multiplication constants. Its \(r\) coefficients are polynomial equations in the \(nr\) variables \(z_{ij}\). Their affine zero scheme over \(A\) represents \(\operatorname{Res}(W)\), on every \(A\)-algebra, and is of finite presentation. Different bases give the same represented functor and glue. The rank-zero case has no variables and gives the terminal scheme over that base piece.

The inclusion \(\operatorname{Res}(W)\subset\operatorname{Res}(Q)\) is an open subfunctor. For a map \(S'_T\to Q\), its image lies in \(W\) precisely off the image in \(T\) of the inverse image of \(Q\setminus W\). That inverse image is closed, and its image is closed because \(S'_T\to T\) is finite. On the resulting open the map factors through \(W\), including all nilpotent parameters. These opens cover: over any point \(t\in T\), the finite scheme \(S'_{\kappa(t)}\) has a finite image in \(Q\); the hypothesis supplies an affine \(W\) containing it. The point \(t\) then belongs to its open locus. Thus the affine restrictions glue along their represented open intersections to represent (8.19).

For separatedness, take two maps from \(S'_T\) to \(Q\), with \(T\) affine. Their equality locus is a closed subscheme of \(S'_T\), since \(Q/S'\) is separated. Locally write the finite free algebra of \(S'_T\) in a basis. The condition that the entire finite scheme lie in this equality locus is that every coefficient of its defining ideal vanish. This is a closed condition on \(T\). It proves that the diagonal of the restriction is closed. Finite presentation of these diagonal conditions follows as well, since the affine equality ideals are finitely generated here.

For a finite set of points in one fibre of the restriction, take the finite union of the images in \(Q\) of their finite parameter schemes \(S'_{\kappa(t)}\). All lie over the same point of \(S\). A common affine \(W\) contains this union, so the affine open \(\operatorname{Res}(W)\) contains the original finite set. Finally, evaluation on a scheme over any new base gives exactly the right-hand side of (8.19) with both bases pulled back. The same scheme therefore represents the base-changed functor. \(\square\)

The Picard scheme of Theorem 8.1 satisfies the hypothesis of Lemma 8.7 even when the finite set lies over several points of \(S'\) above one point of \(S\). Indeed, work over affine \(S\), so \(S'\) is affine. The increasing opens \(P_+-m\xi\) of Lemma 5.1 contain any finite set for one sufficiently large \(m\). Inside that open only finitely many inverse Hilbert polynomials, and finitely many base Hilbert-polynomial pieces, occur at the chosen points. The corresponding charts form a finite disjoint union of schemes quasi-projective over \(S'\). It has an ample line bundle: take the ample bundles on the finitely many charts, and extend a section by zero on the other charts. Quotients and torsors, Lemma 5.4 supplies an affine open containing the entire finite set. Translate it back by \(-m\xi\). This proves exactly the required common-neighbourhood property, rather than assuming that a locally quasi-projective scheme automatically has it.

Theorem 8.8 (finite-flat Stein reduction). Let \(S\) be Noetherian and \(f:X\to S\) projective and flat. Suppose its Stein factorization is

where \(g\) is flat with geometrically integral fibres and \(\pi\) is finite flat. Then its fppf and étale Picard sheaves agree and are represented by the separated, locally finite-type group scheme

It has affine neighbourhoods of every finite set in a fibre over an affine base neighbourhood, and formation commutes with arbitrary base change.

Proof. The morphism \(g\) is projective. Its graph into \(X\times_S S'\) is closed because \(S'/S\) is separated, and that product is projective over \(S'\). Thus Theorem 8.1 constructs \(P_{X/S'}\). Its universal-functions condition follows from proper flatness and geometric integrality, as in the degree-zero argument of Theorem 8.3. The preceding paragraph and Lemma 8.7 represent the right-hand side of (8.21) by a separated scheme with the asserted finiteness and neighbourhood properties.

We now prove the sheaf identity, including a local-representative issue caused by the two different bases. A line bundle on a finite locally free \(S'_T\) is trivial Zariski locally on \(T\). To verify this, take \(T=\operatorname{Spec}A\). Its pushforward is a finite locally free \(A\)-module of the same rank as \(\pi_*\mathcal O_{S'_T}\). On a fibre, an invertible module over a finite Artinian algebra is free of rank one. Lift a generator to a section on \(S'_T\). Multiplication by it is a map between finite locally free \(A\)-modules of equal rank; its determinant is a unit on a neighbourhood of the chosen base point. It is an isomorphism there. These neighbourhoods prove the assertion.

Restriction of relative classes defines a map from the left-hand side of (8.21) to the right. If an actual bundle \(L\) maps to zero, the universal-functions condition for \(g\) and the injectivity argument in Section 1 of the curve-Picard lesson imply \(L\simeq g_T^*N\) for an actual line bundle \(N\) on \(S'_T\). This conclusion is global on \(S'_T\): the local identifications descend through their scalar transition units. The preceding determinant argument makes \(N\) trivial locally on \(T\). Thus \(L\) has zero Picard-sheaf class there. Since every class has a representative locally on \(T\), the map is injective.

For surjectivity, let \(T\) be affine and take a point of \(P_{X/S'}(S'_T)\). It is represented by a bundle after an étale cover of \(S'_T\), by Theorem 8.1 and its sheaf comparison. Choose a finite collection of affine members of this cover whose images cover the affine scheme \(S'_T\), and let their disjoint union be \(U\). Then \(U\to S'_T\) is affine, étale, surjective and of finite presentation, and carries a representative bundle on \(X_U\).

Its affine Weil restriction over \(T\) exists by the coefficient calculation in Lemma 8.7. It is étale: on a square-zero extension, a map from the finite affine parameter scheme into \(U\) lifts uniquely, by formal étaleness of \(U/S'_T\); together with finite presentation this is the étale criterion. It is surjective as well. Over an algebraically closed point of \(T\), the finite parameter algebra is a product of local Artinian algebras with algebraically closed residue field. Every surjective étale cover has a section over each of these strictly henselian Artinian algebras. One direct justification is to lift a residue-field point successively through the nilpotent maximal-ideal powers by formal étaleness. Taking the sections on all factors gives a section over the finite parameter algebra. Thus every geometric fibre of the affine restriction is nonempty.

After this étale surjective cover of \(T\), there is consequently a map \(S'_T\to U\). Pull its representative bundle back to \(X_T\). It represents the chosen restriction point. This proves surjectivity, even in the étale topology on \(T\), and proves agreement of the two Picard sheaves in (8.21). The group operations are induced by tensor product, and base change and the remaining scheme properties follow from Lemma 8.7. No étaleness of \(\pi\) was assumed. \(\square\)

Lemma 8.9 (the étale Stein algebra). Let \(f:X\to S\) be proper, flat and finitely presented over a Noetherian scheme, with geometrically reduced fibres. Then \(C=f_*\mathcal O_X\) is a finite locally free étale \(\mathcal O_S\)-algebra, and its formation commutes with every base change. In the Stein factorization \(X\to S'\to S\), the first map is flat with geometrically connected and geometrically reduced fibres, and the second map is finite étale.

Proof. Proper coherent finiteness makes \(C\) finite. Work at a local ring \((A,\mathfrak m,\kappa)\) of \(S\). The finite algebra \(H^0(X_\kappa,\mathcal O)\) is étale over \(\kappa\). Indeed after an algebraic closure it is the product of one copy of the field for each connected component of the reduced proper fibre, by the global-functions calculation of Proper coherent cohomology, Section 5. Field extension commutes with this cohomology, so reducedness after algebraic closure makes the original finite algebra étale.

Choose a finite separable extension \(\ell/\kappa\) splitting this algebra. Lift a monic minimal polynomial for a primitive generator of \(\ell\) to \(A\). Its discriminant is a unit because its residue polynomial is separable. It therefore defines a finite free étale faithfully flat local algebra \(A'\), with residue field \(\ell\); the simple-root lifting criterion is Formally smooth, unramified and étale ring maps, Proposition 6.1. It is local because its reduction modulo \(\mathfrak m\) is a field and it is finite over the local ring. Flat base change identifies \(C\otimes_A A'\) with the global functions on \(X_{A'}\). It suffices to prove the étale assertion after this faithfully flat extension.

Write \(A_n=A'/\mathfrak m_{A'}^n\), and \(X_n=X_{A_n}\). The closed fibre has global-function algebra \(\ell^r\). Its \(r\) idempotents give its open and closed connected components. They lift uniquely through every nilpotent thickening \(X_n\), and compatibly in \(n\). For completeness, an idempotent lifts through a square-zero ring ideal by correcting a lift using the inverse of \(2e-1\); this element is a unit since its square is \(1\) modulo that ideal. The same calculation makes the lift unique. Iterate the nilpotent filtration and glue the local lifts by uniqueness. It follows that \(X_n\) is the disjoint union of \(r\) proper flat \(A_n\)-schemes.

Each of their sole residue fibres is geometrically reduced and geometrically connected. The connectedness follows also from its global functions \(\ell\): the field case of Stein factorization, Theorem 3.1 gives geometric connectedness when the unit is an isomorphism. Semicontinuity and Grauert's theorem, Theorem 5.2 then gives, with the compatible labelling,

The degree-zero theorem on formal functions, Theorem 4.1 identifies the completed finite algebra with the limit of (8.22):

These are algebra isomorphisms: the formal-functions comparison in degree zero is induced by restriction of functions and preserves multiplication. Completion of a Noetherian local ring is faithfully flat by Completion, Theorem 3.2. Finite étaleness descends through that map and then through \(A'/A\), so \(C\) is finite étale at the original point. In particular it is locally free.

Reduction of the same comparison (8.23) modulo the maximal ideal is the actual degree-zero fibre comparison and identifies it with \(\ell^r\to H^0(X_\ell,\mathcal O)\). It is an isomorphism. Faithful flatness of \(\ell/\kappa\) proves surjectivity of the original comparison at every point of \(S\). The degree-zero cohomology/base-change criterion of Theorem 3.1 of the semicontinuity lesson now gives arbitrary base change for \(C\), including nonreduced parameters.

Thus every geometric Stein fibre selects one of the connected components of \(X_{\bar s}\), with its reduced scheme structure. The Stein connected-fibre assertion is also supplied by the cited Stein theorem. To check flatness of \(g:X\to S'\), work étale locally on \(S\) where the finite étale \(S'\) is a disjoint union of copies of \(S\). The corresponding parts of \(X\) are open and closed, so their flatness follows from that of \(X/S\). Flatness descends. Their geometric fibres are the selected geometrically reduced connected components, proving the remaining assertions. \(\square\)

Corollary 8.10 (geometrically locally integral fibres). Let \(S\) be Noetherian and \(X/S\) projective and flat with geometrically locally integral fibres. Then its étale and fppf Picard sheaves are represented by a separated group scheme locally of finite type, with arbitrary base-change compatibility. With \(\xi\) the class of a chosen relatively very ample bundle, it has an open \(U\), a disjoint union of schemes quasi-projective over \(S\), such that

The displayed union is increasing. Empty fibres are allowed, with the trivial relative Picard scheme on their open and closed base locus.

Proof. A geometrically locally integral fibre is geometrically reduced. Its irreducible components are disjoint: two components through a point would give two distinct minimal primes of that local ring, contradicting its being a domain. Since the fibre is Noetherian, its components are finitely many and open and closed. Consequently every geometric connected component is integral. Lemma 8.9 makes the Stein morphism finite étale and its first map flat with geometrically integral fibres. Theorem 8.8 applies.

It remains to retain the positive-chart structure in (8.24). On \(S'\), let \(U'=P_+\) for \(g\), using the given very ample bundle, and let \(\xi'\) be its class. Lemma 5.1 and Theorem 8.1 give the disjoint quasi-projective charts of \(U'\), its stability under \(+\xi'\), and the increasing cover by \(U'-m\xi'\). Set

This is open by the finite-image argument of Lemma 8.7 and is stable under \(+\xi\). A finite Stein fibre has only finitely many image points in \(P_{X/S'}\), so one sufficiently large \(m\) puts all of them in \(U'-m\xi'\). The same open-locus argument puts a neighbourhood of the parameter point in \(U-m\xi\). This proves the increasing cover.

We verify the disjoint quasi-projective decomposition, rather than presuming that arbitrary scheme descent preserves quasi-projectivity. Work on an open and closed base piece where \(\pi\) has constant rank \(r>0\). The ordering scheme \(E\) of its \(r\) sheets is a finite étale surjection, with the constant symmetric group \(\mathfrak S_r\) acting simply transitively on the orderings. Explicitly \(E\) is the open and closed part of \((S')^r\) where all coordinates are distinct; its geometric fibres are precisely the orderings of \(r\) points. It splits \(S'_E\) into \(r\) copies of \(E\).

Index the disjoint charts of \(U'\) by their inverse Hilbert polynomials and the open and closed base pieces used in their construction, taking an empty chart off its base piece. Over \(E\), the scheme \(U_E\) is the product of the \(r\) pulled-back unions of these charts. Decompose this product by the multiplicities of its finitely many chart labels. For each multiset of \(r\) labels, the resulting piece is a finite disjoint union of products of quasi-projective charts, hence is quasi-projective over \(E\). It is preserved by permutation of the sheets. These pieces descend to disjoint open and closed subschemes \(U_a\subset U\) and cover \(U\).

To prove \(U_a/S\) quasi-projective, choose a relatively ample line bundle \(L\) on \((U_a)_E\). Form

It is relatively ample and has the canonical descent datum which permutes its tensor factors. For ampleness, locally on the affine base take a common very ample power of the finitely many factors; the product embeddings and the Segre embedding make the corresponding power of their tensor product very ample. Effective line-bundle descent gives a bundle \(N\) on \(U_a\) whose pullback is \(M\).

Here is why \(N\) is relatively ample. Work over an affine open in \(S\), so its finite étale ordering cover is affine. For any point of \(U_a\), its finite preimage in \((U_a)_E\) lies in an affine nonvanishing open of a section \(t\) of a positive power of \(M\), by the ample-section proof of Lemma 5.4 cited above. The product of its \(\mathfrak S_r\)-translates is an invariant section of a power of \(M\), so descends to a section of a power of \(N\). Its nonvanishing open upstairs is the finite intersection of the translated affine opens; it is affine since the scheme is separated over the affine base. It contains the entire preimage of our point. Affine descent along the finite étale cover makes the corresponding nonvanishing open downstairs affine as well. Such sections cover \(U_a\), proving that \(N\) is relatively ample. Finite type of \(U_a/S\) descends from the same finite cover. Hence \(U_a\) is quasi-projective over \(S\). The rank-zero piece is trivial. This proves every assertion of (8.24). \(\square\)

The representability, affine-morphism and base-change assertions of Sections 8.2–8.4 are local on the base. They therefore also hold over locally Noetherian bases: apply the proofs on Noetherian affine opens. The universal functors identify the representing schemes and maps on overlaps and force the cocycle condition, so these glue.

Theorem 8.8 and Corollary 8.10 give the finite-flat Stein and geometrically locally integral reductions of FGA 232, §6. The possibly nonétale Stein factor in the first theorem is retained; the second obtains étaleness from geometric reducedness and proves the full positive-chart structure.

9 Examples and the boundary of the theorem

Hypersurfaces as divisors in projective space

The base variety in this example is \(\mathbb P^n_S\), with \(n\geq1\). Its relative Picard scheme is the constant integer scheme, proved in the previous lesson. Let \(V_d\) be the free module of homogeneous degree-\(d\) forms in its \(n+1\) coordinates. For \(d\geq0\), the component of its divisor scheme of bundle class \(\mathcal O(d)\) is

Thus it is the projective space of degree-\(d\) homogeneous forms, whose dimension over a field is \(\binom{n+d}{n}-1\). A quotient of the dual section module gives a form with coefficients generating the unit ideal locally on the base. Exactly this condition keeps its fibre polynomial nonzero. Each nonzero polynomial is a nonzerodivisor on the integral projective-space fibre, so Theorem 3.1 gives a flat relative hypersurface. When \(d=0\), the sole divisor is empty; negative degrees have no effective divisors.

This example says nothing like \(\operatorname{Pic}(Z)=\mathbb Z\) for every hypersurface \(Z\subset\mathbb P^n\). That assertion would be false: a smooth elliptic plane cubic already has a positive-dimensional degree-zero Picard scheme. It is the Picard scheme of the ambient projective space that indexes (9.1).

Curves and a Brauer–Severi fibre

For a smooth projective geometrically connected curve, the relative divisor scheme of degree \(d\) is its symmetric power, as proved in Hilbert and Quot schemes. A pointed curve has a universal normalized bundle on its Picard scheme. For \(d\geq0\) with \(d>2g-2\), Serre duality and Riemann–Roch give

For a curve without a point, the same statement holds after an étale cover supplying a representative of the universal class. The global conclusion can be a Brauer–Severi fibration.

For example let \(C/\mathbb R\) be the conic \(x^2+y^2+z^2=0\). Its degree-one Picard component is \(\operatorname{Spec}\mathbb R\), while

has no real section. After extending to \(\mathbb C\) it is \(\mathbb P^1\). It cannot be the projectivization of a two-dimensional real vector space, which would have real points. The class in its target exists as an étale Picard class although there is no real degree-one line bundle on \(C\). This is the same descent obstruction analysed in the previous lesson.

A family of elliptic curves

Let \(E\to S\) be a smooth proper projective geometrically connected genus-one family over a Noetherian base, with section \(e\). Degree-one line bundles have \(H^1=0\) and one section on every fibre. Their sections therefore form a line bundle on every test scheme. Evaluation cuts out a unique relative divisor of degree one. Such a divisor is finite locally free of rank one and therefore is the graph of a unique section of \(E_T\to T\). The construction and its inverse commute with all base changes, exactly as in the degree-one argument of the previous lesson. Hence

and translation by \(-[\mathcal O(e)]\) gives

The tensor group law on the target transports to the elliptic group law with origin \(e\). Here the degree-zero piece is the fibrewise identity component: it is isomorphic to the given smooth connected proper family. No assertion that all higher-dimensional Picard schemes are smooth is used.

Why the historical erratum matters

The existence theorem uses geometric integrality, not just properness, flatness and (3.1). The broader scheme-existence conjecture in FGA 232, Remark 5.2, was explicitly withdrawn in its erratum, p. 303. That erratum also adds the algebraically closed residue-field qualification to the complete-local-base remark. The full algebraic-space theorem under proper flat finite presentation and the universal condition (3.1) is proved in Moduli stacks are algebraic, Theorem 7.3. It rigidifies the algebraic stack of line bundles after an fppf cover supplying a section, and then descends its algebraic-space coordinates. This is a different conclusion from scheme representability and is consistent with the historical corrections; its Stacks comparison is Tag 0D2C.

One mechanism is visible already in our proof. On a reducible fibre a nonzero section need not be regular. Regular sections form an open inside the projective parameter scheme, and that open need not be proper. The relation projections then lose the properness needed in Theorem 7.1. The historical nonexistence of a Picard scheme in some such cases means failure of representation by a scheme; it does not assert failure of the applicable modern algebraic-space theorem. The affine maps of Section 8 describe closed unions and nilpotent thickenings. The finite-flat Stein reduction and the geometrically locally integral case, including their scheme coordinates and positive-chart structure, are proved in Section 8.4.

10 Exercises

1. Fibrewise flatness. Let \(X\to S\) be flat and finitely presented, and let an invertible ideal locally generated by \(a\) have fibrewise nonzerodivisor equations. Show its divisor is flat over \(S\), and show every base change remains a Cartier divisor. Explain why an absolute Cartier divisor in a flat family need not suffice.

2. The open divisor locus. Give the all-test-scheme proof that \(\operatorname{Div}_{X/S}\) is open in the Hilbert scheme for a flat projective family. Identify the proper morphism used to remove the bad locus. Is geometric integrality required in this exercise?

3. A section bundle. Assume (3.1), integral geometric fibres, and that \(f_*L=V\) is locally free with arbitrary base change. Prove that the complete linear system is \(\mathbb P(V^\vee)\), including the inverse construction on a nonreduced test scheme. Explain what changes if the Picard class has no global representative.

4. Separatedness. Prove the diagonal assertion for the étale Picard sheaf of the main theorem. Do not assume in advance that the target is a scheme. Explain why merely proving uniqueness of geometric specializations would miss a required part of the argument.

5. The curve quotient. Let \(C/k\) be smooth projective geometrically connected, with a point, and let \(d\geq0\), \(d\geq2g-1\). Construct \(\operatorname{Pic}^d_C\) completely as \(\operatorname{Sym}^dC\) modulo linear equivalence. Show the equivalence relation is proper and flat, construct its Hilbert-scheme quotient, identify its sheaf of points and compute the relative dimension of the quotient map. Include \(g=0,d=0\).

6. Two components and a multiplicative parameter. Over \(k=\mathbf Q\), let \(X\subset\mathbb P^1_k\times\mathbb P^1_k\) be the reduced union of the graphs of \(t\mapsto t\) and \(t\mapsto2t\). Identify their scheme-theoretic intersection. Compute the kernel of restriction \(P_{X/k}\to\underline{\mathbf Z}_k^2\) as a group scheme, on arbitrary parameter schemes. Give compatible choices of gluing frames which split this map. Explain why independent scalar automorphisms on the two branches remove one parameter.

7. A square-zero curve and an additive parameter. Let \(d\geq2\), let \(C=\mathbb P^1_k\) over any field, and put \(X=\operatorname{Spec}_C(\mathcal O_C\oplus\mathcal O_C(-d))\), with the second summand a square-zero ideal. Compute \(P_{X/k}\to P_{C/k}\), including its splitting and its kernel on nonreduced parameter schemes. Why does the kernel have an additive, rather than a multiplicative, group law?

8. A nonétale Stein factor. Let \(E/k\) be an elliptic curve with origin, put \(A=k[\epsilon]/(\epsilon^2)\), and view \(X=E_A\) as a proper \(k\)-scheme. Identify its Stein factor and use Theorem 8.8 to compute \(P_{X/k}\). Show that restriction to \(E\) has kernel the vector group of \(H^1(E,\mathcal O_E)\), and determine its group law on arbitrary \(k\)-algebras. Explain why the extra factor cannot be replaced by a second copy of \(P_{E/k}\).

9. Two conjugate components. Regard \(X=\mathbb P^1_{\mathbf C}\) as a projective \(\mathbf R\)-scheme. Compute its Picard scheme over \(\mathbf R\), including its decomposition into copies of \(\operatorname{Spec}\mathbf R\) and \(\operatorname{Spec}\mathbf C\). Determine its real points and its geometric component group. Why would its group of actual line bundles alone give an incorrect constant-scheme answer?

11 Solutions

1. At a point of the divisor, the fibrewise Cartier criterion [Stacks, Tag 062Y] applies to the flat finitely presented ambient family: multiplication by \(a\) is injective and its cokernel is flat over the base. The latter is the divisor's structure sheaf. At points outside the divisor, \(a\) is a unit and the quotient is zero, hence flat. Therefore the divisor is flat everywhere. Tensor the exact sequence \(0\to\mathcal O_X\xrightarrow{a}\mathcal O_X\to\mathcal O_D\to0\) with any base algebra. Flatness of the quotient keeps the first arrow injective. Thus the pulled-back equation remains a nonzerodivisor, and its quotient remains flat over the new base. For a failure of the absolute condition, take \(X=S=\operatorname{Spec}k[t]\) with its identity map and \(D=(t)\). The equation is an absolute nonzerodivisor, but on the fibre over zero it is zero; \(k[t]/(t)\) is not flat over \(k[t]\).

2. Use the universal flat closed subscheme \(Z\subset X_H\). Its Cartier-good locus is open by the fibrewise criterion; let \(B\) be its closed complement in \(Z\). The morphism \(Z\to H\), not a morphism from an affine chart of \(Z\), is proper, so its image of \(B\) is closed. On the complementary open \(U\), the universal family is a relative divisor. A divisor over any \(T\) has Cartier geometric fibres. For its Hilbert map \(T\to H\), faithful flatness of residue-field extensions and the criterion for the flat universal family show that each image point avoids \(B\). Hence the map factors through \(U\). The universal divisor pulls back to the original divisor, and uniqueness is Hilbert uniqueness. This works on nonreduced \(T\), because factorization through an open subscheme is determined by its image. Geometric integrality is unnecessary here; it becomes necessary for turning every nonzero section into a regular section later.

3. A quotient \(V_T^\vee\twoheadrightarrow N\) corresponds to a section of \(L_T\otimes f_T^*N\), by the assumed base-change isomorphism and duality for the vector bundle \(V_T\). Surjectivity means its fibre section is nonzero at every point. Integral fibres make those sections regular; the Cartier criterion supplies a flat divisor \(D\), with \(\mathcal O(D)=L_T\otimes f_T^*N\). Conversely the canonical section of such a divisor corresponds to a fibrewise nonzero map \(V_T^\vee\to N\), which is surjective by Nakayama. Two choices of its displayed bundle isomorphism differ by a base unit under (3.1), so give the same quotient-line point. Taking zero schemes and canonical sections are inverse, including the nilpotent structure of their ideals, and commute with pullback. If the class lacks a global representative, perform this construction on an étale cover. The resulting projective spaces glue as the divisor fibre, but may give a Brauer–Severi scheme. The real conic in (9.3) demonstrates the distinction.

4. For actual bundles \(L,M\) over \(T\), use the locally constant Hilbert polynomials to take the open and closed piece where the degree-\((n-1)\) coefficient of their difference is zero. On an integral \(n\)-dimensional fibre, a nonzero nonisomorphism of line bundles has a nonempty Cartier-divisor cokernel with positive leading coefficient. Thus every nonzero homomorphism on this piece is an isomorphism. The complete linear system of \(M\otimes L^{-1}\) parametrizes precisely equality of their relative classes. It is projective and a monomorphism: any two representing isomorphisms differ only by a base factor isomorphism and a scalar. Properness and finite fibres make it finite, and a finite monomorphism is closed. Off that polynomial piece equality is impossible. For zero-dimensional fibres the family is an isomorphism and the relative functor is zero. For arbitrary classes, the closed equality subschemes from an étale cover descend by their ideals and identical universal property. This proves representability of the diagonal by closed immersions before representability of Picard itself. A statement only about geometric points would not construct this equality subscheme or verify its nilpotent conditions on test schemes.

5. Put \(Y=\operatorname{Sym}^dC\), with its universal divisor. For every degree-\(d\) bundle \(L\), Serre duality gives \(H^1(L)=0\), since \(\deg(\omega_C\otimes L^{-1})=2g-2-d<0\). Riemann–Roch gives \(h^0(L)=d+1-g>0\). The point on the curve identifies Picard classes with normalized actual bundles on all test schemes. Their section modules are base-change-compatible bundles of that rank. Thus the Abel map's pullback by any class is the projective-space bundle of section lines, smooth proper and surjective of relative dimension \(d-g\).

Define \(R\subset Y\times Y\) by equality of the two divisor-bundle classes. Proposition 4.1 makes it closed. Its projections are the just described projective-space bundles, hence smooth, projective and flat. Apply the construction of Theorem 7.1: the fibres of \(R\to Y\) define a map \(u:Y\to H\) to a finite union of Hilbert schemes of \(Y\); the closed graph lies in the flat universal family \(D\to H\); equivalence of the relation makes that graph invariant on \(D\times_HD\); its ideal descends to a closed \(Q\subset H\). The descended map \(Y\to Q\) is faithfully flat projective and smooth, and its kernel pair is \(R\).

Every Picard class acquires an effective divisor étale locally, because its positive-rank section bundle locally has a quotient line of its dual. The Abel map is therefore an epimorphism of étale sheaves with kernel pair \(R\). The universal local-lift argument identifies its target with \(h_Q\). Thus \(Q\) is exactly \(\operatorname{Pic}^d_C\), constructed without presupposing its existence. The quotient map has relative dimension \(d-g\); as \(Y\) has dimension \(d\), the quotient has dimension \(g\). In the boundary case \(g=d=0\), the section rank is one, \(Y=\operatorname{Sym}^0C=\operatorname{Spec}k\), the relation is the identity and the quotient map has relative dimension zero. Properness of \(Q\) can also be read from the projective \(Y\): for any base change, the inverse image of a closed subset of \(Q\) is closed in \(Y\), and surjectivity makes their images in the base equal. This proves universal closedness; finite type and separatedness were already established.

6. In homogeneous coordinates, the two graph equations restrict on the first graph to a nonzero scalar multiple of \(x_0x_1\). Thus they meet in the two reduced rational points \(0,\infty\). Each component has global-function algebra \(k\), and the intersection has algebra \(k\times k\). Formula (8.8) maps the two scalar units to the same ratio on both intersection points. Its image is the diagonal subgroup of \(\mathbf G_m^2\). Consequently (8.9) is \(\mathbf G_m^2/\mathbf G_m\simeq\mathbf G_m\), with coordinate the quotient of the two gluing multipliers. This is a scheme identity on all test algebras, with inverses and scalar changes computed in their unit groups.

More explicitly, for the trivial bundle on each branch normalize the gluing at \(0\) to \(1\). The remaining gluing value at \(\infty\) is an arbitrary unit, uniquely determining its relative class. For a splitting over \(\underline{\mathbf Z}_k^2\), choose frames at \(0,\infty\) for \(\mathcal O(1)\) on each branch. Glue \((\mathcal O(1),\mathcal O)\) using these frames, and similarly glue \((\mathcal O,\mathcal O(1))\). Tensor powers, with the dual frames for negative exponents, provide compatible lifts of every degree pair. Lemma 8.2 supplies the bundles and their tensor identities. Hence \(P_{X/k}\simeq\mathbf G_m\times\underline{\mathbf Z}_k^2\), after these choices. Without dividing by branch automorphisms one would incorrectly retain two gluing parameters.

7. The finite projection \(r:X\to C\) is a retraction of the closed immersion \(C\subset X\), so pullback along \(r\) splits restriction of Picard sheaves. On the common underlying space, units fit into the split sequence \[ 0\longrightarrow\mathcal O_C(-d) \xrightarrow{a\mapsto1+a}\mathcal O_X^\times \longrightarrow\mathcal O_C^\times\longrightarrow1. \] Here \((1+a)(1+b)=1+(a+b)\). For an affine parameter \(T=\operatorname{Spec}B\), its cohomology kernel is \(H^1(C,\mathcal O_C(-d))\otimes_kB\); there is no quotient by global units because the unit sequence is split. The standard two-chart computation on \(\mathbb P^1\) identifies this vector space with \(B^{d-1}\): in a Laurent coordinate the surviving cocycles may be represented by \(t^{-1},\ldots,t^{-(d-1)}\). This tensor calculation holds for every \(B\), including rings with nilpotents. Sheafification and the splitting therefore give \[ P_{X/k}\simeq\mathbf G_a^{d-1}\times\underline{\mathbf Z}_k. \] The projection to the second factor is the affine restriction morphism of Theorem 8.6. Its additive law follows from the square-zero multiplication just computed, in every characteristic.

8. The constants are \(A\), so the Stein factor is \(\operatorname{Spec}A\to\operatorname{Spec}k\). The first map is the smooth projective genus-one family \(E_A/A\); the second is finite flat of rank two and is not étale. Let \(P=P_{E/k}\). Base change for the Picard scheme gives \(P_{X/A}=P_A\), and Theorem 8.8 yields \[ P_{X/k}=\operatorname{Res}_{A/k}(P_A). \] For a \(k\)-algebra \(B\), reduction \(P(B[\epsilon])\to P(B)\) is split by the constant inclusion \(B\to B[\epsilon]\). At any \(p\in P(B)\), the lifts are derivations of its coordinate functions into \(\epsilon B\), hence are sections of the pulled-back tangent bundle. Translation by \(p^{-1}\) identifies that bundle with \(\operatorname{Lie}(P)\otimes_k B\). It follows on all affine tests, and then by descent, that \[ \begin{gathered} P_{X/k}\simeq P\times\mathbf V(\operatorname{Lie}(P)),\\ \operatorname{Lie}(P)=H^1(E,\mathcal O_E),\qquad \dim_k H^1(E,\mathcal O_E)=1. \end{gathered} \] Here \(\mathbf V(W)(B)=W\otimes_kB\). The tangent-space identification is Proposition 2.1 of the Picard-structure lesson. For a commutative group the translated tangent vectors add under multiplication: differentiate the multiplication map, whose restrictions along both identity axes are the identity. Thus the isomorphism is one of group schemes, with direct-product law. Choosing a basis gives \(P_{X/k}\simeq P\times\mathbf G_a\). The finite dual-number algebra has one underlying point and an infinitesimal direction; its maps encode tangent vectors, whereas a split étale rank-two algebra encodes two independent points. Replacing it by \(k\times k\) would therefore give the wrong group scheme.

9. The Stein factor is \(\operatorname{Spec}\mathbf C\to\operatorname{Spec}\mathbf R\), finite étale of degree two, and the geometric fibre is two disjoint projective lines. Thus \[ P_{X/\mathbf R}= \operatorname{Res}_{\mathbf C/\mathbf R} (\underline{\mathbf Z}_{\mathbf C}). \] After extension to \(\mathbf C\), this is the constant scheme on \(\mathbf Z^2\). Conjugation interchanges the two coordinates, since it interchanges the two sheets of the Stein factor. Its fixed pairs \((a,a)\) each descend to \(\operatorname{Spec}\mathbf R\); every two-element orbit \(\{(a,b),(b,a)\}\), \(a\ne b\), descends to \(\operatorname{Spec}\mathbf C\). Their disjoint union is the required étale real group scheme, with addition descended from coordinatewise addition on \(\mathbf Z^2\). In particular its real points are the diagonal \(\mathbf Z\), while its geometric component group is \(\mathbf Z^2\). Actual bundles on \(\mathbb P^1_{\mathbf C}\) also form \(\mathbf Z\), but that real-point calculation does not determine the represented functor or its conjugate geometric components.

What this lesson does not prove

The divisor locus, module of sections, complete-linear-system correspondence, Picard diagonal, positive-locus construction, effective quotient, closed-union affine maps, affine nilpotent restriction and finite-flat Stein/Weil-restriction reductions are established here. In particular the proper-flat quotient is not an unproved input.

The foundational results used are these precise programme proofs:

• Hilbert and Quot schemes, Lemma 7.B proves the fibrewise Cartier criterion over an arbitrary base, openness of its locus, flatness of the quotient and compatibility with every base change [Stacks, Tag 062Y]. Its Sections 3–6 construct the projective Hilbert schemes and the proper-support open for quasi-projective ambient schemes; Section 7 proves the all-characteristic curve/symmetric-power assertion.
• Moduli stacks are algebraic, “Perfect tensor pushforwards over an arbitrary base” proves perfect proper cohomology with tensor base change for finitely presented base-flat sheaves [Tag 0A1H]. Base change and the Grothendieck complex, Sections 3–4, Flattening stratifications, Section 4, and Semicontinuity and Grauert's theorem, Sections 1–3 give the finite-complex, cancellation and base-change arguments used in Sections 2, 4 and 5. Euler characteristic is the alternating sum of the finite projective ranks, hence locally constant [Tag 0B9T]. The universal-functions theorem for geometrically reduced connected fibres is the latter lesson's Theorem 5.2, including arbitrary base change and nonreduced bases [Tag 0E0L].
• Serre's theorems on projective schemes, Sections 1–2 and Corollary 2.3 proves relative vanishing and generation over a Noetherian base [Tags 01XO, 0B5T and 02O1]. Lemma 5.1 here gives the flat tensor-base-change step and the finite transition-function descent to every affine test algebra [Tag 01ZR].
• Faithfully flat descent, Sections 2–5 and 7 proves effective descent of quasi-coherent modules, algebras and closed ideals [Tag 023R]. Section 7 here gives the actual quotient-ideal datum. Descending properties of schemes and morphisms, Theorem 2.1 proves descent of finite presentation, properness and smoothness; Smooth morphisms, Theorem 5.2 proves the étale-local section assertion, including arbitrary residue fields.
• Zariski's Main Theorem, Theorem 5.1 proves proper quasi-finite morphisms finite over every base [Tag 02LS]. Proposition 4.1 here proves the finite-monomorphism closed-immersion step directly [Tag 03BB]; the chart argument following (0.1) proves the Proj quotient-line property on all test schemes.
• The affine Isom construction is Moduli stacks are algebraic, Section 1 and Corollary 1.3, with both sheaves flat and finitely presented and of proper support. Affine normal quotients and closed flat images of finite-type field groups are Quotients and torsors, Theorem 11.1d and Corollary 11.10. Proper-field Picard schemes and their finite frames are proved in The structure of the Picard scheme, Theorem 1.2. The proofs above supply the patching, obstruction, translation action and quotient identifications used with these inputs.
• The degree-zero theorem on formal functions is The theorem on formal functions, Sections 1–4 and Theorem 4.1. Its Rees-module and cofinal-filtration proof supplies the completed scalar algebra in (8.23). Completion, Theorems 3.1–3.2 proves the exactness and faithful flatness used there; Formally smooth, unramified and étale ring maps, Proposition 6.1 and Lemma 6.2 gives the finite-étale field splitting and unique idempotent lifting. Finite-type Stein factorization and geometric connectedness are Zariski connectedness and Stein factorization, Theorem 3.1. Lemma 8.9 here proves the further finite-étale and arbitrary-base-change assertions using the degree-zero comparison and constants theorem.
• The Picard functor and the Picard scheme of a curve, Sections 1–3 and Theorem 3.1 proves the sheaf comparison and injection of the raw relative quotient under (3.1). Castelnuovo–Mumford regularity and boundedness, Section 3 proves the dimension and positive-leading-coefficient facts used in the diagonal argument.

References

• Grothendieck, Technique de descente et théorèmes d'existence en géométrie algébrique V: les schémas de Picard, théorèmes d'existence, Bourbaki Exposé 232, §§3–6; Proposition 4.1, Theorem 4.3 and §5. Original text. The historical qualifications are in the erratum, p. 303.
• Kleiman, The Picard scheme, §3, Theorems 3.7 and 3.13; §4, Theorem 4.8 and Lemma 4.9. Author's survey.
• Nitsure, Construction of Hilbert and Quot Schemes, §6, especially Theorem 6.8. Author's exposition.
• The Stacks project, Divisors, More on Morphisms, Derived Categories of Schemes, and Quot and Hilbert Spaces, cited by the tags above and read in the AI Integrated Stacks Project edition. Relative Cartier divisors, Picard algebraic spaces.
• AI Integrated Stacks Project, Moduli, the sections on relative effective Cartier divisors, complete linear systems and the positive Picard locus; labels section-relative-effective-Cartier-divisors, theorem-complete-linear-system, section-positive-picard-locus, remark-classical-positive-divisor-quotient and remark-corrected-picard-existence-boundary.