Moduli stacks are algebraic

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

A coherent sheaf can move on a space which is neither proper nor flat over the base. Its moduli problem is nevertheless algebraic if the sheaf itself is flat over the parameter space and has proper support. Quotients and closed subschemes then inherit algebraic coordinates from this stack. Line bundles require a further distinction: their stack remembers scalar automorphisms, whereas the Picard sheaf removes them by descent.

Read Artin's axioms first. We use its proved representable-diagonal criterion, product obstruction criterion and strong-effectiveness criterion. The projective Quot and Hilbert schemes of AG-HP-05, and the curve Picard scheme of AG-HP-09, provide familiar special cases. The coherent-cohomology and derived-category inputs needed here are stated precisely in Section 10. No algebraicity theorem for the moduli problems proved below is taken as an input.

Throughout, \(X\) and \(B\) are algebraic spaces over a scheme \(S\), and \(f:X\to B\) is a morphism. A parameter scheme \(T\to B\) gives \(X_T=X\times_B T\). A sheaf is finitely presented as an \(\mathcal O_{X_T}\)-module; it is flat over \(T\) as a relative module. “Coherent” in the notation for the stack means these two conditions, even when the rings are not Noetherian.

1. Families of homomorphisms

1.1. Proper support, rather than proper ambient space

For a finitely presented sheaf \(G\), its topological support is the closed set where its stalk is nonzero. Its scheme-theoretic support is defined by \(\operatorname{Ann}(G)\). Proper support means that this closed subspace is proper over the base. In the finite-presentation setting used here, an equivalent test uses \[ Z(G)=V(\operatorname{Fitt}_0(G)). \tag{1.1} \] The Fitting ideal is of finite type and commutes with every base change. Its vanishing set is the support. If \(G\) is generated by \(n\) elements locally, then \(\operatorname{Ann}(G)^n\subset\operatorname{Fitt}_0(G)\subset\operatorname{Ann}(G)\). Thus these two closed subspaces differ by a nilpotent thickening. Since \(Z(G)\) is of finite presentation over \(X\), properness of either support gives universal closedness of the other; finite type and separatedness then give properness. This formulation also proves that proper support is stable under arbitrary base change and descends under a faithfully flat cover. It avoids claiming that annihilator ideals themselves always commute with base change.

Define the functor \[ \underline{\operatorname{Hom}}_{X/B}(F,G)(T) =\operatorname{Hom}_{\mathcal O_{X_T}}(F_T,G_T). \tag{1.2} \] Here \(F\) may be any quasi-coherent sheaf. Pullback of a homomorphism defines the functor on all \(B\)-schemes. Quasi-coherent descent makes it an fppf sheaf.

Theorem 1.1 (Hom). Suppose \(f\) is of finite presentation and \(G\) is finitely presented, flat over \(B\), with proper support over \(B\). Then (1.2) is represented by an algebraic space affine over \(B\). If \(F\) is finitely presented, that affine morphism is of finite presentation. Neither separatedness nor flatness of \(f\) is required. This is [Stacks, Tag 08K6].

We first prove the finite calculation behind the theorem.

Lemma 1.2 (a finite complex represents its degree-zero kernel). Let \(A\) be a ring and let \(K\) be perfect over \(A\). Suppose \[ H^i(K\otimes_A^{\mathbf L}C)=0\quad(i<0) \tag{1.3} \] for every \(A\)-algebra \(C\). Locally on \(\operatorname{Spec}A\), the functor \(C\mapsto H^0(K\otimes_A^{\mathbf L}C)\) is represented by an affine scheme of finite presentation.

Proof. Locally represent \(K\) by a bounded complex of finite projectives, and shrink further to make them free. If its first nonzero term \(P^a\) has \(a<0\), then the differential \(P^a\to P^{a+1}\) is injective on every residue-field fibre, by (1.3). At any chosen point an appropriate maximal minor is a unit. On that open set the map is a split injection and the corresponding two equal free summands form a contractible complex. Remove them. Repeating this finite operation removes all terms in negative degrees. It suffices to work with \[ P^0\xrightarrow{d}P^1\longrightarrow\cdots\longrightarrow P^e. \] For every \(C\), degree-zero cohomology is \(\ker(d\otimes C)\). The vector scheme associated to \(P^0\) represents \(P^0\otimes C\): its coordinate algebra is \(\operatorname{Sym}_A((P^0)^\vee)\). Cut it out by the finitely many linear equations expressing \(d(v)=0\), namely the image of \((P^1)^\vee\to(P^0)^\vee\). The resulting affine scheme represents the kernel and is of finite presentation. The local constructions agree because they represent the same functor, so they glue. \(\square\)

Proof of Theorem 1.1, with \(B=\operatorname{Spec}A\) Noetherian and \(F\) finitely presented. The space \(X\) is Noetherian and quasi-compact and quasi-separated. Perfect approximation gives a perfect complex \(P\to F[0]\) with \[ P\in D^{\leq0}(X),\qquad H^0(P)\simeq F. \tag{1.4} \] For clarity, the approximation is an isomorphism in degrees \(>-1\) and surjective in degree \(-1\); as \(F\) has no negative cohomology, these conditions give (1.4). This is the exact ordinary approximation input [Stacks, Tags 08HI, 08HJ and 08HP].

For any base change \(h:X_T\to X\), derived pullback is right \(t\)-exact. Therefore \(\mathbf Lh^*P\) remains in degrees at most zero and its degree-zero cohomology is \(F_T\). Maps from such a complex to the sheaf \(G_T[0]\) in degree zero are exactly maps from its degree-zero cohomology. Indeed, the truncation triangle has a term in \(D^{\leq-1}\), and both that term and its shift by \(1\) have no degree-zero maps to \(G_T[0]\). Consequently \[ \operatorname{Hom}(F_T,G_T) =H^0R\operatorname{Hom}_{X_T}(\mathbf Lh^*P,G_T). \tag{1.5} \]

Set \(K=Rf_*R\mathcal H om(P,G)\). The precise proper-support direct-image theorem says that \(K\) is perfect, and the flatness of \(G\) gives arbitrary derived base change: \[ \mathbf Lg^*K\simeq Rf_{T,*}R\mathcal H om(\mathbf Lh^*P,G_T). \tag{1.6} \] Its hypotheses are a Noetherian base, a quasi-separated locally finite type morphism, a perfect source, and a bounded coherent target flat over the base with proper support. They hold here; properness of \(X\) and flatness of \(X/B\) are absent. See [Stacks, Tags 0DKK and 08JQ].

If \(T=\operatorname{Spec}C\), equations (1.5)–(1.6) identify (1.2) with \(H^0(K\otimes_A^{\mathbf L}C)\). There is no negative cohomology: derived Hom from a complex in degrees at most zero to a sheaf has none, nor does its derived global-section complex. Lemma 1.2 represents this functor affinely with finite presentation. The equalities extend from affine \(T\) to arbitrary \(T\) by the sheaf condition. Equivalently, the degree-zero kernel sheaf of the finite complex has the same sections after gluing an affine cover; no assertion that higher cohomology vanishes on an arbitrary \(T\) is needed.

Passage to arbitrary \(B\). Affineness and finite presentation descend fpqc locally on \(B\), so use affine étale charts and take \(B=\operatorname{Spec}A\). Write \(A\) as the filtered colimit of its finitely generated \(\mathbf Z\)-subalgebras. Finite presentation descends \(X\), \(f\), \(F\), and \(G\) to one of them. After increasing that index, \(G\) is flat over the descended base and has proper support: these are the ordinary finite-presentation flatness and proper-support limit theorems, with the latter in [Stacks, Tag 08K2]. The Noetherian argument supplies an affine finitely presented scheme over that base. Its pullback represents (1.2) over \(A\), because every \(A\)-scheme is also a scheme over the descended base and the sheaves pulled back to its \(X_T\) are exactly \(F_T,G_T\). The identifications commute with pullback. Descent glues these affine schemes on the charts of \(B\).

Passage to arbitrary \(F\). On a quasi-compact quasi-separated algebraic space, a quasi-coherent sheaf is a filtered colimit of finitely presented quasi-coherent sheaves. Write \(F=\operatorname{colim}_iF_i\). Pullback preserves this colimit, and \[ \operatorname{Hom}(F_T,G_T) =\lim_i\operatorname{Hom}((F_i)_T,G_T). \tag{1.7} \] The transition maps on the right come from precomposition. An inverse limit of affine spaces over \(B\) is affine over \(B\): on each affine base chart its algebra is the colimit of their coordinate algebras. Thus (1.7) is represented by an affine space. We make no finite-presentation claim in this last step. This completes the proof. \(\square\)

1.2. Isom requires the hypotheses in both directions

Corollary 1.3 (Isom). If \(f\) is of finite presentation and both \(F\) and \(G\) are finitely presented, flat over \(B\), with proper support, then \(\underline{\operatorname{Isom}}_{X/B}(F,G)\) is affine of finite presentation over \(B\) [Stacks, Tag 08K9].

Proof. All four Hom spaces with source and target in \(\{F,G\}\) satisfy Theorem 1.1. In \(\underline{\operatorname{Hom}}(F,G)\times_B\underline{\operatorname{Hom}}(G,F)\) impose the equations \[ v\circ u=1_F,\qquad u\circ v=1_G. \tag{1.8} \] The composition map has target \(\underline{\operatorname{Hom}}(F,F)\times_B\underline{\operatorname{Hom}}(G,G)\). Its identity section is a closed immersion of finite presentation, since the target is affine of finite presentation. The inverse image is therefore affine of finite presentation. Its points are exactly mutually inverse pairs, and the inverse of an isomorphism is unique. Projection to \(u\) identifies it with Isom, functorially on every base change. \(\square\)

The requirement on \(F\) is substantive. Finite presentation of \(F\) alone does not suffice for this conclusion.

Example 1.4 (an omitted proper-support hypothesis). Let \[ B=\operatorname{Spec}k[t],\quad X=\operatorname{Spec}k[t,x],\quad G=\mathcal O_X/(x),\quad H=\mathcal O_X/(tx-1),\quad F=G\oplus H. \tag{1.9} \] Both \(F\) and \(G\) are finitely presented and flat over \(B\). The support of \(G\) is the zero section, hence proper. The support of \(H\) maps to \(D(t)\) by an isomorphism and is not proper over \(B\). On every \(T\), the element \(x\) annihilates \(G_T\) and is invertible on \(H_T\). Thus an isomorphism \(F_T\simeq G_T\) forces \(H_T=0\); conversely this condition makes \(F_T=G_T\). But \(H_T=0\) means that \(t\) is locally nilpotent on \(T\). The Isom functor has a copy of \(\mathbf G_m(T)\) on such tests and no points on the other tests.

It cannot be affine of finite presentation over \(B\). If it were \(\operatorname{Spec}C\) with \(C\) a finitely presented \(k[t]\)-algebra, then \(C\) would be Noetherian. Every prime of \(C\), tested by its residue field, would contain \(t\). Hence \(t^N=0\) in \(C\) for some \(N\geq1\). Nevertheless \(T=\operatorname{Spec}(k[t]/(t^{N+1}))\) has the identity isomorphism \(F_T=G_T\), giving a \(k[t]\)-algebra map \(C\to k[t]/(t^{N+1})\). This contradicts \(t^N=0\). Formula (1.9) explains why Corollary 1.3 retains flatness and proper support for both sheaves.

2. The coherent-sheaf groupoid and its diagonal

From now on assume that \(f:X\to B\) is separated and of finite presentation. Define \(\mathcal C=\mathcal{Coh}_{X/B}\) over \(B\) as follows. An object over \(T\to B\) is a finitely presented quasi-coherent sheaf \(F\) on \(X_T\), flat over \(T\), with proper support over \(T\). An arrow over \(T'\to T\) is an isomorphism \(F'\simeq F|_{X_{T'}}\). Thus cartesian arrows are part of the definition, and pullback of sheaves makes \(\mathcal C\) a category fibred in groupoids [Stacks, Tags 08KB and 08W5].

Lemma 2.1 (descent and diagonal). The category \(\mathcal C\) is an fppf stack. Its relative diagonal over \(B\) is affine of finite presentation.

Proof. A descent datum consists of quasi-coherent sheaves and isomorphisms satisfying the cocycle, not just their isomorphism classes. Effective quasi-coherent descent on algebraic spaces glues the sheaf and all its arrows. Finite presentation and relative flatness descend faithfully flat. The Fitting-support test (1.1) and descent of properness descend proper support. Hence the descended sheaf is an object of \(\mathcal C\).

For two objects \(F,G\) over \(T\), the fibre of the diagonal is \(\underline{\operatorname{Isom}}_{X_T/T}(F,G)\), including the specified identifying arrow. Both sheaves satisfy Corollary 1.3. This proves the diagonal assertion. \(\square\)

Viewed over \(S\), the diagonal is still representable by algebraic spaces. Given \(g,h:T\to B\) and two families, first require \(g=h\): this is the base change of \(\Delta_{B/S}\), an algebraic space over \(T\). Over that equality space apply the preceding Isom construction. We do not claim that the absolute diagonal over \(S\) is affine unless the needed hypothesis on \(B/S\) is also present [Stacks, Tags 08W6 and 08KC].

There is no size obstruction in applying Artin's criterion. On a fixed \(T\), finitely presented modules can be described on étale charts by finite matrices, gluing matrices and finitely many equality conditions on a covering refinement. These data belong to a set in the chosen universe. Morphisms between two sheaves are a set by the ordinary module category. Choose a set of representatives of the resulting isomorphism classes. This gives the small fibres required in the criterion without bounding the ranks of all sheaves simultaneously.

Lemma 2.2 (full limit preservation). If \(A=\operatorname{colim}_iA_i\) is a filtered colimit of \(B\)-algebras, then \[ \operatorname{colim}_i\mathcal C(A_i)\ \longrightarrow\ \mathcal C(A) \tag{2.1} \] is an equivalence of groupoids.

Proof. A finitely presented sheaf on \(X_A\) descends to some \(X_{A_i}\): use finitely many affine étale charts, finite presentations of the modules, and finite presentations of the gluing maps; their equations and cocycle equalities hold at some finite index. Relative flatness descends after increasing the index. Proper support descends by the finite-type support limit theorem used in Theorem 1.1. Thus an object descends.

For two descended objects, each homomorphism descends by finite presentation and the same finite-chart calculation. An isomorphism descends together with its inverse. The two inverse equalities, or the equality of two descended arrows, hold after a further increase because the diagram is filtered. This proves full faithfulness as well as essential surjectivity. The assertions are local on \(B\), so apply the affine argument on an étale chart and descend. \(\square\)

This is the full groupoid form of [Stacks, Tag 08KD]. In particular the relative morphism \(\mathcal C\to B\), once algebraicity is known, will be locally of finite presentation.

Numerical pieces. Suppose \(X/B\) is projective with a fixed relatively ample line bundle \(L\). For a family \(F\), the Euler characteristic \(\chi(X_t,F_t\otimes L_t^n)\), for each integer \(n\), is locally constant in \(t\). On a Noetherian affine parameter chart this follows by representing the proper flat coherent direct image by a perfect complex: its fibrewise Euler characteristic is the alternating sum of its locally constant finite-projective ranks. It is compatible with field extension. All such functions are constant on each connected component. A Noetherian affine has finitely many connected components, each open, so the entire Hilbert polynomial of the fibres is constant on each of them. For an arbitrary affine parameter chart, descend \(X,L,F\), including relative flatness and proper support, to a Noetherian model and pull these pieces back. Thus a prescribed Hilbert polynomial defines an open and closed substack of \(\mathcal C\). The same construction gives the usual numerical pieces of Quot and Hilbert. This explains the numerical decompositions discussed in [Stacks, Tags 0DLX, 0DM1 and 0DM5], without treating an arbitrary infinite intersection of open sets as open.

3. Patching sheaves without a flat ambient space

The ring square in this section is \[ A_1\longrightarrow A_0\longleftarrow A_2,\qquad A_2\twoheadrightarrow A_0,\quad I^2=0,\qquad P=A_1\times_{A_0}A_2. \tag{3.1} \] Let \(J=\ker(P\to A_1)\). Projection identifies \(J\) with \(I\), and \(J^2=0\). The following algebra is why a nonflat \(X/P\) causes no problem.

Lemma 3.1 (flat module patching). A pair of flat modules \(M_i/A_i\), \(i=1,2\), with a specified isomorphism \[ M_1\otimes_{A_1}A_0\simeq M_2\otimes_{A_2}A_0=M_0 \] patches to the flat \(P\)-module \(M=M_1\times_{M_0}M_2\). Its two base changes recover the given modules. Every flat \(P\)-module is recovered this way, including all its homomorphisms.

Proof. We first record the reduction calculation. The map \(M\to M_1\) is surjective, with kernel \(IM_2\), viewed as pairs \((0,m_2)\). The image of \(M_1\) generates \(M_0\) over \(A_0\). Lifting coefficients to \(A_2\) shows that the image of \(M\to M_2\), together with \(IM_2\), generates \(M_2\). Since \(I^2=0\), this image generates \(M_2\) by the nilpotent Nakayama calculation. Multiplication by \(J=I\) then gives \(JM=IM_2\). Hence \(M\otimes_P A_1=M_1\).

The map \(M\otimes_P A_2\to M_2\) is surjective by this generation statement. A kernel element reduces to zero in \(M_0=M_1\otimes_{A_1}A_0\); tensor right exactness expresses it using pairs in \(IM_2\) and tensors \(m\otimes i\). Each such tensor can be moved to the first factor, using the corresponding element of \(J\). Thus it is \((0,z)\otimes1\) for \(z\in IM_2\). Its image in \(M_2\) is \(z\), so it is zero when its image is zero. This proves the second base-change identity. This elementary calculation is [Stacks, Tag 07RU].

To prove flatness, use the ideal criterion. For any ideal \(\mathfrak a\subset P\), its image \((\mathfrak a+J)/J\) is an ideal of \(A_1\), so its tensor map into \(M_1\) is injective. Therefore a kernel element of \(\mathfrak a\otimes_PM\to M\) comes from \((\mathfrak a\cap J)\otimes_PM\). Also \[ J\otimes_PM=I\otimes_{A_2}M_2\ \longrightarrow\ M_2 \] is injective, by flatness of \(M_2\). It follows that \(\operatorname{Tor}_1^P(A_1,M)=0\). As \(M_1\) is flat over \(A_1\), the change-of-rings Tor sequence gives \(\operatorname{Tor}_1^P(N,M)=0\) for every \(A_1\)-module \(N\).

Set \(L=\mathfrak a\cap J\). Enlarge it to the \(A_2\)-submodule \(L'=A_2L\subset I=J\). The quotient \(L'/L\) is annihilated by \(J\). The Tor vanishing just proved shows that \(L\otimes_PM\to L'\otimes_PM\) is injective. But \[ L'\otimes_PM=L'\otimes_{A_2}M_2\ \longrightarrow\ M_2 \] is injective, since \(L'\) is an ideal of \(A_2\) and \(M_2\) is flat. It follows that \(L\otimes_PM\to M\) is injective. This eliminates the original kernel and proves flatness. This is the ideal argument of [Stacks, Tag 0D2I].

Finally, tensoring the exact sequence \[ 0\longrightarrow P\longrightarrow A_1\oplus A_2 \longrightarrow A_0\longrightarrow0 \] with a flat \(P\)-module recovers it as the stated fibre product. Maps into a fibre product are exactly compatible pairs of maps. The base-change identities make these statements inverse functors, proving full faithfulness. \(\square\)

Lemma 3.2 (relative patching and finite presentation). Let \(C\) be a finitely presented \(P\)-algebra, \(C_i=C\otimes_PA_i\). Compatible \(C_i\)-modules \(M_i\), flat over \(A_i\), patch to a \(C\)-module \(M\) flat over \(P\). If the \(M_i\) are finitely presented over \(C_i\), then \(M\) is finitely presented over \(C\). No flatness of \(C/P\) is assumed.

Proof. Apply Lemma 3.1 to the underlying base modules. The \(C\)-action on the fibre product is componentwise through \(C\to C_i\); it is well defined because the two actions agree over \(C_0\). The recovered base changes recover the \(C_i\)-actions. Full faithfulness for \(C\)-linear maps follows by patching the underlying maps and testing \(C\)-linearity on the two restrictions; the target injects into their product.

For finite presentation choose a finite presentation \(C=D/K\), where \(D=P[z_1,\ldots,z_n]\) and \(K\) is finitely generated. The algebra \(D\) is flat over \(P\), although \(C\) need not be. The module \(M/JM=M_1\) is finite over \(D_1\). Lift its finitely many generators to \(M\), and let \(L\) be their \(D\)-span. Then \(M/L=J(M/L)=J^2(M/L)=0\). Thus \(D^r\twoheadrightarrow M\) for some finite \(r\).

Write \(H\) for its kernel. Since \(M\) and \(D^r\) are flat over \(P\), \(H\) is flat over \(P\), and base change preserves the exact sequence. Hence \[ H\otimes_PA_i=\ker(D_i^r\to M_i). \] These kernels are finite over \(D_i\), since \(C_i\) is finitely presented over \(D_i\) and \(M_i\) is finitely presented over \(C_i\). Lift finitely many generators of \(H/JH\) to \(H\). The identical nilpotent Nakayama calculation proves that \(H\) is finite over \(D\). Therefore \(M\) is finitely presented over \(D\), and consequently over its quotient \(C\), which already acts on \(M\). \(\square\)

Proposition 3.3 (the full Rim–Schlessinger condition). The stack \(\mathcal C\) satisfies (RS\(^*\)): \[ \mathcal C(P)\ \xrightarrow{\sim}\ \mathcal C(A_1)\times_{\mathcal C(A_0)}\mathcal C(A_2) \tag{3.2} \] for every diagram (3.1). In particular it satisfies Artin's Artinian condition (RS) [Stacks, Tag 08LQ].

Proof. On an affine étale chart \(\operatorname{Spec}C\to X_P\), pull the two sheaves and their given identification back to \(C_i\). These are exactly the modules of Lemma 3.2. Patch them to a finitely presented \(C\)-module, flat over \(P\). On overlaps patch the descent isomorphisms by full faithfulness. The cocycle equality holds because it holds on both restrictions and the patched target injects into their product. Étale descent then gives a sheaf on \(X_P\). The same argument patches all arrows; inverse arrows patch too, so the equivalence is one of groupoids.

Its support is proper over \(P\). Indeed, \(P\to A_1\) is a square-zero quotient, so \(\operatorname{Spec}A_1\to\operatorname{Spec}P\) is a universal homeomorphism. By (1.1), the support of the patched sheaf is a separated finite type space whose restriction to this thickening is the proper support of the first sheaf. After any base change, the thickening is still a homeomorphism and that restricted support is closed in the base. Hence the support upstairs is universally closed. Separatedness and finite type make it proper.

Conversely, apply the final recovery assertion of Lemma 3.1 on each chart to a sheaf flat over \(P\). It recovers the sheaf, its identifications and all arrows. This proves (3.2), without inserting flatness of \(X_P/P\) anywhere. \(\square\)

The finite-presentation part of Lemma 3.2 also explains [Stacks, Tag 08W7]: a relatively flat lift through a square-zero base thickening is finitely presented if its restriction is. When the ambient algebra is not flat, the polynomial algebra \(D\), rather than that ambient algebra, provides the flat presentation used in the proof.

4. Tangents and effective formal families

In this section work locally on \(B\) over a Noetherian affine base. When computing a relative tangent, the map to \(B\) is fixed. Artin's criterion can be applied with that affine base as \(S\); there is no need to add deformations of a moving base map.

Proposition 4.1 (tangent and infinitesimal automorphism spaces). For a field \(k\to B\) and an object \(F\in\mathcal C(k)\), \[ T_F\mathcal C=\operatorname{Ext}^1_{X_k}(F,F),\qquad \operatorname{Inf}_F\mathcal C=\operatorname{Hom}_{X_k}(F,F). \tag{4.1} \] If \(k\) is a finite type field over the Noetherian base, these are finite-dimensional \(k\)-vector spaces [Stacks, Tag 08W8].

Proof. The first-order space is \(X_k\times_k\operatorname{Spec}k[\epsilon]\). A marked sheaf \(F'\), flat over \(k[\epsilon]\), gives \[ 0\longrightarrow F\longrightarrow F'\longrightarrow F\longrightarrow0 \tag{4.2} \] as sheaves on \(X_k\): multiplication by \(\epsilon\) identifies the first \(F\) with the kernel of reduction. Conversely, an extension (4.2) becomes a module over \(\mathcal O_{X_k}[\epsilon]\) by making \(\epsilon\) the composite of quotient and inclusion. Its square is zero, and its kernel equals its image. This is the square-zero flatness criterion over \(k[\epsilon]\), so it is a relatively flat deformation. It is finitely presented: on Noetherian charts its middle module is coherent, hence finitely presented; equivalently use the finite-presentation lifting argument of Section 3. Its support is a nilpotent thickening of the original proper support. Thus marked deformations and extensions correspond, and Baer addition agrees with tangent addition.

An automorphism reducing to the identity is \(1+\epsilon a\), where \(a:F\to F\); its inverse is \(1-\epsilon a\), and composition adds \(a\)'s. This proves the second formula.

For finiteness, the sheaves \(\mathcal E xt^q(F,F)\) are coherent on the Noetherian space \(X_k\) and supported on the proper support of \(F\). Coherent cohomology with proper support is finite-dimensional. The local-to-global spectral sequence \[ H^p(X_k,\mathcal E xt^q(F,F)) \ \Longrightarrow\ \operatorname{Ext}^{p+q}_{X_k}(F,F) \] has only finitely many terms in any fixed total degree \(p+q\geq0\). It makes the two groups in (4.1) finite. This argument works after every finite extension of \(k\), including inseparable extensions. It does not replace proper support by a false properness assumption on \(X_k\). \(\square\)

Proposition 4.2 (formal existence with every arrow). Let \(R\) be a complete Noetherian local ring, with maximal ideal \(\mathfrak m\), and a map \(\operatorname{Spec}R\to B\). Then \[ \mathcal C(R)\ \longrightarrow\ \lim_n\mathcal C(R/\mathfrak m^{n+1}) \tag{4.3} \] is an equivalence of groupoids [Stacks, Tag 08W9].

Proof. The proper-support Grothendieck existence theorem on the separated finite type algebraic space \(X_R\) is an equivalence between coherent sheaves with proper support and compatible coherent formal sheaves with proper support. Its morphisms are all compatible homomorphisms. Apply it to the given system \((F_n,\phi_n)\) to obtain a coherent sheaf \(F\) with proper support and the prescribed identifications. Since the space is Noetherian, \(F\) is finitely presented.

It remains to check relative flatness; ordinary existence alone does not include that conclusion. On an affine étale chart the completed flatness criterion says that a finite module whose reductions are flat over every \(R/\mathfrak m^{n+1}\) is flat over \(R\) at every point of the closed fibre. This follows from the ideal criterion for flatness and Krull intersection, and its exact relative-space formulation is [Stacks, Tag 08VP]. The relative flat locus of \(F\) is open. Its closed complement lies in the proper support of \(F\), since a zero sheaf is flat. The image of that complement in \(\operatorname{Spec}R\) is closed and misses the closed point. Any nonempty closed subset of the spectrum of a local ring contains its closed point. The complement is therefore empty, proving flatness on all of \(X_R\).

Existence is fully faithful on coherent sheaves with proper support. Hence every compatible arrow in (4.3) is a unique homomorphism upstairs. If its reductions are isomorphisms, apply full faithfulness also to their compatible inverses; the two composites are identities by faithfulness. Thus all isomorphisms and their compositions are recovered. This proves the equivalence, not just effectivity on isomorphism classes. \(\square\)

The ordinary existence input used here is [Stacks, Tags 08B7 and 08BE], with the complete Noetherian ring and proper-support hypotheses just stated. It allows \(X_R\) to be an algebraic space and to be nonproper.

5. Openness and the algebraicity theorem

5.1. Flat ambient morphisms: a perfect complex of low Ext groups

First suppose \(f\) is flat. Over a Noetherian affine parameter space \(\operatorname{Spec}A\), let \(F\) be a family. For a square-zero extension \(A'\twoheadrightarrow A\), with kernel \(M\), the flat-module deformation theorem gives an obstruction \[ o_F(A')\in \operatorname{Ext}^2_{X_A}(F,F\otimes_AM). \tag{5.1} \] It is zero exactly when there is a marked relatively flat lift. When lifts exist, their classes form a torsor under \(\operatorname{Ext}^1(F,F\otimes_AM)\), and their marked automorphisms are \(\operatorname{Ext}^0(F,F\otimes_AM)\). The obstruction commutes with maps of extensions with fixed quotient \(A\). These are the precise deformation inputs [Stacks, Tags 08VW and 0CYE]; they apply because both \(X_A/A\) and \(X_{A'}/A'\) are flat. Finite presentation of a lift is automatic by the lifting argument of Section 3, and its support is proper by nilpotent invariance. Hence (5.1) detects lifts in \(\mathcal C\), rather than lifts in a larger module category.

We explain why these groups commute with products of arbitrary \(A\)-modules. This is stronger than their finite generation for finite modules.

Lemma 5.1 (one complex computes the low Ext functors). There is a perfect \(K\in D(A)\), depending on \(F\), with functorial isomorphisms \[ H^i(K\otimes_A^{\mathbf L}M) \simeq \operatorname{Ext}^i_{X_A}(F,F\otimes_AM) \quad(i\leq2) \tag{5.2} \] for every \(A\)-module \(M\), compatible with the boundary maps of short exact module sequences.

Proof. Choose a perfect approximation \(P\to F\) which is an isomorphism in degrees \(>-3\) and surjective in degree \(-3\). Its cone \(E\) lies in \(D^{\leq-4}(X_A)\). The target \(F\otimes_AM\) is a sheaf in degree zero, so \(\operatorname{Ext}^i(E,F\otimes_AM)=0\) for \(i\leq3\). The Hom long exact sequence therefore gives \[ \operatorname{Ext}^i(F,F\otimes_AM) \simeq\operatorname{Ext}^i(P,F\otimes_AM) \quad(i\leq2). \] Put \(K=Rf_*R\mathcal H om(P,F)\). It is perfect by the proper-support direct-image theorem used in Section 1. Its projection-formula calculation for an arbitrary module gives \[ R\Gamma(X_A,R\mathcal H om(P,F\otimes_AM)) \simeq K\otimes_A^{\mathbf L}M. \] Indeed \(P\) is perfect, so internal Hom is \(P^\vee\otimes-\); \(F\) is \(A\)-flat, so its ordinary tensor with \(M\) is its derived tensor. The derived projection formula then gives the displayed equality. Taking cohomology proves (5.2). All maps are induced by exact or derived functors, so the connecting maps agree. This is the low-degree construction in [Stacks, Tag 08JR], with its approximation and direct-image ingredients exposed. \(\square\)

Locally represent \(K\) by a bounded complex of finite projective \(A\)-modules. Finite projectives tensor-commute with products: first prove this for a finite free module and then take a direct summand. Thus \[ K\otimes_A\prod_jM_j\simeq\prod_j(K\otimes_AM_j) \] term by term. Products of module sequences are exact, and kernels and images of the componentwise differential are the products of the respective kernels and images. Consequently cohomology also commutes with these products. Formula (5.2) proves \[ T_F(\prod_jM_j)\simeq\prod_jT_F(M_j),\qquad O_F(\prod_jM_j)\simeq\prod_jO_F(M_j), \tag{5.3} \] where \(T_F=\operatorname{Ext}^1(F,F\otimes_A-)\) and \(O_F=\operatorname{Ext}^2(F,F\otimes_A-)\).

The proved product criterion of Lesson 7, Theorem 5.3, now applies: the diagonal is representable, (RS\(^*\)) and limits hold, and (5.1) is a vanishing-if-and-only-if obstruction compatible with maps of extensions. Its proof only uses the fixed-quotient functoriality present here. Equations (5.3) supply its two product conditions, so versal loci are open. This verifies [Stacks, Tag 08WA] in the flat case by the product route; it does not assume that \(F\) is a perfect sheaf.

5.2. Nonflat ambient morphisms: stronger coherent existence

If \(f\) is not flat, the ambient thickenings used in (5.1) need not be flat, so the preceding obstruction formula is not a valid justification for openness. A different input supplies it.

We use the following precise coherent-existence theorem, distinct from Proposition 4.2. Let \(A_n\) be an inverse sequence of rings with surjective transition maps whose kernels are locally nilpotent, put \(A=\lim_nA_n\), and let \(Y/A\) be separated and of finite presentation. A compatible system of finitely presented sheaves \(G_n\) on \(Y_{A_n}\), flat over \(A_n\) with proper support, has a finitely presented sheaf \(G\) on \(Y\), flat over \(A\) with proper support, together with isomorphisms to all the \(G_n\) respecting their specified transition isomorphisms. There is no Noetherian or adic hypothesis on this sequence, and no flatness hypothesis on \(Y/A\). This is [Stacks, Tags 0CX4 and 0CXB]. It is an ordinary coherent-existence prerequisite, not an algebraicity assertion about \(\mathcal C\).

Apply this theorem to an inverse tower used in the strong-effectiveness criterion of Lesson 7, Section 5.1. In that criterion the kernel of every map \(A_m\to A_n\), \(m\geq n\), is square zero, so in particular it is locally nilpotent. The given objects are precisely the \(G_n\) in the theorem. The compatible maps \(\operatorname{Spec}A_n\to B\) give a map \(\operatorname{Spec}A\to B\) in the local affine calculation, and \(Y=X_A\) is separated of finite presentation. Thus the compatible groupoid object is effective, with its required markings, over the inverse-limit ring.

The already proved strong-effectiveness criterion requires only this object effectivity with compatible identifications, a representable diagonal, (RS\(^*\)), and limit preservation. Lemmas 2.1–2.2 and Proposition 3.3 establish the other hypotheses. The criterion therefore proves openness of versality for general \(f\). The formal axiom (4.3) alone would not justify this step; the stronger existence theorem supplies exactly the missing tower hypothesis.

5.3. Every axiom, and arbitrary bases

Theorem 5.2 (coherent-sheaf algebraicity). For every separated morphism \(f:X\to B\) of finite presentation, \(\mathcal{Coh}_{X/B}\) is an algebraic stack, locally of finite presentation over \(B\). Its relative diagonal is affine of finite presentation. In particular it is an algebraic stack over \(S\). These are [Stacks, Tags 08WC and 09DS]; the first is the flat case and the second drops flatness.

Proof. First suppose \(B=\operatorname{Spec}\Lambda\), where \(\Lambda\) is a finitely generated \(\mathbf Z\)-algebra. It is Noetherian and a G-ring, as are its finite type extensions [Stacks, Tag 07PX]. We apply the representable-diagonal version of Artin's criterion proved in Lesson 7, Theorem 4.1. Here is every hypothesis, with its verification:

Criterion input Verification for \(\mathcal C\)
Representable diagonal The affine Isom space of Corollary 1.3
G-ring hypothesis at finite type points Finite type \(\Lambda\)-algebras are G-rings
[−1] Small fibres The finite-presentation descriptions following Lemma 2.1 bound object isomorphism classes and arrow sets in the chosen universe
[0] Effective étale descent The stronger fppf descent of Lemma 2.1, including all arrows
[1] Limit preservation The groupoid equivalence of Lemma 2.2
[2] Rim–Schlessinger Proposition 3.3 proves the stronger arbitrary-ring (RS\(^*\))
[3] Finite tangent and infinitesimal automorphism spaces Proposition 4.1, after every finite residue-field extension
[4] Formal effectivity Proposition 4.2 proves an equivalence over every complete Noetherian local test ring
[5] Openness of versality Section 5.1 if \(f\) is flat; Section 5.2 without that assumption

Because the first diagonal is already representable, Theorem 4.1 requires openness of versality for \(\mathcal C\) itself. The criterion gives a smooth scheme atlas and hence algebraicity. Its local finite-presentation conclusion, or the equivalence between local finite presentation and full limit preservation for algebraic stacks, gives \(\mathcal C\to B\) locally of finite presentation. Lemma 2.1 supplies the stronger diagonal assertion.

Now let \(B=\operatorname{Spec}A\) be arbitrary. Finite presentation and separatedness descend \(X/A\) to a separated finitely presented space \(X_0/A_0\), where \(A_0\) is finitely generated over \(\mathbf Z\). If proving the flat case separately, descend flatness too by increasing the index. For every \(A\)-scheme \(T\), the space \((X_0)_T\) is \(X_T\), and the conditions defining its sheaves are the same. Therefore there is an equivalence of stacks \[ \mathcal{Coh}_{X/A}\simeq \mathcal{Coh}_{X_0/A_0}\times_{\operatorname{Spec}A_0}\operatorname{Spec}A. \tag{5.4} \] This is an equality of moduli conditions for all \(T\), not a claim that every family comes from \(A_0\). Pull back the smooth atlas just constructed. Algebraicity and local finite presentation are preserved by base change.

Finally choose an étale scheme cover \(B'\to B\) and cover \(B'\) by affines. On them (5.4) gives algebraic stacks and smooth scheme atlases. Their disjoint union is a smooth scheme cover of \(\mathcal C\): the map from the pulled-back stack to \(\mathcal C\) is representable étale, and the covers are surjective. The relative diagonal is representable by Lemma 2.1, so the smooth-atlas recognition theorem of Lesson 5 proves algebraicity over \(B\) and over \(S\). Local finite presentation over \(B\) is étale local and follows from the affine case. No finiteness, Noetherian or G-ring condition on \(B/S\) remains in the conclusion. \(\square\)

The proof establishes the flat-case theorem by its own obstruction calculation and the general theorem by strong existence. It never uses either theorem as a shortcut to the other axioms.

6. Quotients and closed subschemes

6.1. The quotient functor

For a quasi-coherent \(F\) on \(X\), a point of \(\operatorname{Quot}_{F/X/B}(T)\) is an equivalence class of surjections \[ q\colon F_T\twoheadrightarrow Q,\qquad Q\in\mathcal C(T). \tag{6.1} \] Two surjections are equivalent if an isomorphism of their targets carries one to the other. Such an isomorphism is unique because \(q\) is surjective. Hence the groupoid of pairs \((Q,q)\) has no nontrivial automorphisms; its isomorphism classes form an fppf sheaf by descent of sheaves and maps. We retain the pair groupoid during the proof, so that no descent identification is lost [Stacks, Tag 09TQ].

Lemma 6.1 (surjectivity is an open condition). For a family \(u:F_T\to Q\), with \(Q\) finitely presented and of proper support, there is an open \(T^\circ\subset T\) such that a map \(T'\to T\) factors through it exactly when \(u_{T'}\) is surjective. This holds without finite presentation of \(F\).

Proof. The cokernel \(H\) is of finite type because it is a quotient of \(Q\). Its support is a closed subset of the proper support of \(Q\), so its image in \(T\) is closed. Take the complement for \(T^\circ\). For a finite type module, vanishing on a fibre is detected at its residue fields by Nakayama. It follows that after any base change its support is the inverse image of its original support. The cokernel of a pulled-back map is the pullback of its cokernel by tensor right exactness. Thus \(H_{T'}=0\) exactly when \(T'\) avoids that closed image. This proves the universal property. \(\square\)

This is the proper-support surjectivity test [Stacks, Tag 09TP]. One can also test it in the universal Hom family: the preceding proof works over its parameter space.

Theorem 6.2 (Quot). For separated \(f\) of finite presentation and arbitrary quasi-coherent \(F\), \(\operatorname{Quot}_{F/X/B}\) is an algebraic space. It is locally of finite presentation over \(B\) if \(F\) is finitely presented [Stacks, Tag 09TU].

Proof. Forget \(q\) to obtain a morphism from the pair stack to \(\mathcal C\). Pull it back by an object \(Q\) over \(T\), retaining the specified identifying isomorphism to \(Q\). Its fibre is exactly the open subspace of \(\underline{\operatorname{Hom}}_{X_T/T}(F_T,Q)\) where the universal map is surjective. Theorem 1.1 and Lemma 6.1 make this an algebraic space; thus the forgetful morphism is representable.

The pair stack is algebraic, since a representable morphism to an algebraic stack has representable diagonal and an atlas obtained by pulling back an atlas of the target and then taking scheme coordinates. It has trivial automorphisms, as observed after (6.1). The proved algebraic-stack-to-space recognition in Lesson 5 therefore makes its sheaf of isomorphism classes an algebraic space. More explicitly, on a smooth atlas the relation is an equivalence relation with at most one arrow between any two objects; the algebraic-space quotient theorem and flat-space bootstrap of Lessons 1–2 identify its sheaf quotient with this stack. This is the desired Quot space.

If \(F\) is finitely presented, its Hom space over \(T\) is of finite presentation, and the surjectivity locus is open, hence locally of finite presentation. The forgetful morphism is then locally of finite presentation. Compose it with the locally finitely presented morphism \(\mathcal C\to B\) from Theorem 5.2 to obtain the final assertion. \(\square\)

The forgetful map need not be a monomorphism: the same \(Q\) can receive different quotients of \(F_T\). Its representability follows by fixing \(Q\) and the identifying arrow, not by pretending these quotients are equal.

There is also a useful separatedness check. Given two quotients \(q_i:F_T\to Q_i\) with kernels \(K_i\), they are equivalent exactly when \[ K_1\longrightarrow Q_2\text{ is zero},\qquad K_2\longrightarrow Q_1\text{ is zero}. \tag{6.2} \] The first equation makes \(q_2\) factor through \(Q_1\), and the second supplies the reverse factorization. Surjectivity makes their composites identities. Each equation is a closed condition: \(K_i\) is quasi-coherent and its Hom into \(Q_j\) is affine by Theorem 1.1; equality to the zero section is a closed immersion. Under a base change, \(Q_i\)'s relative flatness identifies the pulled-back kernel with the kernel of the pulled-back quotient, so this construction really represents the diagonal on all tests. Their intersection proves that Quot is separated over \(B\). Testing just the first kernel inclusion would prove only factorization, not isomorphism.

For example, over a field take \(F=k^2\), \(q_1=1_{k^2}\) and \(q_2:k^2\twoheadrightarrow k\) a projection. Then \(K_1=0\), so the first equation holds, but the quotients are not isomorphic. Both kernels in (6.2) are needed. If \(F\) is of finite type, both \(K_i\) are of finite type because the \(Q_i\) are finitely presented. The proper-support zero-map criterion [Stacks, Tag 083M] makes each of these closed conditions finitely presented, so the Quot diagonal is then a closed immersion of finite presentation. The two-condition argument proves the claim of [Stacks, Tag 0DM2] at its stated generality.

6.2. The Hilbert functor

Define \(\operatorname{Hilb}_{X/B}(T)\) to be closed subspaces \(Z\subset X_T\) whose structure morphisms \(Z\to T\) are proper, flat and of finite presentation. They are subspaces with their closed immersion, rather than abstract proper spaces [Stacks, Tag 0CZX].

Corollary 6.3 (Hilbert). For separated \(f\) of finite presentation, \(\operatorname{Hilb}_{X/B}\) is an algebraic space, separated and locally of finite presentation over \(B\) [Stacks, Tag 0D01].

Proof. Identify it with \(\operatorname{Quot}_{\mathcal O_X/X/B}\). A surjection \(\mathcal O_{X_T}\to Q\) has an ideal kernel \(I\), and it identifies \(Q\) with \(\mathcal O_{X_T}/I\). The multiplication and unit on this quotient are forced by the surjection; no extra algebra structure is being chosen. Since \(Q\) is finitely presented as a module, \(I\) is of finite type. Thus \(Z=V(I)\) is a closed subspace of finite presentation over \(X_T\), hence over \(T\). Relative flatness of \(Q\) is exactly flatness of \(Z/T\), and its support is \(Z\), so proper support is exactly properness of \(Z/T\).

Conversely, if \(Z\) is such a closed subspace, its closed immersion into \(X_T\) is of finite presentation: both spaces are of finite presentation over \(T\), and an immersion between them is locally of finite presentation; here it is also quasi-compact. Its ideal is therefore of finite type, making \(\mathcal O_Z\) a finitely presented \(\mathcal O_{X_T}\)-module. It has the required flatness and support. These operations preserve pullback and identify the equivalence of quotient maps with equality of closed subspaces. Apply Theorem 6.2 and the separatedness test (6.2). \(\square\)

For projective \(X/B\) with a fixed relatively ample line bundle, the locus with a prescribed Hilbert polynomial is the usual projective Hilbert scheme of AG-HP-05. The present argument gives the whole Hilbert space for separated finitely presented algebraic spaces, without a polarization or a projective ambient space. The projective theorem's further properness assertion is a separate result.

6.3. Morphisms as graphs

Corollary 6.4 (morphism spaces). If \(Z/B\) is proper, flat and of finite presentation, and \(Y/B\) is separated of finite presentation, then the functor \(T\mapsto\operatorname{Mor}_T(Z_T,Y_T)\) is an algebraic space locally of finite presentation over \(B\) [Stacks, Tags 0D19 and 0D1C].

Proof. A morphism gives its closed graph in \(Z_T\times_TY_T\). This graph is isomorphic to \(Z_T\), so it is proper, flat and finitely presented over \(T\). It gives a point of the Hilbert space of \(Z\times_BY/B\). Conversely a member \(W\) of that Hilbert space is a graph precisely when its projection \(p:W\to Z_T\) is an isomorphism; the inverse of that projection then recovers the morphism to \(Y_T\). In particular two maps with the same graph are equal.

We verify that this is an open condition on \(T\). Both \(W\) and \(Z_T\) are proper, flat and finitely presented over \(T\), so \(p\) is proper and finitely presented. Near a fibre on which it is an isomorphism, first remove the image in \(T\) of the closed non-quasi-finite locus of \(p\). That image is closed by properness of \(W/T\) and misses the chosen point. The restricted \(p\) is proper and quasi-finite, hence finite, by the ordinary space form of Zariski main used in Lesson 2.

Now \(\mathcal O_{Z_T}\to p_*\mathcal O_W\) is a map of finitely presented modules on \(Z_T\). On affine charts, both are flat over \(T\), since \(Z_T\) and \(W\) are. Its cokernel is finite and vanishes on the chosen fibre; remove its proper closed support image. The map becomes surjective. Its kernel is finite, and the flatness of the target over \(T\) identifies its fibre with the kernel of the fibre map, which is zero. Nakayama kills that kernel near the whole chosen fibre; remove its proper closed support image as well. The algebra map is then an isomorphism, so \(p\) is an isomorphism.

These neighbourhoods for all isomorphism fibres define an open \(T^\circ\). Fibrewise isomorphism is preserved and detected by field extension. The argument just given, applied after any base change, proves that \(p_{T'}\) is an isomorphism exactly when \(T'\) factors through \(T^\circ\). Thus the graph functor is an open subspace of the Hilbert space, proving the corollary. \(\square\)

7. Picard stacks and Picard spaces

7.1. The open substack of line bundles

Assume now that \(f\) is flat, proper and of finite presentation. Let \(\mathcal{Pic}_{X/B}(T)\) be the groupoid of line bundles on \(X_T\), with all their isomorphisms. Such bundles are flat over \(T\) because \(X_T/T\) is flat, and their support is proper. They therefore form a full substack of \(\mathcal C\) [Stacks, Tag 0D02].

Lemma 7.1 (the fibrewise locally free locus). Let \(Y\to T\) be locally of finite presentation and let \(E\) be a finitely presented sheaf flat over \(T\). The set \[ W_r=\{y\in Y:E|_{Y_{t}}\text{ is locally free of rank }r \text{ at }y,\ t=\operatorname{image}(y)\} \tag{7.1} \] is open and commutes with arbitrary base change. On it \(E\) is locally free of rank \(r\) as an \(\mathcal O_Y\)-module.

Proof. At a point in (7.1), choose \(r\) generators lifting a basis on the fibre to a map \(\mathcal O_Y^r\to E\), locally on an affine chart. Its cokernel is finite, so it vanishes near the point by Nakayama. Let \(H\) be the kernel. Relative flatness of \(E\) ensures that reduction to the fibre preserves injectivity of \(H\to\mathcal O_Y^r\); thus \(H\) has zero fibre at the chosen point. Since \(E\) is finitely presented, \(H\) is finite. Nakayama kills \(H\) locally too. The map is an isomorphism on a neighbourhood. Conversely local freeness immediately gives the fibre condition. Fibrewise finite free modules of fixed rank descend and ascend under field extensions, so the same criterion describes the inverse-image open under every base change. The chart argument descends étale locally. This is [Stacks, Tag 0CZT]. \(\square\)

Theorem 7.2 (Picard stack). The stack \(\mathcal{Pic}_{X/B}\) is an open substack of \(\mathcal C\), hence algebraic and locally of finite presentation over \(B\) [Stacks, Tag 0D04].

Proof. For a family \(E\) over \(T\), let \(W_1\subset X_T\) be Lemma 7.1's open. The complement \(Z=X_T\setminus W_1\) has closed image in \(T\), since \(X_T\to T\) is proper. Its complement \(T^\circ\) consists exactly of parameters whose whole fibre belongs to \(W_1\). The lemma and proper base change of this support test show that \(T'\to T\) factors through \(T^\circ\) exactly when \(E_{T'}\) is a line bundle on all of \(X_{T'}\). Thus the full-substack inclusion is represented by the open immersion \(T^\circ\hookrightarrow T\). Apply Theorem 5.2. \(\square\)

This construction similarly gives the open substack \(\mathcal{Bun}_{r,X/B}\) of bundles of a specified rank. Here properness of the ambient \(X\) is used to test freeness on all of a fibre. The zero sheaf at a point is flat but is not a line bundle there, so the complement must be taken in \(X_T\), not just in the support of \(E\).

7.2. Removing scalar automorphisms by rigidification

The relative Picard functor is the fppf sheaf associated to \[ T\longmapsto\operatorname{Pic}(X_T). \tag{7.2} \] Equivalently, it is the sheafification of \(\operatorname{Pic}(X_T)/f_T^*\operatorname{Pic}(T)\), since a base line bundle becomes trivial on an fppf cover. Denote it by \(P=\operatorname{Pic}_{X/B}\). This sheaf should not be confused with the line-bundle groupoid.

Theorem 7.3 (Picard space). In addition to the hypotheses of Theorem 7.2, assume \[ \mathcal O_T\xrightarrow{\sim}f_{T,*}\mathcal O_{X_T} \quad\text{for every }B\text{-scheme }T. \tag{7.3} \] Then \(P\) is an algebraic space [Stacks, Tag 0D2C].

Proof with a section. Suppose first that \(f\) has a section \(\sigma:B\to X\). Define the stack \(\mathcal R_\sigma\) of pairs \[ (L,\alpha),\qquad L\text{ a line bundle on }X_T,\quad \alpha:\sigma_T^*L\xrightarrow{\sim}\mathcal O_T. \tag{7.4} \] Forget \(\alpha\). Over a fixed \(L/T\), the fibre is the scheme of frames of the line bundle \(\sigma_T^*L\), a \(\mathbf G_m\)-torsor. Thus \(\mathcal R_\sigma\to\mathcal{Pic}_{X/B}\) is representable, smooth and surjective. The same atlas and diagonal recognition used for Quot proves that \(\mathcal R_\sigma\) is algebraic and locally of finite presentation.

Every automorphism of \(L\) is an invertible global function on \(X_T\). By (7.3) it is a unique invertible function on \(T\). To preserve \(\alpha\), its restriction under \(\sigma_T\) must be \(1\), so it is \(1\). Hence \(\mathcal R_\sigma\) has no nontrivial automorphisms and is an algebraic space.

We identify its sheaf with \(P\), including the descent needed for this assertion. Send \(L\) to its normalization \[ N_\sigma(L)=L\otimes f_T^*(\sigma_T^*L)^\vee, \tag{7.5} \] equipped with the canonical frame along \(\sigma_T\). Tensoring \(L\) with \(f_T^*M\), for any line bundle \(M/T\), leaves (7.5) canonically unchanged. Conversely, two normalized framed bundles are isomorphic exactly when their original classes differ by a base line bundle. An unframed isomorphism between normalized bundles can be adjusted by the unique base unit which makes it preserve the frames. Therefore framed isomorphisms, when they exist, are unique.

The presheaf \[ T\longmapsto \ker\bigl(\sigma_T^*:\operatorname{Pic}(X_T)\to\operatorname{Pic}(T)\bigr) \tag{7.6} \] is already an fppf sheaf. To see this directly, representatives of a compatible family in (7.6) admit frames along the section. On overlaps, compatibility gives unframed isomorphisms; adjust them by their scalar restriction to obtain framed isomorphisms. Their uniqueness gives the cocycle equality, so the framed line bundles descend. The same uniqueness proves that equal local classes give equal global classes. Normalization identifies (7.6) with the quotient presheaf by base line bundles and hence with its sheafification (7.2). Thus the algebraic space \(\mathcal R_\sigma\) represents \(P\).

Removal of the section hypothesis. Equation (7.3), applied to geometric fields, makes every fibre of \(f\) nonempty; otherwise its ring of global functions would be zero rather than that field. Since \(f\) is flat and of finite presentation, it is faithfully flat and locally of finite presentation. After the fppf base change \(B'=X\to B\), the projection \(X\times_BX\to X\) has the diagonal section. All hypotheses, including (7.3), survive this base change. One may take an étale scheme cover of \(X\) to obtain a covering by schemes on which the preceding argument applies.

The Picard sheaf commutes with base change: restricting its fppf definition to \(B'\)-schemes gives exactly the sheaf for \(X_{B'}/B'\). Hence \(P\times_BB'\) is an algebraic space by the section argument. The flat sheaf bootstrap of Lesson 2 now descends algebraicity of \(P\). Explicitly, a scheme atlas of \(P\times_BB'\) is a representable faithfully flat locally finitely presented cover of \(P\); its relation is represented by the corresponding fibre products after the cover. The proved bootstrap replaces this flat presentation by étale coordinates. It also descends local finite presentation over \(B\). \(\square\)

The morphism \(\mathcal{Pic}_{X/B}\to P\) is a gerbe banded by \(\mathbf G_m\). Objects and comparisons exist fppf locally by the definition of the sheafification, and (7.3) identifies the automorphism sheaf of any line bundle with \(\mathbf G_{m,T}\). With a section, the universal normalized bundle on \(\mathcal R_\sigma=P\) neutralizes this gerbe. Without a section, a Picard class need not be represented by a line bundle on its given \(X_T\), and there need not be a universal line bundle on \(X\times_BP\). The sheaf and its algebraicity do not assert either stronger conclusion. In particular (7.3) is universal, rather than an equality checked just on \(B\).

7.3. Two curve examples

Let \(C/k\) be a smooth projective geometrically connected curve.

Vector bundles. The rank-\(r\) bundles form the open algebraic stack \(\mathcal{Bun}_{r,C/k}\subset\mathcal{Coh}_{C/k}\). At \(E\), the formulas of Proposition 4.1 become \[ T_E\mathcal{Bun}_{r,C/k}=H^1(C,\mathcal E nd(E)),\qquad \operatorname{Inf}_E\mathcal{Bun}_{r,C/k}=H^0(C,\mathcal E nd(E)). \tag{7.7} \] Indeed a locally free source has no higher internal Ext, so global Ext is coherent cohomology. For a family of bundles on \(C_A\), the obstruction group for a square-zero lift is \(H^2(C_A,\mathcal E nd(E)\otimes_AM)=0\); the relative coherent-cohomology dimension bound for the projective curve holds for every \(M\). Every marked infinitesimal lift therefore exists. The stack is locally of finite presentation, so this formal smoothness proves smoothness over \(k\). Scalar automorphisms already give a positive-dimensional infinitesimal automorphism group for nonzero rank. Thus smoothness does not make this a Deligne–Mumford stack.

Picard classes. Geometric connectedness and properness give \(H^0(C_{\bar k},\mathcal O)=\bar k\); the coherent base-change theorem gives (7.3) on every \(k\)-scheme. Theorem 7.3 therefore makes \(\operatorname{Pic}_{C/k}\) an algebraic space. The curve Picard theorem of AG-HP-09 gives the stronger scheme result: it is a smooth group scheme locally of finite type, its degree-zero component is the Jacobian, and its degree-\(d\) part is a torsor under that Jacobian. A rational point supplies a normalization and universal bundle; without such a point the scheme assertion remains valid while the universal-bundle assertion needs its own hypothesis. The scheme result is imported from AG-HP-09, rather than inferred merely from algebraic-space representability.

8. Two larger moduli stacks, stated

The methods above are part of a larger pattern, but the following two theorems are statements in this lesson. Their deformation theory concerns the underlying space itself.

Stated theorem 8.1 (polarized schemes). The stack whose objects over \(T\) are proper flat finitely presented schemes \(Y/T\) with a chosen relatively ample invertible sheaf \(L\) is algebraic. An arrow includes both the isomorphism of schemes and the specified isomorphism of the polarizations. The version formed by base change to any algebraic space \(B\) is algebraic too [Stacks, Tags 0D1L and 0D4X].

Thus the polarization is data, rather than a property saying that some ample bundle exists. An automorphism of the chosen bundle covering the identity of \(Y\) is still an arrow in this stack; it has not been removed by passing to a numerical class. The finite-presentation, flatness and properness conditions belong to the definition.

Stated theorem 8.2 (curves). The stack whose objects over \(T\) are proper flat finitely presented algebraic spaces \(Y/T\) with all fibres of dimension at most \(1\), and whose arrows are isomorphisms over the parameter map, is algebraic. Its version over any algebraic space \(B\) is algebraic [Stacks, Tags 0D4Y and 0D5A].

This statement allows zero-dimensional fibres, nonreduced curves and reducible curves. It imposes neither stability nor smoothness nor geometric connectedness. The source locator for 0D5A also points to de Jong–He–Starr, Proposition 3.3, and Jack Hall's Appendix B, Theorem B.1, in Smyth. The later lessons impose nodal and stability conditions to obtain the moduli stacks of stable curves; those restrictions are additional mathematics, not implicit hypotheses of Theorem 8.2.

9. Exercises with complete solutions

Exercise 9.1 (easy: openness of Picard). For \(f\) flat, proper and of finite presentation, prove that \(\mathcal{Pic}_{X/B}\) is open in \(\mathcal{Coh}_{X/B}\).

Solution. Fix a coherent family \(E\) over \(T\). On an affine chart, choose generators of a rank-one free fibre module and lift them to a map \(\mathcal O_{X_T}\to E\). Its finite cokernel vanishes near the chosen fibre point. Relative flatness of \(E\) makes the kernel's reduction the kernel of the fibre map, which is zero. The kernel is finite by finite presentation, so Nakayama kills it near that point. Thus the fibrewise rank-one free locus \(W_1\) is the open locally free rank-one locus. It is stable under arbitrary base change because freeness of a finite module over a local fibre ring is faithfully detected after residue-field extension.

Let \(Z=X_T\setminus W_1\). Properness of \(X_T/T\) makes its image closed. On \(T^\circ=T\setminus f_T(Z)\), every point of every fibre belongs to \(W_1\), so \(E\) is invertible everywhere. If \(T'\to T\) meets the closed image, its base change has a fibre point outside \(W_1\), so the pulled-back sheaf is not invertible there. Hence the pullback of the full-substack inclusion is exactly \(T^\circ\hookrightarrow T\), with the same isomorphisms as in the coherent stack. This is the required representable open immersion. Properness is used for the whole ambient space in this exercise; proper support alone would not force a sheaf to have rank one at points outside its support.

Exercise 9.2 (medium: Rim–Schlessinger). Verify (RS\(^*\)) for \(\mathcal{Coh}_{X/B}\), including the case in which \(X/B\) is not flat [Stacks, Tag 08LQ].

Solution. Use (3.1). A point of the fibre-product groupoid includes sheaves \(F_1,F_2\) and an actual isomorphism between their restrictions to \(X_{A_0}\). On an affine étale chart \(C/P\), use that isomorphism to form the module fibre product \(M_1\times_{M_0}M_2\). Lemma 3.1's ideal argument proves its \(P\)-flatness and its two base-change identities; no property of the ambient algebra \(C/P\) is used there.

To verify finite presentation, write \(C=P[z_1,\ldots,z_n]/K\) with \(K\) finite. The polynomial algebra is \(P\)-flat. Lift finitely many generators from \(M/JM=M_1\) to get a finite free polynomial-module surjection onto \(M\); its kernel is \(P\)-flat because the quotient \(M\) is. Its reduction is a finite kernel of a presentation of \(M_1\). Lift these kernel generators and use \(J^2=0\) to kill the remaining quotient. This proves finite presentation over the polynomial algebra and hence over \(C\).

On chart intersections full faithfulness for module fibre products gives the unique gluing maps. Testing the cocycle on both restrictions gives the equality upstairs, so the sheaf descends. Its Fitting support is separated of finite type and is proper after the nilpotent base thickening to \(A_1\); all base changes of that thickening are homeomorphisms, so the support is universally closed and therefore proper. Compatible arrows patch by exactly the same fibre-product rule. A flat module originally over \(P\) is recovered by tensoring \(0\to P\to A_1\oplus A_2\to A_0\to0\), so the construction is inverse on both objects and arrows. This proves the groupoid equivalence. Filtering an Artinian surjection by square-zero kernels gives (RS).

Exercise 9.3 (medium: Isom from Hom). Deduce Corollary 1.3 from Theorem 1.1. Identify the hypotheses needed for every Hom space used in the proof [Stacks, Tag 08K9].

Solution. On the product of Hom spaces parameterizing \(u:F\to G\) and \(v:G\to F\), compose to obtain a morphism to \(\underline{\operatorname{Hom}}(F,F)\times_B\underline{\operatorname{Hom}}(G,G)\). The source factors require the targets \(G\) and \(F\), respectively, to be flat with proper support; both sources must be finitely presented for finite presentation of the representing schemes. The same conditions apply to the two endomorphism spaces. Thus both \(F,G\) must be finitely presented, flat over \(B\), with proper support.

The identity pair is a section of an affine finitely presented morphism and therefore a closed immersion of finite presentation. Pull it back by composition. Its equations are \(vu=1_F\) and \(uv=1_G\), with the domains as written. The resulting affine finitely presented space has points exactly mutually inverse pairs. The inverse of \(u\) is unique, so projection to \(u\) gives a functorial identification with Isom on every \(B\)-scheme. Example 1.4 shows that just requiring finite presentation of \(F\) while retaining the other conditions only for \(G\) would be a false statement.

Exercise 9.4 (hard: the effectivity axiom). Verify Artin's formal effectivity axiom for \(\mathcal{Coh}_{X/B}\) using Grothendieck existence, and prove full faithfulness on isomorphisms [Stacks, Tag 08W9].

Solution. Let \(R\) be complete Noetherian local and take a compatible system of families over \(R_n=R/\mathfrak m^{n+1}\), with its specified transition identifications. Since \(X_R\) is separated of finite type, the proper-support existence equivalence algebraizes this system to a coherent sheaf \(F\) with proper support. It includes all coherent morphisms, so the reductions of \(F\) have exactly the prescribed identifications.

On affine étale charts, the flatness criterion for all powers of \(\mathfrak m\) shows that \(F\) is \(R\)-flat along the closed fibre, because each \(F_n\) is \(R_n\)-flat. The nonflat locus is closed, lies in the proper support, and therefore has closed image in \(\operatorname{Spec}R\). It misses the closed point. A nonempty closed subset of a local spectrum contains that point, so the image, and hence the nonflat locus, is empty. No flatness of \(X_R/R\) is required. Coherence on the Noetherian space gives finite presentation, so \(F\) is an object of the coherent stack.

Given compatible isomorphisms between two systems, full faithfulness of existence yields a unique homomorphism between their algebraizations. Algebraize the compatible inverse homomorphisms as well. Their composites reduce to the identities at all orders, so faithfulness makes the composites the identities upstairs. Equality and composition of arbitrary compatible arrows are preserved for the same reason. Completion is therefore an equivalence of groupoids. This proves more than the essential surjectivity required by the axiom. The arbitrary inverse-ring-system theorem used for nonflat openness in Section 5.2 is a stronger input and is not being deduced from this exercise.

10. Precisely imported inputs and scope

The Hom and Isom theorems, both coherent-stack algebraicity theorems, and the Quot, Hilbert, Picard-stack and Picard-space consequences have been proved. The following are prerequisites about modules, coherent cohomology, spaces and schemes, rather than imported moduli algebraicity conclusions:

The finite-presentation module patching, the finite-complex Hom representation, the relative free-locus test and the product calculations were proved here. Each moduli algebraicity assertion follows after its own descent, diagonal and deformation or presentation argument.

References