Moduli stacks are algebraic

Source proofs by the Stacks Project authors, as distributed in the AI Integrated Stacks Project. Source copyright: Copyright (C) 2005 -- 2025 Johan de Jong. This modified course edition is published by the Open Math Courses project, KokunoYumeto/open-math-courses. Writing, adaptation and integration: GPT-6.1 Sol (OpenAI), Codex, Ultra, October 2026. Permission is granted to copy, distribute and modify this modified chapter under the GNU Free Documentation License, Version 1.2 or any later version, with no Invariant Sections, Front-Cover Texts or Back-Cover Texts. Eligible independently written additions retain their CC0 1.0 dedication. Self-checked by the writing AI. The complete licence accompanies this edition.

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.

Appendix A §A.6 proves 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:

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. Polarized schemes and all curve spaces

The following theorems concern the deformation of the underlying space itself. Appendix A §§A.7–A.8 supplies their complete constructions, including all the marked algebraization and approximation comparisons.

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].

Proof of Theorem 8.1. Appendix A §A.7 constructs the diagonal through Hom/Isom and graph Hilbert spaces, patches marked polarized families, proves the graded algebraization and finite-presentation statements, and verifies Artin's axioms. The arbitrary-base return is included there. The full proof gives the chosen line bundle and its specified arrow, with no quotient by scalar automorphisms.

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.

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].

Proof of Theorem 8.2. Appendix A §A.8 proves the dimension-at-most-one stack theorem using the preceding polarized construction and its infinitesimal/projective lifting argument. Its field scheme recognition and projectivity foundations are proved in §§A.12.10–A.12.11. The proof returns from its charts to the original algebraic spaces and base, rather than restricting the theorem to projective schemes.

This theorem 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. Appendix A supplies the coherent-existence, space cohomology and perfect-test extensions below at their exact scope. Its individual lower native references remain explicitly conditional when no included proof or exact programme binding has been established. The genuine projective-scheme and Picard examples retain their named programme providers:

• On quasi-compact quasi-separated algebraic spaces, finite-presentation descent along affine limits, quasi-coherent sheaves as filtered colimits of finitely presented sheaves, and descent of relative flatness and proper support. The exact proper-support limit statement used above is [Stacks, Tag 08K2]. Finite presentation of spaces and sheaves permits descent to a finitely generated \(\mathbf Z\)-algebra.
• Perfect approximation on a Noetherian algebraic space: for a bounded-above coherent complex \(E\) and any integer \(m\), there is a perfect \(P\to E\) inducing isomorphisms in degrees \(>m\) and a surjection in degree \(m\) [Stacks, Tags 08HI, 08HJ and 08HP]. The proper-support direct-image theorem makes \(Rf_*R\mathcal H om(P,G)\) perfect for perfect \(P\) and bounded coherent \(G\) flat over a Noetherian base with proper support, when \(f\) is locally finite type and quasi-separated; arbitrary derived base change holds [Stacks, Tags 0DKK and 08JQ]. Section 5.1 proves the needed low-Ext reduction, also recorded in Tag 08JR.
• Coherent cohomology finiteness with proper support, internal Ext coherence over Noetherian charts, the local-to-global Ext spectral sequence and the relative cohomological dimension bound for projective curves. These imply the finite tangent calculation; a related proper-space finiteness statement is [Stacks, Tag 0D0T].
• Grothendieck existence with all morphisms, for a separated finite type algebraic space over a complete Noetherian ring and coherent sheaves of proper support [Stacks, Tags 08B7 and 08BE]. Its flatness application uses the local criterion along all powers of an ideal and the openness of relative flatness [Stacks, Tag 08VP]. Proposition 4.2 proves the additional flatness and groupoid assertions.
• Strong coherent existence exactly as stated in Section 5.2 [Stacks, Tags 0CX4 and 0CXB], for arbitrary surjective inverse ring sequences with locally nilpotent kernels, separated finitely presented ambient space, and compatible finitely presented relatively flat sheaves with proper support. This is explicitly stronger than Noetherian formal existence.
• Flat-module square-zero deformation theory with its obstruction in Ext \(2\), marked-lift torsor in Ext \(1\), automorphisms in Ext \(0\), and functoriality for maps with fixed quotient [Stacks, Tags 08VW and 0CYE]. It is used only where the ambient base changes are flat. G-ring permanence and the G-ring property of finite type \(\mathbf Z\)-algebras are [Stacks, Tag 07PX] and the inputs already isolated in Lesson 7.
• The finite-type zero-map criterion used for the additional finite-presentation assertion on the Quot diagonal: a map from a finite type quasi-coherent sheaf to a finitely presented relatively flat sheaf with proper support has a finitely presented closed vanishing locus on the base [Stacks, Tag 083M]. For arbitrary quasi-coherent source its closedness was derived here directly from Hom. The morphism-space corollary uses the ordinary proper quasi-finite-to-finite theorem of Lesson 2.
• The projective Quot and Hilbert theorems from AG-HP-05 and the smooth projective curve Picard scheme theorem from AG-HP-09. These stronger scheme assertions are used only to identify the projective and curve examples. The full polarized-scheme and dimension-at-most-one curve-space algebraicity proofs are in Appendix A §§A.7–A.8. They retain their genuinely assigned deformation prerequisites and explicitly recorded lower references.

The ordinary proper-support existence and direct-image support are written in Appendix A, and strong existence is proved there for the entire stated inverse-system scope. 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

• [Stacks] The Stacks project, Quot and Hilbert Spaces: Hom and Isom, Tags 08JS, 08K6, 08K7 and 08K9; coherent sheaves and their Artin axioms, Tags 08KA, 08KB, 08W5, 08W6, 08KC, 08KD, 08LQ, 08W7, 08W8, 08W9, 08WA, 08WC, 08WB and 09DS; Quot and Hilbert, Tags 09TQ, 09TU, 0CZX and 0D01; Picard stack and functor, Tags 0D02, 0D04, 0D24 and 0D2C; morphism spaces, polarized schemes and curves, Tags 0D19, 0D1C, 0D1L, 0D4X, 0D4Y and 0D5A. Read AI Integrated Stacks Project, Quot and Hilbert Spaces.
• [Stacks] Moduli Stacks, Tags 0DLU, 0DLW, 0DLX, 0DM1, 0DM5, 0DM9 and 0DMD, for the collected properties of these moduli stacks. See AI Integrated Stacks Project, Moduli Stacks.
• [Stacks] Derived Categories of Spaces, Tags 08HI, 08HJ, 08HP, 08JQ, 08JR, 0D0T and 0DKK; More on Morphisms of Spaces, Tags 08B7, 08BE and 08VP; and More on Flatness of Spaces, Tags 0CX4 and 0CXB, for the precise ordinary inputs listed in Section 10. AI Integrated Stacks Project retains the upstream tags; its additions and corrections are not reviewed by the Stacks project's maintainers. Source readings for this lesson were made at the recorded immutable revision.

Appendix A. Coherent existence and moduli of proper spaces

Section and statement numbers within this appendix are local to the appendix; an explicit course or provider title qualifies every outside reference.

This chapter supplies the coherent-existence and moduli arguments used in Lesson 8. The theorem statements retain arbitrary inverse ring sequences, algebraic-space ambient objects, and the full dimension-at-most-one curve stack. The mathematical proofs below are adaptations of the human Stacks Project treatment in the AI Integrated Stacks edition. The mathematical adaptation is distributed under GNU FDL 1.2 or any later version, with no Invariant Sections, Front-Cover Texts, or Back-Cover Texts. The full licence is included below. Human authorship belongs to the Stacks Project authors; the AI Integrated Stacks Project and this AI adaptation are additional, separate credits. Independently written programme integration notes and verification records remain CC0 1.0. No AI rights holder is invented.

A compatible family over finite quotients carries both objects and specified comparison maps. The central problem is to recover that marked family over the inverse-limit ring. We first construct finite cohomological tests, then prove coherent existence, and finally use it to construct the two moduli stacks. The supporting mathematical constructions are proved below at their stated scope.

A.1. The inverse system and its markings

Let

be a sequence of surjective ring homomorphisms with locally nilpotent kernels. Here locally nilpotent means that every finite collection of elements of the kernel generates a nilpotent ideal. Put $A=\varprojlim_n A_n$ and $I_n=\ker(A\to A_n)$. The projections $A\to A_n$ are surjective: lift an element successively through the surjective transitions.

Every composite transition has a locally nilpotent kernel. Indeed, for a finite collection in its kernel, a power of the ideal they generate maps to zero at the preceding stage, and a further power is zero in the upper ring. Induction on the number of stages proves the assertion. Consequently the schemes $\operatorname{Spec}A_n$ have the same underlying topological space. Their common image in $\operatorname{Spec}A$ is $V(I_1)=V(I_n)$.

Each pair $(A,I_n)$ is henselian, and $I_n$ is contained in the Jacobson radical. To prove the radical assertion, an element which is a unit in $A_n$ is a unit at every later stage, because units lift through a locally nilpotent ideal. Its inverses are compatible and define an inverse in $A$. Apply this to $1-ax$, $a\in A$, $x\in I_n$. For the henselian assertion, start with a coprime monic factorization of a monic polynomial over $A_n$. Coprime factorizations lift uniquely through a locally nilpotent ideal: their finitely many coefficients and corrections reduce the question to a nilpotent ideal, where the usual linear correction, using a Bezout relation between the two factors, raises the order of the error until it is zero. Lift recursively and take the coefficientwise limit. This is precisely the factorization definition of a henselian pair.

Let $Y\to\operatorname{Spec}A$ be a separated morphism of finite presentation of algebraic spaces. Write $Y_n=Y\times_A A_n$, with closed immersion $i_n:Y_n\to Y$. We are given finitely presented quasi-coherent modules $G_n$ on $Y_n$, flat over $A_n$, with proper support over $A_n$, and specified isomorphisms

All comparisons below use these specified maps. Replacing them by an equality of isomorphism classes would lose the required data.

The conclusion to be proved is a finitely presented module $G$ on $Y$, flat over $A$, with proper support, and isomorphisms

whose transition composites are the $\phi_n$. This is the strong coherent-existence statement of Tags 0CX4/0CXB. It does not assert an equivalence of categories for every such tower; that stronger assertion requires its own morphism-recovery argument.

A.2. Proper support and the finite complexes used below

For a finitely presented module $F$ on an algebraic space, use the closed subspace

Its defining ideal is of finite type and its formation commutes with arbitrary base change. On a chart on which $F$ has $r$ generators,

To check the base-change assertion, present $F$ on an étale chart by $B^q\xrightarrow{M}B^r\to F\to0$. The ideal is generated by the $r$-by-$r$ minors of $M$; if $q<r$, it is zero. Tensoring with any chart algebra after any base change preserves this presentation, and determinants commute with ring homomorphisms. Therefore

for every $g:X'\to X$, with no flatness assumption.

For the containments, the adjugate identity says that each maximal minor kills the cokernel. Conversely, if $a_1,\ldots,a_r$ annihilate $F$, then each vector $a_j e_j$ is in the image of $M$. Expressing these $r$ columns through $M$ and using the determinant expansion puts $\prod_j a_j$ in the maximal-minor ideal. These products generate $(\operatorname{Ann}F)^r$. The identities are étale-local and hence give the stated ideal containments on the space.

Thus the annihilator support is a closed nilpotent subspace of $Z_F$. Its immersion is a universal homeomorphism, also after arbitrary base change. When the ambient space is separated and finitely presented over the base, both supports are separated and of finite type over the base: $Z_F$ is finitely presented as a closed subspace of the finitely presented ambient space, while the annihilator support is its closed subspace. Their universal closedness, and therefore properness, are equivalent. In particular the Fitting support of every $G_n$ is proper exactly when its usual annihilator support is proper. The displayed base-change identity identifies $Z_F\times_A A_n$ with this proper Fitting support of $G_n$. This supplies the exact proper-support consumer, as well as the convention already proved in Lesson 8, Section 1.1.

We require four bridges, each at algebraic-space scope:

1. On a quasi-compact quasi-separated algebraic space, pseudo-coherent objects can be approximated arbitrarily far to the left by perfect objects. The support-preserving version permits a closed support whose complement is quasi-compact.
2. For $f:X\to T$ of finite presentation, a perfect $P$ and a bounded complex $F^\bullet$ of finitely presented modules flat over $T$, with proper support, give a perfect object

on $T$, with the canonical comparison isomorphism after every base change. Flatness of $X/T$ is unnecessary. 3. If $K\in D^-_{\mathrm{QCoh}}(Y)$ has pseudo-coherent $R\Gamma(Y,E\otimes^{\mathbf L}K)$ for every pseudo-coherent $E$, then $K$ is pseudo-coherent relative to $A$. Relative pseudo-coherence means that, on an affine étale chart $\operatorname{Spec}B\to Y$ and a smooth presentation $P\to B$, the pushforward is represented by a bounded-above complex of finite free $P$-modules. 4. A compatible tower of perfect complexes over the $A_n$, with derived transition isomorphisms, has perfect derived limit over $A$, and derived base change of that limit recovers every level.

Sections 3–5 prove the bridges and explain the exact geometric constructions behind them. Their incorporated source proof chain is retained in the appendix. In particular, a scheme proof over a Noetherian base is used at its proved scope; it is not silently promoted to the arbitrary-base algebraic-space statement.

A.3. Coherent direct images with proper support

First let $T$ be Noetherian and $f:X\to T$ locally of finite type and quasi-separated. A coherent module $F$ with proper support is the pushforward from a proper closed support $Z\to T$. Over a Noetherian chart its annihilator is coherent, so this is an actual closed subspace. A closed immersion is affine and has no higher quasi-coherent direct images. Leray therefore reduces finiteness of $R^qf_*F$ to proper coherent cohomology on $Z$.

Here is the algebraic-space extension of the scheme proof. Coherence, the ascending chain condition, and Artin–Rees are checked on a finite affine étale cover. Localization and flat étale restriction preserve all module intersections used in the Artin–Rees calculation. The extension lemma for a morphism on $U=X\setminus V(J)$ also has the same proof: form its quasi-coherent graph inside $G\oplus F$, take the inverse image of this graph, and apply Artin–Rees to its projection kernel. A sufficiently high ideal power then yields a map $J^aG\to F$. Two maps agreeing on $U$ agree on a higher source power.

Consequently the generic-lattice and filtration arguments of AG-QC, Proper Direct Images, Sections 1–3, extend to Noetherian algebraic spaces. More explicitly, for an integral closed subspace $Z$ and a coherent module of generic rank $r>0$, choose a basis on an affine open of its schematic dense locus, clear denominators, and extend the resulting map by the preceding graph argument. Its source is $I^{\oplus r}$, $I\subset\mathcal O_Z$ a nonzero coherent ideal. Its kernel is zero because $I$ embeds in the function field and the kernel has zero generic stalk. Its cokernel has smaller closed support. Induction on closed supports, followed by the finite filtration by powers of the ideal of the reduced support, gives a filtration of any coherent module by ideals on integral closed subspaces.

For the cohomology property, a witness of any finite positive generic rank suffices. If $I^{\oplus r}$ has finite higher direct images, each $I$ does, because its cohomology is a direct summand. Comparing $I$ with another nonzero ideal through their intersection makes the two quotients have smaller support. Two out of three in the cohomology long exact sequence and Noetherian induction therefore transfer the property to all filtration factors and then to every coherent module.

A witness is supplied by the weak Chow lemma for algebraic spaces: there is a proper surjection $\pi:Z'\to Z$, with $Z'$ a scheme admitting an immersion into projective space over the affine base. Since $Z$ is proper over that base, the immersion is closed, and both $\pi$ and $Z'\to T$ are projective. With $L$ the restricted hyperplane bundle, take $d$ sufficiently large that higher direct images of $L^d$ vanish for both morphisms. Then $H=\pi_*L^d$ is coherent, Leray gives

and these are finite, zero for positive $q$. Its generic stalk is nonzero: a globally generating hyperplane section not vanishing at a point over the generic point has a nonzero image under adjunction. The support of $H$ is therefore all of integral $Z$. This is the required finite-positive-rank witness. It proves the proper coherent direct-image theorem for spaces and hence the proper-support version.

The weak Chow construction is included below. Its empty-space case is $Z'=\varnothing$. For a nonempty space, choose an affine étale cover with a bound $d$ on its geometric fibre sizes. The open locus of $d$ distinct lifts gives an immersion into the $d$-fold product of a projective closure of the cover. Take its scheme-theoretic image and the union of the opens on which one projection lands in the original cover. The projections agree as maps to $Z$ on the schematically dense locus, hence agree everywhere by separatedness. The valuative criterion proves properness: after an extension of valuation rings a lift to the original cover selects one of the $d$ generic lifts, and properness of the corresponding projective projection extends it into that open. The uncovered closed locus has fibre bound $<d$, so induction supplies the remaining proper cover. This retains the full algebraic-space construction rather than assuming a birational scheme modification exists.

For bounded coherent $E$ with proper cohomological supports, the spectral sequence

now gives coherent cohomology. A finite cohomological-dimension bound on quasi-coherent direct image gives boundedness. If $E$ has finite Tor amplitude over $T$, the projection formula

gives a uniform lower bound for every quasi-coherent $M$; the upper bound comes from the bounded-above complex and the cohomological-dimension bound. Thus $Rf_*E$ is pseudo-coherent of finite Tor dimension, hence perfect.

Apply this to $E=P\otimes^{\mathbf L}F^\bullet$. On each chart a perfect $P$ is a finite complex of finite projectives. Its tensor with the bounded relatively flat complex has finite relative Tor amplitude and coherent cohomology, supported on the union of the proper supports of the terms. This proves perfection over a Noetherian base.

A.4. Approximation and passage to arbitrary bases

The approximation statement is the following. For a closed subset $Z\subset |X|$ with quasi-compact complement, there is an integer $r_Z$ such that an $(m-r_Z)$-pseudo-coherent $E$, with $H^i(E)$ supported on $Z$ for $i\geq m-r_Z$, admits a perfect $P$ supported on $Z$ and a map $P\to E$ inducing isomorphisms in degrees $>m$ and a surjection in degree $m$.

On an affine $\operatorname{Spec}B$, write $Z=V(f_1,\ldots,f_r)$. The Koszul complexes on powers of the $f_i$ represent every cohomology class supported on $Z$, once a sufficiently large power is chosen. Here one uses the exact cohomological support range: after the appropriate shift, the cohomology in degrees $-r+1,\ldots,0$ is supported on $Z$. The complement is covered by the $r$ principal affines $D(f_i)$, so the finite Čech spectral sequence makes $H^0$ of the restricted complex zero. Write $K_e$ for the Koszul complex on $f_1^e,\ldots,f_r^e$, and $I_e\to B\to K_e$ for its triangle obtained by separating the term in degree zero. There is a canonical isomorphism

Indeed, the term indexed by a nonempty subset $J$ in the left colimit is $M[(\prod_{j\in J}f_j)^{-1}]$: its transition multiplies by $\prod_{j\in J}f_j$. These are exactly the terms of the finite open Čech complex, with the usual total-complex signs. Finite free $I_e$ compute the derived Hom, and filtered module colimits are exact. Thus, if a map $B\to M$ restricts to zero on the complement, its composite with $I_e\to B$ is already zero for some finite $e$. The triangle lifts it to a map $K_e\to M$. This is the closed-support Koszul representative lemma incorporated below; localization of a single cocycle without the stated cohomological range would not suffice. Choose finitely many generators in the largest nonzero cohomological degree $t$, and take a finite sum of these maps from shifted Koszul complexes. Its cone has zero cohomology in degrees $\geq t$ and retains pseudo-coherence and the specified support range. Repeat until $t<m$. Splicing the resulting triangles gives the required perfect approximation. When $Z=X$, ordinary finite-free truncation of a bounded-above pseudo-coherent resolution gives the same argument without any Koszul variables. When $E$ has no nonzero degree $\geq m$, the zero perfect complex is sufficient.

To pass to algebraic spaces use elementary distinguished squares $(U\subset X,V\to X)$, with $V$ affine and an isomorphism over the complement of $U$. Their finite étale filtration gives an induction principle. The derived Mayer–Vietoris triangle glues two complexes and two maps agreeing on $U\times_XV$: take the homotopy fibre of

Restriction to $U$ cancels the second summand. On $V$, the cone of its map to $B$ is supported on the distinguished complement; its pushforward is zero, and pushforward there is faithful, so that cone is zero. This proves both gluing assertions.

The lifting steps on the affine member of the square retain support. A perfect object on a quasi-compact open extends after adding its shift. First extend a bounded-above finite-free resolution and truncate below its Tor amplitude; the extra kernel is a vector bundle. Lift the projector on that extra bundle, after clearing denominators by a perfect Koszul modification of the source, and take its cone. This extends $P\oplus P[1]$. To keep support $V(g_1,\ldots,g_s)$, take successive cones of multiplication by powers of $g_j$ that act trivially on $P$. Each cone is supported on the added zero locus. The restriction is a finite direct sum of shifts of $P$, containing $P$. Lift the endomorphism equal to zero on that $P$ and the identity on the other summands; its cone restricts to $P\oplus P[1]$ and retains support. The source modification in the map-lifting construction is a tensor with a perfect complex, so it also retains support. All these constructions and the Koszul representative lemmas are retained with their complete proofs in the appendix.

Now approximate on $U$, extend that approximation and its map across $V$ by the preceding construction, and glue. The map kills the highest nonzero cohomology on $U$. Its cone has a strictly smaller highest degree there. Repeat. When this degree is below the support threshold, all relevant cohomology is supported on the distinguished complement and the affine support approximation on $V$, extended by zero, finishes. Splicing the triangles completes the induction. The integer used for the pair is the maximum of the integer on $U$ and the number of affines covering the relevant complement in $V$. Thus the process is finite for every requested $m$.

For arbitrary $T$, the canonical base-change comparison for $P\otimes^{\mathbf L}F^\bullet$ is an isomorphism without any Noetherian hypothesis. The assertion is étale-local on $T$. On affine $X=\operatorname{Spec}B$, represent $P$ by finite projectives and tensor with the relatively flat terms of $F^\bullet$; this is a bounded complex flat over the affine base ring, so ordinary tensor computes its derived tensor after every ring map. The resulting complex is exactly the one computing the base-changed tensor with the derived pullback of $P$ and the ordinary pullback of $F^\bullet$. For an elementary distinguished square, its Mayer–Vietoris triangle remains the corresponding triangle after base change. If the comparisons are isomorphisms on its three smaller pieces, the triangle proves the comparison on the whole space. The finite étale induction principle completes the proof. The comparisons are the adjunction/projection-formula maps, so successive base changes and morphisms of the input complexes give the same isomorphisms, not merely abstract complexes with equal cohomology dimensions.

Finally write an affine base ring as the filtered colimit of its finitely generated $\mathbf Z$-subalgebras. Descend $X$, the bounded complex $F^\bullet$, its differentials, and $P$ to some Noetherian stage. Increase the stage so that the terms of $F^\bullet$ are relatively flat and have proper support. The finite-presentation and proper-support limit theorems justify this step; only finitely many terms and equations occur. Perfect complexes descend by their finite chart presentations and transition maps. Section 3 proves that the descended direct image is perfect. The arbitrary base-change comparison just proved identifies its pullback with the original direct image. A pullback of a perfect complex is perfect. This proves Bridge 2 over arbitrary algebraic-space bases.

For a pseudo-coherent source and a bounded-above relatively flat target, approximate the source sufficiently far left, then truncate the target sufficiently far left. The cohomological-dimension bound makes both replacements isomorphisms in any chosen high cohomological range after pushforward. The resulting direct image is perfect by the preceding result. This proves pseudo-coherence of the original direct image one degree range at a time.

A.5. Recovering local information from perfect tests

For later use, perfect tests also detect relative pseudo-coherence. Suppose $K\in D^-_{\mathrm{QCoh}}(Y)$ and that $R\Gamma(Y,P\otimes^{\mathbf L}K)$ is pseudo-coherent over $A$ for every perfect $P$. Given a pseudo-coherent $E$, approximate $E$ far enough left. If $K\in D^{\leq b}$ and cohomological dimension is at most $N$, an approximation whose cone is in degrees $<m-N-1-b$ changes the tensor direct image only below degree $m$. Thus the perfect-test hypothesis implies the same hypothesis for every pseudo-coherent $E$.

Fix $y\in Y$. Derived Chow supplies a closed subspace

an open $V\subset\mathbf P^d_A$, and a pseudo-coherent $E$ supported on $Z$, such that $W=Z\cap(Y\times_AV)\to Y$ is étale and meets $y$, $W\to V$ is a closed immersion, and $E|_{Y\times_AV}=\mathcal O_W$ pushed forward.

Here is its construction and the return from a Noetherian model. Descend $Y/A$ to a separated finitely presented $Y'/A'$ with $A'$ finitely generated over $\mathbf Z$. Choose an affine étale chart of $Y'$ through the image of $y$, embed that chart in affine space, and take the scheme-theoretic closure of its graph in $Y'\times\mathbf P^d_{A'}$. Separatedness makes the closure a graph over a suitable open of the projective closure of the chart. Over this open it is the original étale chart. The coherent structure sheaf of that closed graph is pseudo-coherent over the Noetherian model. Pull it back derivedly to $A$. The restriction is still the ordinary structure sheaf of the base-changed graph because the chart is flat over $Y'$ and projective space is flat over $A'$; these give the needed Tor independence. Choose a point in the base-changed chart over $y$ by surjectivity on the relevant fibre. This point need not be unique.

Let $p,q$ be the projections of $Y\times\mathbf P^d_A$. For every twist $j$, the projection formula gives

The complex inside the right tensor is pseudo-coherent because $p$ is flat, proper and finitely presented and the pseudo-coherent direct-image result of Section 4 applies. The right side is therefore pseudo-coherent by the hypothesis. The projective-space test for pseudo-coherence, proved immediately below by the finite Koszul resolution of the diagonal, now says $Rq_*(Lp^*K\otimes^{\mathbf L}E)$ is pseudo-coherent. Over $V$ it is the pushforward of $K|_W$ by the closed immersion $W\to V$. Hence $K|_W$ is pseudo-coherent relative to $A$. The étale charts $W\to Y$ so obtained prove Bridge 3.

Here is a complete proof of that projective-space test over an arbitrary ring; it needs no equivalence with a differential graded module category. Put $P=\mathbf P^d_A$. If $M\in D_{\mathrm{QCoh}}(P)$ and $R\Gamma(P,M(j))$ is pseudo-coherent over $A$ for $j=0,\ldots,d$, then $M$ is pseudo-coherent on $P$. On $P\times_AP$, the tautological inclusion on the first factor followed by the tautological quotient on the second gives a section of the rank-$d$ vector bundle

Its zero scheme is the diagonal. On a chart meeting the diagonal its equations are $x_i-y_i$, $i=1,\ldots,d$. They form a regular sequence over every $A$: eliminate one variable at a time, each by a monic linear equation. Away from the diagonal one equation is a unit locally. Thus its Koszul complex is a finite locally free resolution of $\mathcal O_\Delta$, with terms

Tensor this resolution by $p_1^*\mathcal O(d)\otimes p_2^*\mathcal O(-d)$. Its restriction to the diagonal is canonically trivial, so it still resolves $\mathcal O_\Delta$. After derived tensor with $Lp_1^*M$ and applying $Rp_{2,*}$, its term of Koszul degree $-i$ is

The comparison follows by flat base change along $P/A$ and projection formula. These formulas hold even for unbounded $M$: the finite standard affine cover of $P$, with its affine intersections, computes the comparison by a finite Čech total complex, and the Koszul complex is finite and locally free.

Each displayed term is pseudo-coherent by the assumption and tensoring with a vector bundle. The finite Koszul filtration expresses the pushed-forward total complex as finitely many cones of these terms; pseudo-coherence is closed under these cones and shifts. That total complex is $M$: derived tensor with $\mathcal O_\Delta$ restricts $Lp_1^*M$ to the diagonal, where $p_1$ is the identity, and $p_2\circ\Delta$ is the identity as well. This proves the test. For $d=0$ the resolution has one term and the assertion is tautological. Applied above, its hypotheses hold for all twists and therefore in particular for $0,\ldots,d$.

One further observation gives access to any affine étale chart $U\to Y$. There are perfect $E_m$ on $Y$ such that

for every $K\in D_{\mathrm{QCoh}}(Y)$. To construct them, descend $U\to Y$ to a separated finite-type $\mathbf Z$-model $U'\to Y'$. The model ring and all affine étale chart rings are countable. Put $M=R(U'\to Y')_*\mathcal O_{U'}$. Its cohomology has countable sections on affine étale charts: after base change to such a chart, use finite cohomological dimension and an elementary distinguished-square induction; affine terms are countable modules, and kernels, cokernels and finite extensions of countable groups are countable.

Choose a perfect generator $G$ of $D_{\mathrm{QCoh}}(Y')$, whose full algebraic-space construction is incorporated below. Perfect objects are compact: derived Hom from one commutes with arbitrary direct sums, as follows for affines from a finite projective complex and for spaces by the finite Mayer–Vietoris induction. The same induction shows that every $\operatorname{Hom}(G[t],M)$, and every $\operatorname{Hom}(G[t],E)$ for perfect $E$, is countable. Their union over $t\in\mathbf Z$ is therefore countable.

Here is the complete sequential construction, avoiding any differential graded equivalence. Begin with $E_0=0$ and $a_0:E_0\to M$. Enumerate all classes in all the groups $\operatorname{Hom}(G[t],M)$. Given $a_n:E_n\to M$, enumerate the kernels of $\operatorname{Hom}(G[t],E_n)\to\operatorname{Hom}(G[t],M)$, for all $t$. At stage $n+1$, add the first $n+1$ target classes by replacing the current perfect object with its direct sum with the corresponding finite number of shifts of $G$, mapping by those classes. Next, for each $m\leq n$, kill the first $n+1$ enumerated kernel classes from stage $m$, viewed in the current object via its transition map. To kill $b:G[t]\to E$ with $ab=0$, use

Exactness of $\operatorname{Hom}(-,M)$ for this triangle extends $a$ to a map $\operatorname{Cone}(b)\to M$. A finite number of these operations yields a perfect $E_{n+1}$ and a map extending $a_n$. The enumerations are set-theoretic choices; no functorial cell selection is required.

Every target class is eventually added, and every kernel class arising at any fixed stage is eventually killed. Compactness of $G$ gives

For clarity, the equality follows by applying $\operatorname{Hom}(G[t],-)$ to the telescope triangle; the map $1-\mathrm{shift}$ is injective on the direct sums of the preceding Hom groups, so its cokernel is the displayed colimit. The cone of $\operatorname{hocolim}E_n\to M$ has zero Hom from every shift of the generator, hence is zero. Thus $M=\operatorname{hocolim}E_n$.

Pull this sequence back to $A$. The canonical arbitrary-base comparison for the flat finitely presented morphism $U'\to Y'$, computed on affine pieces and then by the finite Mayer–Vietoris induction, identifies its colimit with $R(U\to Y)_*\mathcal O_U$. Derived pullback preserves perfect objects and homotopy colimits. The projection formula now proves the displayed sequential perfect-test formula. The incorporated native countable lemma gives an alternative proof through its differential graded equivalence, but that equivalence and its cell-construction lemma are not prerequisites of this reader proof.

For Bridge 4, stabilize finite-free pseudo-coherent resolutions of the compatible perfect complexes $C_n$. The stabilization lemma lifts a bounded-above free resolution at level $n$ to one at $n+1$, adding contractible pairs if necessary, with an actual isomorphism after reduction. Thus their degreewise inverse limit is a bounded-above finite-free complex $C$ over $A$, and $C\otimes_A A_n=C_n$ as complexes. Surjectivity of the degreewise maps computes its derived limit. This proves pseudo-coherence and every derived base-change comparison. Every maximal ideal of $A$ contains $I_1$, so the residue-field complex of $C$ is the corresponding residue-field complex of perfect $C_1$, whose Tor amplitude has a uniform finite bound. The fibrewise perfectness criterion for a pseudo-coherent complex makes $C$ perfect. These statements include arbitrary locally nilpotent kernels; a fixed nilpotence exponent is not assumed.

A.6. Strong coherent existence

View each $G_n$ as a module on $Y$ by $i_{n,*}$, and form

Here $DQ_Y$ is the right adjoint to inclusion of complexes with quasi-coherent cohomology. Its construction by elementary distinguished squares, its finite boundedness property, and the ordinary derived-limit calculation on affine étale charts show $K\in D^b_{\mathrm{QCoh}}(Y)$ and $K\in D^{\geq0}$. The sheaf derived limit has hypercohomology on any affine étale chart only in degrees zero and one; the boundedness of $DQ_Y$ increases only the upper bound. We do not identify $K$ with the naive sheaf inverse limit.

For a perfect $P$ on $Y$, put

Bridge 2 makes $C_n$ perfect over $A_n$, and identifies its derived transition map with the given reduction of $G_n$. Bridge 4 makes $C=\operatorname{holim}C_n$ perfect over $A$, with $C\otimes_A^{\mathbf L}A_n=C_n$. Derived Hom from $P$, regarded as a right adjoint on the derived quasi-coherent category, commutes with its products and homotopy limits. Hence

Duality for perfect objects gives the same statement for $R\Gamma(Y,K\otimes^{\mathbf L}P)$. Section 5 proves that $K$ is pseudo-coherent relative to $A$.

For an affine étale $U\to Y$, its sequential perfect-test formula and commutation of derived tensor with homotopy colimits give the canonical comparison

The comparison is induced by the limit projections, so its composites at consecutive levels are exactly the maps coming from $\phi_n$.

We next make $K$ a finitely presented sheaf near the common closed fibre. Let $u\in U$ lie over $V(I_1)$. Choose a finite-presentation smooth algebra $P$ surjecting onto the affine chart algebra $B$. Relative pseudo-coherence represents $K|_U$, pushed to $\operatorname{Spec}P$, by a bounded-above finite-free $P$-complex $F^\bullet$. Its derived reduction to $A_n$ has cohomology only in degree zero, equal to the module of $G_n|_{U_n}$. Since $P$ is $A$-flat, ordinary tensor of this complex computes that reduction.

If its largest nonzero term has degree $t>0$, the last differential is surjective on the residue-field fibre at $u$. A maximal minor is invertible on a neighbourhood of $u$. Split off this surjective finite-free pair, replacing the complex by its complement. Repeat until the largest degree is at most zero. Since $K$ has no negative cohomology, on this neighbourhood it is precisely $H^0(K)$. The cokernel of the last retained finite-free differential is finitely presented over $P$ and, as $B$ acts on it, finitely presented over the quotient $B$. Tensor right exactness identifies every ordinary reduction with $G_n$. These neighbourhoods descend through the open étale chart maps. Their union is an open $W\subset Y$ containing every $Y_n$, on which

with $F$ finitely presented and marked reductions $G_n$. The space $Y_1$ is quasi-compact, and all the $Y_n$ have its underlying closed-fibre set. Choose finitely many quasi-compact open neighbourhoods within this union which cover that set, and replace $W$ by their union. Now $W$ is a quasi-compact open in the finitely presented $Y/A$, so $W/A$ is finitely presented. This supplies the finite-type hypothesis for the proper-part theorem.

Use the Fitting closed support $Z=V(\operatorname{Fitt}_0F)$ in $W$. Its base change to $A_1$ is proper by Section 2. The henselian proper-part theorem for separated finite-type algebraic spaces splits

with $Z_{\mathrm{pr}}$ proper over $A$, and $Z_{\mathrm{away}}$ disjoint from the common closed fibre. Its proof reduces by the weak Chow cover to a quasi-projective scheme, then to its projective closure; general Stein factorization and henselian lifting of idempotents split the projective closed fibre. The unwanted boundary has proper closed image missing $V(I_1)$, so is empty. Under the proper cover, the two images are closed; their intersection has proper closed image missing $V(I_1)$, hence is empty. They therefore descend to complementary open-and-closed subspaces. This proves the algebraic-space proper-part extension. Remove $Z_{\mathrm{away}}$ from $W$. The support of $F$ is now proper over $A$.

It remains to establish flatness; finite presentation and the marked ordinary reductions alone have not established it. Proper support makes $F$ universally pure over $A$: the closure of an associated fibre point has proper closed image, so it meets every specialization of its base point. The pure universal-flattening theorem gives a monomorphism of finite presentation

whose pullback tests are exactly those on which $F$ is relatively flat. Every $\operatorname{Spec}A_n$ factors through $T$, because $F|_{Y_n}=G_n$ is flat.

The elementary monomorphism lemma proves $T=\operatorname{Spec}A$. Here are its details. Take a quasi-compact open of $T$ containing the common image of the $\operatorname{Spec}A_n$. It is separated and quasi-finite over $\operatorname{Spec}A$, hence embeds openly into a finite $A$-scheme $\operatorname{Spec}B$ by Zariski main. The unique compatible maps $B\to A_n$ define $B\to A$. Thus its section $\operatorname{Spec}A\to\operatorname{Spec}B$ pulls this open back to an open of $\operatorname{Spec}A$ containing $V(I_1)$. Every closed point lies in $V(I_1)$; a closed complement would have a closed point, so that open is all of $\operatorname{Spec}A$. The section factors through $T$. A monomorphism admitting a section is an isomorphism. Hence $F$ is $A$-flat.

Finally its proper support $Z_{\mathrm{pr}}\to Y$ is closed: a proper map to $A$ mapping to a separated $A$-space is proper, and this map is an immersion. Extend $F$ by zero along $W\to Y$, equivalently push it from that closed support. In a neighbourhood outside the support it is zero; on $W$ it is finitely presented. Thus the resulting $G$ is finitely presented on all of $Y$, remains $A$-flat, has proper support, and retains every marked comparison with $G_n$. This proves the theorem.

The exact universal-flattening proof is incorporated below at algebraic-space scope, including its reduction to complete dévissage. Its transitive scheme dévissage inputs are explicitly retained in Section 11; their mere source locators do not certify programme closure.

For the coherent-stack use in Lesson 8, the towers in Lesson 7's strong-effectivity criterion have square-zero kernels for every composite transition, so they satisfy the present hypotheses. The limit family $Y=X_A$ need not be flat. The theorem gives the required marked object, and the existing diagonal, limit and (RS*) proofs then give openness of versality for the nonflat coherent stack.

A.7. Polarized proper schemes

Let $\mathcal P$ be the stack whose objects over a scheme $T$ are proper flat finitely presented schemes $X/T$ and a relatively ample invertible module $L$. An arrow over $T'\to T$ includes a cartesian isomorphism $X'\cong X\times_TT'$ and an actual isomorphism between the pulled-back polarization and $L'$. Scalar automorphisms are retained.

Descent is effective. Descend the proper flat finitely presented algebraic space and the invertible module separately, with their cocycles. Relative ampleness descends. An algebraic space with a relatively ample invertible module over a scheme is a scheme: its standard opens defined by sufficiently positive sections are affine over affine base charts and cover it, and the sheaf identifications give their open immersions. Thus the descended object has exactly the scheme condition required in the definition.

The diagonal is representable. For two pairs $(X,L),(Y,N)$, first form the space of scheme isomorphisms by imposing the two inverse equations in

These morphism spaces are the graph spaces of Lesson 8, Corollary 6.4. Over its universal isomorphism $a:X\to Y$, the additional datum is $\operatorname{Isom}(L,a^*N)$, affine of finite presentation by Lesson 8, Corollary 1.3. Its composite over $T$ represents the polarized Isom functor.

Finite presentations of spaces, line bundles and arrows descend along filtered ring colimits. Properness, flatness and ampleness descend after increasing the index. Descending an arrow together with its inverse, then its two inverse equalities, proves full groupoid limit preservation.

The (RS*) check retains the bundle and its arrow. Flat locally finitely presented schemes patch over an affine pushout along a nilpotent thickening. On affine charts their structure algebras are the flat fibre-product algebras; charts and their intersections glue by full faithfulness. Relatively flat modules patch by the fibre-product module argument of Lesson 8, Section 3. A pair of line bundles therefore patches to a finitely presented relatively flat module whose fibres are free of rank one, so the local free-locus lemma makes it invertible. Properness and ampleness are recovered across the nilpotent thickening. This gives the entire fibre-product groupoid, including the specified identification on the overlap.

For the finite tangent check there is a shorter route than the graded-deformation argument with omitted details in the source. Deformations of proper finite-type $X/k$ have tangent $\operatorname{Ext}^1(\mathrm{NL}_{X/k},\mathcal O_X)$ and infinitesimal automorphisms $\operatorname{Ext}^0(\mathrm{NL}_{X/k},\mathcal O_X)$. The naive cotangent complex has coherent cohomology only in degrees $-1,0$, so coherent proper cohomology and the local-to-global Ext spectral sequence make both spaces finite. Forgetting $L$ induces a linear map from the pair tangent to this finite tangent. Its kernel is a quotient of the tangent for line bundles on fixed $X$, namely $H^1(X,\mathcal O_X)$: any deformation in the kernel is isomorphic, after a marked trivialization of $X$, to a deformation of $L$ on the trivial family. Similarly the kernel on infinitesimal automorphisms is $H^0(X,\mathcal O_X)$. Both are finite. The images are subspaces of finite spaces, so the pair tangent and automorphism spaces are finite. The marking changes act by the relevant infinitesimal automorphisms; no unjustified identification of the kernel with the entire $H^1$ is asserted.

Here is strong effectivity at full non-Noetherian scope. Let $R=\varprojlim R_n$ be any surjective locally nilpotent tower and let $(X_n,L_n)$ be a compatible system of objects. Choose one integer $d_0>0$ such that the sufficiently positive section algebra of $(X_1,L_1)$ has vanishing positive cohomology and finite projective degree pieces. The uniform projective-deformation lemma, proved in the appendix, applies to every thickening $R_n\to R_1$, with the same $d_0$. It gives

a finitely presented graded $R_n$-algebra with finite projective degree pieces, compatible with every transition, and canonical isomorphisms

Set $Q_d=\varprojlim_n(Q_n)_d$ and $Q=\bigoplus_dQ_d$. Lifting finite projectives over the henselian pair $(R,\ker(R\to R_1))$, followed by successive lifts of module maps, proves $Q_d$ finite projective and $Q_d\otimes_RR_n=(Q_n)_d$.

We prove finite presentation of the whole algebra; it does not follow merely from finite projectivity of its separate pieces. Choose a finite homogeneous presentation

with all $x_i$ of positive degree. Lift these homogeneous generators into $Q$, giving $\Psi:R[x_1,\ldots,x_s]\to Q$. In each degree its cokernel is finite and has zero reduction modulo the Jacobson ideal $\ker(R\to R_1)$, so Nakayama makes $\Psi$ surjective degreewise. Each homogeneous polynomial degree piece is finite free and $Q_d$ is projective, so $(\ker\Psi)_d$ is finite projective and reduction is exact. Thus lift each $f_j$ to a homogeneous $g_j\in\ker\Psi$. For each degree the quotient of $(\ker\Psi)_d$ by the degree-$d$ part of the ideal $(g_1,\ldots,g_t)$ is finite and has zero reduction. Nakayama kills it. Therefore

is finitely presented. The empty family gives $Q=R$ and $\operatorname{Proj}Q=\varnothing$; it causes no exception.

Take $X=\operatorname{Proj}Q$, $L=\mathcal O_X(1)$. Localizing the graded algebra and taking degree-zero components shows $X$ flat over $R$; the same calculation makes $L$ relatively flat. Finite homogeneous presentation makes $X/R$ proper and finitely presented and makes $L$ finitely presented. Proj and its twisting module commute with base change, giving exactly the marked $X_n,L_n$. The fibrewise free-locus lemma makes $L$ invertible near $X_1$. Its complement has proper closed image in $\operatorname{Spec}R$ missing all closed points, so is empty. The relative ample locus is open, contains $\operatorname{Spec}R_1$, and similarly is the whole base. Thus $(X,L)$ is the required compatible polarized family.

This strong effectivity yields ordinary formal effectivity over complete Noetherian local rings and yields openness of versality through Lesson 7's strong-effectivity criterion. We have checked small fibres (finite chart descriptions in the chosen universe), descent, limits, the diagonal, (RS*), finite tangent and automorphism spaces, ordinary effectivity, and openness. Finite-type $\mathbf Z$-algebras are G-rings. Lesson 7's representable-diagonal Artin theorem therefore proves that $\mathcal P$ is algebraic. This proves Tag 0D4X, at the scope of the definition corresponding to 0D1L.

For any algebraic space $B$, the stack adding a parameter map $T\to B$ is $\mathcal P\times_{\operatorname{Spec}\mathbf Z}B$. Pull back its smooth atlas and diagonal. This proves algebraicity over arbitrary $B$, without an added Noetherian or G-ring hypothesis on $B$.

A.8. All proper flat curve spaces of dimension at most one

Let $\mathcal C_{\leq1}$ have objects proper flat finitely presented algebraic spaces $X/T$, $T$ a scheme, with every fibre of dimension at most one; arrows are cartesian isomorphisms. Fibres may be empty, zero-dimensional, disconnected, reducible or nonreduced. No smoothness, nodality, stability, polarization or geometric connectedness is imposed.

First consider the stack $\mathcal S$ of all proper flat finitely presented spaces. This stack has effective fppf descent by the algebraic-space descent theorem of Lessons 1–2. Its diagonal is the Isom space constructed from the two inverse equations in the morphism spaces of Lesson 8, Corollary 6.4. Finite-presentation descent and descent of properness and flatness give full groupoid limit preservation. Flat algebraic spaces patch across affine pushouts along nilpotent thickenings, giving (RS*); its source proof is included below. Properness is recovered across the thickening. The finite tangent and infinitesimal-automorphism calculation is the naive cotangent calculation from Section 7, now on the small étale site of $X$, with proper coherent cohomology from Section 3.

No algebraicity or ordinary effectivity assertion for $\mathcal S$ is needed. Openness of versality follows from its much more limited square-zero-tower effectivity. Suppose $R_n\to R_m$ has square-zero kernel for every $n\geq m$, and $X_n/R_n$ is a compatible system. Put $I_n=\ker(R_n\to R_1)$, $R=\varprojlim R_n$, $I=\varprojlim I_n$; then $I^2=0$. On the common small étale site of $X_1$ there are marked square-zero extensions

The left modules have surjective transition maps. Their sheaf inverse limit has no higher derived-limit cohomology on affine étale charts. Moreover

becomes an isomorphism under $DQ_{X_1}$. To verify this, test against a perfect object and use the proper flat perfect direct-image bridge of Section 4; its finite-projective complex tensor-commutes with this surjective module limit. A perfect generator then detects the comparison.

The exact square-zero-extension comparison lemma, proved in the appendix, now yields an extension of sheaves of $R$-algebras

which maps, with all markings, to each preceding extension. The construction is an extension in the ringed étale topos, not an assertion that the ordinary inverse limit of chart algebras is quasi-coherent. A square-zero ringed-topos thickening with this quasi-coherent ideal comes from a unique algebraic-space thickening $X_1\subset X$. The square-zero flatness criterion gives $X/R$ flat. Finite presentation and properness lift over this square-zero thickening. Base change of the marked extension to each $R_n$ recovers $X_n$, by the extension equivalence. Thus $\mathcal S$ satisfies the strong-effectivity condition used solely for openness.

The condition on fibre dimension cuts out an open substack: for a proper finitely presented morphism, the set of points where the fibre dimension is at most one is open, and formation of this open is preserved and reflected by every residue-field extension and hence every base change. Equivalently, the closed locus where fibre dimension is at least two has proper closed image. Pullback of $\mathcal C_{\leq1}\subset\mathcal S$ by a family is exactly this open immersion. Consequently its diagonal, descent, (RS*) and infinitesimal tangent comparisons are those for $\mathcal S$. Dimension descends along filtered limits after increasing the index; together with the space-limit argument this proves full limit preservation for $\mathcal C_{\leq1}$. Since its inclusion is open, a versal neighbourhood at a point in this substack can be intersected with that open, so openness of versality for $\mathcal S$ restricts to it.

Ordinary formal effectivity is a separate argument. Let $R$ be complete Noetherian local, with maximal ideal $\mathfrak m$, and let $X_n/(R/\mathfrak m^n)$ be a compatible proper flat finitely presented family with special fibre dimension at most one. A proper algebraic space of dimension at most one over a field is a scheme; a proper scheme of that dimension is projective, including its nonreduced and disconnected cases. The exact source proofs and their scope are retained below. To cover the endpoint carefully, a zero-dimensional proper finite-type field scheme is quasi-finite and proper, hence finite; its Artinian affine structure has ample trivial bundle. If one-dimensional and isolated zero-dimensional components coexist, the latter form an open-and-closed Artinian part, and the finite decomposition glues the ample bundles. The reduced one-dimensional part has ample bundles on its integral components; choose positive powers and sections nonvanishing at their finite intersections, glue their trivializations there, and use the finite affine closed-cover lemma to obtain an affine section-open around the intersections. Away from them, component sections extend through the complementary component ideal. These opens prove ampleness. Lift the bundle through the nilpotent reduction using $H^2=0$, and use invariance of ampleness under nilpotent thickening. This is the full reducible/nonreduced route in the incorporated proof chain; no normalization or smooth-curve argument is substituted.

Choose an ample invertible module $L_1$ on $X_1$. Nilpotent thickenings of a scheme in algebraic spaces are schemes, so all $X_n$ are schemes.

Lift $L_n$ recursively through

The ideal in the left term is square-zero, so multiplication identifies that sheaf with its additive quasi-coherent module. Its underlying Noetherian topological space has dimension at most one; coherent cohomological dimension gives $H^2=0$. Therefore the obstruction vanishes, and there is an invertible $L_{n+1}$ together with a specified isomorphism to $L_n$ on restriction. Ampleness is invariant under a nilpotent thickening, so every $L_n$ is ample. Apply the proved polarized strong-effectivity theorem in Section 7 to the tower $R/\mathfrak m^n$. It gives a proper flat finitely presented scheme $X/R$ with exactly the marked formal reductions $X_n$.

All fibres of this algebraization have dimension at most one. For a proper finitely presented morphism the locus of fibre dimension at least two is closed. Any nonempty closed subset of the spectrum of a local ring contains its closed point; here that point has dimension at most one. The locus is therefore empty. This verifies the exact curve-stack object conditions and proves ordinary formal effectivity. Choosing an ample bundle is part of the proof only; the final object of the unpolarized groupoid is $X$ with its original markings.

The full Artin check now gives algebraicity of $\mathcal C_{\leq1}$: small fibres, effective descent, representable diagonal, full limits, (RS*), finite tangent and automorphism spaces, the just-proved ordinary effectivity, and the square-zero argument giving openness of versality. The G-ring base is again $\operatorname{Spec}\mathbf Z$. This proves Tag 0D5A, with definition 0D4Y. Its product with any algebraic space $B$ is algebraic by base change of the atlas and diagonal. A scheme-only total-space moduli problem would have been a different statement.

A.9. The order of the moduli arguments

The Hom and Isom constructions precede coherent-sheaf algebraicity. Quot and Hilbert then represent graphs, giving the diagonal needed for the polarized and curve-space stacks. Ordinary adic existence, strong existence for arbitrary locally nilpotent towers, and the flat-case low-Ext calculation each retain their own hypotheses and proofs. Fixed Hilbert-polynomial components and stable or smooth curve loci are further moduli problems; the two stacks proved here retain their full stated definitions.

A.10. Support, base change and graded relations

• The strong-existence proper-support step uses Fitting support. Annihilator supports need not commute with nonflat base change. Only the Fitting closed support equality, and the properness equivalence with annihilator support, are used.
• The printed formal dimension-at-most-one algebraization lemma omits properness in its statement although its proof uses a proper special fibre. The lemma as printed is false: the constant tower of affine lines over a field is not the reduction of a projective scheme. Here the statement has the properness hypotheses of the actual curve-stack consumer, and the proof is supplied through Section 7.
• The arbitrary direct-image corollary in spaces-perfect.tex labels its module “flat over S” although the cited theorem needs flatness over Y. Here flatness is over the direct-image target throughout; the broader printed corollary is not used.
• In derived Chow, the chosen point over y is not asserted unique. The projective direct image restricts to V, not to Y times V; the comparison uses the graph's structure-sheaf property with its correctly typed source.
• In the support-preserving perfect lift, successive cones use g_1 through g_s, each once. Starting with P plus P[1] yields the corresponding binomial multiplicities with s+1 factors; the proof only needs a distinguished P summand. Taking the projector cone then gives exactly P plus P[1].
• The source pushout category uses the fibre-product category over X, and its flat-module statement requires the second module to be flat over X prime. The printed category over Y prime and “flat over X” are type errors. The reader proof uses the correctly typed affine patching equivalence.
• The polarized finite-presentation argument explicitly uses projectivity of every graded piece to lift relations and apply Nakayama degreewise. Right exactness alone would not identify the polynomial presentation kernel after reduction.
• The source's graded tangent equivalence contains omitted details. The complete tangent finiteness argument in Section 7 uses the marked forgetful tangent map, its H^1 and H^0 kernels, and the naive cotangent finite-space calculation instead.

A.11. Scheme foundations and algebraic-space extensions

The three target theorem proofs have been supplied at their full local mathematical scope, conditional on the explicitly identified foundational proof material below. Incorporation of the supporting native proofs is local evidence, not a certificate of recursive closure for every external label in those proofs.

The following actual AG-QC programme bodies were inspected in the current edition, not merely read from a course title:

The current Noetherian scheme complex is a genuine written provider. Section 3 supplies the space extension; Section 4 supplies the arbitrary-base extension. The projective adic scheme theorem is genuine written programme mathematics, but it is not the arbitrary sequence theorem proved in Section 6.

The algebraic-space pure-flattening argument, complete dévissage, generic presentations, support-preserving perfect lifting, perfect generators, and square-zero ringed-topos comparisons are supplied in the following proofs. The sequential perfect-test construction in Section 5 is an independent route; its complete differential graded alternative is retained with its own prerequisites. The diagonal Koszul resolution gives the projective-space pseudo-coherence test without requiring that alternative.

Lesson 3, Section 5.6.7, C.1, supplies the actually written finite flat-module model theorem at arbitrary filtered-system scope. Its blowup-flattening theorem is a different assertion from universal pure-flattening representability and is not substituted for it. Existing Lessons 1–7 and the genuine AG-QC scheme proofs retain their recorded hypotheses and lower dependencies. The separate Artin reader supplies desingularization and polynomial approximation. The uncovered prerequisite record below preserves the remaining exact lower obligations.

A.12. The supporting cohomological and deformation constructions

These are the supporting mathematical statements and their complete proofs. The independent reader proofs in Sections 3–8 supply the scope extensions, repaired statements and two alternate constructions. The converted proofs retain the exact statement scope and the explicit prerequisites. The known statement and map corrections have been applied to these reader proofs. The unchanged human arguments retain their original mathematical scope and source credit.

The projective-space pseudo-coherence criterion and sequential approximation in Section 5 are complete reader alternatives. The native countable proof and the second quasi-coherator proof are retained as sound optional alternatives; their additional unbound prerequisites are not dependencies of the selected reader routes. The source proper-support lemma is omitted because its nonflat annihilator-base-change claim requires the Fitting replacement proved in Sections 2 and 6. The printed nonproper formal algebraization statement is also omitted; Section 8 proves its actual proper consumer.

A.12.1 Henselian limits and compatible finite complexes

The following arguments adapt the human Stacks Project treatment in more-algebra.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. Henselian pairs from locally nilpotent inverse systems

Let $A = \varprojlim A_n$ where $(A_n)$ is an inverse system of rings whose transition maps are surjective and have locally nilpotent kernels. Then $(A, I_n)$ is a henselian pair, where $I_n = \operatorname{Ker}(A \to A_n)$.

Proof. Fix $n$. Let $a \in A$ be an element which maps to $1$ in $A_n$. By Algebra, Lemma Nilpotent thickenings and local algebra we see that $a$ maps to a unit in $A_m$ for all $m \geq n$. Hence $a$ is a unit in $A$. Thus by Algebra, Lemma Containment in the Jacobson radical the ideal $I_n$ is contained in the Jacobson radical of $A$. Let $f \in A[T]$ be a monic polynomial and let $\overline{f} = g_nh_n$ be a factorization of $\overline{f} = f \bmod I_n$ with $g_n, h_n \in A_n[T]$ monic generating the unit ideal in $A_n[T]$. By Lemma Henselian rings and nilpotent thickenings we can successively lift this factorization to $f \bmod I_m = g_m h_m$ with $g_m, h_m$ monic in $A_m[T]$ for all $m \geq n$. At each step we have to verify that our lifts $g_m, h_m$ generate the unit ideal in $A_n[T]$; this follows from the corresponding fact for $g_n, h_n$ and the fact that $\operatorname{Spec}(A_n[T]) = \operatorname{Spec}(A_m[T])$ because the kernel of $A_m \to A_n$ is locally nilpotent. As $A = \varprojlim A_m$ this finishes the proof. $\square$

Lemma. Compatible finite projectives over an inverse limit

Let $A = \varprojlim A_n$ be a limit of an inverse system $(A_n)$ of rings. Suppose given $A_n$-modules $M_n$ and $A_{n + 1}$-module maps $M_{n + 1} \to M_n$. Assume

1. the transition maps $A_{n + 1} \to A_n$ are surjective with locally nilpotent kernels,

2. $M_1$ is a finite projective $A_1$-module,

3. $M_n$ is a finite flat $A_n$-module, and

4. the maps induce isomorphisms $M_{n + 1} \otimes_{A_{n + 1}} A_n \to M_n$.

Then $M = \varprojlim M_n$ is a finite projective $A$-module and $M \otimes_A A_n \to M_n$ is an isomorphism for all $n$.

Proof. By Lemma Henselian pairs from locally nilpotent inverse systems the pair $(A, \operatorname{Ker}(A \to A_1))$ is henselian. By Lemma Lifting projective, locally free modules and finite algebras we can choose a finite projective $A$-module $P$ and an isomorphism $P \otimes_A A_1 \to M_1$. Since $P$ is projective, we can successively lift the $A$-module map $P \to M_1$ to $A$-module maps $P \to M_2$, $P \to M_3$, and so on. Thus we obtain a map $$P \longrightarrow M$$ Since $P$ is finite projective, we can write $A^{\oplus m} = P \oplus Q$ for some $m \geq 0$ and $A$-module $Q$. Since $A = \varprojlim A_n$ we conclude that $P = \varprojlim P \otimes_A A_n$. Hence, in order to show that the displayed $A$-module map is an isomorphism, it suffices to show that the maps $P \otimes_A A_n \to M_n$ are isomorphisms. From Lemma Lifting projective and locally free modules we see that $M_n$ is a finite projective module. By Lemma Projective, locally free modules and finite algebras the maps $P \otimes_A A_n \to M_n$ are isomorphisms. $\square$

Lemma. Pseudo-coherence under a nilpotent reduction

Let $R' \to R$ be a surjective ring map whose kernel is a nilpotent ideal. Let $K' \in D(R')$ and set $K = K' \otimes_{R'}^\mathbf{L} R$. Then $K$ is pseudo-coherent if and only if $K'$ is pseudo-coherent.

Proof. One direction follows from Lemma Pullback of pseudo-coherent complexes and coherent sheaves. For the other direction, assume $K$ is pseudo-coherent. Then by Lemma Pseudo-coherent complexes and coherent sheaves we can represent $K$ by a bounded above complex $E^\bullet$ of finite free $R$-modules. By Lemma Lifting projective, locally free modules and derived categories we can represent $K'$ by a bounded above complex $P^\bullet$ of projective $R'$-modules such that $P^n \otimes_{R'} R = E^n$. By Nakayama's lemma we see that $P^n$ is finite free and we conclude that $K'$ is pseudo-coherent as well. $\square$

Lemma. Lifting a complex after adjoining free summands

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $E^\bullet$ be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that

1. $E^\bullet$ is a bounded above complex of finite stably free $R/I$-modules,

2. $K \otimes_R^\mathbf{L} R/I$ is represented by $E^\bullet$ in $D(R/I)$,

3. $K^\bullet$ is pseudo-coherent, and

4. every element of $1 + I$ is invertible.

Then there exists a bounded above complex $P^\bullet$ of finite stably free $R$-modules representing $K$ in $D(R)$ such that $P^\bullet \otimes_R R/I$ is isomorphic to $E^\bullet$. Moreover, if $E^i$ is free, then $P^i$ is free.

Proof. We apply Lemma Lifting derived categories using the class $\mathcal{P}$ of all finite stably free $R$-modules. Property (1) of the lemma is immediate. Property (2) follows from Lemma Projective and locally free modules. Property (3) follows from Nakayama's lemma (Algebra, Lemma Nakayama's lemma). Property (4) follows from the fact that we can lift finite stably free $R/I$-modules to finite stably free $R$-modules, see Lemma Lifting projective and locally free modules. Part (5) holds because a pseudo-coherent complex can be represented by a bounded above complex of finite free $R$-modules. The final assertion of the lemma follows from Lemma Projective, locally free modules and finite algebras. $\square$

Lemma. Pseudo-coherent inverse limits

Let $A = \varprojlim A_n$ be a limit of an inverse system $(A_n)$ of rings. Suppose given $K_n \in D(A_n)$ and maps $K_{n + 1} \to K_n$ in $D(A_{n + 1})$. Assume

1. the transition maps $A_{n + 1} \to A_n$ are surjective with locally nilpotent kernels,

2. either all $K_n$ are pseudo-coherent or there exists an integer $n_0$ such that $K_{n_0}$ is pseudo-coherent and the kernels of $A_{n + 1} \to A_n$ are nilpotent ideals for $n \geq n_0$,

3. the maps induce isomorphisms $K_{n + 1} \otimes_{A_{n + 1}}^\mathbf{L} A_n \to K_n$.

Then $K = R\varprojlim K_n$ is a pseudo-coherent object of $D(A)$ and $K \otimes_A^\mathbf{L} A_n \to K_n$ is an isomorphism for all $n$.

Proof. By assumption we can find a bounded above complex of finite free $A_1$-modules $P_1^\bullet$ representing $K_1$, see Definition Pseudo-coherent complexes. By Lemma Pseudo-coherence under a nilpotent reduction we conclude that $K_n$ is pseudo-coherent for all $n$. Then by Lemma Lifting a complex after adjoining free summands we can, by induction on $n > 1$, find complexes $P_n^\bullet$ of finite free $A_n$-modules representing $K_n$ and maps $P_n^\bullet \to P_{n - 1}^\bullet$ representing the maps $K_n \to K_{n - 1}$ inducing isomorphisms (!) of complexes $P_n^\bullet \otimes_{A_n} A_{n - 1} \to P_{n - 1}^\bullet$. Thus $K = R\varprojlim K_n$ is represented by $P^\bullet = \varprojlim P_n^\bullet$, see Lemma Modules and Remark Uniqueness in the lifting construction. Since $P_n^i$ is a finite free $A_n$-module for each $n$ and $A = \varprojlim A_n$ we see that $P^i$ is finite free of the same rank as $P_1^i$ for each $i$. This means that $K$ is pseudo-coherent. It also follows that $K \otimes_A^\mathbf{L} A_n$ is represented by $P^\bullet \otimes_A A_n = P_n^\bullet$ which proves the final assertion. $\square$

Lemma. Perfect inverse limits

Source credit: the original source citation Bhatt-Algebraize (Lemma 4.2)

Let $A = \varprojlim A_n$ be a limit of an inverse system $(A_n)$ of rings. Suppose given $K_n \in D(A_n)$ and maps $K_{n + 1} \to K_n$ in $D(A_{n + 1})$. Assume

1. the transition maps $A_{n + 1} \to A_n$ are surjective with locally nilpotent kernels,

2. either all $K_n$ are perfect or there exists an integer $n_0$ such that $K_{n_0}$ is perfect and the kernels $A_{n + 1} \to A_n$ are nilpotent for $n \geq n_0$, and

3. the maps induce isomorphisms $K_{n + 1} \otimes_{A_{n + 1}}^\mathbf{L} A_n \to K_n$.

Then $K = R\varprojlim K_n$ is a perfect object of $D(A)$ and $K \otimes_A^\mathbf{L} A_n \to K_n$ is an isomorphism for all $n$.

Proof. We already know that $K$ is pseudo-coherent and that $K \otimes_A^\mathbf{L} A_n \to K_n$ is an isomorphism for all $n$ by Lemma Pseudo-coherent inverse limits. Consider a surjective map $A \to \kappa$ whose kernel is a maximal ideal $\mathfrak m$. Any element of $A$ which maps to a unit in $A_1$ is a unit in $A$ by Algebra, Lemma Nilpotent thickenings and local algebra and hence $\operatorname{Ker}(A \to A_1)$ is contained in the Jacobson radical of $A$ by Algebra, Lemma Containment in the Jacobson radical. Hence $A \to \kappa$ factors as $A \to A_1 \to \kappa$. Hence $$K \otimes_A^\mathbf{L} \kappa = K \otimes_A^\mathbf{L} A_1 \otimes_{A_1}^\mathbf{L} \kappa = K_1 \otimes_{A_1}^\mathbf{L} \kappa$$ Note that $K_1$ is perfect by assumption (2). Hence $K_1$ has finite tor dimension by Lemma Perfect complexes. Thus there exist $a, b \in \mathbf{Z}$ such that $H^i(K \otimes_A^\mathbf{L} \kappa) = 0$ for all $i \not \in [a, b]$. By Lemma Perfect complexes we conclude that $K$ is perfect. $\square$

Lemma. Modules and tensor products and direct sums

Let $A, A', B, B', C, C', D, D', I, M', M, N, \varphi$ be as in Lemma Modules and tensor products and direct sums. If $N$ finite over $D$ and $M'$ finite over $C'$, then $N' = N \times_{\varphi, M} M'$ is finite over $D'$.

Proof. Recall that $D' \to D \times_C C'$ is surjective by Lemma Modules and tensor products and direct sums. Observe that $N' = N \times_{\varphi, M} M'$ is a module over $D \times_C C'$. We can apply Lemma Modules and tensor products and direct sums to the data $C, C', D, D', IC', M', M, N, \varphi$ to see that $N' = N \times_{\varphi, M} M'$ is finite over $D \times_C C'$. Thus it is finite over $D'$. $\square$

Lemma. Relative flat modules over a ring fibre product

With $A, A', B, B', C, C', D, D', I$ as in Situation Modules and tensor products and direct sums.

1. Let $(N, M', \varphi)$ be an object of $\text{Mod}_D \times_{\text{Mod}_C} \text{Mod}_{C'}$. If $M'$ is flat over $A'$ and $N$ is flat over $B$, then $N' = N \times_{\varphi, M} M'$ is flat over $B'$.

2. If $L'$ is a $D'$-module flat over $B'$, then $L' = (L \otimes_{D'} D) \times_{(L \otimes_{D'} C)} (L \otimes_{D'} C')$.

3. The category of $D'$-modules flat over $B'$ is equivalent to the categories of objects $(N, M', \varphi)$ of $\text{Mod}_D \times_{\text{Mod}_C} \text{Mod}_{C'}$ with $N$ flat over $B$ and $M'$ flat over $A'$.

Proof. Part (1) follows from part (1) of Lemma Flatness and modules.

Part (2) follows from part (2) of Lemma Flatness and modules using that $L' \otimes_{D'} D = L' \otimes_{B'} B$, $L' \otimes_{D'} C' = L' \otimes_{B'} A'$, and $L' \otimes_{D'} C = L' \otimes_{B'} A$, see discussion in Situation Modules and tensor products and direct sums.

Part (3) is an immediate consequence of (1) and (2). $\square$

Lemma. Finite presentation under flat module patching

Let $A, A', B, B', C, C', D, D', I, M', M, N, \varphi$ be as in Lemma Modules and tensor products and direct sums. If

1. $N$ is finitely presented over $D$ and flat over $B$,

2. $M'$ finitely presented over $C'$ and flat over $A'$, and

3. the ring map $B' \to D'$ factors as $B' \to D'' \to D'$ with $B' \to D''$ flat and $D'' \to D'$ of finite presentation,

then $N' = N \times_M M'$ is finitely presented over $D'$.

Proof. Choose a surjection $D''' = D''[x_1, \ldots, x_n] \to D'$ with finitely generated kernel $J$. By Algebra, Lemma Finite presentation and finite algebras it suffices to show that $N'$ is finitely presented as a $D'''$-module. Moreover, $D''' \otimes_{B'} B \to D' \otimes_{B'} B = D$ and $D''' \otimes_{B'} A' \to D' \otimes_{B'} A' = C'$ are surjections whose kernels are generated by the image of $J$, hence $N$ is a finitely presented $D''' \otimes_{B'} B$-module and $M'$ is a finitely presented $D''' \otimes_{B'} A'$-module by Algebra, Lemma Finite presentation and finite algebras again. Thus we may replace $D'$ by $D'''$ and $D$ by $D''' \otimes_{B'} B$, etc. Since $D'''$ is flat over $B'$, it follows that we may assume that $B' \to D'$ is flat.

Assume $B' \to D'$ is flat. By Lemma Modules and tensor products and direct sums the module $N'$ is finite over $D'$. Choose a surjection $(D')^{\oplus n} \to N'$ with kernel $K'$. By base change we obtain maps $D^{\oplus n} \to N$, $(C')^{\oplus n} \to M'$, and $C^{\oplus n} \to M$ with kernels $K_D$, $K_{C'}$, and $K_C$. There is a canonical map $$K' \longrightarrow K_D \times_{K_C} K_{C'}$$ On the other hand, since $N' = N \times_M M'$ and $D' = D \times_C C'$ (by Lemma Flatness and modules; applied to the flat $B'$-module $D'$) there is also a canonical map $K_D \times_{K_C} K_{C'} \to K'$ inverse to the displayed arrow. Hence the displayed map is an isomorphism. By Algebra, Lemma Commutative algebra the modules $K_D$ and $K_{C'}$ are finite. We conclude from Lemma Modules and tensor products and direct sums that $K'$ is a finite $D'$-module provided that $K_D \to K_C$ and $K_{C'} \to K_C$ induce isomorphisms $K_D \otimes_B A = K_C = K_{C'} \otimes_{A'} A$. This is true because the flatness assumptions implies the sequences $$0 \to K_D \to D^{\oplus n} \to N \to 0 \quad\text{and}\quad 0 \to K_{C'} \to (C')^{\oplus n} \to M' \to 0$$ stay exact upon tensoring, see Algebra, Lemma Tor vanishing for a flat module. $\square$

A.12.2 Koszul representatives and support-preserving perfect approximation

The following arguments adapt the human Stacks Project treatment in perfect.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Situation. A complex and a finite principal-open cover

Here $A$ is a ring and $f_1, \ldots, f_r$ is a sequence of elements of $A$. We set $X = \operatorname{Spec}(A)$ and $U = D(f_1) \cup \ldots \cup D(f_r) \subset X$. We denote $\mathcal{U} : U = \bigcup_{i = 1, \ldots, r} D(f_i)$ the given open covering of $U$.

Lemma. The alternating Čech complex

In Situation A complex and a finite principal-open cover. Let $M$ be an $A$-module and denote $\mathcal{F}$ the associated $\mathcal{O}_X$-module. Then there is a canonical isomorphism of complexes $$\Psi : \mathop{\operatorname{colim}}_e \operatorname{Hom}_A(I^\bullet(f_1^e, \ldots, f_r^e), M) \longrightarrow \check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})$$ functorial in $M$ where the differentials on the $\operatorname{Hom}$-complex are the contragredients of the differentials on $I^\bullet(f_1^e, \ldots, f_r^e)$.

Proof. Recall that the alternating Čech complex is the subcomplex of the usual Čech complex given by alternating cochains, see Cohomology, Section Sheaf cohomology. As usual we view a $p$-cochain in $\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})$ as an alternating function $s$ on $\{1, \ldots, r\}^{p + 1}$ whose value $s_{i_0\ldots i_p}$ at $(i_0, \ldots, i_p)$ lies in $M_{f_{i_0}\ldots f_{i_p}} = \mathcal{F}(U_{i_0\ldots i_p})$. On the other hand, a $p$-cochain $t$ in $\operatorname{Hom}^\bullet(I^\bullet(f_1^e, \ldots, f_r^e), M)$ is a map $t : \wedge^{p + 1}(A^{\oplus r}) \to M$. Write $[i] \in A^{\oplus r}$ for the $i$th basis element and write $$[i_0, \ldots, i_p] = [i_0] \wedge \ldots \wedge [i_p] \in \wedge^{p + 1}(A^{\oplus r})$$ For $t$ as above we set $$\Psi(t)_{i_0 \ldots i_p} = (-1)^p \frac{t([i_0, \ldots, i_p])}{f_{i_0}^e\ldots f_{i_p}^e}$$ It is clear that $\Psi(t)$ is an alternating cochain. The rule above is compatible with the transition maps of the system as the transition map $$I^\bullet(f_1^e, \ldots, f_r^e) \leftarrow I^\bullet(f_1^{e + 1}, \ldots, f_r^{e + 1}),$$ of (Perfect complexes) sends $[i_0, \ldots, i_p]$ to $f_{i_0}\ldots f_{i_p}[i_0, \ldots, i_p]$. It is clear from the description of the localizations $M_{f_{i_0} \ldots f_{i_p}}$ in Algebra, Lemma Localization as a filtered colimit that the rule $\Psi$ defines an isomorphism of cochain modules in degree $p$ in the colimit. To finish the proof we have to show that the map is compatible with differentials. To see this, for $t$ as above we compute $$\begin{aligned} d(\Psi(t))_{i_0 \ldots i_{p + 1}} & = \sum\nolimits_{j = 0}^{p + 1} (-1)^j \Psi(t)_{i_0\ldots \hat i_j \ldots i_{p + 1}} \\ & = (-1)^p \sum\nolimits_{j = 0}^{p + 1} (-1)^j t([i_0 \ldots \hat i_j \ldots i_{p + 1}]) (f_{i_0} \ldots \hat f_{i_j} \ldots f_{i_p})^{-e} \end{aligned}$$ Recall that the differentials on $I^\bullet(f_1^e, \ldots, f_r^e)$ are the negative of the differentials on $K^\bullet(f_1, \ldots, f_r)$. Thus $$\begin{aligned} \Psi(d(t))_{i_0 \ldots i_{p + 1}} & = (-1)^{p + 1} d(t)([i_0, \ldots, i_{p + 1}]) (f_{i_0} \ldots f_{i_{p + 1}})^{-e} \\ & = (-1)^{p + 1} t(d([i_0, \ldots, i_{p + 1}])) (f_{i_0} \ldots f_{i_{p + 1}})^{-e} \\ & = (-1)^{p + 1} t(-\sum\nolimits_{j = 0}^{p + 1} (-1)^j f_{i_j}^e [i_0, \ldots, \hat i_j, \ldots i_{p + 1}]) (f_{i_0} \ldots f_{i_{p + 1}})^{-e} \\ & = -(-1)^{p + 1} \sum\nolimits_{j = 0}^{p + 1} (-1)^j t([i_0, \ldots, \hat i_j, \ldots i_{p + 1}]) (f_{i_0} \ldots \hat f_{i_j} \ldots f_{i_p})^{-e} \end{aligned}$$ The two formulas agree concluding the proof. $\square$

Lemma. The Čech double complex

In Situation A complex and a finite principal-open cover. Let $M^\bullet$ be a complex of $A$-modules and denote $\mathcal{F}^\bullet$ the associated complex of $\mathcal{O}_X$-modules. Then there is a canonical isomorphism of complexes $$\mathop{\operatorname{colim}}_e \operatorname{Hom}^\bullet(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet))$$ functorial in $M^\bullet$.

Proof. Consider the double complex $F^{\bullet, \bullet}$ with terms $F^{p, q} = \mathcal{C}_{alt}^p(\mathcal{U}, \mathcal{F}^q)$ discussed in Cohomology, Section Sheaf cohomology and derived categories. Consider the double complex $G^{\bullet, \bullet}$ with terms $G^{p, q} = \mathop{\operatorname{colim}}_e \operatorname{Hom}_A(I^{-p}(f_1^e, \ldots, f_r^e), M^q)$ and differentials given by functoriality (without the intervention of signs). The maps $\psi^{p, q} : G^{p, q} \to F^{p, q}$ constructed in the proof of Lemma The alternating Čech complex are isomorphisms and compatible with the differentials $d_1$ (by the lemma) and $d_2$ (this is clear). However, the differentials $d$ on the complexes on the left and right hand side of the arrow in the lemma have different signs. Namely, for $g \in G^{p, q}$ is given by $$d(g) = d_2(g) - (-1)^{p + q} d_1(g)$$ (see More on Algebra, Section Derived Hom, Ext and derived categories) and the differential for $f \in F^{p, q}$ is given by $$d(f) = d_1(f) + (-1)^p d_2(f)$$ Thus we can fix the signs by multiplying $\psi^{p, q}$ by $(-1)^{pq + p(p - 1)/2}$. $\square$

Lemma. Čech computation of derived cohomology

In Situation A complex and a finite principal-open cover. Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_X$-modules. Then there is a canonical isomorphism $$\text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet)) \longrightarrow R\Gamma(U, \mathcal{F}^\bullet)$$ in $D(A)$ functorial in $\mathcal{F}^\bullet$.

Proof. Let $\mathcal{B}$ be the set of affine opens of $U$. Since the higher cohomology groups of a quasi-coherent module on an affine scheme are zero (Cohomology of Schemes, Lemma Quasi-coherent complexes and sheaf cohomology (uncovered prerequisite)) this is a special case of Cohomology, Lemma Sheaf cohomology and derived categories. $\square$

Proposition. Koszul representatives on a principal-open cover

In Situation A complex and a finite principal-open cover. For every object $E$ of $D_\mathrm{QCoh}(\mathcal{O}_X)$ the map (Comparison for perfect complexes) is an isomorphism.

Proof. By Lemma Bounded comparison of affine derived categories we may assume that \(E\) is given by a complex of quasi-coherent sheaves \(\mathcal{F}^\bullet\). Let \(M^\bullet = \Gamma(X, \mathcal{F}^\bullet)\) be the corresponding complex of \(A\)-modules. By Lemmas The Čech double complex and Čech computation of derived cohomology we have quasi-isomorphisms

By More on Algebra, Lemma Derived Hom, Ext and projective and locally free modules and Equation (Derived Hom and Ext) taking \(H^0\) of the complex \(\operatorname{Hom}^\bullet(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet)\) computes \(\operatorname{Hom}\) in \(D(A)\). Thus taking \(H^0\) on both sides we obtain

Since \(\operatorname{Hom}_{D(A)}(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet) = \operatorname{Hom}_{D(\mathcal{O}_X)}(I_e, E)\) by Lemma Bounded comparison of affine derived categories the lemma follows. \(\square\)

Lemma. Koszul representatives with closed support

In Situation A complex and a finite principal-open cover. Let $E$ be an object of $D_\mathrm{QCoh}(\mathcal{O}_X)$. Assume that $H^i(E)|_U = 0$ for $i = - r + 1, \ldots, 0$. Then given $s \in H^0(X, E)$ there exists an $e \geq 0$ and a morphism $K_e \to E$ such that $s$ is in the image of $H^0(X, K_e) \to H^0(X, E)$.

Proof. Since $U$ is covered by $r$ affine opens we have $H^j(U, \mathcal{F}) = 0$ for $j \geq r$ and any quasi-coherent module (Cohomology of Schemes, Lemma Vanishing and affine neighbourhoods (uncovered prerequisite)). By Lemma Computing derived Hom with a quasi-coherent K-injective model we see that $H^0(U, E)$ is equal to $H^0(U, \tau_{\geq -r + 1}E)$. There is a spectral sequence $$H^j(U, H^i(\tau_{\geq -r + 1}E)) \Rightarrow H^{i + j}(U, \tau_{\geq -N}E)$$ see Derived Categories, Lemma Derived tensor products, Tor amplitude and derived categories. Hence $H^0(U, E) = 0$ by our assumed vanishing of cohomology sheaves of $E$. We conclude that $s|_U = 0$. Think of $s$ as a morphism $\mathcal{O}_X \to E$ in $D(\mathcal{O}_X)$. By Proposition Koszul representatives on a principal-open cover the composition $I_e \to \mathcal{O}_X \to E$ is zero for some $e$. By the distinguished triangle $I_e \to \mathcal{O}_X \to K_e \to I_e[1]$ we obtain a morphism $K_e \to E$ such that $s$ is the composition $\mathcal{O}_X \to K_e \to E$. $\square$

Lemma. Extending a perfect complex after adjoining a locally free summand

Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. For every perfect object $E$ of $D(\mathcal{O}_U)$ there exists an integer $r$ and a finite locally free sheaf $\mathcal{F}$ on $U$ such that $\mathcal{F}[-r] \oplus E$ is the restriction of a perfect object of $D(\mathcal{O}_X)$.

Proof. Say $X = \operatorname{Spec}(A)$. Recall that a perfect complex is pseudo-coherent, see Cohomology, Lemma Perfect complexes. By Lemma Lifting pseudo-coherent complexes and coherent sheaves we can find a bounded above complex $\mathcal{F}^\bullet$ of finite free $A$-modules such that $E$ is isomorphic to $\mathcal{F}^\bullet|_U$ in $D(\mathcal{O}_U)$. By Cohomology, Lemma Perfect complexes and since $U$ is quasi-compact, we see that $E$ has finite tor dimension, say $E$ has tor amplitude in $[a, b]$. Pick $r < a$ and set $$\mathcal{K} = \operatorname{Ker}(\mathcal{F}^{r} \to \mathcal{F}^{r + 1}) = \operatorname{Im}(\mathcal{F}^{r - 1} \to \mathcal{F}^r).$$ Since $E$ has tor amplitude in $[a, b]$ we see that $\mathcal{F} = \mathcal{K}|_U$ is flat (Cohomology, Lemma Flatness). Hence $\mathcal{F}$ is flat and of finite presentation, thus finite locally free (Properties, Lemma Projective, locally free modules and finite algebras). It follows that $$\mathcal{F} \to \mathcal{F}^r|_U \to \mathcal{F}^{r + 1}|_U \to \ldots$$ is a strictly perfect complex on $U$ representing $E$. On the other hand, the complex $P = (\mathcal{F}^r \to \mathcal{F}^{r + 1} \to \ldots )$ is a perfect complex on $X$. Using stupid truncations we obtain a distinguished triangle $$P|_U \to E \to \mathcal{F}[-r - 1] \to (P|_U)[1]$$ If the map $E \to \mathcal{F}[-r - 1]$ is zero in $D(\mathcal{O}_U)$, then $P|_U = \mathcal{F}[-r - 2] \oplus E$, see Derived Categories, Lemma Splitting an exact triangle. This will be true for $r \ll 0$ for example by Lemma Ext from a perfect complex to bounded quasi-coherent cohomology. $\square$

Lemma. Extending a morphism after finite denominators are cleared

Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. Let $E, E'$ be objects of $D_\mathrm{QCoh}(\mathcal{O}_X)$ with $E$ perfect. For every map $\alpha : E|_U \to E'|_U$ there exist maps $$E \xleftarrow{\beta} E_1 \xrightarrow{\gamma} E'$$ of complexes on $X$ with $E_1$ perfect such that $\beta : E_1 \to E$ restricts to an isomorphism on $U$ and such that $\alpha = \gamma|_U \circ \beta|_U^{-1}$. Moreover we can assume $E_1 = E \otimes_{\mathcal{O}_X}^\mathbf{L} I$ for some perfect complex $I$ on $X$.

Proof. Write $X = \operatorname{Spec}(A)$. Write $U = D(f_1) \cup \ldots \cup D(f_r)$. Choose finite complex of finite projective $A$-modules $M^\bullet$ representing $E$ (Lemma Perfect complexes on an affine scheme). Choose a complex of $A$-modules $(M')^\bullet$ representing $E'$ (Lemma Bounded comparison of affine derived categories). In this case the complex $H^\bullet = \operatorname{Hom}_A(M^\bullet, (M')^\bullet)$ is a complex of $A$-modules whose associated complex of quasi-coherent $\mathcal{O}_X$-modules represents $R\mathcal{H}om(E, E')$, see Cohomology, Lemma Perfect complexes and derived Hom and Ext. Then $\alpha$ determines an element $s$ of $H^0(U, R\mathcal{H}om(E, E'))$, see Cohomology, Lemma Derived Hom and Ext. There exists an $e$ and a map $$\xi : I^\bullet(f_1^e, \ldots, f_r^e) \to \operatorname{Hom}_A(M^\bullet, (M')^\bullet)$$ corresponding to $s$, see Proposition Koszul representatives on a principal-open cover. Letting $E_1$ be the object corresponding to complex of quasi-coherent $\mathcal{O}_X$-modules associated to $$\text{Tot}(I^\bullet(f_1^e, \ldots, f_r^e) \otimes_A M^\bullet)$$ we obtain $E_1 \to E$ using the canonical map $I^\bullet(f_1^e, \ldots, f_r^e) \to A$ and $E_1 \to E'$ using $\xi$ and Cohomology, Lemma Derived Hom and Ext. $\square$

Lemma. Extending a perfect complex together with its shift

Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. For every perfect object $F$ of $D(\mathcal{O}_U)$ the object $F \oplus F[1]$ is the restriction of a perfect object of $D(\mathcal{O}_X)$.

Proof. By Lemma Extending a perfect complex after adjoining a locally free summand we can find a perfect object $E$ of $D(\mathcal{O}_X)$ such that $E|_U = \mathcal{F}[r] \oplus F$ for some finite locally free $\mathcal{O}_U$-module $\mathcal{F}$. By Lemma Extending a morphism after finite denominators are cleared we can find a morphism of perfect complexes $\alpha : E_1 \to E$ such that $(E_1)|_U \cong E|_U$ and such that $\alpha|_U$ is the map $$\left( \begin{matrix} \text{id}_{\mathcal{F}[r]} & 0 \\ 0 & 0 \end{matrix} \right) : \mathcal{F}[r] \oplus F \to \mathcal{F}[r] \oplus F$$ Then the cone on $\alpha$ is a solution. $\square$

Lemma. A perfect test complex with prescribed support

Let $X$ be a quasi-compact and quasi-separated scheme. Let $f \in \Gamma(X, \mathcal{O}_X)$. For any morphism $\alpha : E \to E'$ in $D_\mathrm{QCoh}(\mathcal{O}_X)$ such that

1. $E$ is perfect, and

2. $E'$ is supported on $T = V(f)$

there exists an $n \geq 0$ such that $f^n \alpha = 0$.

Proof. We have Mayer-Vietoris for morphisms in the derived category, see Cohomology, Lemma Derived Hom and Ext. Thus if $X = U \cup V$ and the result of the lemma holds for $f|_U$, $f|_V$, and $f|_{U \cap V}$, then the result holds for $f$. Thus it suffices to prove the lemma when $X$ is affine, see Cohomology of Schemes, Lemma Induction by elementary distinguished squares (uncovered prerequisite).

Let $X = \operatorname{Spec}(A)$. Then $f \in A$. We will use the equivalence $D(A) = D_\mathrm{QCoh}(X)$ of Lemma Bounded comparison of affine derived categories without further mention. Represent $E$ by a finite complex of finite projective $A$-modules $P^\bullet$. This is possible by Lemma Perfect complexes on an affine scheme. Let $t$ be the largest integer such that $P^t$ is nonzero. The distinguished triangle $$P^t[-t] \to P^\bullet \to \sigma_{\leq t - 1}P^\bullet \to P^t[-t + 1]$$ shows that by induction on the length of the complex $P^\bullet$ we can reduce to the case where $P^\bullet$ has a single nonzero term. This and the shift functor reduces us to the case where $P^\bullet$ consists of a single finite projective $A$-module $P$ in degree $0$. Represent $E'$ by a complex $M^\bullet$ of $A$-modules. Then $\alpha$ corresponds to a map $P \to H^0(M^\bullet)$. Since the module $H^0(M^\bullet)$ is supported on $V(f)$ by assumption (2) we see that every element of $H^0(M^\bullet)$ is annihilated by a power of $f$. Since $P$ is a finite $A$-module the map $f^n\alpha : P \to H^0(M^\bullet)$ is zero for some $n$ as desired. $\square$

Lemma. Lifting perfect complexes while retaining support

Let $X$ be an affine scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. Let $U \subset X$ be a quasi-compact open. For every perfect object $F$ of $D(\mathcal{O}_U)$ supported on $T \cap U$ the object $F \oplus F[1]$ is the restriction of a perfect object $E$ of $D(\mathcal{O}_X)$ supported in $T$.

Proof. If the principal-open cover of the complement has no members, then the support is all of the ambient space. The preceding extension lemma applies directly to $F \oplus F[1]$. Hence we may assume $s \geq 1$ in the induction below.

Say $T = V(g_1, \ldots, g_s)$. After replacing $g_j$ by a power we may assume multiplication by $g_j$ is zero on $F$, see Lemma A perfect test complex with prescribed support. Choose $E$ as in Lemma Extending a perfect complex together with its shift. Note that $g_j : E \to E$ restricts to zero on $U$. Choose a distinguished triangle $$E \xrightarrow{g_1} E \to C_1 \to E[1]$$ By Derived Categories, Lemma Splitting an exact triangle the object $C_1$ restricts to $F \oplus F[1] \oplus F[1] \oplus F[2]$ on $U$. Moreover, $g_1 : C_1 \to C_1$ has square zero by Derived Categories, Lemma Nilpotent thickenings. Namely, the diagram $$\begin{gathered}\begin{matrix}E & C_1 & E[1] \\ E & C_1 & E[1]\end{matrix} \\[6pt] \begin{aligned}E & \longrightarrow C_1 \\ E & \xrightarrow{0} E \\ C_1 & \xrightarrow{g_1} C_1 \\ C_1 & \longrightarrow E[1] \\ E[1] & \xrightarrow{0} E[1] \\ E & \longrightarrow C_1 \\ C_1 & \longrightarrow E[1]\end{aligned}\end{gathered}$$ is commutative since the compositions $E \xrightarrow{g_1} E \to C_1$ and $C_1 \to E[1] \xrightarrow{g_1} E[1]$ are zero. Continuing, setting $C_{i + 1}$ equal to the cone of the map $g_{i + 1} : C_i \to C_i$ we obtain a perfect complex $C_s$ on $X$ supported on $T$ whose restriction to $U$ gives $$\bigoplus_{j=0}^{s+1} F[j]^{\oplus {s+1 \choose j}}$$ Choose morphisms of perfect complexes $\beta : C' \to C_s$ and $\gamma : C' \to C_s$ as in Lemma Extending a morphism after finite denominators are cleared such that $\beta|_U$ is an isomorphism and such that $\gamma|_U \circ \beta|_U^{-1}$ is the morphism $$\bigoplus_{j=0}^{s+1} F[j]^{\oplus {s+1 \choose j}} \to \bigoplus_{j=0}^{s+1} F[j]^{\oplus {s+1 \choose j}}$$ which is the identity on all summands except for $F$ where it is zero. By Lemma Extending a morphism after finite denominators are cleared we also have $C' = C_s \otimes^\mathbf{L} I$ for some perfect complex $I$ on $X$. Hence the nullity of $g_j^2\text{id}_{C_s}$ implies the same thing for $C'$. Thus $C'$ is supported on $T$ as well. Then $\text{Cone}(\gamma)$ is a solution. $\square$

Lemma. Lifting maps from support-preserving perfect complexes

Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open. Let $T \subset X$ be a closed subset with $X \setminus T$ retro-compact in $X$. Let $E$ be an object of $D_\mathrm{QCoh}(\mathcal{O}_X)$. Let $\alpha : P \to E|_U$ be a map where $P$ is a perfect object of $D(\mathcal{O}_U)$ supported on $T \cap U$. Then there exists a map $\beta : R \to E$ where $R$ is a perfect object of $D(\mathcal{O}_X)$ supported on $T$ such that $P$ is a direct summand of $R|_U$ in $D(\mathcal{O}_U)$ compatible $\alpha$ and $\beta|_U$.

Proof. Since $X$ is quasi-compact there exists an integer $m$ such that $X = U \cup V_1 \cup \ldots \cup V_m$ for some affine opens $V_j$ of $X$. Arguing by induction on $m$ we see that we may assume $m = 1$. In other words, we may assume that $X = U \cup V$ with $V$ affine. By Lemma Lifting perfect complexes while retaining support we can choose a perfect object $Q$ in $D(\mathcal{O}_V)$ supported on $T \cap V$ and an isomorphism $Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}$. By Lemma Extending a morphism after finite denominators are cleared we can replace $Q$ by $Q \otimes^\mathbf{L} I$ (still supported on $T \cap V$) and assume that the map $$Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V} \longrightarrow P|_{U \cap V} \longrightarrow E|_{U \cap V}$$ lifts to $Q \to E|_V$. By Cohomology, Lemma Derived gluing across an elementary distinguished square we find an morphism $a : R \to E$ of $D(\mathcal{O}_X)$ such that $a|_U$ is isomorphic to $P \oplus P[1] \to E|_U$ and $a|_V$ isomorphic to $Q \to E|_V$. Thus $R$ is perfect and supported on $T$ as desired. $\square$

Lemma. Perfect approximation on an affine chart

Let $X$ be an affine scheme. Then approximation holds for every triple $(T, E, m)$ as in Definition Bounds for perfect approximation such that there exists an integer $r \geq 0$ with

1. $E$ is $m$-pseudo-coherent,

2. $H^i(E)$ is supported on $T$ for $i \geq m - r + 1$,

3. $X \setminus T$ is the union of $r$ affine opens.

In particular, approximation by perfect complexes holds for affine schemes.

Proof. Say $X = \operatorname{Spec}(A)$. Write $T = V(f_1, \ldots, f_r)$. (The case $r = 0$, i.e., $T = X$ follows immediately from Lemma Pseudo-coherent complexes on an affine scheme and the definitions.) Let $(T, E, m)$ be a triple as in the lemma. Let $t$ be the largest integer such that $H^t(E)$ is nonzero. We will proceed by induction on $t$. The base case is $t < m$; in this case the result is trivial. Now suppose that $t \geq m$. By Cohomology, Lemma Finiteness of cohomology groups the sheaf $H^t(E)$ is of finite type. Since it is quasi-coherent it is generated by finitely many sections (Properties, Lemma Modules and finite algebras). For every $s \in \Gamma(X, H^t(E)) = H^t(X, E)$ (see proof of Lemma Bounded comparison of affine derived categories) we can find an $e > 0$ and a morphism $K_e[-t] \to E$ such that $s$ is in the image of $H^0(K_e) = H^t(K_e[-t]) \to H^t(E)$, see Lemma Koszul representatives with closed support. Taking a finite direct sum of these maps we obtain a map $P \to E$ where $P$ is a perfect complex supported on $T$, where $H^i(P) = 0$ for $i > t$, and where $H^t(P) \to E$ is surjective. Choose a distinguished triangle $$P \to E \to E' \to P[1]$$ Then $E'$ is $m$-pseudo-coherent (Cohomology, Lemma Pseudo-coherent complexes and coherent sheaves), $H^i(E') = 0$ for $i \geq t$, and $H^i(E')$ is supported on $T$ for $i \geq m - r + 1$. By induction we find an approximation $P' \to E'$ of $(T, E', m)$. Fit the composition $P' \to E' \to P[1]$ into a distinguished triangle $P \to P'' \to P' \to P[1]$ and extend the morphisms $P' \to E'$ and $P[1] \to P[1]$ into a morphism of distinguished triangles $$\begin{gathered}\begin{matrix}P & P'' & P' & P[1] \\ P & E & E' & P[1]\end{matrix} \\[6pt] \begin{aligned}P & \longrightarrow P'' \\ P & \longrightarrow P \\ P'' & \longrightarrow E \\ P'' & \longrightarrow P' \\ P' & \longrightarrow E' \\ P' & \longrightarrow P[1] \\ P[1] & \longrightarrow P[1] \\ P & \longrightarrow E \\ E & \longrightarrow E' \\ E' & \longrightarrow P[1]\end{aligned}\end{gathered}$$ using TR3. Then $P''$ is a perfect complex (Cohomology, Lemma Perfect complexes) supported on $T$. An easy diagram chase shows that $P'' \to E$ is the desired approximation. $\square$

A.12.3 Proper coherent cohomology for algebraic spaces

The following arguments adapt the human Stacks Project treatment in spaces-cohomology.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. The weak Chow lemma for algebraic spaces

Let $A$ be a ring. Let $X$ be an algebraic space over $\operatorname{Spec}(A)$ whose structure morphism $X \to \operatorname{Spec}(A)$ is separated of finite type. Then there exists a proper surjective morphism $X' \to X$ where $X'$ is a scheme which is H-quasi-projective over $\operatorname{Spec}(A)$.

Proof. Let $W$ be an affine scheme and let $f : W \to X$ be a surjective étale morphism. There exists an integer $d$ such that all geometric fibres of f have $\leq d$ points (because $X$ is a separated algebraic hence reasonable, see Decent Spaces, Lemma Schematic neighbourhoods). Picking $d$ minimal we get a nonempty open $U \subset X$ such that $f^{-1}(U) \to U$ is finite étale of degree $d$, see Decent Spaces, Lemma Schematic neighbourhoods. Let $$V \subset W \times_X W \times_X \ldots \times_X W$$ ($d$ factors in the fibre product) be the complement of all the diagonals. Because $W \to X$ is separated the diagonal $W \to W \times_X W$ is a closed immersion. Since $W \to X$ is étale the diagonal $W \to W \times_X W$ is an open immersion, see Morphisms of Spaces, Lemmas Étale morphisms and unramified morphisms and Unramified morphisms and diagonals and separation. Hence the diagonals are open and closed subschemes of the quasi-compact scheme $W \times_X \ldots \times_X W$. In particular we conclude $V$ is a quasi-compact scheme. Choose an open immersion $W \subset Y$ with $Y$ H-projective over $A$ (this is possible as $W$ is affine and of finite type over $A$; for example we can use Morphisms, Lemmas Affine neighbourhoods and finite algebras (uncovered prerequisite) and Projective and locally free modules (uncovered prerequisite)). Let $$Z \subset Y \times_A Y \times_A \ldots \times_A Y$$ be the scheme theoretic image of the composition $V \to W \times_X \ldots \times_X W \to Y \times_A \ldots \times_A Y$. Observe that this morphism is quasi-compact since $V$ is quasi-compact and $Y \times_A \ldots \times_A Y$ is separated. Note that $V \to Z$ is an open immersion as $V \to Y \times_A \ldots \times_A Y$ is an immersion, see Morphisms, Lemma Diagonals and separation (uncovered prerequisite). The projection morphisms give $d$ morphisms $g_i : Z \to Y$. These morphisms $g_i$ are projective as $Y$ is projective over $A$, see material in Morphisms, Section Projective and locally free modules. We set $$X' = \bigcup g_i^{-1}(W) \subset Z$$ There is a morphism $X' \to X$ whose restriction to $g_i^{-1}(W)$ is the composition $g_i^{-1}(W) \to W \to X$. Namely, these morphisms agree over $V$ hence agree over $g_i^{-1}(W) \cap g_j^{-1}(W)$ by Morphisms of Spaces, Lemma Morphisms of algebraic spaces. Claim: the morphism $X' \to X$ is proper.

If the claim holds, then the lemma follows by induction on $d$. Namely, by construction $X'$ is H-quasi-projective over $\operatorname{Spec}(A)$. The image of $X' \to X$ contains the open $U$ as $V$ surjects onto $U$. Denote $T$ the reduced induced algebraic space structure on $X \setminus U$. Then $T \times_X W$ is a closed subscheme of $W$, hence affine. Moreover, the morphism $T \times_X W \to T$ is étale and every geometric fibre has $< d$ points. By induction hypothesis there exists a proper surjective morphism $T' \to T$ where $T'$ is a scheme H-quasi-projective over $\operatorname{Spec}(A)$. Since $T$ is a closed subspace of $X$ we see that $T' \to X$ is a proper morphism. Thus the lemma follows by taking the proper surjective morphism $X' \amalg T' \to X$.

Proof of the claim. By construction the morphism $X' \to X$ is separated and of finite type. We will check conditions (1) -- (4) of Morphisms of Spaces, Lemma Morphisms of algebraic spaces for the morphisms $V \to X'$ and $X' \to X$. Conditions (1) and (2) we have seen above. Condition (3) holds as $X' \to X$ is separated (as a morphism whose source is a separated algebraic space). Thus it suffices to check liftability to $X'$ for diagrams $$\begin{gathered}\begin{matrix}\operatorname{Spec}(K) & V \\ \operatorname{Spec}(R) & X\end{matrix} \\[6pt] \begin{aligned}\operatorname{Spec}(K) & \longrightarrow V \\ \operatorname{Spec}(K) & \longrightarrow \operatorname{Spec}(R) \\ V & \longrightarrow X \\ \operatorname{Spec}(R) & \longrightarrow X\end{aligned}\end{gathered}$$ where $R$ is a valuation ring with fraction field $K$. Note that the top horizontal map is given by $d$ pairwise distinct $K$-valued points $w_1, \ldots, w_d$ of $W$. In fact, this is a complete set of inverse images of the point $x \in X(K)$ coming from the diagram. Since $W \to X$ is surjective, we can, after possibly replacing $R$ by an extension of valuation rings, lift the morphism $\operatorname{Spec}(R) \to X$ to a morphism $w : \operatorname{Spec}(R) \to W$, see Morphisms of Spaces, Lemma Lifting flatness. Since $w_1, \ldots, w_d$ is a complete collection of inverse images of $x$ we see that $w|_{\operatorname{Spec}(K)}$ is equal to one of them, say $w_i$. Thus we see that we get a commutative diagram $$\begin{gathered}\begin{matrix}\operatorname{Spec}(K) & Z \\ \operatorname{Spec}(R) & Y\end{matrix} \\[6pt] \begin{aligned}\operatorname{Spec}(K) & \longrightarrow Z \\ \operatorname{Spec}(K) & \longrightarrow \operatorname{Spec}(R) \\ Z & \xrightarrow{g_i} Y \\ \operatorname{Spec}(R) & \xrightarrow{w} Y\end{aligned}\end{gathered}$$ By the valuative criterion of properness for the projective morphism $g_i$ we can lift $w$ to $z : \operatorname{Spec}(R) \to Z$, see Morphisms, Lemma Projective, locally free modules and proper morphisms (uncovered prerequisite) and Schemes, Proposition Criteria for the geometric construction (uncovered prerequisite). The image of $z$ is in $g_i^{-1}(W) \subset X'$ and the proof is complete. $\square$

Lemma. Sheaf cohomology and proper morphisms

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$. Assume

1. For any short exact sequence of coherent sheaves on $X$ if two out of three have property $\mathcal{P}$ so does the third.

2. If $\mathcal{P}$ holds for $\mathcal{F}^{\oplus r}$ for some $r \geq 1$, then it holds for $\mathcal{F}$.

3. For every reduced closed subspace $i : Z \to X$ with $|Z|$ irreducible there exists a coherent sheaf $\mathcal{G}$ on $X$ whose scheme theoretic support is $Z$ such that $\mathcal{P}$ holds for $\mathcal{G}$.

Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.

Proof. We will show that conditions (1) and (2) of Lemma The initial cohomological property hold. This is clear for condition (1). To show that (2) holds, let $$\mathcal{T} = \left\{ \begin{matrix} i : Z \to X \text{ reduced closed subspace with }|Z|\text{ irreducible such}\\ \text{ that }i_*\mathcal{I}\text{ does not have }\mathcal{P} \text{ for some quasi-coherent }\mathcal{I} \subset \mathcal{O}_Z \end{matrix} \right\}$$ If $\mathcal{T}$ is nonempty, then since $X$ is Noetherian, we can find an $i : Z \to X$ which is minimal in $\mathcal{T}$. We will show that this leads to a contradiction.

Let $\mathcal{G}$ be the sheaf whose scheme theoretic support is $Z$ whose existence is assumed in assumption (3). Let $\varphi : i_*\mathcal{I}^{\oplus r} \to \mathcal{G}$ be as in Lemma Filtering a coherent sheaf by irreducible supports. Let $$0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \operatorname{Coker}(\varphi)$$ be a filtration as in Lemma Coherent sheaves. By minimality of $Z$ and assumption (1) we see that $\operatorname{Coker}(\varphi)$ has property $\mathcal{P}$. As $\varphi$ is injective we conclude using assumption (1) once more that $i_*\mathcal{I}^{\oplus r}$ has property $\mathcal{P}$. Using assumption (2) we conclude that $i_*\mathcal{I}$ has property $\mathcal{P}$.

Finally, if $\mathcal{J} \subset \mathcal{O}_Z$ is a second quasi-coherent sheaf of ideals, set $\mathcal{K} = \mathcal{I} \cap \mathcal{J}$ and consider the short exact sequences $$0 \to \mathcal{K} \to \mathcal{I} \to \mathcal{I}/\mathcal{K} \to 0 \quad \text{and} \quad 0 \to \mathcal{K} \to \mathcal{J} \to \mathcal{J}/\mathcal{K} \to 0$$ Arguing as above, using the minimality of $Z$, we see that $i_*\mathcal{I}/\mathcal{K}$ and $i_*\mathcal{J}/\mathcal{K}$ satisfy $\mathcal{P}$. Hence by assumption (1) we conclude that $i_*\mathcal{K}$ and then $i_*\mathcal{J}$ satisfy $\mathcal{P}$. In other words, $Z$ is not an element of $\mathcal{T}$ which is the desired contradiction. $\square$

Lemma. Proper coherent direct images for algebraic spaces

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $R^if_*\mathcal{F}$ is a coherent $\mathcal{O}_Y$-module for all $i \geq 0$.

Proof. We first remark that $X$ is a locally Noetherian algebraic space by Morphisms of Spaces, Lemma Noetherian rings and finite algebras. Hence the statement of the lemma makes sense. Moreover, computing $R^if_*\mathcal{F}$ commutes with étale localization on $Y$ (Properties of Spaces, Lemma Base change for étale morphisms and modules) and checking whether $R^if_*\mathcal{F}$ coherent can be done étale locally on $Y$ (Lemma Coherent sheaves and Noetherian rings). Hence we may assume that $Y = \operatorname{Spec}(A)$ is a Noetherian affine scheme.

Assume $Y = \operatorname{Spec}(A)$ is an affine scheme. Note that $f$ is locally of finite presentation (Morphisms of Spaces, Lemma Finite presentation and Noetherian rings). Thus it is of finite presentation, hence $X$ is Noetherian (Morphisms of Spaces, Lemma Finite presentation and Noetherian rings). Thus Lemma Sheaf cohomology and proper morphisms applies to the category of coherent modules of $X$. For a coherent sheaf $\mathcal{F}$ on $X$ we say $\mathcal{P}$ holds if and only if $R^if_*\mathcal{F}$ is a coherent module on $\operatorname{Spec}(A)$. We will show that conditions (1), (2), and (3) of Lemma Sheaf cohomology and proper morphisms hold for this property thereby finishing the proof of the lemma.

Verification of condition (1). Let $$0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$$ be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of higher direct images $$R^{p - 1}f_*\mathcal{F}_3 \to R^pf_*\mathcal{F}_1 \to R^pf_*\mathcal{F}_2 \to R^pf_*\mathcal{F}_3 \to R^{p + 1}f_*\mathcal{F}_1$$ Then it is clear that if 2-out-of-3 of the sheaves $\mathcal{F}_i$ have property $\mathcal{P}$, then the higher direct images of the third are sandwiched in this exact complex between two coherent sheaves. Hence these higher direct images are also coherent by Lemmas Coherent sheaves and Noetherian rings and Quasi-coherent complexes and coherent sheaves. Hence property $\mathcal{P}$ holds for the third as well.

Verification of condition (2). This follows immediately from the fact that $R^if_*(\mathcal{F}_1 \oplus \mathcal{F}_2) = R^if_*\mathcal{F}_1 \oplus R^if_*\mathcal{F}_2$ and that a summand of a coherent module is coherent (see lemmas cited above).

Verification of condition (3). Let $i : Z \to X$ be a closed immersion with $Z$ reduced and $|Z|$ irreducible. Set $g = f \circ i : Z \to \operatorname{Spec}(A)$. Let $\mathcal{G}$ be a coherent module on $Z$ whose scheme theoretic support is equal to $Z$ such that $R^pg_*\mathcal{G}$ is coherent for all $p$. Then $\mathcal{F} = i_*\mathcal{G}$ is a coherent module on $X$ whose scheme theoretic support is $Z$ such that $R^pf_*\mathcal{F} = R^pg_*\mathcal{G}$. To see this use the Leray spectral sequence (Cohomology on Sites, Lemma Sheaf cohomology) and the fact that $R^qi_*\mathcal{G} = 0$ for $q > 0$ by Lemma Affine neighbourhoods and the fact that a closed immersion is affine. (Morphisms of Spaces, Lemma Diagonals, separation and affine neighbourhoods). Thus we reduce to finding a coherent sheaf $\mathcal{G}$ on $Z$ with support equal to $Z$ such that $R^pg_*\mathcal{G}$ is coherent for all $p$.

We apply Lemma The weak Chow lemma for algebraic spaces to the morphism $Z \to \operatorname{Spec}(A)$. Thus we get a diagram $$\begin{gathered}\begin{matrix}Z & Z' & \mathbf{P}^n_A \\ \phantom{X} & \operatorname{Spec}(A) & \phantom{X}\end{matrix} \\[6pt] \begin{aligned}Z & \xrightarrow{g} \operatorname{Spec}(A) \\ Z' & \xrightarrow{g'} \operatorname{Spec}(A) \\ Z' & \xrightarrow{\pi} Z \\ Z' & \xrightarrow{i} \mathbf{P}^n_A \\ \mathbf{P}^n_A & \longrightarrow \operatorname{Spec}(A)\end{aligned}\end{gathered}$$ with $\pi : Z' \to Z$ proper surjective and $i$ an immersion. Since $Z \to \operatorname{Spec}(A)$ is proper we conclude that $g'$ is proper (Morphisms of Spaces, Lemma Composition and proper morphisms). Hence $i$ is a closed immersion (Morphisms of Spaces, Lemmas Morphisms of algebraic spaces and Diagonals and separation). It follows that the morphism $i' = (i, \pi) : \mathbf{P}^n_A \times_{\operatorname{Spec}(A)} Z' = \mathbf{P}^n_Z$ is a closed immersion (Morphisms of Spaces, Lemma Diagonals and separation). Set $$\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_A}(1) = (i')^*\mathcal{O}_{\mathbf{P}^n_Z}(1)$$ We may apply Lemma Quasi-coherent cohomology to $\mathcal{L}$ and $\pi$ as well as $\mathcal{L}$ and $g'$. Hence for all $d \gg 0$ we have $R^p\pi_*\mathcal{L}^{\otimes d} = 0$ for all $p > 0$ and $R^p(g')_*\mathcal{L}^{\otimes d} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi_*\mathcal{L}^{\otimes d}$. By the Leray spectral sequence (Cohomology on Sites, Lemma Sheaf cohomology) we have $$E_2^{p, q} = R^pg_* R^q\pi_*\mathcal{L}^{\otimes d} \Rightarrow R^{p + q}(g')_*\mathcal{L}^{\otimes d}$$ and by choice of $d$ the only nonzero terms in $E_2^{p, q}$ are those with $q = 0$ and the only nonzero terms of $R^{p + q}(g')_*\mathcal{L}^{\otimes d}$ are those with $p = q = 0$. This implies that $R^pg_*\mathcal{G} = 0$ for $p > 0$ and that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes d}$. Applying Cohomology of Schemes, Lemma Projective, locally free modules and local algebra (uncovered prerequisite) we see that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes d}$ is coherent.

We still have to check that the support of $\mathcal{G}$ is $Z$. This follows from the fact that $\mathcal{L}^{\otimes d}$ has lots of global sections. We spell it out here. Note that $\mathcal{L}^{\otimes d}$ is globally generated for all $d \geq 0$ because the same is true for $\mathcal{O}_{\mathbf{P}^n}(d)$. Pick a point $z \in Z'$ mapping to the generic point $\xi$ of $Z$ which we can do as $\pi$ is surjective. (Observe that $Z$ does indeed have a generic point as $|Z|$ is irreducible and $Z$ is Noetherian, hence quasi-separated, hence $|Z|$ is a sober topological space by Properties of Spaces, Lemma Diagonals and separation.) Pick $s \in \Gamma(Z', \mathcal{L}^{\otimes d})$ which does not vanish at $z$. Since $\Gamma(Z, \mathcal{G}) = \Gamma(Z', \mathcal{L}^{\otimes d})$ we may think of $s$ as a global section of $\mathcal{G}$. Choose a geometric point $\overline{z}$ of $Z'$ lying over $z$ and denote $\overline{\xi} = g' \circ \overline{z}$ the corresponding geometric point of $Z$. The adjunction map $$(g')^*\mathcal{G} = (g')^*g'_*\mathcal{L}^{\otimes d} \longrightarrow \mathcal{L}^{\otimes d}$$ induces a map of stalks $\mathcal{G}_{\overline{\xi}} \to \mathcal{L}_{\overline{z}}$, see Properties of Spaces, Lemma Quasi-coherent complexes and coherent sheaves. Moreover the adjunction map sends the pullback of $s$ (viewed as a section of $\mathcal{G}$) to $s$ (viewed as a section of $\mathcal{L}^{\otimes d}$). Thus the image of $s$ in the vector space which is the source of the arrow $$\mathcal{G}_{\overline{\xi}} \otimes \kappa(\overline{\xi}) \longrightarrow \mathcal{L}^{\otimes d}_{\overline{z}} \otimes \kappa(\overline{z})$$ isn't zero since by choice of $s$ the image in the target of the arrow is nonzero. Hence $\xi$ is in the support of $\mathcal{G}$ (Morphisms of Spaces, Lemma Closed support and finite algebras). Since $|Z|$ is irreducible and $Z$ is reduced we conclude that the scheme theoretic support of $\mathcal{G}$ is all of $Z$ as desired. $\square$

A.12.4 Étale gluing, direct images and perfect approximation

The following arguments adapt the human Stacks Project treatment in spaces-perfect.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. Quasi-coherent complexes and coherent sheaves

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $(K_n)$ be an inverse system of $D_\mathrm{QCoh}(\mathcal{O}_X)$ with derived limit $K = R\varprojlim K_n$ in $D(\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \to H^q(K_n)$ is surjective for all $q \in \mathbf{Z}$ and $n \geq 1$. Then

1. $H^q(K) = \varprojlim H^q(K_n)$,

2. $R\varprojlim H^q(K_n) = \varprojlim H^q(K_n)$, and

3. for every affine open $U \subset X$ we have $H^p(U, \varprojlim H^q(K_n)) = 0$ for $p > 0$.

Proof. Let $\mathcal{B} \subset \operatorname{Ob}(X_\mathrm{\acute{e}tale})$ be the set of affine objects. Since $H^q(K_n)$ is quasi-coherent we have $H^p(U, H^q(K_n)) = 0$ for $U \in \mathcal{B}$ by the discussion in Cohomology of Spaces, Section Quasi-coherent cohomology and Cohomology of Schemes, Lemma Quasi-coherent complexes and sheaf cohomology (uncovered prerequisite). Moreover, the maps $H^0(U, H^q(K_{n + 1})) \to H^0(U, H^q(K_n))$ are surjective for $U \in \mathcal{B}$ by similar reasoning. Part (1) follows from Cohomology on Sites, Lemma Derived categories whose conditions we have just verified. Parts (2) and (3) follow from Cohomology on Sites, Lemma Derived categories. $\square$

Lemma. Quasi-coherent complexes and coherent sheaves

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The functor $Lf^*$ sends $D_\mathrm{QCoh}(\mathcal{O}_X)$ into $D_\mathrm{QCoh}(\mathcal{O}_Y)$.

Proof. Choose a diagram $$\begin{gathered}\begin{matrix}U & V \\ X & Y\end{matrix} \\[6pt] \begin{aligned}U & \xrightarrow{a} X \\ U & \xrightarrow{h} V \\ V & \xrightarrow{b} Y \\ X & \xrightarrow{f} Y\end{aligned}\end{gathered}$$ where $U$ and $V$ are schemes, the vertical arrows are étale, and $a$ is surjective. Since $a^* \circ Lf^* = Lh^* \circ b^*$ the result follows from Lemma Quasi-coherent complexes and coherent sheaves and the case of schemes which is Derived Categories of Schemes, Lemma Quasi-coherent complexes and coherent sheaves. $\square$

Lemma. Quasi-coherent complexes and coherent sheaves

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$.

1. The functor $Rf_*$ sends $D_\mathrm{QCoh}(\mathcal{O}_X)$ into $D_\mathrm{QCoh}(\mathcal{O}_Y)$.

2. If $Y$ is quasi-compact, there exists an integer $N = N(X, Y, f)$ such that for an object $E$ of $D_\mathrm{QCoh}(\mathcal{O}_X)$ with $H^m(E) = 0$ for $m > 0$ we have $H^m(Rf_*E) = 0$ for $m \geq N$.

3. In fact, if $Y$ is quasi-compact we can find $N = N(X, Y, f)$ such that for every morphism of algebraic spaces $Y' \to Y$ the same conclusion holds for the functor $R(f')_*$ where $f' : X' \to Y'$ is the base change of $f$.

Proof. Let $E$ be an object of $D_\mathrm{QCoh}(\mathcal{O}_X)$. To prove (1) we have to show that $Rf_*E$ has quasi-coherent cohomology sheaves. This question is local on $Y$, hence we may assume $Y$ is quasi-compact. Pick $N = N(X, Y, f)$ as in Cohomology of Spaces, Lemma Vanishing and quasi-coherent cohomology. Thus $R^pf_*\mathcal{F} = 0$ for all quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}$ and all $p \geq N$. Moreover $R^pf_*\mathcal{F}$ is quasi-coherent for all $p$ by Cohomology of Spaces, Lemma Quasi-coherent cohomology. These statements remain true after base change.

First, assume $E$ is bounded below. We will show (1) and (2) and (3) hold for such $E$ with our choice of $N$. In this case we can for example use the spectral sequence $$R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E$$ (Derived Categories, Lemma Derived tensor products, Tor amplitude and derived categories), the quasi-coherence of $R^pf_*H^q(E)$, and the vanishing of $R^pf_*H^q(E)$ for $p \geq N$ to see that (1), (2), and (3) hold in this case.

Next we prove (2) and (3). Say $H^m(E) = 0$ for $m > 0$. Let $V$ be an affine object of $Y_\mathrm{\acute{e}tale}$. We have $H^p(V \times_Y X, \mathcal{F}) = 0$ for $p \geq N$, see Cohomology of Spaces, Lemma Quasi-coherent complexes and coherent sheaves. Hence we may apply Lemma Computing derived Hom with a quasi-coherent K-injective model to the functor $\Gamma(V \times_Y X, -)$ to see that $$R\Gamma(V, Rf_*E) = R\Gamma(V \times_Y X, E)$$ has vanishing cohomology in degrees $\geq N$. Since this holds for all $V$ affine in $Y_\mathrm{\acute{e}tale}$ we conclude that $H^m(Rf_*E) = 0$ for $m \geq N$.

Next, we prove (1) in the general case. Recall that there is a distinguished triangle $$\tau_{\leq -n - 1}E \to E \to \tau_{\geq -n}E \to (\tau_{\leq -n - 1}E)[1]$$ in $D(\mathcal{O}_X)$, see Derived Categories, Remark Derived categories. By (2) we see that $Rf_*\tau_{\leq -n - 1}E$ has vanishing cohomology sheaves in degrees $\geq -n + N$. Thus, given an integer $q$ we see that $R^qf_*E$ is equal to $R^qf_*\tau_{\geq -n}E$ for some $n$ and the result above applies. $\square$

Lemma. Proper morphisms and closed support

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ locally Noetherian. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module with support proper over $Y$. Then $R^pf_*\mathcal{F}$ is a coherent $\mathcal{O}_Y$-module for all $p \geq 0$.

Proof. By Lemma Proper morphisms and modules there exists a closed immersion $i : Z \to X$ with $g = f \circ i : Z \to Y$ proper and $\mathcal{F} = i_*\mathcal{G}$ for some coherent module $\mathcal{G}$ on $Z$. We see that $R^pg_*\mathcal{G}$ is coherent on $S$ by Cohomology of Spaces, Lemma Proper coherent direct images for algebraic spaces. On the other hand, $R^qi_*\mathcal{G} = 0$ for $q > 0$ (Cohomology of Spaces, Lemma Coherent sheaves and finite algebras). By Cohomology on Sites, Lemma Sheaf cohomology we get $R^pf_*\mathcal{F} = R^pg_*\mathcal{G}$ and the lemma follows. $\square$

Lemma. Direct images and coherent sheaves

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian. Let $E$ be an object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ such that the support of $H^i(E)$ is proper over $Y$ for all $i$. Then $Rf_*E$ is an object of $D^b_{\textit{Coh}}(\mathcal{O}_Y)$.

Proof. Consider the spectral sequence $$R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E$$ see Derived Categories, Lemma Derived tensor products, Tor amplitude and derived categories. By assumption and Lemma Proper morphisms and closed support the sheaves $R^pf_*H^q(E)$ are coherent. Hence $R^{p + q}f_*E$ is coherent, i.e., $E \in D_{\textit{Coh}}(\mathcal{O}_Y)$. Boundedness from below is trivial. Boundedness from above follows from Cohomology of Spaces, Lemma Vanishing and quasi-coherent cohomology or from Lemma Quasi-coherent complexes and coherent sheaves. $\square$

Lemma. Induction by elementary distinguished squares

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P$ be a property of the quasi-compact and quasi-separated objects of $X_{spaces, \mathrm{\acute{e}tale}}$. Assume that

1. $P$ holds for every affine object of $X_{spaces, \mathrm{\acute{e}tale}}$,

2. for every elementary distinguished square $(U \subset W, f : V \to W)$ such that

   1. $W$ is a quasi-compact and quasi-separated object of $X_{spaces, \mathrm{\acute{e}tale}}$,

   2. $U$ is quasi-compact,

   3. $V$ is affine, and

   4. $P$ holds for $U$, $V$, and $U \times_W V$,

   then $P$ holds for $W$.

Then $P$ holds for every quasi-compact and quasi-separated object of $X_{spaces, \mathrm{\acute{e}tale}}$ and in particular for $X$.

Proof. We first claim that $P$ holds for every representable quasi-compact and quasi-separated object of $X_{spaces, \mathrm{\acute{e}tale}}$. Namely, suppose that $U \to X$ is étale and $U$ is a quasi-compact and quasi-separated scheme. By assumption (1) property $P$ holds for every affine open of $U$. Moreover, if $W, V \subset U$ are quasi-compact open with $V$ affine and $P$ holds for $W$, $V$, and $W \cap V$, then $P$ holds for $W \cup V$ by (2) (as the pair $(W \subset W \cup V, V \to W \cup V)$ is an elementary distinguished square). Thus $P$ holds for $U$ by the induction principle for schemes, see Cohomology of Schemes, Lemma Induction by elementary distinguished squares (uncovered prerequisite).

To finish the proof it suffices to prove $P$ holds for $X$ (because we can simply replace $X$ by any quasi-compact and quasi-separated object of $X_{spaces, \mathrm{\acute{e}tale}}$ we want to prove the result for). We will use the filtration $$\emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X$$ and the morphisms $f_p : V_p \to U_p$ of Decent Spaces, Lemma Diagonals and separation. We will prove that $P$ holds for $U_p$ by descending induction on $p$. Note that $P$ holds for $U_{n + 1}$ by (1) as an empty algebraic space is affine. Assume $P$ holds for $U_{p + 1}$. Note that $(U_{p + 1} \subset U_p, f_p : V_p \to U_p)$ is an elementary distinguished square, but (2) may not apply as $V_p$ may not be affine. However, as $V_p$ is a quasi-compact scheme we may choose a finite affine open covering $V_p = V_{p, 1} \cup \ldots \cup V_{p, m}$. Set $W_{p, 0} = U_{p + 1}$ and $$W_{p, i} = U_{p + 1} \cup f_p(V_{p, 1} \cup \ldots \cup V_{p, i})$$ for $i = 1, \ldots, m$. These are quasi-compact open subspaces of $X$. Then we have $$U_{p + 1} = W_{p, 0} \subset W_{p, 1} \subset \ldots \subset W_{p, m} = U_p$$ and the pairs $$(W_{p, 0} \subset W_{p, 1}, f_p|_{V_{p, 1}}), (W_{p, 1} \subset W_{p, 2}, f_p|_{V_{p, 2}}),\ldots, (W_{p, m - 1} \subset W_{p, m}, f_p|_{V_{p, m}})$$ are elementary distinguished squares by Lemma Derived quasi-coherent complexes. Note that $P$ holds for each $V_{p, i}$ (as affine schemes) and for $W_{p, i} \times_{W_{p, i + 1}} V_{p, i + 1}$ as this is a quasi-compact open of $V_{p, i + 1}$ and hence $P$ holds for it by the first paragraph of this proof. Thus (2) applies to each of these and we inductively conclude $P$ holds for $W_{p, 1}, \ldots, W_{p, m} = U_p$. $\square$

Lemma. Derived gluing across an elementary distinguished square

Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Suppose given

1. an object $A$ of $D(\mathcal{O}_U)$,

2. an object $B$ of $D(\mathcal{O}_V)$, and

3. an isomorphism $c : A|_{U \times_X V} \to B|_{U \times_X V}$.

Then there exists an object $F$ of $D(\mathcal{O}_X)$ and isomorphisms $f : F|_U \to A$, $g : F|_V \to B$ such that $c = g|_{U \times_X V} \circ f^{-1}|_{U \times_X V}$. Moreover, given

1. an object $E$ of $D(\mathcal{O}_X)$,

2. a morphism $a : A \to E|_U$ of $D(\mathcal{O}_U)$,

3. a morphism $b : B \to E|_V$ of $D(\mathcal{O}_V)$,

such that $$a|_{U \times_X V} = b|_{U \times_X V} \circ c.$$ Then there exists a morphism $F \to E$ in $D(\mathcal{O}_X)$ whose restriction to $U$ is $a \circ f$ and whose restriction to $V$ is $b \circ g$.

Proof. Denote $j_U$, $j_V$, $j_{U \times_X V}$ the corresponding morphisms towards $X$. Choose a distinguished triangle $$F \to Rj_{U, *}A \oplus Rj_{V, *}B \to Rj_{U \times_X V, *}(B|_{U \times_X V}) \to F[1]$$ Here the map $Rj_{V, *}B \to Rj_{U \times_X V, *}(B|_{U \times_X V})$ is the obvious one. The map $Rj_{U, *}A \to Rj_{U \times_X V, *}(B|_{U \times_X V})$ is the composition of $Rj_{U, *}A \to Rj_{U \times_X V, *}(A|_{U \times_X V})$ with $Rj_{U \times_X V, *}c$. Restricting to $U$ we obtain $$F|_U \to A \oplus (Rj_{V, *}B)|_U \to (Rj_{U \times_X V, *}(B|_{U \times_X V}))|_U \to F|_U[1]$$ Denote $j : U \times_X V \to U$. Compatibility of restriction and total direct image (Lemma Derived quasi-coherent complexes) shows that both $(Rj_{V, *}B)|_U$ and $(Rj_{U \times_X V, *}(B|_{U \times_X V}))|_U$ are canonically isomorphic to $Rj_*(B|_{U \times_X V})$. Hence the second arrow of the last displayed equation has a section, and we conclude that the morphism $F|_U \to A$ is an isomorphism.

To see that the morphism $F|_V \to B$ is an isomorphism we will use a trick. Namely, choose a distinguished triangle $$F|_V \to B \to B' \to F[1]|_V$$ in $D(\mathcal{O}_V)$. Since $F|_U \to A$ is an isomorphism, and since we have the isomorphism $c : A|_{U \times_X V} \to B|_{U \times_X V}$ the restriction of $F|_V \to B$ is an isomorphism over $U \times_X V$. Thus $B'$ is supported on $j_V^{-1}(T)$ where $T = |X| \setminus |U|$. On the other hand, there is a morphism of distinguished triangles $$\begin{gathered}\begin{matrix}F & Rj_{U, *}F|_U \oplus Rj_{V, *}F|_V & Rj_{U \times_X V, *}F|_{U \times_X V} & F[1] \\ F & Rj_{U, *}A \oplus Rj_{V, *}B & Rj_{U \times_X V, *}(B|_{U \times_X V}) & F[1]\end{matrix} \\[6pt] \begin{aligned}F & \longrightarrow Rj_{U, *}F|_U \oplus Rj_{V, *}F|_V \\ F & \longrightarrow F \\ Rj_{U, *}F|_U \oplus Rj_{V, *}F|_V & \longrightarrow Rj_{U \times_X V, *}F|_{U \times_X V} \\ Rj_{U, *}F|_U \oplus Rj_{V, *}F|_V & \longrightarrow Rj_{U, *}A \oplus Rj_{V, *}B \\ Rj_{U \times_X V, *}F|_{U \times_X V} & \longrightarrow F[1] \\ Rj_{U \times_X V, *}F|_{U \times_X V} & \longrightarrow Rj_{U \times_X V, *}(B|_{U \times_X V}) \\ F[1] & \longrightarrow F[1] \\ F & \longrightarrow Rj_{U, *}A \oplus Rj_{V, *}B \\ Rj_{U, *}A \oplus Rj_{V, *}B & \longrightarrow Rj_{U \times_X V, *}(B|_{U \times_X V}) \\ Rj_{U \times_X V, *}(B|_{U \times_X V}) & \longrightarrow F[1]\end{aligned}\end{gathered}$$ The all of the vertical maps in this diagram are isomorphisms, except for the map $Rj_{V, *}F|_V \to Rj_{V, *}B$, hence that is an isomorphism too (Derived Categories, Lemma Derived categories). This implies that $Rj_{V, *}B' = 0$. Hence $B' = 0$ by Lemma Direct images and closed support.

The existence of the morphism $F \to E$ follows from the Mayer-Vietoris sequence for $\operatorname{Hom}$, see Lemma Derived Hom and Ext. $\square$

Definition. Bounds for perfect approximation

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider triples $(T, E, m)$ where

1. $T \subset |X|$ is a closed subset,

2. $E$ is an object of $D_\mathrm{QCoh}(\mathcal{O}_X)$, and

3. $m \in \mathbf{Z}$.

We say approximation holds for the triple $(T, E, m)$ if there exists a perfect object $P$ of $D(\mathcal{O}_X)$ supported on $T$ and a map $\alpha : P \to E$ which induces isomorphisms $H^i(P) \to H^i(E)$ for $i > m$ and a surjection $H^m(P) \to H^m(E)$.

Definition. Perfect approximation with prescribed cohomology

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say approximation by perfect complexes holds on $X$ if for any closed subset $T \subset |X|$ such that the morphism $X \setminus T \to X$ is quasi-compact there exists an integer $r$ such that for every triple $(T, E, m)$ as in Definition Bounds for perfect approximation with

1. $E$ is $(m - r)$-pseudo-coherent, and

2. $H^i(E)$ is supported on $T$ for $i \geq m - r$

approximation holds.

Lemma. Direct images and perfect complexes

Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic space over $S$. Let $E$ be a perfect object of $D(\mathcal{O}_V)$ supported on $j^{-1}(T)$ where $T = |X| \setminus |U|$. Then $Rj_*E$ is a perfect object of $D(\mathcal{O}_X)$.

Proof. Being perfect is local on $X_\mathrm{\acute{e}tale}$. Thus it suffices to check that $Rj_*E$ is perfect when restricted to $U$ and $V$. We have $Rj_*E|_V = E$ by Lemma Direct images and closed support which is perfect. We have $Rj_*E|_U = 0$ because $E|_{V \setminus j^{-1}(T)} = 0$ (use Lemma Derived quasi-coherent complexes). $\square$

Lemma. Restriction to an open subspace

Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Let $T$ be a closed subset of $|X| \setminus |U|$ and let $(T, E, m)$ be a triple as in Definition Bounds for perfect approximation. If

1. approximation holds for $(j^{-1}T, E|_V, m)$, and

2. the sheaves $H^i(E)$ for $i \geq m$ are supported on $T$,

then approximation holds for $(T, E, m)$.

Proof. Let $P \to E|_V$ be an approximation of the triple $(j^{-1}T, E|_V, m)$ over $V$. Then $Rj_*P$ is a perfect object of $D(\mathcal{O}_X)$ by Lemma Direct images and perfect complexes. On the other hand, $Rj_*P = j_!P$ by Lemma Direct images and closed support. We see that $j_!P$ is supported on $T$ for example by (Sheaves on ringed sites). Hence we obtain an approximation $Rj_*P = j_!P \to j_!(E|_V) \to E$. $\square$

Lemma. Perfect approximation on an affine chart

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ which is representable by an affine scheme. Then approximation holds for every triple $(T, E, m)$ as in Definition Bounds for perfect approximation such that there exists an integer $r \geq 0$ with

1. $E$ is $m$-pseudo-coherent,

2. $H^i(E)$ is supported on $T$ for $i \geq m - r + 1$,

3. $X \setminus T$ is the union of $r$ affine opens.

In particular, approximation by perfect complexes holds for affine schemes.

Proof. Let $X_0$ be an affine scheme representing $X$. Let $T_0 \subset X_0$ by the closed subset corresponding to $T$. Let $\epsilon : X_\mathrm{\acute{e}tale} \to X_{0, Zar}$ be the morphism (Derived quasi-coherent complexes). We may write $E = \epsilon^*E_0$ for some object $E_0$ of $D_\mathrm{QCoh}(\mathcal{O}_{X_0})$, see Lemma Étale morphisms and quasi-coherent complexes. Then $E_0$ is $m$-pseudo-coherent, see Lemma Descent of pseudo-coherent complexes and coherent sheaves. Comparing stalks of cohomology sheaves (see proof of Lemma Flatness) we see that $H^i(E_0)$ is supported on $T_0$ for $i \geq m - r + 1$. By Derived Categories of Schemes, Lemma Perfect approximation on an affine chart there exists an approximation $P_0 \to E_0$ of $(T_0, E_0, m)$. By Lemma Descent of perfect complexes we see that $P = \epsilon^*P_0$ is a perfect object of $D(\mathcal{O}_X)$. Pulling back we obtain an approximation $P = \epsilon^*P_0 \to \epsilon^*E_0 = E$ as desired. $\square$

Lemma. Extending perfect approximation across a distinguished square

Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Assume $U$ quasi-compact, $V$ affine, and $U \times_X V$ quasi-compact. If approximation by perfect complexes holds on $U$, then approximation by perfect complexes holds on $X$.

Proof. Let $T \subset |X|$ be a closed subset with $X \setminus T \to X$ quasi-compact. Let $r_U$ be the integer of Definition Perfect approximation with prescribed cohomology adapted to the pair $(U, T \cap |U|)$. Set $T' = T \setminus |U|$. Endow $T'$ with the induced reduced subspace structure. Since $|T'|$ is contained in $|X| \setminus |U|$ we see that $j^{-1}(T') \to T'$ is an isomorphism. Moreover, $V \setminus j^{-1}(T')$ is quasi-compact as it is the fibre product of $U \times_X V$ with $X \setminus T$ over $X$ and we've assumed $U \times_X V$ quasi-compact and $X \setminus T \to X$ quasi-compact. Let $r'$ be the number of affines needed to cover $V \setminus j^{-1}(T')$. We claim that $r = \max(r_U, r')$ works for the pair $(X, T)$.

To see this choose a triple $(T, E, m)$ such that $E$ is $(m - r)$-pseudo-coherent and $H^i(E)$ is supported on $T$ for $i \geq m - r$. Let $t$ be the largest integer such that $H^t(E)|_U$ is nonzero. (Such an integer exists as $U$ is quasi-compact and $E|_U$ is $(m - r)$-pseudo-coherent.) We will prove that $E$ can be approximated by induction on $t$.

Base case: $t \leq m - r'$. This means that $H^i(E)$ is supported on $T'$ for $i \geq m - r'$. Hence Lemma Perfect approximation on an affine chart guarantees the existence of an approximation $P \to E|_V$ of $(T', E|_V, m)$ on $V$. Applying Lemma Restriction to an open subspace we see that $(T', E, m)$ can be approximated. Such an approximation is also an approximation of $(T, E, m)$.

Induction step. Choose an approximation $P \to E|_U$ of $(T \cap |U|, E|_U, m)$. This in particular gives a surjection $H^t(P) \to H^t(E|_U)$. In the rest of the proof we will use the equivalence of Lemma Étale morphisms and quasi-coherent complexes (and the compatibilities of Remark Derived quasi-coherent complexes) for the representable algebraic spaces $V$ and $U \times_X V$. We will also use the fact that $(m - r)$-pseudo-coherence, resp. perfectness on the Zariski site and étale site agree, see Lemmas Descent of pseudo-coherent complexes and coherent sheaves and Descent of perfect complexes. Thus we can use the results of Derived Categories of Schemes, Section Lifting perfect complexes for the open immersion $U \times_X V \subset V$. In this way Derived Categories of Schemes, Lemma Lifting perfect complexes while retaining support implies there exists a perfect object $Q$ in $D(\mathcal{O}_V)$ supported on $j^{-1}(T)$ and an isomorphism $Q|_{U \times_X V} \to (P \oplus P[1])|_{U \times_X V}$. By Derived Categories of Schemes, Lemma Extending a morphism after finite denominators are cleared we can replace $Q$ by $Q \otimes^\mathbf{L} I$ and assume that the map $$Q|_{U \times_X V} \longrightarrow (P \oplus P[1])|_{U \times_X V} \longrightarrow P|_{U \times_X V} \longrightarrow E|_{U \times_X V}$$ lifts to $Q \to E|_V$. By Lemma Derived gluing across an elementary distinguished square we find an morphism $a : R \to E$ of $D(\mathcal{O}_X)$ such that $a|_U$ is isomorphic to $P \oplus P[1] \to E|_U$ and $a|_V$ isomorphic to $Q \to E|_V$. Thus $R$ is perfect and supported on $T$ and the map $H^t(R) \to H^t(E)$ is surjective on restriction to $U$. Choose a distinguished triangle $$R \to E \to E' \to R[1]$$ Then $E'$ is $(m - r)$-pseudo-coherent (Cohomology on Sites, Lemma Pseudo-coherent complexes and coherent sheaves), $H^i(E')|_U = 0$ for $i \geq t$, and $H^i(E')$ is supported on $T$ for $i \geq m - r$. By induction we find an approximation $R' \to E'$ of $(T, E', m)$. Fit the composition $R' \to E' \to R[1]$ into a distinguished triangle $R \to R'' \to R' \to R[1]$ and extend the morphisms $R' \to E'$ and $R[1] \to R[1]$ into a morphism of distinguished triangles $$\begin{gathered}\begin{matrix}R & R'' & R' & R[1] \\ R & E & E' & R[1]\end{matrix} \\[6pt] \begin{aligned}R & \longrightarrow R'' \\ R & \longrightarrow R \\ R'' & \longrightarrow E \\ R'' & \longrightarrow R' \\ R' & \longrightarrow E' \\ R' & \longrightarrow R[1] \\ R[1] & \longrightarrow R[1] \\ R & \longrightarrow E \\ E & \longrightarrow E' \\ E' & \longrightarrow R[1]\end{aligned}\end{gathered}$$ using TR3. Then $R''$ is a perfect complex (Cohomology on Sites, Lemma Perfect complexes) supported on $T$. An easy diagram chase shows that $R'' \to E$ is the desired approximation. $\square$

Theorem. Perfect approximation with prescribed closed support

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Then approximation by perfect complexes holds on $X$.

Proof. This follows from the induction principle of Lemma Induction by elementary distinguished squares and Lemmas Extending perfect approximation across a distinguished square and Perfect approximation on an affine chart. $\square$

Theorem. A perfect generator for a quasi-compact algebraic space

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The category $D_\mathrm{QCoh}(\mathcal{O}_X)$ can be generated by a single perfect object. More precisely, there exists a perfect object $P$ of $D(\mathcal{O}_X)$ such that for $E \in D_\mathrm{QCoh}(\mathcal{O}_X)$ the following are equivalent

1. $E = 0$, and

2. $\operatorname{Hom}_{D(\mathcal{O}_X)}(P[n], E) = 0$ for all $n \in \mathbf{Z}$.

Proof. We will prove this using the induction principle of Lemma Induction by elementary distinguished squares.

If $X$ is affine, then $\mathcal{O}_X$ is a perfect generator. This follows from Lemma Étale morphisms and quasi-coherent complexes and Derived Categories of Schemes, Lemma Bounded comparison of affine derived categories.

Assume that $(U \subset X, f : V \to X)$ is an elementary distinguished square with $U$ quasi-compact such that the theorem holds for $U$ and $V$ is an affine scheme. Let $P$ be a perfect object of $D(\mathcal{O}_U)$ which is a generator for $D_\mathrm{QCoh}(\mathcal{O}_U)$. Using Lemma Tensor products and direct sums we may choose a perfect object $Q$ of $D(\mathcal{O}_X)$ whose restriction to $U$ is a direct sum one of whose summands is $P$. Say $V = \operatorname{Spec}(A)$. Let $Z \subset V$ be the reduced closed subscheme which is the inverse image of $X \setminus U$ and maps isomorphically to it (see Definition Derived quasi-coherent complexes). This is a retrocompact closed subset of $V$. Choose $f_1, \ldots, f_r \in A$ such that $Z = V(f_1, \ldots, f_r)$. Let $K \in D(\mathcal{O}_V)$ be the perfect object corresponding to the Koszul complex on $f_1, \ldots, f_r$ over $A$. Note that since $K$ is supported on $Z$, the pushforward $K' = Rf_*K$ is a perfect object of $D(\mathcal{O}_X)$ whose restriction to $V$ is $K$ (see Lemmas Direct images and perfect complexes and Direct images and closed support). We claim that $Q \oplus K'$ is a generator for $D_\mathrm{QCoh}(\mathcal{O}_X)$.

Let $E$ be an object of $D_\mathrm{QCoh}(\mathcal{O}_X)$ such that there are no nontrivial maps from any shift of $Q \oplus K'$ into $E$. By Lemma Direct images and closed support we have $K' = f_! K$ and hence $$\operatorname{Hom}_{D(\mathcal{O}_X)}(K'[n], E) = \operatorname{Hom}_{D(\mathcal{O}_V)}(K[n], E|_V)$$ Thus by Derived Categories of Schemes, Lemma Koszul complexes, regular sequences and derived categories (using also Lemma Étale morphisms and quasi-coherent complexes) the vanishing of these groups implies that $E|_V$ is isomorphic to $R(U \times_X V \to V)_*E|_{U \times_X V}$. This implies that $E = R(U \to X)_*E|_U$ (small detail omitted). If this is the case then $$\operatorname{Hom}_{D(\mathcal{O}_X)}(Q[n], E) = \operatorname{Hom}_{D(\mathcal{O}_U)}(Q|_U[n], E|_U)$$ which contains $\operatorname{Hom}_{D(\mathcal{O}_U)}(P[n], E|_U)$ as a direct summand. Thus by our choice of $P$ the vanishing of these groups implies that $E|_U$ is zero. Whence $E$ is zero. $\square$

Lemma. The derived quasi-coherator

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The inclusion functor $D_\mathrm{QCoh}(\mathcal{O}_X) \to D(\mathcal{O}_X)$ has a right adjoint.

First proof. We will use the induction principle in Lemma Induction by elementary distinguished squares to prove this. If $D(\mathrm{QCoh}(\mathcal{O}_X)) \to D_\mathrm{QCoh}(\mathcal{O}_X)$ is an equivalence, then the lemma is true because the functor $RQ_X$ of Section Derived tensor products and Tor amplitude is a right adjoint to the functor $D(\mathrm{QCoh}(\mathcal{O}_X)) \to D(\mathcal{O}_X)$. In particular, our lemma is true for affine algebraic spaces, see Lemma Derived tensor products, Tor amplitude and affine neighbourhoods. Thus we see that it suffices to show: if $(U \subset X, f : V \to X)$ is an elementary distinguished square with $U$ quasi-compact and $V$ affine and the lemma holds for $U$, $V$, and $U \times_X V$, then the lemma holds for $X$.

The adjoint exists if and only if for every object $E$ of $D(\mathcal{O}_X)$ we can find a distinguished triangle $$E' \to E \to K \to E'[1]$$ in $D(\mathcal{O}_X)$ such that $E'$ is in $D_\mathrm{QCoh}(\mathcal{O}_X)$ and such that $\operatorname{Hom}(M, K) = 0$ for all $M$ in $D_\mathrm{QCoh}(\mathcal{O}_X)$. See Derived Categories, Lemma Existence of a right adjoint. Consider the distinguished triangle $$E \to Rj_{U, *}E|_U \oplus Rj_{V, *}E|_V \to Rj_{U \times_X V, *}E|_{U \times_X V} \to E[1]$$ in $D(\mathcal{O}_X)$ of Lemma Derived quasi-coherent complexes. By Derived Categories, Lemma Preparing the adjunction for derived functors it suffices to construct the desired distinguished triangles for $Rj_{U, *}E|_U$, $Rj_{V, *}E|_V$, and $Rj_{U \times_X V, *}E|_{U \times_X V}$. This reduces us to the statement discussed in the next paragraph.

Let $j : U \to X$ be an étale morphism corresponding with $U$ quasi-compact and quasi-separated and the lemma is true for $U$. Let $L$ be an object of $D(\mathcal{O}_U)$. Then there exists a distinguished triangle $$E' \to Rj_*L \to K \to E'[1]$$ in $D(\mathcal{O}_X)$ such that $E'$ is in $D_\mathrm{QCoh}(\mathcal{O}_X)$ and such that $\operatorname{Hom}(M, K) = 0$ for all $M$ in $D_\mathrm{QCoh}(\mathcal{O}_X)$. To see this we choose a distinguished triangle $$L' \to L \to Q \to L'[1]$$ in $D(\mathcal{O}_U)$ such that $L'$ is in $D_\mathrm{QCoh}(\mathcal{O}_U)$ and such that $\operatorname{Hom}(N, Q) = 0$ for all $N$ in $D_\mathrm{QCoh}(\mathcal{O}_U)$. This is possible because the statement in Derived Categories, Lemma Existence of a right adjoint is an if and only if. We obtain a distinguished triangle $$Rj_*L' \to Rj_*L \to Rj_*Q \to Rj_*L'[1]$$ in $D(\mathcal{O}_X)$. Observe that $Rj_*L'$ is in $D_\mathrm{QCoh}(\mathcal{O}_X)$ by Lemma Quasi-coherent complexes and coherent sheaves. On the other hand, if $M$ in $D_\mathrm{QCoh}(\mathcal{O}_X)$, then $$\operatorname{Hom}(M, Rj_*Q) = \operatorname{Hom}(Lj^*M, Q) = 0$$ because $Lj^*M$ is in $D_\mathrm{QCoh}(\mathcal{O}_U)$ by Lemma Quasi-coherent complexes and coherent sheaves. This finishes the proof. $\square$

Second proof. The adjoint exists by Derived Categories, Proposition Brown representability for a triangulated category. The hypotheses are satisfied: First, note that $D_\mathrm{QCoh}(\mathcal{O}_X)$ has direct sums and direct sums commute with the inclusion functor (Lemma Quasi-coherent complexes and coherent sheaves). On the other hand, $D_\mathrm{QCoh}(\mathcal{O}_X)$ is compactly generated because it has a perfect generator Theorem A perfect generator for a quasi-compact algebraic space and because perfect objects are compact by Proposition Compact objects are perfect. $\square$

Lemma. Direct images and the derived quasi-coherator

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. If the right adjoints $DQ_X$ and $DQ_Y$ of the inclusion functors $D_\mathrm{QCoh} \to D$ exist for $X$ and $Y$, then $$Rf_* \circ DQ_X = DQ_Y \circ Rf_*$$

Proof. The statement makes sense because $Rf_*$ sends $D_\mathrm{QCoh}(\mathcal{O}_X)$ into $D_\mathrm{QCoh}(\mathcal{O}_Y)$ by Lemma Quasi-coherent complexes and coherent sheaves. The statement is true because $Lf^*$ similarly maps $D_\mathrm{QCoh}(\mathcal{O}_Y)$ into $D_\mathrm{QCoh}(\mathcal{O}_X)$ (Lemma Quasi-coherent complexes and coherent sheaves) and hence both $Rf_* \circ DQ_X$ and $DQ_Y \circ Rf_*$ are right adjoint to $Lf^* : D_\mathrm{QCoh}(\mathcal{O}_Y) \to D(\mathcal{O}_X)$. $\square$

Lemma. Bounds for the derived quasi-coherator

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The functor $DQ_X$ of Lemma The derived quasi-coherator has the following boundedness property: there exists an integer $N = N(X)$ such that, if $K$ in $D(\mathcal{O}_X)$ with $H^i(U, K) = 0$ for $U$ affine étale over $X$ and $i \not \in [a, b]$, then the cohomology sheaves $H^i(DQ_X(K))$ are zero for $i \not \in [a, b + N]$.

Proof. We will prove this using the induction principle of Lemma Induction by elementary distinguished squares.

If $X$ is affine, then the lemma is true with $N = 0$ because then $RQ_X = DQ_X$ is given by taking the complex of quasi-coherent sheaves associated to $R\Gamma(X, K)$. See Lemma Derived tensor products, Tor amplitude and affine neighbourhoods.

Let $(U \subset W, f : V \to W)$ be an elementary distinguished square with $W$ quasi-compact and quasi-separated, $U \subset W$ quasi-compact open, $V$ affine such that the lemma holds for $U$, $V$, and $U \times_W V$. Say with integers $N(U)$, $N(V)$, and $N(U \times_W V)$. Now suppose $K$ is in $D(\mathcal{O}_X)$ with $H^i(W, K) = 0$ for all affine $W$ étale over $X$ and all $i \not \in [a, b]$. Then $K|_U$, $K|_V$, $K|_{U \times_W V}$ have the same property. Hence we see that $RQ_U(K|_U)$ and $RQ_V(K|_V)$ and $RQ_{U \cap V}(K|_{U \times_W V})$ have vanishing cohomology sheaves outside the interval $[a, b + \max(N(U), N(V), N(U \times_W V))$. Since the functors $Rj_{U, *}$, $Rj_{V, *}$, $Rj_{U \times_W V, *}$ have finite cohomological dimension on $D_\mathrm{QCoh}$ by Lemma Quasi-coherent complexes and coherent sheaves we see that there exists an $N$ such that $Rj_{U, *}DQ_U(K|_U)$, $Rj_{V, *}DQ_V(K|_V)$, and $Rj_{U \cap V, *}DQ_{U \times_W V}(K|_{U \times_W V})$ have vanishing cohomology sheaves outside the interval $[a, b + N]$. Then finally we conclude by the distinguished triangle of Remark Derived quasi-coherent complexes. $\square$

Lemma. Canonical arbitrary base change for relatively flat tensors

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $E \in D_\mathrm{QCoh}(\mathcal{O}_X)$. Let $\mathcal{G}^\bullet$ be a bounded above complex of quasi-coherent $\mathcal{O}_X$-modules flat over $Y$. Then formation of $$Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet)$$ commutes with arbitrary base change (see proof for precise statement).

Proof. The statement means the following. Let $g : Y' \to Y$ be a morphism of algebraic spaces and consider the base change diagram $$\begin{gathered}\begin{matrix}X' & X \\ Y' & Y\end{matrix} \\[6pt] \begin{aligned}X' & \xrightarrow{g'} X \\ X' & \xrightarrow{f'} Y' \\ X & \xrightarrow{f} Y \\ Y' & \xrightarrow{g} Y\end{aligned}\end{gathered}$$ in other words $X' = Y' \times_Y X$. The lemma asserts that $$Lg^*Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet) \longrightarrow Rf'_*(L(g')^*E \otimes^\mathbf{L}_{\mathcal{O}_{X'}} (g')^*\mathcal{G}^\bullet)$$ is an isomorphism. Observe that on the right hand side we do not use derived pullback on $\mathcal{G}^\bullet$. To prove this, we apply Lemmas Base change for derived categories and Inheritance of a cohomological base-change condition to see that it suffices to prove the canonical map $$L(g')^*\mathcal{G}^\bullet \to (g')^*\mathcal{G}^\bullet$$ satisfies the equivalent conditions of Lemma Base change for derived categories. This follows by checking the condition on stalks, where it immediately follows from the fact that $\mathcal{G}^\bullet_{\overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{Y', \overline{y}'}$ computes the derived tensor product by our assumptions on the complex $\mathcal{G}^\bullet$. $\square$

Lemma. Canonical arbitrary base change for derived Hom

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $\mathcal{G}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_X$-modules. If

1. $E$ is perfect, $\mathcal{G}^\bullet$ is a bounded above, and $\mathcal{G}^n$ is flat over $Y$, or

2. $E$ is pseudo-coherent, $\mathcal{G}^\bullet$ is bounded, and $\mathcal{G}^n$ is flat over $Y$,

then formation of $$Rf_*R\mathcal{H}om(E, \mathcal{G}^\bullet)$$ commutes with arbitrary base change (see proof for precise statement).

Proof. The statement means the following. Let $g : Y' \to Y$ be a morphism of algebraic spaces and consider the base change diagram $$\begin{gathered}\begin{matrix}X' & X \\ Y' & Y\end{matrix} \\[6pt] \begin{aligned}X' & \xrightarrow{h} X \\ X' & \xrightarrow{f'} Y' \\ X & \xrightarrow{f} Y \\ Y' & \xrightarrow{g} Y\end{aligned}\end{gathered}$$ in other words $X' = Y' \times_Y X$. The lemma asserts that $$Lg^*Rf_*R\mathcal{H}om(E, \mathcal{G}^\bullet) \longrightarrow R(f')_*R\mathcal{H}om(L(g')^*E, (g')^*\mathcal{G}^\bullet)$$ is an isomorphism. Observe that on the right hand side we do not use the derived pullback on $\mathcal{G}^\bullet$. To prove this, we apply Lemmas Base change for derived categories and Inheritance of a cohomological base-change condition to see that it suffices to prove the canonical map $$L(g')^*\mathcal{G}^\bullet \to (g')^*\mathcal{G}^\bullet$$ satisfies the equivalent conditions of Lemma Base change for derived categories. This was shown in the proof of Lemma Canonical arbitrary base change for relatively flat tensors. $\square$

Lemma. Perfectness of a proper direct image

Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \in D(\mathcal{O}_X)$ such that

1. $E \in D^b_{\textit{Coh}}(\mathcal{O}_X)$,

2. the support of $H^i(E)$ is proper over $Y$ for all $i$,

3. $E$ has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_Y)$.

Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_Y)$.

Proof. By Lemma Direct images and coherent sheaves we see that $Rf_*E$ is an object of $D^b_{\textit{Coh}}(\mathcal{O}_Y)$. Hence $Rf_*E$ is pseudo-coherent (Lemma Pseudo-coherent complexes and coherent sheaves). Hence it suffices to show that $Rf_*E$ has finite tor dimension, see Cohomology on Sites, Lemma Perfect complexes. By Lemma Tor amplitude on a quasi-compact quasi-separated scheme it suffices to check that $Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}$ has universally bounded cohomology for all quasi-coherent sheaves $\mathcal{F}$ on $Y$. Bounded from above is clear as $Rf_*(E)$ is bounded from above. Let $T \subset |X|$ be the union of the supports of $H^i(E)$ for all $i$. Then $T$ is proper over $Y$ by assumptions (1) and (2) and Lemma Proper morphisms. In particular there exists a quasi-compact open subspace $X' \subset X$ containing $T$. Setting $f' = f|_{X'}$ we have $Rf_*(E) = Rf'_*(E|_{X'})$ because $E$ restricts to zero on $X \setminus T$. Thus we may replace $X$ by $X'$ and assume $f$ is quasi-compact. We have assumed $f$ is quasi-separated. Thus $$Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F} = Rf_*\left(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}\right) = Rf_*\left(E \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} f^{-1}\mathcal{F}\right)$$ by Lemma Base change for sheaf cohomology and Cohomology on Sites, Lemma Derived categories. By assumption (3) the complex $E \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} f^{-1}\mathcal{F}$ has cohomology sheaves in a given finite range, say $[a, b]$. Then $Rf_*$ of it has cohomology in the range $[a, \infty)$ and we win. $\square$

Lemma. Perfect complexes and tensor products and direct sums

Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$. Let $f : X \to B$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \in D(\mathcal{O}_X)$ be perfect. Let $\mathcal{G}^\bullet$ be a bounded complex of coherent $\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then $K = Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet)$ is a perfect object of $D(\mathcal{O}_B)$.

Proof. The object $K$ is perfect by Lemma Perfectness of a proper direct image. We check the lemma applies: Locally $E$ is isomorphic to a finite complex of finite free $\mathcal{O}_X$-modules. Hence locally $E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet$ is isomorphic to a finite complex whose terms are of the form $$\bigoplus\nolimits_{i = a, \ldots, b} (\mathcal{G}^i)^{\oplus r_i}$$ for some integers $a, b, r_a, \ldots, r_b$. This immediately implies the cohomology sheaves $H^i(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G})$ are coherent. The hypothesis on the tor dimension also follows as $\mathcal{G}^i$ is flat over $f^{-1}\mathcal{O}_Y$. $\square$

Lemma. Perfect complexes and derived Hom and Ext

Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$. Let $f : X \to B$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \in D(\mathcal{O}_X)$ be perfect. Let $\mathcal{G}^\bullet$ be a bounded complex of coherent $\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then $K = Rf_*R\mathcal{H}om(E, \mathcal{G})$ is a perfect object of $D(\mathcal{O}_B)$.

Proof. Since $E$ is a perfect complex there exists a dual perfect complex $E^\vee$, see Cohomology on Sites, Lemma Perfect complexes and derived categories. Observe that $R\mathcal{H}om(E, \mathcal{G}^\bullet) = E^\vee \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet$. Thus the perfectness of $K$ follows from Lemma Perfect complexes and tensor products and direct sums. $\square$

Lemma. Perfect complexes and tensor products and direct sums

Assumptions and notation as in Lemma Perfect complexes and tensor products and direct sums. Then there are functorial isomorphisms $$H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \longrightarrow H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}))$$ for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof).

Proof. We have $$\mathcal{G}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F} = \mathcal{G}^\bullet \otimes_{f^{-1}\mathcal{O}_B}^\mathbf{L} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes_{f^{-1}\mathcal{O}_B} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}$$ the first equality by Cohomology on Sites, Lemma Derived categories, the second as $\mathcal{G}^n$ is a flat $f^{-1}\mathcal{O}_B$-module, and the third by definition of pullbacks. Hence we obtain $$\begin{aligned} H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})) & = H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}) \\ & = H^i(B, Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet \otimes^\mathbf{L}_{\mathcal{O}_X} Lf^*\mathcal{F})) \\ & = H^i(B, Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet) \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \\ & = H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \end{aligned}$$ The first equality by the above, the second by Leray (Cohomology on Sites, Remark Sheaf cohomology), and the third equality by Lemma Base change for sheaf cohomology. The statement on boundary maps means the following: Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ then the isomorphisms fit into commutative diagrams $$\begin{gathered}\begin{matrix}H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3) & H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3)) \\ H^{i + 1}(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1) & H^{i + 1}(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1))\end{matrix} \\[6pt] \begin{aligned}H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3) & \longrightarrow H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3)) \\ H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3) & \xrightarrow{\delta} H^{i + 1}(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1) \\ H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3)) & \xrightarrow{\delta} H^{i + 1}(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1)) \\ H^{i + 1}(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1) & \longrightarrow H^{i + 1}(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1))\end{aligned}\end{gathered}$$ where the boundary maps come from the distinguished triangle $$K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1 \to K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_2 \to K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3 \to K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1[1]$$ and the distinguished triangle in $D(\mathcal{O}_X)$ associated to the short exact sequence $$0 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3 \to 0$$ of complexes. This sequence is exact because $\mathcal{G}^n$ is flat over $B$. We omit the verification of the commutativity of the displayed diagram. $\square$

Lemma. Perfect complexes and derived Hom and Ext

Assumption and notation as in Lemma Perfect complexes and derived Hom and Ext. Then there are functorial isomorphisms $$H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \longrightarrow \operatorname{Ext}^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})$$ for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof).

Proof. As in the proof of Lemma Perfect complexes and derived Hom and Ext let \(E^\vee\) be the dual perfect complex and recall that \(K = Rf_*(E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\). Since we also have

by construction of \(E^\vee\), the existence of the isomorphisms follows from Lemma Perfect complexes and tensor products and direct sums applied to \(E^\vee\) and \(\mathcal{G}^\bullet\). The statement on boundary maps means the following: Given a short exact sequence \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) then the isomorphisms fit into commutative diagrams

where the boundary maps come from the distinguished triangle

and the distinguished triangle in \(D(\mathcal{O}_X)\) associated to the short exact sequence

of complexes. This sequence is exact because \(\mathcal{G}^n\) is flat over \(B\). We omit the verification of the commutativity of the displayed diagram. \(\square\)

Lemma. Derived Hom and Ext

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$, $E \in D(\mathcal{O}_X)$, and $\mathcal{F}^\bullet$ a complex of $\mathcal{O}_X$-modules. Assume

1. $B$ is Noetherian,

2. $f$ is locally of finite type and quasi-separated,

3. $E \in D^-_{\textit{Coh}}(\mathcal{O}_X)$,

4. $\mathcal{G}^\bullet$ is a bounded complex of coherent $\mathcal{O}_X$-module flat over $B$ with support proper over $B$.

Then the following two statements are true

1. for every $m \in \mathbf{Z}$ there exists a perfect object $K$ of $D(\mathcal{O}_B)$ and functorial maps $$\alpha^i_\mathcal{F} : \operatorname{Ext}^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \longrightarrow H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F})$$ for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof) such that $\alpha^i_\mathcal{F}$ is an isomorphism for $i \leq m$, and

2. there exists a pseudo-coherent $L \in D(\mathcal{O}_B)$ and functorial isomorphisms $$\operatorname{Ext}^i_{\mathcal{O}_B}(L, \mathcal{F}) \longrightarrow \operatorname{Ext}^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})$$ for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps.

Proof. Proof of (A). Suppose $\mathcal{G}^i$ is nonzero only for $i \in [a, b]$. We may replace $X$ by a quasi-compact open neighbourhood of the union of the supports of $\mathcal{G}^i$. Hence we may assume $X$ is Noetherian. In this case $X$ and $f$ are quasi-compact and quasi-separated. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, -m - 1 + a)$ (possible by Theorem Perfect approximation with prescribed closed support). Then the induced map $$\operatorname{Ext}^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \longrightarrow \operatorname{Ext}^i_{\mathcal{O}_X}(P, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})$$ is an isomorphism for $i \leq m$. Namely, the kernel, resp. cokernel of this map is a quotient, resp. submodule of $$\operatorname{Ext}^i_{\mathcal{O}_X}(C, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \quad\text{resp.}\quad \operatorname{Ext}^{i + 1}_{\mathcal{O}_X}(C, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})$$ where $C$ is the cone of $P \to E$. Since $C$ has vanishing cohomology sheaves in degrees $\geq -m - 1 + a$ these $\operatorname{Ext}$-groups are zero for $i \leq m + 1$ by Derived Categories, Lemma Vanishing of negative Ext groups. This reduces us to the case that $E$ is a perfect complex which is Lemma Perfect complexes and derived Hom and Ext. The statement on boundaries is explained in the proof of Lemma Perfect complexes and derived Hom and Ext.

Proof of (B). As in the proof of (A) we may assume $X$ is Noetherian. Observe that $E$ is pseudo-coherent by Lemma Pseudo-coherent complexes and coherent sheaves. By Lemma Pseudo-coherent complexes and coherent sheaves we can write $E = \text{hocolim} E_n$ with $E_n$ perfect and $E_n \to E$ inducing an isomorphism on truncations $\tau_{\geq -n}$. Let $E_n^\vee$ be the dual perfect complex (Cohomology on Sites, Lemma Perfect complexes and derived categories). We obtain an inverse system $\ldots \to E_3^\vee \to E_2^\vee \to E_1^\vee$ of perfect objects. This in turn gives rise to an inverse system $$\ldots \to K_3 \to K_2 \to K_1\quad\text{with}\quad K_n = Rf_*(E_n^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)$$ perfect on $Y$, see Lemma Perfect complexes and tensor products and direct sums. By Lemma Perfect complexes and derived Hom and Ext and its proof and by the arguments in the previous paragraph (with $P = E_n$) for any quasi-coherent $\mathcal{F}$ on $Y$ we have functorial canonical maps $$\begin{gathered}\begin{matrix}\phantom{X} & \operatorname{Ext}^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \\ H^i(Y, K_{n + 1} \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}) & \phantom{X} & H^i(Y, K_n \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F})\end{matrix} \\[6pt] \begin{aligned}\operatorname{Ext}^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) & \longrightarrow H^i(Y, K_{n + 1} \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}) \\ \operatorname{Ext}^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) & \longrightarrow H^i(Y, K_n \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}) \\ H^i(Y, K_{n + 1} \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}) & \longrightarrow H^i(Y, K_n \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F})\end{aligned}\end{gathered}$$ which are isomorphisms for $i \leq n + a$. Let $L_n = K_n^\vee$ be the dual perfect complex. Then we see that $L_1 \to L_2 \to L_3 \to \ldots$ is a system of perfect objects in $D(\mathcal{O}_Y)$ such that for any quasi-coherent $\mathcal{F}$ on $Y$ the maps $$\operatorname{Ext}^i_{\mathcal{O}_Y}(L_{n + 1}, \mathcal{F}) \longrightarrow \operatorname{Ext}^i_{\mathcal{O}_Y}(L_n, \mathcal{F})$$ are isomorphisms for $i \leq n + a - 1$. This implies that $L_n \to L_{n + 1}$ induces an isomorphism on truncations $\tau_{\geq -n - a + 2}$ (hint: take cone of $L_n \to L_{n + 1}$ and look at its last nonvanishing cohomology sheaf). Thus $L = \text{hocolim} L_n$ is pseudo-coherent, see Lemma Pseudo-coherent complexes and coherent sheaves. The mapping property of homotopy colimits gives that $\operatorname{Ext}^i_{\mathcal{O}_Y}(L, \mathcal{F}) = \operatorname{Ext}^i_{\mathcal{O}_Y}(L_n, \mathcal{F})$ for $i \leq n + a - 3$ which finishes the proof. $\square$

Lemma. Perfect proper-support direct images over arbitrary bases

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \in D(\mathcal{O}_X)$ be a perfect object. Let $\mathcal{G}^\bullet$ be a bounded complex of finitely presented $\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then $$K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)$$ is a perfect object of $D(\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma Canonical arbitrary base change for relatively flat tensors. Thus it suffices to show that $K$ is a perfect object. If $Y$ is Noetherian, then this follows from Lemma Perfect complexes and tensor products and direct sums. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is local on $Y$, hence we may assume $Y$ is affine. Say $Y = \operatorname{Spec}(R)$. We write $R = \mathop{\operatorname{colim}} R_i$ as a filtered colimit of Noetherian rings $R_i$. By Limits of Spaces, Lemma Descent of finite presentation and finite algebras there exists an $i$ and an algebraic space $X_i$ of finite presentation over $R_i$ whose base change to $R$ is $X$. By Limits of Spaces, Lemma Descent of finite presentation and modules we may assume after increasing $i$, that there exists a bounded complex of finitely presented $\mathcal{O}_{X_i}$-modules $\mathcal{G}_i^\bullet$ whose pullback to $X$ is $\mathcal{G}^\bullet$. After increasing $i$ we may assume $\mathcal{G}_i^n$ is flat over $R_i$, see Limits of Spaces, Lemma Descent of flatness. After increasing $i$ we may assume the support of $\mathcal{G}_i^n$ is proper over $R_i$, see Limits of Spaces, Lemma Proper morphisms and closed support. Finally, by Lemma Perfect complexes we may, after increasing $i$, assume there exists a perfect object $E_i$ of $D(\mathcal{O}_{X_i})$ whose pullback to $X$ is $E$. By Lemma Perfect complexes and tensor products and direct sums we have that $K_i = Rf_{i, *}(E_i \otimes_{\mathcal{O}_{X_i}}^\mathbf{L} \mathcal{G}_i^\bullet)$ is perfect on $\operatorname{Spec}(R_i)$ where $f_i : X_i \to \operatorname{Spec}(R_i)$ is the structure morphism. By the base change result (Lemma Canonical arbitrary base change for relatively flat tensors) the pullback of $K_i$ to $Y = \operatorname{Spec}(R)$ is $K$ and we conclude. $\square$

Lemma. Base change for pseudo-coherent complexes and coherent sheaves

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \in D(\mathcal{O}_X)$ be a pseudo-coherent object. Let $\mathcal{G}^\bullet$ be a bounded above complex of finitely presented $\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then $$K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)$$ is a pseudo-coherent object of $D(\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma Canonical arbitrary base change for relatively flat tensors. Thus it suffices to show that $K$ is a pseudo-coherent object. This will follow from Lemma Perfect proper-support direct images over arbitrary bases by approximation by perfect complexes. We encourage the reader to skip the rest of the proof.

The question is étale local on $Y$, hence we may assume $Y$ is affine. Then $X$ is quasi-compact and quasi-separated. Moreover, there exists an integer $N$ such that total direct image $Rf_* : D_\mathrm{QCoh}(\mathcal{O}_X) \to D_\mathrm{QCoh}(\mathcal{O}_Y)$ has cohomological dimension $N$ as explained in Lemma Quasi-coherent complexes and coherent sheaves. Choose an integer $b$ such that $\mathcal{G}^i = 0$ for $i > b$. It suffices to show that $K$ is $m$-pseudo-coherent for every $m$. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, m - N - 1 - b)$. This is possible by Theorem Perfect approximation with prescribed closed support. Choose a distinguished triangle $$P \to E \to C \to P[1]$$ in $D_\mathrm{QCoh}(\mathcal{O}_X)$. The cohomology sheaves of $C$ are zero in degrees $\geq m - N - 1 - b$. Hence the cohomology sheaves of $C \otimes^\mathbf{L} \mathcal{G}^\bullet$ are zero in degrees $\geq m - N - 1$. Thus the cohomology sheaves of $Rf_*(C \otimes^\mathbf{L} \mathcal{G})$ are zero in degrees $\geq m - 1$. Hence $$Rf_*(P \otimes^\mathbf{L} \mathcal{G}) \to Rf_*(E \otimes^\mathbf{L} \mathcal{G})$$ is an isomorphism on cohomology sheaves in degrees $\geq m$. Next, suppose that $H^i(P) = 0$ for $i > a$. Then $P \otimes^\mathbf{L} \sigma_{\geq m - N - 1 - a}\mathcal{G}^\bullet \longrightarrow P \otimes^\mathbf{L} \mathcal{G}^\bullet$ is an isomorphism on cohomology sheaves in degrees $\geq m - N - 1$. Thus again we find that $$Rf_*(P \otimes^\mathbf{L} \sigma_{\geq m - N - 1 - a}\mathcal{G}^\bullet) \to Rf_*(P \otimes^\mathbf{L} \mathcal{G}^\bullet)$$ is an isomorphism on cohomology sheaves in degrees $\geq m$. By Lemma Perfect proper-support direct images over arbitrary bases the source is a perfect complex. We conclude that $K$ is $m$-pseudo-coherent as desired. $\square$

Lemma. Pullback of derived quasi-coherent complexes

Let $R$ be a ring. Let $X$ be an algebraic space and let $f : X \to \operatorname{Spec}(R)$ be proper, flat, and of finite presentation. Let $(M_n)$ be an inverse system of $R$-modules with surjective transition maps. Then the canonical map $$\mathcal{O}_X \otimes_R (\varprojlim M_n) \longrightarrow \varprojlim \mathcal{O}_X \otimes_R M_n$$ induces an isomorphism from the source to $DQ_X$ applied to the target.

Proof. The statement means that for any object $E$ of $D_\mathrm{QCoh}(\mathcal{O}_X)$ the induced map $$\operatorname{Hom}(E, \mathcal{O}_X \otimes_R (\varprojlim M_n)) \longrightarrow \operatorname{Hom}(E, \varprojlim \mathcal{O}_X \otimes_R M_n)$$ is an isomorphism. Since $D_\mathrm{QCoh}(\mathcal{O}_X)$ has a perfect generator (Theorem A perfect generator for a quasi-compact algebraic space) it suffices to check this for perfect $E$. By Lemma Quasi-coherent complexes and coherent sheaves we have $\varprojlim \mathcal{O}_X \otimes_R M_n = R\varprojlim \mathcal{O}_X \otimes_R M_n$. The exact functor $R\operatorname{Hom}_X(E, -) : D_\mathrm{QCoh}(\mathcal{O}_X) \to D(R)$ of Cohomology on Sites, Section Derived Hom and Ext commutes with products and hence with derived limits, whence $$R\operatorname{Hom}_X(E, \varprojlim \mathcal{O}_X \otimes_R M_n) = R\varprojlim R\operatorname{Hom}_X(E, \mathcal{O}_X \otimes_R M_n)$$ Let $E^\vee$ be the dual perfect complex, see Cohomology on Sites, Lemma Perfect complexes and derived categories. We have $$R\operatorname{Hom}_X(E, \mathcal{O}_X \otimes_R M_n) = R\Gamma(X, E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*M_n) = R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} M_n$$ by Lemma Base change for sheaf cohomology. From Lemma Perfect direct images for proper morphisms of finite presentation we see $R\Gamma(X, E^\vee)$ is a perfect complex of $R$-modules. In particular it is a pseudo-coherent complex and by More on Algebra, Lemma Pseudo-coherent complexes and coherent sheaves we obtain $$R\varprojlim R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} M_n = R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} \varprojlim M_n$$ as desired. $\square$

Lemma. Detecting a complex by perfect tests

Let $A$ be a ring. Let $X$ be an algebraic space over $A$ which is quasi-compact and quasi-separated. Let $K \in D^-_\mathrm{QCoh}(\mathcal{O}_X)$. If $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every perfect $E$ in $D(\mathcal{O}_X)$, then $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_X)$.

Proof. There exists an integer $N$ such that $R\Gamma(X, -) : D_\mathrm{QCoh}(\mathcal{O}_X) \to D(A)$ has cohomological dimension $N$ as explained in Lemma Quasi-coherent complexes and coherent sheaves. Let $b \in \mathbf{Z}$ be such that $H^i(K) = 0$ for $i > b$. Let $E$ be pseudo-coherent on $X$. It suffices to show that $R\Gamma(X, E \otimes^\mathbf{L} K)$ is $m$-pseudo-coherent for every $m$. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, m - N - 1 - b)$. This is possible by Theorem Perfect approximation with prescribed closed support. Choose a distinguished triangle $$P \to E \to C \to P[1]$$ in $D_\mathrm{QCoh}(\mathcal{O}_X)$. The cohomology sheaves of $C$ are zero in degrees $\geq m - N - 1 - b$. Hence the cohomology sheaves of $C \otimes^\mathbf{L} K$ are zero in degrees $\geq m - N - 1$. Thus the cohomology of $R\Gamma(X, C \otimes^\mathbf{L} K)$ are zero in degrees $\geq m - 1$. Hence $$R\Gamma(X, P \otimes^\mathbf{L} K) \to R\Gamma(X, E \otimes^\mathbf{L} K)$$ is an isomorphism on cohomology in degrees $\geq m$. By assumption the source is pseudo-coherent. We conclude that $R\Gamma(X, E \otimes^\mathbf{L} K)$ is $m$-pseudo-coherent as desired. $\square$

Lemma. Perfect proper-support derived Hom over arbitrary bases

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \in D(\mathcal{O}_X)$ be a perfect object. Let $\mathcal{G}^\bullet$ be a bounded complex of finitely presented $\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then $$K = Rf_*R\mathcal{H}om(E, \mathcal{G}^\bullet)$$ is a perfect object of $D(\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma Canonical arbitrary base change for derived Hom. Thus it suffices to show that $K$ is a perfect object. If $Y$ is Noetherian, then this follows from Lemma Perfect complexes and derived Hom and Ext. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is local on $Y$, hence we may assume $Y$ is affine. Say $Y = \operatorname{Spec}(R)$. We write $R = \mathop{\operatorname{colim}} R_i$ as a filtered colimit of Noetherian rings $R_i$. By Limits of Spaces, Lemma Descent of finite presentation and finite algebras there exists an $i$ and an algebraic space $X_i$ of finite presentation over $R_i$ whose base change to $R$ is $X$. By Limits of Spaces, Lemma Descent of finite presentation and modules we may assume after increasing $i$, that there exists a bounded complex of finitely presented $\mathcal{O}_{X_i}$-module $\mathcal{G}_i^\bullet$ whose pullback to $X$ is $\mathcal{G}$. After increasing $i$ we may assume $\mathcal{G}_i^n$ is flat over $R_i$, see Limits of Spaces, Lemma Descent of flatness. After increasing $i$ we may assume the support of $\mathcal{G}_i^n$ is proper over $R_i$, see Limits of Spaces, Lemma Proper morphisms and closed support. Finally, by Lemma Descent of perfect complexes we may, after increasing $i$, assume there exists a perfect object $E_i$ of $D(\mathcal{O}_{X_i})$ whose pullback to $X$ is $E$. Applying Lemma Perfect complexes and derived Hom and Ext to $X_i \to \operatorname{Spec}(R_i)$, $E_i$, $\mathcal{G}_i^\bullet$ and using the base change property already shown we obtain the result. $\square$

Lemma. Sheaf cohomology

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K$ be an object of $D_\mathrm{QCoh}(\mathcal{O}_X)$ such that the cohomology sheaves $H^i(K)$ have countable sets of sections over affine schemes étale over $X$. Then for any quasi-compact and quasi-separated étale morphism $U \to X$ and any perfect object $E$ in $D(\mathcal{O}_X)$ the sets $$H^i(U, K \otimes^\mathbf{L} E),\quad \operatorname{Ext}^i(E|_U, K|_U)$$ are countable.

Proof. Using Cohomology on Sites, Lemma Perfect complexes and derived categories we see that it suffices to prove the result for the groups $H^i(U, K \otimes^\mathbf{L} E)$. We will use the induction principle to prove the lemma, see Lemma Induction by elementary distinguished squares.

When $U = \operatorname{Spec}(A)$ is affine the result follows from the case of schemes, see Derived Categories of Schemes, Lemma Sheaf cohomology.

To finish the proof it suffices to show: if $(U \subset W, V \to W)$ is an elementary distinguished triangle and the result holds for $U$, $V$, and $U \times_W V$, then the result holds for $W$. This is an immediate consequence of the Mayer-Vietoris sequence, see Lemma Mayer–Vietoris for unbounded quasi-coherent complexes. $\square$

Lemma. Countable approximation by perfect objects

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume the sets of sections of $\mathcal{O}_X$ over affines étale over $X$ are countable. Let $K$ be an object of $D_\mathrm{QCoh}(\mathcal{O}_X)$. The following are equivalent

1. $K = \text{hocolim} E_n$ with $E_n$ a perfect object of $D(\mathcal{O}_X)$, and

2. the cohomology sheaves $H^i(K)$ have countable sets of sections over affines étale over $X$.

Proof. If (1) is true, then (2) is true because homotopy colimits commutes with taking cohomology sheaves (by Derived Categories, Lemma Sheaf cohomology) and because a perfect complex is locally isomorphic to a finite complex of finite free $\mathcal{O}_X$-modules and therefore satisfies (2) by assumption on $X$.

Assume (2). Choose a K-injective complex $\mathcal{K}^\bullet$ representing $K$. Choose a perfect generator $E$ of $D_\mathrm{QCoh}(\mathcal{O}_X)$ and represent it by a K-injective complex $\mathcal{I}^\bullet$. According to Theorem Quasi-coherent complexes and its proof there is an equivalence of triangulated categories $F : D_\mathrm{QCoh}(\mathcal{O}_X) \to D(A, \text{d})$ where $(A, \text{d})$ is the differential graded algebra $$(A, \text{d}) = \operatorname{Hom}_{\text{Comp}^{dg}(\mathcal{O}_X)} (\mathcal{I}^\bullet, \mathcal{I}^\bullet)$$ which maps $K$ to the differential graded module $$M = \operatorname{Hom}_{\text{Comp}^{dg}(\mathcal{O}_X)} (\mathcal{I}^\bullet, \mathcal{K}^\bullet)$$ Note that $H^i(A) = \operatorname{Ext}^i(E, E)$ and $H^i(M) = \operatorname{Ext}^i(E, K)$. Moreover, since $F$ is an equivalence it and its quasi-inverse commute with homotopy colimits. Therefore, it suffices to write $M$ as a homotopy colimit of compact objects of $D(A, \text{d})$. By Differential Graded Algebra, Lemma Countable approximation by perfect objects it suffices show that $\operatorname{Ext}^i(E, E)$ and $\operatorname{Ext}^i(E, K)$ are countable for each $i$. This follows from Lemma Sheaf cohomology. $\square$

Lemma. Derived quasi-coherent complexes

Let $A$ be a ring. Let $f : U \to X$ be a flat morphism of algebraic spaces of finite presentation over $A$. Then

1. there exists an inverse system of perfect objects $L_n$ of $D(\mathcal{O}_X)$ such that $$R\Gamma(U, Lf^*K) = \text{hocolim}\ R\operatorname{Hom}_X(L_n, K)$$ in $D(A)$ functorially in $K$ in $D_\mathrm{QCoh}(\mathcal{O}_X)$, and

2. there exists a system of perfect objects $E_n$ of $D(\mathcal{O}_X)$ such that $$R\Gamma(U, Lf^*K) = \text{hocolim}\ R\Gamma(X, E_n \otimes^\mathbf{L} K)$$ in $D(A)$ functorially in $K$ in $D_\mathrm{QCoh}(\mathcal{O}_X)$.

Proof. By Lemma Base change for sheaf cohomology we have $$R\Gamma(U, Lf^*K) = R\Gamma(X, Rf_*\mathcal{O}_U \otimes^\mathbf{L} K)$$ functorially in $K$. Observe that $R\Gamma(X, -)$ commutes with homotopy colimits because it commutes with direct sums by Lemma Quasi-coherent complexes and coherent sheaves. Similarly, $- \otimes^\mathbf{L} K$ commutes with derived colimits because $- \otimes^\mathbf{L} K$ commutes with direct sums (because direct sums in $D(\mathcal{O}_X)$ are given by direct sums of representing complexes). Hence to prove (2) it suffices to write $Rf_*\mathcal{O}_U = \text{hocolim} E_n$ for a system of perfect objects $E_n$ of $D(\mathcal{O}_X)$. Once this is done we obtain (1) by setting $L_n = E_n^\vee$, see Cohomology on Sites, Lemma Perfect complexes and derived categories.

Write $A = \mathop{\operatorname{colim}} A_i$ with $A_i$ of finite type over $\mathbf{Z}$. By Limits of Spaces, Lemma Descent of finite presentation and finite algebras we can find an $i$ and morphisms $U_i \to X_i \to \operatorname{Spec}(A_i)$ of finite presentation whose base change to $\operatorname{Spec}(A)$ recovers $U \to X \to \operatorname{Spec}(A)$. After increasing $i$ we may assume that $f_i : U_i \to X_i$ is flat, see Limits of Spaces, Lemma Descent of flatness. By Lemma Base change for derived quasi-coherent complexes the derived pullback of $Rf_{i, *}\mathcal{O}_{U_i}$ by $g : X \to X_i$ is equal to $Rf_*\mathcal{O}_U$. Since $Lg^*$ commutes with derived colimits, it suffices to prove what we want for $f_i$. Hence we may assume that $U$ and $X$ are of finite type over $\mathbf{Z}$.

Assume $f : U \to X$ is a morphism of algebraic spaces of finite type over $\mathbf{Z}$. To finish the proof we will show that $Rf_*\mathcal{O}_U$ is a homotopy colimit of perfect complexes. To see this we apply Lemma Countable approximation by perfect objects. Thus it suffices to show that $R^if_*\mathcal{O}_U$ has countable sets of sections over affines étale over $X$. This follows from Lemma Sheaf cohomology applied to the structure sheaf. $\square$

A.12.5 Derived Chow and the henselian proper part

The following arguments adapt the human Stacks Project treatment in spaces-more-morphisms.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. Separating the proper part over a henselian pair

Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space separated and of finite type over $A$. Set $X_0 = X \times_{\operatorname{Spec}(A)} \operatorname{Spec}(A/I)$. Let $Y \subset X_0$ be an open and closed subspace such that $Y \to \operatorname{Spec}(A/I)$ is proper. Then there exists an open and closed subspace $W \subset X$ which is proper over $A$ with $W \times_{\operatorname{Spec}(A)} \operatorname{Spec}(A/I) = Y$.

Proof. We will denote $T \mapsto T_0$ the base change by $\operatorname{Spec}(A/I) \to \operatorname{Spec}(A)$. By a weak version of Chow's lemma (in the form of Cohomology of Spaces, Lemma The weak Chow lemma for algebraic spaces) there exists a surjective proper morphism $\varphi : X' \to X$ such that $X'$ admits an immersion into $\mathbf{P}^n_A$. Set $Y' = \varphi^{-1}(Y)$. This is an open and closed subscheme of $X'_0$. The lemma holds for $(X', Y')$ by More on Morphisms, Lemma Separating the proper part over a henselian pair. Let $W' \subset X'$ be the open and closed subscheme proper over $A$ such that $Y' = W'_0$. By Morphisms of Spaces, Lemma Morphisms of algebraic spaces $Q_1 = \varphi(|W'|) \subset |X|$ and $Q_2 = \varphi(|X' \setminus W'|) \subset |X|$ are closed subsets and by Morphisms of Spaces, Lemma Proper morphisms any closed subspace structure on $Q_1$ is proper over $A$. The image of $Q_1 \cap Q_2$ in $\operatorname{Spec}(A)$ is closed. Since $(A, I)$ is henselian, if $Q_1 \cap Q_2$ is nonempty, then we find that $Q_1 \cap Q_2$ has a point lying over $\operatorname{Spec}(A/I)$. This is impossible as $W'_0 = Y' = \varphi^{-1}(Y)$. We conclude that $Q_1$ is open and closed in $|X|$. Let $W \subset X$ be the corresponding open and closed subspace. Then $W$ is proper over $A$ with $W_0 = Y$. $\square$

Lemma. Derived tensor products and Tor amplitude

Let $S$ be a scheme. Consider a commutative diagram of algebraic spaces $$\begin{gathered}\begin{matrix}Z' & Y' \\ X' & B'\end{matrix} \\[6pt] \begin{aligned}Z' & \longrightarrow X' \\ Z' & \longrightarrow Y' \\ Y' & \longrightarrow B' \\ X' & \longrightarrow B'\end{aligned}\end{gathered}$$ over $S$. Let $B \to B'$ be a morphism. Denote by $X$ and $Y$ the base changes of $X'$ and $Y'$ to $B$. Assume $Y' \to B'$ and $Z' \to X'$ are flat. Then $X \times_B Y$ and $Z'$ are Tor independent over $X' \times_{B'} Y'$.

Proof. By Derived Categories of Spaces, Lemma Derived tensor products and Tor amplitude we may check tor independence étale locally on $X \times_B Y$ and $Z'$. This[^1] reduces the lemma to the case of schemes which is More on Morphisms, Lemma Derived tensor products and Tor amplitude. $\square$

Lemma. A derived Chow neighbourhood (Derived Chow's lemma)

Let $A$ be a ring. Let $X$ be a separated algebraic space of finite presentation over $A$. Let $x \in |X|$. Then there exist an $n \geq 0$, a closed subspace $Z \subset X \times_A \mathbf{P}^n_A$, a point $z \in |Z|$, an open $V \subset \mathbf{P}^n_A$, and an object $E$ in $D(\mathcal{O}_{X \times_A \mathbf{P}^n_A})$ such that

1. $Z \to X \times_A \mathbf{P}^n_A$ is of finite presentation,

2. $c : Z \to \mathbf{P}^n_A$ is a closed immersion over $V$, set $W = c^{-1}(V)$,

3. the restriction of $b : Z \to X$ to $W$ is étale, $z \in W$, and $b(z) = x$,

4. $E|_{X \times_A V} \cong (b, c)_*\mathcal{O}_Z|_{X \times_A V}$,

5. $E$ is pseudo-coherent and supported on $Z$.

Proof. We can find a finite type $\mathbf{Z}$-subalgebra $A' \subset A$ and an algebraic space $X'$ separated and of finite presentation over $A'$ whose base change to $A$ is $X$. See Limits of Spaces, Lemmas Descent of finite presentation and finite algebras and Descent of diagonals and separation. Let $x' \in |X'|$ be the image of $x$. If we can prove the lemma for $(X'/A', x')$, then the lemma follows for $(X/A, x)$. Namely, if $n', Z', z', V', E'$ provide the solution for $(X'/A', x')$, then we can let $n = n'$, let $Z \subset X \times \mathbf{P}^n$ be the inverse image of $Z'$, choose a point $z$ of the base-changed étale neighbourhood over $x$, let $V \subset \mathbf{P}^n_A$ be the inverse image of $V'$, and let $E$ be the derived pullback of $E'$. Observe that $E$ is pseudo-coherent by Cohomology on Sites, Lemma Pseudo-coherent complexes and coherent sheaves. It remains to check the graph restriction in (4) and support in (5). To see this set $W = c^{-1}(V)$ and $W' = (c')^{-1}(V')$ and consider the cartesian square $$\begin{gathered}\begin{matrix}W & W' \\ X \times_A V & X' \times_{A'} V'\end{matrix} \\[6pt] \begin{aligned}W & \xrightarrow{(b, c)} X \times_A V \\ W & \longrightarrow W' \\ W' & \xrightarrow{(b', c')} X' \times_{A'} V' \\ X \times_A V & \longrightarrow X' \times_{A'} V'\end{aligned}\end{gathered}$$ By Lemma Derived tensor products and Tor amplitude $X \times_A V$ and $W'$ are tor-independent over $X' \times_{A'} V'$. Thus the derived pullback of $(b', c')_*\mathcal{O}_{W'}$ to $X \times_A V$ is $(b, c)_*\mathcal{O}_W$ by Derived Categories of Spaces, Lemma Base change for derived quasi-coherent complexes. This also uses that $R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'}$ because $(b', c')$ is a closed immersion and similarly for $(b, c)_*\mathcal{O}_Z$. Since $E'|_{X' \times_{A'} V'} = (b', c')_*\mathcal{O}_{W'}$ we obtain $E|_{X \times_A V} = (b, c)_*\mathcal{O}_W$ and (4) holds. Derived pullback restricts to zero off the inverse image of the support, so (5) also holds. This reduces us to the situation described in the next paragraph.

Assume $A$ is of finite type over $\mathbf{Z}$. Choose an étale morphism $U \to X$ where $U$ is an affine scheme and a point $u \in U$ mapping to $x$. Then $U$ is of finite type over $A$. Choose a closed immersion $U \to \mathbf{A}^n_A$ and denote $j : U \to \mathbf{P}^n_A$ the immersion we get by composing with the open immersion $\mathbf{A}^n_A \to \mathbf{P}^n_A$. Let $Z$ be the scheme theoretic closure of $$(\text{id}_U, j) : U \longrightarrow X \times_A \mathbf{P}^n_A$$ Let $z \in Z$ be the image of $u$. Let $Y \subset \mathbf{P}^n_A$ be the scheme theoretic closure of $j$. Then it is clear that $Z \subset X \times_A Y$ is the scheme theoretic closure of $(\text{id}_U, j) : U \to X \times_A Y$. As $X$ is separated, the morphism $X \times_A Y \to Y$ is separated as well. Hence we see that $Z \to Y$ is an isomorphism over the open subscheme $j(U) \subset Y$ by Morphisms of Spaces, Lemma Morphisms of algebraic spaces. Choose $V \subset \mathbf{P}^n_A$ open with $V \cap Y = j(U)$. Then we see that (2) holds, that $W = (\text{id}_U, j)(U)$, and hence that (3) holds. Part (1) holds because $A$ is Noetherian.

Because $A$ is Noetherian we see that $X$ and $X \times_A \mathbf{P}^n_A$ are Noetherian algebraic spaces. Hence we can take $E = (b, c)_*\mathcal{O}_Z$ in this case: (4) is clear and for (5) see Derived Categories of Spaces, Lemma Pseudo-coherent complexes and coherent sheaves. This finishes the proof. $\square$

Lemma. The restriction comparison in a derived Chow neighbourhood

Let $X/A$, $x \in |X|$, and $n, Z, z, V, E$ be as in Lemma A derived Chow neighbourhood. For any $K \in D_\mathrm{QCoh}(\mathcal{O}_X)$ we have $$Rq_*(Lp^*K \otimes^\mathbf{L} E)|_V = R(W \to V)_*K|_W$$ where $p : X \times_A \mathbf{P}^n_A \to X$ and $q : X \times_A \mathbf{P}^n_A \to \mathbf{P}^n_A$ are the projections and where the morphism $W \to V$ is the finitely presented closed immersion $c|_W : W \to V$.

Proof. Since $W = c^{-1}(V)$ and since $c$ is a closed immersion over $V$, we see that $c|_W$ is a closed immersion. It is of finite presentation because $W$ and $V$ are of finite presentation over $A$, see Morphisms of Spaces, Lemma Finite presentation and finite algebras. First we have $$Rq_*(Lp^*K \otimes^\mathbf{L} E)|_V = Rq'_*\left((Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V}\right)$$ where $q' : X \times_A V \to V$ is the projection because formation of total direct image commutes with localization. Denote $i = (b, c)|_W : W \to X \times_A V$ the given closed immersion. Then $$Rq'_*\left((Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V}\right) = Rq'_*(Lp^*K|_{X \times_A V} \otimes^\mathbf{L} i_*\mathcal{O}_W)$$ by property (4). Since $i$ is a closed immersion we have $i_*\mathcal{O}_W = Ri_*\mathcal{O}_W$. Using Derived Categories of Spaces, Lemma Base change for sheaf cohomology we can rewrite this as $$Rq'_* Ri_* Li^* Lp^*K|_{X \times_A V} = R(q' \circ i)_* Lb^*K|_W = R(W \to V)_* K|_W$$ which is what we want. (Note that restricting to $W$ and derived pulling back via $W \to X$ is the same thing as $W$ is étale over $X$.) $\square$

Lemma. Detecting relative pseudo-coherence by global tests

Let $A$ be a ring. Let $X$ be an algebraic space separated and of finite presentation over $A$. Let $K \in D_\mathrm{QCoh}(\mathcal{O}_X)$. If $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_X)$, then $K$ is pseudo-coherent relative to $A$ (Definition Pseudo-coherent complexes and coherent sheaves).

Proof. Assume $K \in D_\mathrm{QCoh}(\mathcal{O}_X)$ and $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_X)$. Let $x \in |X|$. We will show that $K$ is pseudo-coherent relative to $A$ in an étale neighbourhood of $x$. This will prove the lemma by our definition of relative pseudo-coherence.

Choose $n, Z, z, V, E$ as in Lemma A derived Chow neighbourhood. Denote $p : X \times \mathbf{P}^n \to X$ and $q : X \times \mathbf{P}^n \to \mathbf{P}^n_A$ the projections. Then for any $i \in \mathbf{Z}$ we have $$\begin{aligned} & R\Gamma(\mathbf{P}^n_A, Rq_*(Lp^*K \otimes^\mathbf{L} E) \otimes^\mathbf{L} \mathcal{O}_{\mathbf{P}^n_A}(i)) \\ & = R\Gamma(X \times \mathbf{P}^n, Lp^*K \otimes^\mathbf{L} E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i)) \\ & = R\Gamma(X, K \otimes^\mathbf{L} Rp_*(E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i))) \end{aligned}$$ by Derived Categories of Spaces, Lemma Base change for sheaf cohomology. By Derived Categories of Spaces, Lemma The arbitrary-base proper-support perfect direct-image bridge the complex $Rp_*(E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i))$ is pseudo-coherent on $X$. Hence the assumption tells us the expression in the displayed formula is a pseudo-coherent object of $D(A)$. By Derived Categories of Schemes, Lemma Detecting pseudo-coherence on projective space we conclude that $Rq_*(Lp^*K \otimes^\mathbf{L} E)$ is pseudo-coherent on $\mathbf{P}^n_A$. By Lemma The restriction comparison in a derived Chow neighbourhood we have $$Rq_*(Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V} = R(W \to V)_*K|_W$$ Since $W \to V$ is a closed immersion into an open subscheme of $\mathbf{P}^n_A$ this means $K|_W$ is pseudo-coherent relative to $A$ for example by More on Morphisms, Lemma Pseudo-coherent complexes and coherent sheaves. $\square$

[^1]: Here is the argument in more detail. Choose a surjective étale morphism $W' \to B'$ with $W'$ a scheme. Choose a surjective étale morphism $W \to B \times_{B'} W'$ with $W$ a scheme. Choose a surjective étale morphism $U' \to X' \times_{B'} W'$ with $U'$ a scheme. Choose a surjective étale morphism $V' \to Y' \times_{B'} W'$ with $V'$ a scheme. Observe that $U' \times_{W'} V' \to X' \times_{B'} Y'$ is surjective étale. Choose a surjective étale morphism $T' \to Z' \times_{X' \times_{B'} Y'} U' \times_{W'} V'$ with $T'$ a scheme. Denote $U$ and $V$ the base changes of $U'$ and $V'$ to $W$. Then the lemma says that $X \times_B Y$ and $Z'$ are Tor independent over $X' \times_{B'} Y'$ as algebraic spaces if and only if $U \times_W V$ and $T'$ are Tor independent over $U' \times_{W'} V'$ as schemes. Thus it suffices to prove the lemma for the square with corners $T', U', V', W'$ and base change by $W \to W'$. The flatness of $Y' \to B'$ and $Z' \to X'$ implies flatness of $V' \to W'$ and $T' \to U'$.

A.12.6 Uniform section algebras under deformation

The following arguments adapt the human Stacks Project treatment in more-morphisms.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. Uniform section algebras under nilpotent deformation (Deformations of projective schemes)

Let $f : X \to S$ be a morphism of schemes which is proper, flat, and of finite presentation. Let $\mathcal{L}$ be $f$-ample. Assume $S$ is quasi-compact. There exists a $d_0 \geq 0$ such that for every cartesian diagram $$\begin{gathered}\begin{matrix}X & X' \\ S & S'\end{matrix} \\[6pt] \begin{aligned}X & \xrightarrow{i'} X' \\ X & \xrightarrow{f} S \\ X' & \xrightarrow{f'} S' \\ S & \xrightarrow{i} S'\end{aligned}\end{gathered} \quad\text{and}\quad \begin{matrix} \text{invertible }\mathcal{O}_{X'}\text{-module}\\ \mathcal{L}'\text{ with }\mathcal{L} \cong (i')^*\mathcal{L}' \end{matrix}$$ where $S \subset S'$ is a thickening and $f'$ is proper, flat, of finite presentation we have

1. $R^p(f')_*(\mathcal{L}')^{\otimes d} = 0$ for all $p > 0$ and $d \geq d_0$,

2. $\mathcal{A}'_d = (f')_*(\mathcal{L}')^{\otimes d}$ is finite locally free for $d \geq d_0$,

3. $\mathcal{A}' = \mathcal{O}_{S'} \oplus \bigoplus_{d \geq d_0} \mathcal{A}'_d$ is a quasi-coherent $\mathcal{O}_{S'}$-algebra of finite presentation,

4. there is a canonical isomorphism $r' : X' \to \underline{\text{Proj}}_{S'}(\mathcal{A}')$, and

5. there is a canonical isomorphism $\theta' : (r')^*\mathcal{O}_{\underline{\text{Proj}}_{S'}(\mathcal{A}')}(1) \to \mathcal{L}'$.

The construction of $\mathcal{A}'$, $r'$, $\theta'$ is functorial in the data $(X', S', i, i', f', \mathcal{L}')$.

Proof. We first describe the maps $r'$ and $\theta'$. Observe that $\mathcal{L}'$ is $f'$-ample, see Lemma Proper morphisms and line bundles and ampleness. There is a canonical map of quasi-coherent graded $\mathcal{O}_{S'}$-algebras $\mathcal{A}' \to \bigoplus_{d \geq 0} (f')_*(\mathcal{L}')^{\otimes d}$ which is an isomorphism in degrees $\geq d_0$. Hence this induces an isomorphism on relative Proj compatible with the Serre twists of the structure sheaf, see Constructions, Lemma The geometric construction (uncovered prerequisite). Hence we get the morphism $r'$ by Morphisms, Lemma Criteria for line bundles and ampleness (uncovered prerequisite) (which in turn appeals to the construction given in Constructions, Lemma Line bundles and ampleness (uncovered prerequisite)) and it is an isomorphism by Morphisms, Lemma Proper morphisms and line bundles and ampleness (uncovered prerequisite). We get the map $\theta'$ from Constructions, Lemma Line bundles and ampleness (uncovered prerequisite). By Properties, Lemma Line bundles and ampleness we find that $\theta'$ is an isomorphism (this also uses that the morphism $r'$ over affine opens of $S'$ is the same as the morphism from Properties, Lemma Scheme geometry as is explained in the proof of Morphisms, Lemma Proper morphisms and line bundles and ampleness (uncovered prerequisite)).

Assuming the vanishing and local freeness stated in parts (1) and (2), the functoriality of the construction can be seen as follows. Suppose that $h : T \to S'$ is a morphism of schemes, denote $f_T : X'_T \to T$ the base change of $f'$ and $\mathcal{L}_T$ the pullback of $\mathcal{L}$ to $X'_T$. By cohomology and base change (as formulated in Derived Categories of Schemes, Lemma Base change for perfect complexes for example) we have the corresponding vanishing over $T$ and moreover $h^*\mathcal{A}'_d = f_{T, *}\mathcal{L}_T^{\otimes d}$ (and thus the local freeness of pushforwards as well as the finite generation of the corresponding graded $\mathcal{O}_T$-algebra $\mathcal{A}_T$). Hence the morphism $r_T : X_T \to \underline{\text{Proj}}_T(\bigoplus f_{T, *}\mathcal{L}_T^{\otimes d})$ is simply the base change of $r'$ to $T$ and the pullback of $\theta'$ is the map $\theta_T$.

Having said all of the above, we see that it suffices to prove (1), (2), and (3). Pick $d_0$ such that $R^pf_*\mathcal{L}^{\otimes d} = 0$ for all $d \geq d_0$ and $p > 0$, see Cohomology of Schemes, Lemma Coherent sheaves and proper morphisms (uncovered prerequisite). We claim that $d_0$ works.

By cohomology and base change (Derived Categories of Schemes, Lemma Perfect direct images for proper morphisms of finite presentation) we see that $E'_d = Rf'_*(\mathcal{L}')^{\otimes d}$ is a perfect object of $D(\mathcal{O}_{S'})$ and its formation commutes with arbitrary base change. In particular, $E_d = Li^*E'_d = Rf_*\mathcal{L}^{\otimes d}$. By Derived Categories of Schemes, Lemma Vanishing and projective, locally free modules and local algebra we see that for $d \geq d_0$ the complex $E_d$ is isomorphic to the finite locally free $\mathcal{O}_S$-module $f_*\mathcal{L}^{\otimes d}$ placed in cohomological degree $0$. Then by Derived Categories of Schemes, Lemma Sheaf cohomology we conclude that $E'_d$ is isomorphic to a finite locally free module placed in cohomological degree $0$. Of course this means that $E'_d = \mathcal{A}'_d[0]$, that $R^pf'_*(\mathcal{L}')^{\otimes d} = 0$ for $p > 0$, and that $\mathcal{A}'_d$ is finite locally free. This proves (1) and (2).

The last thing we have to show is finite presentation of $\mathcal{A}'$ as a sheaf of $\mathcal{O}_{S'}$-algebras (this notion was introduced in Properties, Section Quasi-coherent complexes and coherent sheaves). Let $U' = \operatorname{Spec}(R') \subset S'$ be an affine open. Then $A' = \mathcal{A}'(U')$ is a graded $R'$-algebra whose graded parts are finite projective $R'$-modules. We have to show that $A'$ is a finitely presented $R'$-algebra. We will prove this by reduction to the Noetherian case. Namely, we can find a finite type $\mathbf{Z}$-subalgebra $R'_0 \subset R'$ and a pair[^1] $(X'_0, \mathcal{L}'_0)$ over $R'_0$ whose base change is $(X'_{U'}, \mathcal{L}'|_{X'_{U'}})$, see Limits, Lemmas Descent of finite presentation and modules, Descent of finite locally free and invertible modules, Proper morphisms, Descent of finite presentation and flatness, and Filtered limits and line bundles and ampleness. Cohomology of Schemes, Lemma Coherent sheaves and proper morphisms (uncovered prerequisite) implies $A'_0 = \bigoplus_{d \geq 0} H^0(X'_0, (\mathcal{L}'_0)^{\otimes d})$ is a finitely generated graded $R'_0$-algebra and implies there exists a $d'_0$ such that $H^p(X'_0, (\mathcal{L}'_0)^{\otimes d}) = 0$, $p > 0$ for $d \geq d'_0$. By the arguments given above applied to $X'_0 \to \operatorname{Spec}(R'_0)$ and $\mathcal{L}'_0$ we see that $(A'_0)_d$ is a finite projective $R'_0$-module and that $$A'_d = \mathcal{A}'_d(U') = H^0(X'_{U'}, (\mathcal{L}')^{\otimes d}|_{X'_{U'}}) = H^0(X'_0, (\mathcal{L}'_0)^{\otimes d}) \otimes_{R'_0} R' = (A'_0)_d \otimes_{R'_0} R'$$ for $d \geq d'_0$. Now a small twist in the argument is that we don't know that we can choose $d'_0$ equal to $d_0$[^2]. To get around this we use the following sequence of arguments to finish the proof:

1. The algebra $B = R'_0 \oplus \bigoplus_{d \geq \max(d_0, d'_0)} (A'_0)_d$ is an $R'_0$-algebra of finite type: apply the Artin-Tate lemma to $B \subset A'_0$, see Algebra, Lemma The Artin–Tate lemma.

2. As $R'_0$ is Noetherian we see that $B$ is an $R'_0$-algebra of finite presentation.

3. By right exactness of tensor product we see that $B \otimes_{R'_0} R'$ is an $R'$-algebra of finite presentation.

4. By the displayed equalities this exactly says that $C = R' \oplus \bigoplus_{d \geq \max(d_0, d'_0)} A'_d$ is an $R'$-algebra of finite presentation.

5. The quotient $A'/C$ is the direct sum of the finite projective $R'$-modules $A'_d$, $d_0 \leq d \leq \max(d_0, d'_0)$, hence finitely presented as $R'$-module.

6. The quotient $A'/C$ is finitely presented as a $C$-module by Algebra, Lemma Finite presentation and finite algebras.

7. Thus $A'$ is finitely presented as a $C$-module by Algebra, Lemma Commutative algebra.

8. By Algebra, Lemma Finite algebras this implies $A'$ is finitely presented as a $C$-algebra.

9. Finally, by Algebra, Lemma Composition of finite-type ring maps applied to $R' \to C \to A'$ this implies $A'$ is finitely presented as an $R'$-algebra.

This finishes the proof. $\square$

Lemma. Separating the proper part over a henselian pair

Source credit: A reference for the case of an adic Noetherian base is the original source citation EGA (III, Proposition 5.5.1)

Let $(A, I)$ be a henselian pair. Let $X \to \operatorname{Spec}(A)$ be separated and of finite type. Set $X_0 = X \times_{\operatorname{Spec}(A)} \operatorname{Spec}(A/I)$. Let $Y \subset X_0$ be an open and closed subscheme such that $Y \to \operatorname{Spec}(A/I)$ is proper. Then there exists an open and closed subscheme $W \subset X$ which is proper over $A$ with $W \times_{\operatorname{Spec}(A)} \operatorname{Spec}(A/I) = Y$.

Proof. We will denote $T \mapsto T_0$ the base change by $\operatorname{Spec}(A/I) \to \operatorname{Spec}(A)$. By Chow's lemma (in the form of Limits, Lemma Finite algebras) there exists a surjective proper morphism $\varphi : X' \to X$ such that $X'$ admits an immersion into $\mathbf{P}^n_A$. Set $Y' = \varphi^{-1}(Y)$. This is an open and closed subscheme of $X'_0$. Suppose the lemma holds for $(X', Y')$. Let $W' \subset X'$ be the open and closed subscheme proper over $A$ such that $Y' = W'_0$. By Morphisms, Lemma Proper morphisms (uncovered prerequisite) $W = \varphi(W') \subset X$ and $Q = \varphi(X' \setminus W') \subset X$ are closed subsets and by Morphisms, Lemma Proper morphisms (uncovered prerequisite) $W$ is proper over $A$. The image of $W \cap Q$ in $\operatorname{Spec}(A)$ is closed. Since $(A, I)$ is henselian, if $W \cap Q$ is nonempty, then we find that $W \cap Q$ has a point lying over $\operatorname{Spec}(A/I)$. This is impossible as $W'_0 = Y' = \varphi^{-1}(Y)$. We conclude that $W$ is an open and closed subscheme of $X$ proper over $A$ with $W_0 = Y$. Thus we reduce to the case described in the next paragraph.

Assume there exists an immersion $j : X \to \mathbf{P}^n_A$ over $A$. Let $\overline{X}$ be the scheme theoretic image of $j$. Since $j$ is a quasi-compact morphism (Schemes, Lemma The geometric construction (uncovered prerequisite)) we see that $j : X \to \overline{X}$ is an open immersion (Morphisms, Lemma Diagonals and separation (uncovered prerequisite)). Hence the base change $j_0 : X_0 \to \overline{X}_0$ is an open immersion as well. Thus $j_0(Y) \subset \overline{X}_0$ is open. It is also closed by Morphisms, Lemma Proper morphisms (uncovered prerequisite). Suppose that the lemma holds for $(\overline{X}, j_0(Y))$. Let $\overline{W} \subset \overline{X}$ be the corresponding open and closed subscheme proper over $A$ such that $j_0(Y) = \overline{W}_0$. Then $T = \overline{W} \setminus j(X)$ is closed in $\overline{W}$, hence has closed image in $\operatorname{Spec}(A)$ by properness of $\overline{W}$ over $A$. Since $(A, I)$ is henselian, we find that if $T$ is nonempty, then there is a point of $T$ mapping into $\operatorname{Spec}(A/I)$. This is impossible because $j_0(Y) = \overline{W}_0$ is contained in $j(X)$. Hence $\overline{W}$ is contained in $j(X)$ and we can set $W \subset X$ equal to the unique open and closed subscheme mapping isomorphically to $\overline{W}$ via $j$. Thus we reduce to the case described in the next paragraph.

Assume $X \subset \mathbf{P}^n_A$ is a closed subscheme. Then $X \to \operatorname{Spec}(A)$ is a proper morphism. Let $Z = X_0 \setminus Y$. This is an open and closed subscheme of $X_0$ and $X_0 = Y \amalg Z$. Let $X \to X' \to \operatorname{Spec}(A)$ be the Stein factorization as in Theorem The geometric construction. Let $Y' \subset X'_0$ and $Z' \subset X'_0$ be the images of $Y$ and $Z$. Since the fibres of $X \to Z$ are geometrically connected, we see that $Y' \cap Z' = \emptyset$. Hence $X'_0 = Y' \amalg Z'$ as $X \to X'$ is surjective. Since $X' \to \operatorname{Spec}(A)$ is integral, we see that $X'$ is the spectrum of an $A$-algebra integral over $A$. Recall that open and closed subsets of spectra correspond $1$-to-$1$ with idempotents in the corresponding ring, see Algebra, Lemma Product decompositions from disjoint closed subsets. Hence by More on Algebra, Lemma Criteria for henselian rings we see that we may write $X' = W' \amalg V'$ with $W'$ and $V'$ open and closed and with $Y' = W'_0$ and $Z' = V'_0$. Let $W$ be the inverse image in $X$ to finish the proof. $\square$

[^1]: With the same properties as those enjoyed by $X' \to S'$ and $\mathcal{L}'$, i.e., $X'_0 \to \operatorname{Spec}(R'_0)$ is flat and proper and $\mathcal{L}'_0$ is ample.

[^2]: Actually, one can reduce to this case by doing more limit arguments.

A.12.7 Purity, universal flattening and strong coherent existence

The following arguments adapt the human Stacks Project treatment in spaces-flat.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. Universal purity from proper support

In Situation Purity near a specified fibre.

1. If the support of $\mathcal{F}$ is proper over $Y$, then $\mathcal{F}$ is universally pure relative to $Y$.

2. If $f$ is proper, then $\mathcal{F}$ is universally pure relative to $Y$.

3. If $f$ is proper, then $X$ is universally pure relative to $Y$.

Proof. First we reduce (1) to (2). Namely, let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$ (Morphisms of Spaces, Definition Closed support). Let $i : Z \to X$ be the corresponding closed immersion and write $\mathcal{F} = i_*\mathcal{G}$ for some finite type quasi-coherent $\mathcal{O}_Z$-module $\mathcal{G}$. In case (1) $Z \to Y$ is proper by assumption. Thus by Lemma A module supported on a closed subspace case (1) reduces to case (2).

Assume $f$ is proper. Let $(g : T \to Y, t' \leadsto t, \xi)$ be an impurity of $\mathcal{F}$ above $y$. Since $f$ is proper, it is universally closed. Hence $f_T : X_T \to T$ is closed. Since $f_T(\xi) = t'$ this implies that $t \in f(\overline{\{\xi\}})$ which is a contradiction. $\square$

Lemma. Complete dévissage around a fibre

Let $S$ be a scheme. Let $X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $y \in |Y|$ be a point. There exists an étale morphism $(Y', y') \to (Y, y)$ with $Y'$ an affine scheme and étale morphisms $h_i : W_i \to X_{Y'}$, $i = 1, \ldots, n$ such that for each $i$ there exists a complete dévissage of $\mathcal{F}_i/W_i/Y'$ over $y'$, where $\mathcal{F}_i$ is the pullback of $\mathcal{F}$ to $W_i$ and such that $|(X_{Y'})_{y'}| \subset \bigcup h_i(W_i)$.

Proof. The question is étale local on $Y$ hence we may assume $Y$ is an affine scheme. Then $X$ is quasi-compact, hence we can choose an affine scheme $X'$ and a surjective étale morphism $X' \to X$. Then we may apply More on Flatness, Lemma Complete dévissage around a fibre to $X' \to Y$, $(X' \to Y)^*\mathcal{F}$, and $y$ to get what we want. $\square$

Theorem. Universal flattening for a pure finitely presented module

In Situation The universal flatness dimension stratum. Assume moreover that $f$ is of finite presentation, that $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, and that $\mathcal{F}$ is pure relative to $Y$. Then $F_n$ is an algebraic space and $F_n \to Y$ is a monomorphism of finite presentation.

Proof. The functor $F_n$ is a sheaf for the fppf topology by Lemma The flatness dimension stratum. Since $F_n \to Y$ is a monomorphism of sheaves on $(\mathrm{Sch}/S)_{fppf}$ we see that $\Delta : F_n \to F_n \times F_n$ is the pullback of the diagonal $\Delta_Y : Y \to Y \times_S Y$. Hence the representability of $\Delta_Y$ implies the same thing for $F_n$. Therefore it suffices to prove that there exists a scheme $W$ over $S$ and a surjective étale morphism $W \to F_n$.

To construct $W \to F_n$ choose an étale covering $\{Y_i \to Y\}$ with $Y_i$ a scheme. Let $X_i = X \times_Y Y_i$ and let $\mathcal{F}_i$ be the pullback of $\mathcal{F}$ to $X_i$. Then $\mathcal{F}_i$ is pure relative to $Y_i$ either by definition or by Lemma Quasi-finite base change. The other assumptions of the theorem are preserved as well. Finally, the restriction of $F_n$ to $Y_i$ is the functor $F_n$ corresponding to $X_i \to Y_i$ and $\mathcal{F}_i$. Hence it suffices to show: Given $\mathcal{F}$ and $f : X \to Y$ as in the statement of the theorem where $Y$ is a scheme, the functor $F_n$ is representable by a scheme $Z_n$ and $Z_n \to Y$ is a monomorphism of finite presentation.

Observe that a monomorphism of finite presentation is separated and quasi-finite (Morphisms, Lemma Finite algebras (uncovered prerequisite)). Hence combining Descent, Lemma Descent of the geometric construction, More on Morphisms, Lemma Diagonals, separation and finite algebras , and Descent, Lemmas Descent of proper morphisms and Descent of finite presentation and proper morphisms we see that the question is local for the étale topology on $Y$.

In particular the situation is local for the Zariski topology on $Y$ and we may assume that $Y$ is affine. In this case the dimension of the fibres of $f$ is bounded above, hence we see that $F_n$ is representable for $n$ large enough. Thus we may use descending induction on $n$. Suppose that we know $F_{n + 1}$ is representable by a monomorphism $Z_{n + 1} \to Y$ of finite presentation. Consider the base change $X_{n + 1} = Z_{n + 1} \times_Y X$ and the pullback $\mathcal{F}_{n + 1}$ of $\mathcal{F}$ to $X_{n + 1}$. The morphism $Z_{n + 1} \to Y$ is quasi-finite as it is a monomorphism of finite presentation, hence Lemma Quasi-finite base change implies that $\mathcal{F}_{n + 1}$ is pure relative to $Z_{n + 1}$. Since $F_n$ is a subfunctor of $F_{n + 1}$ we conclude that in order to prove the result for $F_n$ it suffices to prove the result for the corresponding functor for the situation $\mathcal{F}_{n + 1}/X_{n + 1}/Z_{n + 1}$. In this way we reduce to proving the result for $F_n$ in case $Y_{n + 1} = Y$, i.e., we may assume that $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over $Y$.

Fix $n$ and assume $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over the affine scheme $Y$. To finish the proof we have to show that $F_n$ is representable by a monomorphism $Z_n \to S$ of finite presentation. Since the question is local in the étale topology on $Y$ it suffices to show that for every $y \in Y$ there exists an étale neighbourhood $(Y', y') \to (Y, y)$ such that the result holds after base change to $Y'$. Thus by Lemma Complete dévissage around a fibre we may assume there exist étale morphisms $h_j : W_j \to X$, $j = 1, \ldots, m$ such that for each $j$ there exists a complete dévissage of $\mathcal{F}_j/W_j/Y$ over $y$, where $\mathcal{F}_j$ is the pullback of $\mathcal{F}$ to $W_j$ and such that $|X_y| \subset \bigcup h_j(W_j)$. Since $h_j$ is étale, by Lemma The local flatness dimension test the sheaves $\mathcal{F}_j$ are still flat over in dimensions $\geq n + 1$ over $Y$. Set $W = \bigcup h_j(W_j)$, which is a quasi-compact open of $X$. As $\mathcal{F}$ is pure along $X_y$ we see that $$E = \{t \in |Y| : \text{Ass}_{X_t}(\mathcal{F}_t) \subset W \}.$$ contains all generalizations of $y$. By Divisors on Spaces, Lemma Prime spectra and associated points $E$ is a constructible subset of $Y$. We have seen that $\operatorname{Spec}(\mathcal{O}_{Y, y}) \subset E$. By Morphisms, Lemma The geometric construction (uncovered prerequisite) we see that $E$ contains an open neighbourhood of $y$. Hence after shrinking $Y$ we may assume that $E = Y$. It follows from Lemma Localization of a flatness dimension stratum that it suffices to prove the lemma for the functor $F_n$ associated to $X = \coprod W_j$ and $\mathcal{F} = \coprod \mathcal{F}_j$. If $F_{j, n}$ denotes the functor for $W_j \to Y$ and the sheaf $\mathcal{F}_j$ we see that $F_n = \prod F_{j, n}$. Hence it suffices to prove each $F_{j, n}$ is representable by some monomorphism $Z_{j, n} \to Y$ of finite presentation, since then $$Z_n = Z_{1, n} \times_Y \ldots \times_Y Z_{m, n}$$ Thus we have reduced the theorem to the special case handled in More on Flatness, Lemma Representing a flatness dimension stratum. $\square$

Lemma. The hypotheses for universal flattening

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.

1. If $f$ is of finite presentation, $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, and $\mathcal{F}$ is pure relative to $Y$, then there exists a universal flattening $Y' \to Y$ of $\mathcal{F}$. Moreover $Y' \to Y$ is a monomorphism of finite presentation.

2. If $f$ is of finite presentation and $X$ is pure relative to $Y$, then there exists a universal flattening $Y' \to Y$ of $X$. Moreover $Y' \to Y$ is a monomorphism of finite presentation.

3. If $f$ is proper and of finite presentation and $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, then there exists a universal flattening $Y' \to Y$ of $\mathcal{F}$. Moreover $Y' \to Y$ is a monomorphism of finite presentation.

4. If $f$ is proper and of finite presentation then there exists a universal flattening $Y' \to Y$ of $X$.

Proof. These statements follow immediately from Theorem Universal flattening for a pure finitely presented module applied to $F_0 = F_{flat}$ and the fact that if $f$ is proper then $\mathcal{F}$ is automatically pure over the base, see Lemma Universal purity from proper support. $\square$

Situation. The marked inverse system for coherent existence

Here we have an inverse system of rings $(A_n)$ with surjective transition maps whose kernels are locally nilpotent. Set $A = \varprojlim A_n$. We have an algebraic space $X$ separated and of finite presentation over $A$. We set $X_n = X \times_{\operatorname{Spec}(A)} \operatorname{Spec}(A_n)$ and we view it as a closed subspace of $X$. We assume further given a system $(\mathcal{F}_n, \varphi_n)$ where $\mathcal{F}_n$ is a finitely presented $\mathcal{O}_{X_n}$-module, flat over $A_n$, with support proper over $A_n$, and $$\varphi_n : \mathcal{F}_n \otimes_{\mathcal{O}_{X_n}} \mathcal{O}_{X_{n - 1}} \longrightarrow \mathcal{F}_{n - 1}$$ is an isomorphism (notation using the equivalence of Morphisms of Spaces, Lemma Groupoids and equivalence relations).

Lemma. The derived quasi-coherent inverse limit

In Situation The marked inverse system for coherent existence consider $$K = R\varprojlim_{D_\mathrm{QCoh}(\mathcal{O}_X)}(\mathcal{F}_n) = DQ_X(R\varprojlim_{D(\mathcal{O}_X)}\mathcal{F}_n)$$ Then $K$ is in $D^b_{\mathrm{QCoh}}(\mathcal{O}_X)$ and in fact $K$ has nonzero cohomology sheaves only in degrees $\geq 0$.

Proof. Special case of Derived Categories of Spaces, Example Quasi-coherent complexes and coherent sheaves. $\square$

Lemma. Computing the inverse limit against perfect complexes

In Situation The marked inverse system for coherent existence let $K$ be as in Lemma The derived quasi-coherent inverse limit. For any perfect object $E$ of $D(\mathcal{O}_X)$ we have

1. $M = R\Gamma(X, K \otimes^\mathbf{L} E)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\Gamma(X_n, \mathcal{F}_n \otimes^\mathbf{L} E|_{X_n}) = M \otimes_A^\mathbf{L} A_n$ in $D(A_n)$,

2. $N = R\operatorname{Hom}_X(E, K)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\operatorname{Hom}_{X_n}(E|_{X_n}, \mathcal{F}_n) = N \otimes_A^\mathbf{L} A_n$ in $D(A_n)$.

In both statements $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$.

Proof. Proof of (2). Write $E_n = E|_{X_n}$ and $N_n = R\operatorname{Hom}_{X_n}(E_n, \mathcal{F}_n)$. Recall that $R\operatorname{Hom}_{X_n}(-, -)$ is equal to $R\Gamma(X_n, R\mathcal{H}om(-, -))$, see Cohomology on Sites, Section Derived Hom and Ext. Hence by Derived Categories of Spaces, Lemma Perfect proper-support derived Hom over arbitrary bases we see that $N_n$ is a perfect object of $D(A_n)$ whose formation commutes with base change. Thus the maps $N_n \otimes_{A_n}^\mathbf{L} A_{n - 1} \to N_{n - 1}$ coming from $\varphi_n$ are isomorphisms. By More on Algebra, Lemma Perfect inverse limits we find that $R\varprojlim N_n$ is perfect and that its base change back to $A_n$ recovers $N_n$. On the other hand, the exact functor $R\operatorname{Hom}_X(E, -) : D_\mathrm{QCoh}(\mathcal{O}_X) \to D(A)$ of triangulated categories commutes with products and hence with derived limits, whence $$R\operatorname{Hom}_X(E, K) = R\varprojlim R\operatorname{Hom}_X(E, \mathcal{F}_n) = R\varprojlim R\operatorname{Hom}_X(E_n, \mathcal{F}_n) = R\varprojlim N_n$$ This proves (2). To see that (1) holds, translate it into (2) using Cohomology on Sites, Lemma Perfect complexes and derived categories. $\square$

Lemma. Relative pseudo-coherence of the inverse limit

In Situation The marked inverse system for coherent existence let $K$ be as in Lemma The derived quasi-coherent inverse limit. Then $K$ is pseudo-coherent relative to $A$.

Proof. Combinging Lemma Computing the inverse limit against perfect complexes and Derived Categories of Spaces, Lemma Detecting a complex by perfect tests we see that $R\Gamma(X, K \otimes^\mathbf{L} E)$ is pseudo-coherent in $D(A)$ for all pseudo-coherent $E$ in $D(\mathcal{O}_X)$. Thus the lemma follows from More on Morphisms of Spaces, Lemma Detecting relative pseudo-coherence by global tests. $\square$

Lemma. The marked comparison on an affine étale chart

In Situation The marked inverse system for coherent existence let $K$ be as in Lemma The derived quasi-coherent inverse limit. For any étale morphism $U \to X$ with $U$ quasi-compact and quasi-separated we have $$R\Gamma(U, K) \otimes_A^\mathbf{L} A_n = R\Gamma(U_n, \mathcal{F}_n)$$ in $D(A_n)$ where $U_n = U \times_X X_n$.

Proof. Fix $n$. By Derived Categories of Spaces, Lemma Derived quasi-coherent complexes there exists a system of perfect complexes $E_m$ on $X$ such that $R\Gamma(U, K) = \text{hocolim} R\Gamma(X, K \otimes^\mathbf{L} E_m)$. In fact, this formula holds not just for $K$ but for every object of $D_\mathrm{QCoh}(\mathcal{O}_X)$. Applying this to $\mathcal{F}_n$ we obtain $$\begin{aligned} R\Gamma(U_n, \mathcal{F}_n) & = R\Gamma(U, \mathcal{F}_n) \\ & = \text{hocolim}_m R\Gamma(X, \mathcal{F}_n \otimes^\mathbf{L} E_m) \\ & = \text{hocolim}_m R\Gamma(X_n, \mathcal{F}_n \otimes^\mathbf{L} E_m|_{X_n}) \end{aligned}$$ Using Lemma Computing the inverse limit against perfect complexes and the fact that $- \otimes_A^\mathbf{L} A_n$ commutes with homotopy colimits we obtain the result. $\square$

Lemma. Finite presentation near the common closed fibre

In Situation The marked inverse system for coherent existence let $K$ be as in Lemma The derived quasi-coherent inverse limit. Denote $X_0 \subset |X|$ the closed subset consisting of points lying over the closed subset $\operatorname{Spec}(A_1) = \operatorname{Spec}(A_2) = \ldots$ of $\operatorname{Spec}(A)$. There exists an open subspace $W \subset X$ containing $X_0$ such that

1. $H^i(K)|_W$ is zero unless $i = 0$,

2. $\mathcal{F} = H^0(K)|_W$ is of finite presentation, and

3. $\mathcal{F}_n = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_{X_n}$.

Proof. Fix $n \geq 1$. By construction there is a canonical map $K \to \mathcal{F}_n$ in $D_\mathrm{QCoh}(\mathcal{O}_X)$ and hence a canonical map $H^0(K) \to \mathcal{F}_n$ of quasi-coherent sheaves. This explains the meaning of part (3).

Let $x \in X_0$ be a point. We will find an open neighbourhood $W$ of $x$ such that (1), (2), and (3) are true. Since $X_0$ is quasi-compact this will prove the lemma. Let $U \to X$ be an étale morphism with $U$ affine and $u \in U$ a point mapping to $x$. Since $|U| \to |X|$ is open it suffices to find an open neighbourhood of $u$ in $U$ where (1), (2), and (3) are true. Say $U = \operatorname{Spec}(B)$. Choose a surjection $P \to B$ with $P$ smooth over $A$. By Lemma Relative pseudo-coherence of the inverse limit and the definition of relative pseudo-coherence there exists a bounded above complex $F^\bullet$ of finite free $P$-modules representing $Ri_*K$ where $i : U \to \operatorname{Spec}(P)$ is the closed immersion induced by the presentation. Let $M_n$ be the $B$-module corresponding to $\mathcal{F}_n|_U$. By Lemma The marked comparison on an affine étale chart $$H^i(F^\bullet \otimes_A A_n) = \left\{ \begin{matrix} 0 & \text{if} & i \not = 0 \\ M_n & \text{if} & i = 0 \end{matrix} \right.$$ Let $i$ be the maximal index such that $F^i$ is nonzero. If $i \leq 0$, then (1), (2), and (3) are true. If not, then $i > 0$ and we see that the rank of the map $$F^{i - 1} \to F^i$$ in the point $u$ is maximal. Hence in an open neighbourhood of $u$ inside $\operatorname{Spec}(P)$ the rank is maximal. Thus after replacing $P$ by a principal localization we may assume that the displayed map is surjective. Since $F^i$ is finite free we may choose a splitting $F^{i - 1} = F' \oplus F^i$. Then we may replace $F^\bullet$ by the complex $$\ldots \to F^{i - 2} \to F' \to 0 \to \ldots$$ and we win by induction on $i$. $\square$

Theorem. Strong coherent existence (Grothendieck Existence Theorem)

In Situation The marked inverse system for coherent existence there exists a finitely presented $\mathcal{O}_X$-module $\mathcal{F}$, flat over $A$, with support proper over $A$, such that $\mathcal{F}_n = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_{X_n}$ for all $n$ compatibly with the maps $\varphi_n$.

Proof. Apply Lemmas The derived quasi-coherent inverse limit, Computing the inverse limit against perfect complexes, Relative pseudo-coherence of the inverse limit, The marked comparison on an affine étale chart, Finite presentation near the common closed fibre, and Fitting support and the proper-support existence argument to get an open subspace $W \subset X$ containing all points lying over $\operatorname{Spec}(A_n)$ and a finitely presented $\mathcal{O}_W$-module $\mathcal{F}$ whose support is proper over $A$ with $\mathcal{F}_n = \mathcal{F} \otimes_{\mathcal{O}_W} \mathcal{O}_{X_n}$ for all $n \geq 1$. (This makes sense as $X_n \subset W$.) By Lemma Universal purity from proper support we see that $\mathcal{F}$ is universally pure relative to $\operatorname{Spec}(A)$. By Theorem Universal flattening for a pure finitely presented module (for explanation, see Lemma The hypotheses for universal flattening) there exists a universal flattening $S' \to \operatorname{Spec}(A)$ of $\mathcal{F}$ and moreover the morphism $S' \to \operatorname{Spec}(A)$ is a monomorphism of finite presentation. In particular $S'$ is a scheme (this follows from the proof of the theorem but it also follows a postoriori by Morphisms of Spaces, Proposition Diagonals, separation and finite algebras). Since the base change of $\mathcal{F}$ to $\operatorname{Spec}(A_n)$ is $\mathcal{F}_n$ we find that $\operatorname{Spec}(A_n) \to \operatorname{Spec}(A)$ factors (uniquely) through $S'$ for each $n$. By More on Flatness, Lemma Recovering a monomorphism from a locally nilpotent tower we see that $S' = \operatorname{Spec}(A)$. This means that $\mathcal{F}$ is flat over $A$. Finally, since the scheme theoretic support $Z$ of $\mathcal{F}$ is proper over $\operatorname{Spec}(A)$, the morphism $Z \to X$ is closed. Hence the pushforward $(W \to X)_*\mathcal{F}$ is supported on $W$ and has all the desired properties. $\square$

Lemma. The local flatness dimension test

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $n \geq 0$. The following are equivalent

1. for some commutative diagram $$\begin{gathered}\begin{matrix}U & V \\ X & Y\end{matrix} \\[6pt] \begin{aligned}U & \xrightarrow{\varphi} X \\ U & \longrightarrow V \\ V & \longrightarrow Y \\ X & \longrightarrow Y\end{aligned}\end{gathered}$$ with surjective, étale vertical arrows where $U$ and $V$ are schemes, the sheaf $\varphi^*\mathcal{F}$ is flat over $V$ in dimensions $\geq n$ (More on Flatness, Definition Flatness and dimension of a fibre module),

2. for every commutative diagram $$\begin{gathered}\begin{matrix}U & V \\ X & Y\end{matrix} \\[6pt] \begin{aligned}U & \xrightarrow{\varphi} X \\ U & \longrightarrow V \\ V & \longrightarrow Y \\ X & \longrightarrow Y\end{aligned}\end{gathered}$$ with étale vertical arrows where $U$ and $V$ are schemes, the sheaf $\varphi^*\mathcal{F}$ is flat over $V$ in dimensions $\geq n$, and

3. for $x \in |X|$ such that $\mathcal{F}$ is not flat at $x$ over $Y$ the transcendence degree of $x/f(x)$ is $< n$ (Morphisms of Spaces, Definition The dimension of a fibre).

If this is true, then it remains true after any base change $Y' \to Y$.

Proof. Suppose that we have a diagram as in (1). Then the equivalence of the conditions in More on Flatness, Lemma The local flatness dimension test shows that (1) and (3) are equivalent. But condition (3) is inherited by $\varphi^*\mathcal{F}$ for any $U \to V$ as in (2). Whence we see that (3) implies (2) by the result for schemes again. The result for schemes also implies the statement on base change. $\square$

Definition. Flatness and dimension of a fibre module

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $n \geq 0$. We say $\mathcal{F}$ is flat over $Y$ in dimensions $\geq n$ if the equivalent conditions of Lemma The local flatness dimension test are satisfied.

Situation. The universal flatness dimension stratum

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. For any scheme $T$ over $Y$ we will denote $\mathcal{F}_T$ the base change of $\mathcal{F}$ to $T$, in other words, $\mathcal{F}_T$ is the pullback of $\mathcal{F}$ via the projection morphism $X_T = X \times_Y T \to X$. Note that $f_T : X_T \to T$ is of finite type and that $\mathcal{F}_T$ is an $\mathcal{O}_{X_T}$-module of finite type (Morphisms of Spaces, Lemma Base change for finite algebras and Modules on Sites, Lemma Local pullback on a ringed site). Let $n \geq 0$. By Definition Flatness and dimension of a fibre module and Lemma The local flatness dimension test we obtain a functor

Lemma. The flatness dimension stratum

In Situation The universal flatness dimension stratum.

1. The functor $F_n$ satisfies the sheaf property for the fpqc topology.

2. If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_n$ is limit preserving.

Proof. Proof of (1). Suppose that $\{T_i \to T\}$ is an fpqc covering of a scheme $T$ over $Y$. We have to show that if $F_n(T_i)$ is nonempty for all $i$, then $F_n(T)$ is nonempty. Choose a diagram as in part (1) of Lemma The local flatness dimension test. Denote $F'_n$ the corresponding functor for $\varphi^*\mathcal{F}$ and the morphism $U \to V$. By More on Flatness, Lemma The flatness dimension stratum we have the sheaf property for $F'_n$. Thus we get the sheaf property for $F_n$ because for $T \to Y$ we have $F_n(T) = F'_n(V \times_Y T)$ by Lemma The local flatness dimension test and because $\{V \times_Y T_i \to V \times_Y T\}$ is an fpqc covering.

Proof of (2). Suppose that $T = \varprojlim_{i \in I} T_i$ is a filtered limit of affine schemes $T_i$ over $Y$ and assume that $F_n(T)$ is nonempty. We have to show that $F_n(T_i)$ is nonempty for some $i$. Choose a diagram as in part (1) of Lemma The local flatness dimension test. Fix $i \in I$ and choose an affine open $W_i \subset V \times_Y T_i$ mapping surjectively onto $T_i$. For $i' \geq i$ let $W_{i'}$ be the inverse image of $W_i$ in $V \times_Y T_{i'}$ and let $W \subset V \times_Y T$ be the inverse image of $W_i$. Then $W = \varprojlim_{i' \geq i} W_i$ is a filtered limit of affine schemes over $V$. By Lemma The local flatness dimension test again it suffices to show that $F'_n(W_{i'})$ is nonempty for some $i' \geq i$. But we know that $F'_n(W)$ is nonempty because of our assumption that $F_n(T) = F'_n(V \times_Y T)$ is nonempty. Thus we can apply More on Flatness, Lemma The flatness dimension stratum to conclude. $\square$

Lemma. Localization of a flatness dimension stratum

In Situation The universal flatness dimension stratum. Let $h : X' \to X$ be an étale morphism. Set $\mathcal{F}' = h^*\mathcal{F}$ and $f' = f \circ h$. Let $F_n'$ be (the displayed identity) associated to $(f' : X' \to Y, \mathcal{F}')$. Then $F_n$ is a subfunctor of $F_n'$ and if $h(X') \supset \text{Ass}_{X/Y}(\mathcal{F})$, then $F_n = F'_n$.

Proof. Choose $U \to X$, $V \to Y$, $U \to V$ as in part (1) of Lemma The local flatness dimension test. Choose a surjective étale morphism $U' \to U \times_X X'$ where $U'$ is a scheme. Then we have the lemma for the two functors $F_{U, n}$ and $F_{U', n}$ determined by $U' \to U$ and $\mathcal{F}|_U$ over $V$, see More on Flatness, Lemma Localization of a flatness dimension stratum. On the other hand, Lemma The local flatness dimension test tells us that given $T \to Y$ we have $F_n(T) = F_{U, n}(V \times_Y T)$ and $F'_n(T) = F_{U', n}(V \times_Y T)$. This proves the lemma. $\square$

A.12.8 Patching flat algebraic spaces and modules

The following arguments adapt the human Stacks Project treatment in spaces-pushouts.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. Algebraic spaces over a pushout

In the situation of Lemma Categories of spaces over a pushout the functor $F \circ G$ is isomorphic to the identity functor.

Proof. We will prove that $F \circ G$ is isomorphic to the identity by reducing this to the corresponding statement of More on Morphisms, Lemma Groupoids and equivalence relations.

Choose a scheme $Y_1$ and a surjective étale morphism $Y_1 \to Y$. Set $X_1 = Y_1 \times_Y X$. This is a scheme affine over $Y_1$ with a surjective étale morphism $X_1 \to X$. By More on Morphisms of Spaces, Lemma Nilpotent thickenings and groupoids and equivalence relations there exists a $X'_1 \to X'$ surjective étale with $X_1 = X_1' \times_{X'} X$. In particular the morphism of schemes $X_1 \to X_1'$ is a thickening too. Apply More on Morphisms, Lemma Pushouts along a thickening of algebraic spaces to obtain a pushout $Y_1' = Y_1 \amalg_{X_1} X_1'$ in the category of schemes. In the proof of Lemma Pushouts along a thickening of algebraic spaces we constructed $Y'$ as a quotient of an étale equivalence relation on $Y_1'$ such that we get a commutative diagram

$$\begin{gathered}\begin{matrix}\phantom{X} & X & \phantom{X} & X' \\ X_1 & \phantom{X} & X_1' & \phantom{X} \\ \phantom{X} & Y & \phantom{X} & Y' \\ Y_1 & \phantom{X} & Y_1'\end{matrix} \\[6pt] \begin{aligned}X & \longrightarrow X' \\ X & \longrightarrow Y \\ X' & \longrightarrow Y' \\ X_1 & \longrightarrow X_1' \\ X_1 & \longrightarrow Y_1 \\ X_1 & \longrightarrow X \\ X_1' & \longrightarrow Y_1' \\ X_1' & \longrightarrow X' \\ Y & \longrightarrow Y' \\ Y_1 & \longrightarrow Y_1' \\ Y_1 & \longrightarrow Y \\ Y_1' & \longrightarrow Y'\end{aligned}\end{gathered}$$ where all squares except the front and back squares are cartesian (the front and back squares are pushouts) and the northeast arrows are surjective étale. Denote $F_1$, $G_1$ the functors constructed in More on Morphisms, Lemma Groupoids and equivalence relations for the front square. Then the diagram of categories $$\begin{gathered}\begin{matrix}(\mathrm{Sch}/Y_1') & (\mathrm{Sch}/Y_1) \times_{(\mathrm{Sch}/X_1)} (\mathrm{Sch}/X_1') \\ (\textit{Spaces}/Y') & (\textit{Spaces}/Y) \times_{(\textit{Spaces}/X)} (\textit{Spaces}/X')\end{matrix} \\[6pt] \begin{aligned}(\mathrm{Sch}/Y_1') & \xrightarrow{F_1} (\mathrm{Sch}/Y_1) \times_{(\mathrm{Sch}/X_1)} (\mathrm{Sch}/X_1') \\ (\mathrm{Sch}/Y_1') & \longrightarrow (\textit{Spaces}/Y') \\ (\mathrm{Sch}/Y_1) \times_{(\mathrm{Sch}/X_1)} (\mathrm{Sch}/X_1') & \longrightarrow (\textit{Spaces}/Y) \times_{(\textit{Spaces}/X)} (\textit{Spaces}/X') \\ (\mathrm{Sch}/Y_1) \times_{(\mathrm{Sch}/X_1)} (\mathrm{Sch}/X_1') & \xrightarrow{G_1} (\mathrm{Sch}/Y_1') \\ (\textit{Spaces}/Y') & \xrightarrow{F} (\textit{Spaces}/Y) \times_{(\textit{Spaces}/X)} (\textit{Spaces}/X') \\ (\textit{Spaces}/Y) \times_{(\textit{Spaces}/X)} (\textit{Spaces}/X') & \xrightarrow{G} (\textit{Spaces}/Y')\end{aligned}\end{gathered}$$ is commutative by simple considerations regarding base change functors and the agreement of pushouts in schemes with pushouts in spaces of Lemma Pushouts along a thickening of schemes.

Let $(V, U', \varphi)$ be an object of $(\textit{Spaces}/Y) \times_{(\textit{Spaces}/X)} (\textit{Spaces}/X')$. Denote $U = U' \times_{X'} X$ so that $G(V, U', \varphi) = V \amalg_U U'$. Choose a scheme $V_1$ and a surjective étale morphism $V_1 \to Y_1 \times_Y V$. Set $U_1 = V_1 \times_Y X$. Then $$U_1 = V_1 \times_Y X \longrightarrow (Y_1 \times_Y V) \times_Y X = X_1 \times_Y V = X_1 \times_X X \times_Y V = X_1 \times_X U$$ is surjective étale too. By More on Morphisms of Spaces, Lemma Nilpotent thickenings and groupoids and equivalence relations there exists a thickening $U_1 \to U_1'$ and a surjective étale morphism $U_1' \to X_1' \times_{X'} U'$ whose base change to $X_1 \times_X U$ is the displayed morphism. At this point $(V_1, U'_1, \varphi_1)$ is an object of $(\mathrm{Sch}/Y_1) \times_{(\mathrm{Sch}/X_1)} (\mathrm{Sch}/X_1')$. In the proof of Lemma Pushouts along a thickening of algebraic spaces we constructed $G(V, U', \varphi) = V \amalg_U U'$ as a quotient of an étale equivalence relation on $G_1(V_1, U_1', \varphi_1) = V_1 \amalg_{U_1} U_1'$ such that we get a commutative diagram

$$\begin{gathered}\begin{matrix}\phantom{X} & U & \phantom{X} & U' \\ U_1 & \phantom{X} & U_1' & \phantom{X} \\ \phantom{X} & V & \phantom{X} & G(V, U', \varphi) \\ V_1 & \phantom{X} & G_1(V_1, U_1', \varphi_1)\end{matrix} \\[6pt] \begin{aligned}U & \longrightarrow U' \\ U & \longrightarrow V \\ U' & \longrightarrow G(V, U', \varphi) \\ U_1 & \longrightarrow U_1' \\ U_1 & \longrightarrow V_1 \\ U_1 & \longrightarrow U \\ U_1' & \longrightarrow G_1(V_1, U_1', \varphi_1) \\ U_1' & \longrightarrow U' \\ V & \longrightarrow G(V, U', \varphi) \\ V_1 & \longrightarrow G_1(V_1, U_1', \varphi_1) \\ V_1 & \longrightarrow V \\ G_1(V_1, U_1', \varphi_1) & \longrightarrow G(V, U', \varphi)\end{aligned}\end{gathered}$$ where all squares except the front and back squares are cartesian (the front and back squares are pushouts) and the northeast arrows are surjective étale. In particular $$G_1(V_1, U_1', \varphi_1) \to G(V, U', \varphi)$$ is surjective étale.

Finally, we come to the proof of the lemma. We have to show that the adjunction mapping $(V, U', \varphi) \to F(G(V, U', \varphi))$ is an isomorphism. We know $(V_1, U_1', \varphi_1) \to F_1(G_1(V_1, U_1', \varphi_1))$ is an isomorphism by More on Morphisms, Lemma Groupoids and equivalence relations. Recall that $F$ and $F_1$ are given by base change. Using the properties of (the displayed identity) and Lemma The cartesian squares defining the pushout we see that $V \to G(V, U', \varphi) \times_{Y'} Y$ and $U' \to G(V, U', \varphi) \times_{Y'} X'$ are isomorphisms, i.e., $(V, U', \varphi) \to F(G(V, U', \varphi))$ is an isomorphism. $\square$

Lemma. Flat module patching over an algebraic-space pushout

Let $S$ be a base scheme. Let $X \to X'$ be a thickening of algebraic spaces over $S$ and let $X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $Y' = Y \amalg_X X'$ be the pushout (see Lemma Pushouts along a thickening of algebraic spaces). Let $V' \to Y'$ be a morphism of algebraic spaces over $S$. Set $V = Y \times_{Y'} V'$, $U' = X' \times_{Y'} V'$, and $U = X \times_{Y'} V'$. There is an equivalence of categories between

1. quasi-coherent $\mathcal{O}_{V'}$-modules flat over $Y'$, and

2. the category of triples $(\mathcal{G}, \mathcal{F}', \varphi)$ where

   1. $\mathcal{G}$ is a quasi-coherent $\mathcal{O}_V$-module flat over $Y$,

   2. $\mathcal{F}'$ is a quasi-coherent $\mathcal{O}_{U'}$-module flat over $X'$, and

   3. $\varphi : (U \to V)^*\mathcal{G} \to (U \to U')^*\mathcal{F}'$ is an isomorphism of $\mathcal{O}_U$-modules.

The equivalence maps $\mathcal{G}'$ to $((V \to V')^*\mathcal{G}', (U' \to V')^*\mathcal{G}', can)$. Suppose $\mathcal{G}'$ corresponds to the triple $(\mathcal{G}, \mathcal{F}', \varphi)$. Then

1. $\mathcal{G}'$ is a finite type $\mathcal{O}_{V'}$-module if and only if $\mathcal{G}$ and $\mathcal{F}'$ are finite type $\mathcal{O}_Y$ and $\mathcal{O}_{U'}$-modules.

2. if $V' \to Y'$ is locally of finite presentation, then $\mathcal{G}'$ is an $\mathcal{O}_{V'}$-module of finite presentation if and only if $\mathcal{G}$ and $\mathcal{F}'$ are $\mathcal{O}_Y$ and $\mathcal{O}_{U'}$-modules of finite presentation.

Proof. A quasi-inverse functor assigns to the triple $(\mathcal{G}, \mathcal{F}', \varphi)$ the fibre product $$(V \to V')_*\mathcal{G} \times_{(U \to V')_*\mathcal{F}} (U' \to V')_*\mathcal{F}'$$ where $\mathcal{F} = (U \to U')^*\mathcal{F}'$. This works, because on affines étale over $V'$ and $Y'$ we recover the equivalence of More on Algebra, Lemma Relative flat modules over a ring fibre product. Details omitted.

Parts (a) and (b) reduce by étale localization (Properties of Spaces, Section Proper morphisms and modules) to the case where $V'$ and $Y'$ are affine in which case the result follows from More on Algebra, Lemmas Modules and tensor products and direct sums and Finite presentation under flat module patching. $\square$

Lemma. Patching flat algebraic spaces

In the situation of Lemma Algebraic spaces over a pushout. If $V' = G(V, U', \varphi)$ for some triple $(V, U', \varphi)$, then

1. $V' \to Y'$ is locally of finite type if and only if $V \to Y$ and $U' \to X'$ are locally of finite type,

2. $V' \to Y'$ is flat if and only if $V \to Y$ and $U' \to X'$ are flat,

3. $V' \to Y'$ is flat and locally of finite presentation if and only if $V \to Y$ and $U' \to X'$ are flat and locally of finite presentation,

4. $V' \to Y'$ is smooth if and only if $V \to Y$ and $U' \to X'$ are smooth,

5. $V' \to Y'$ is étale if and only if $V \to Y$ and $U' \to X'$ are étale, and

6. add more here as needed.

If $W'$ is flat over $Y'$, then the adjunction mapping $G(F(W')) \to W'$ is an isomorphism. Hence $F$ and $G$ define mutually quasi-inverse functors between the category of spaces flat over $Y'$ and the category of triples $(V, U', \varphi)$ with $V \to Y$ and $U' \to X'$ flat.

Proof. Choose a diagram (the displayed identity) as in the proof of Lemma Algebraic spaces over a pushout.

Proof of (1) -- (5). Let $(V, U', \varphi)$ be an object of $(\textit{Spaces}/Y) \times_{(\textit{Spaces}/X)} (\textit{Spaces}/X')$. Construct a diagram (the displayed identity) as in the proof of Lemma Algebraic spaces over a pushout. Then the base change of $G(V, U', \varphi) \to Y'$ to $Y'_1$ is $G_1(V_1, U_1', \varphi_1) \to Y_1'$. Hence (1) -- (5) follow immediately from the corresponding statements of More on Morphisms, Lemma Flatness and groupoids and equivalence relations for schemes.

Suppose that $W' \to Y'$ is flat. Choose a scheme $W'_1$ and a surjective étale morphism $W'_1 \to Y_1' \times_{Y'} W'$. Observe that $W'_1 \to W'$ is surjective étale as a composition of surjective étale morphisms. We know that $G_1(F_1(W_1')) \to W_1'$ is an isomorphism by More on Morphisms, Lemma Flatness and groupoids and equivalence relations applied to $W'_1$ over $Y'_1$ and the front of the diagram (with functors $G_1$ and $F_1$ as in the proof of Lemma Algebraic spaces over a pushout). Then the construction of $G(F(W'))$ (as a pushout, i.e., as constructed in Lemma Pushouts along a thickening of algebraic spaces) shows that $G_1(F_1(W'_1)) \to G(F(W))$ is surjective étale. Whereupon we conclude that $G(F(W)) \to W$ is étale, see for example Properties of Spaces, Lemma Étale morphisms and local algebra. But $G(F(W)) \to W$ is an isomorphism on underlying reduced algebraic spaces (by construction), hence it is an isomorphism. $\square$

A.12.9 Square-zero extensions and the quasi-coherator

The following arguments adapt the human Stacks Project treatment in defos.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. Deformations of algebraic spaces

Let $S$ be a scheme. Let $Y \subset Y'$ be a first order thickening of algebraic spaces over $S$. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. If there exists a flat morphism $f' : X' \to Y'$ of algebraic spaces over $S$ and an isomorphsm $a : X \to X' \times_{Y'} Y$ over $Y$, then

1. the set of isomorphism classes of pairs $(f' : X' \to Y', a)$ is principal homogeneous under $\operatorname{Ext}^1_{\mathcal{O}_X}(\mathrm{NL}_{X/Y}, f^*\mathcal{C}_{Y/Y'})$, and

2. the set of automorphisms of $\varphi : X' \to X'$ over $Y'$ which reduce to the identity on $X' \times_{Y'} Y$ is $\operatorname{Ext}^0_{\mathcal{O}_X}(\mathrm{NL}_{X/Y}, f^*\mathcal{C}_{Y/Y'})$.

Proof. We will apply the material on deformations of ringed topoi to the small étale topoi of the algebraic spaces in the lemma. We may think of $X$ as a closed subspace of $X'$ so that $(f, f') : (X \subset X') \to (Y \subset Y')$ is a morphism of first order thickenings. By Lemma Matching square-zero thickenings and their markings this translates into a morphism of thickenings of ringed topoi. Then we see from More on Morphisms of Spaces, Lemma Square-zero deformations and their comparison maps (or from the more general Lemma Deformations of modules on ringed topoi) that the ideal sheaf of $X$ in $X'$ is equal to $f^*\mathcal{C}_{Y'/Y}$ and this is in fact equivalent to flatness of $X'$ over $Y'$. Hence we have a commutative diagram $$\begin{gathered}\begin{matrix}0 & f^*\mathcal{C}_{Y/Y'} & \mathcal{O}_{X'} & \mathcal{O}_X & 0 \\ 0 & f_{small}^{-1}\mathcal{C}_{Y/Y'} & f_{small}^{-1}\mathcal{O}_{Y'} & f_{small}^{-1}\mathcal{O}_Y & 0\end{matrix} \\[6pt] \begin{aligned}0 & \longrightarrow f^*\mathcal{C}_{Y/Y'} \\ f^*\mathcal{C}_{Y/Y'} & \longrightarrow \mathcal{O}_{X'} \\ \mathcal{O}_{X'} & \longrightarrow \mathcal{O}_X \\ \mathcal{O}_X & \longrightarrow 0 \\ 0 & \longrightarrow f_{small}^{-1}\mathcal{C}_{Y/Y'} \\ f_{small}^{-1}\mathcal{C}_{Y/Y'} & \longrightarrow f^*\mathcal{C}_{Y/Y'} \\ f_{small}^{-1}\mathcal{C}_{Y/Y'} & \longrightarrow f_{small}^{-1}\mathcal{O}_{Y'} \\ f_{small}^{-1}\mathcal{O}_{Y'} & \longrightarrow \mathcal{O}_{X'} \\ f_{small}^{-1}\mathcal{O}_{Y'} & \longrightarrow f_{small}^{-1}\mathcal{O}_Y \\ f_{small}^{-1}\mathcal{O}_Y & \longrightarrow \mathcal{O}_X \\ f_{small}^{-1}\mathcal{O}_Y & \longrightarrow 0\end{aligned}\end{gathered}$$ Please compare with (Sheaves on ringed sites). Observe that automorphisms $\varphi$ as in (2) give automorphisms $\varphi^\sharp : \mathcal{O}_{X'} \to \mathcal{O}_{X'}$ fitting in the diagram above. Conversely, an automorphism $\alpha : \mathcal{O}_{X'} \to \mathcal{O}_{X'}$ fitting into the diagram of sheaves above is equal to $\varphi^\sharp$ for some automorphism $\varphi$ as in (2) by More on Morphisms of Spaces, Lemma Nilpotent thickenings. Finally, by More on Morphisms of Spaces, Lemma Nilpotent thickenings if we find another sheaf of rings $\mathcal{A}$ on $X_\mathrm{\acute{e}tale}$ fitting into the diagram $$\begin{gathered}\begin{matrix}0 & f^*\mathcal{C}_{Y/Y'} & \mathcal{A} & \mathcal{O}_X & 0 \\ 0 & f_{small}^{-1}\mathcal{C}_{Y/Y'} & f_{small}^{-1}\mathcal{O}_{Y'} & f_{small}^{-1}\mathcal{O}_Y & 0\end{matrix} \\[6pt] \begin{aligned}0 & \longrightarrow f^*\mathcal{C}_{Y/Y'} \\ f^*\mathcal{C}_{Y/Y'} & \longrightarrow \mathcal{A} \\ \mathcal{A} & \longrightarrow \mathcal{O}_X \\ \mathcal{O}_X & \longrightarrow 0 \\ 0 & \longrightarrow f_{small}^{-1}\mathcal{C}_{Y/Y'} \\ f_{small}^{-1}\mathcal{C}_{Y/Y'} & \longrightarrow f^*\mathcal{C}_{Y/Y'} \\ f_{small}^{-1}\mathcal{C}_{Y/Y'} & \longrightarrow f_{small}^{-1}\mathcal{O}_{Y'} \\ f_{small}^{-1}\mathcal{O}_{Y'} & \longrightarrow \mathcal{A} \\ f_{small}^{-1}\mathcal{O}_{Y'} & \longrightarrow f_{small}^{-1}\mathcal{O}_Y \\ f_{small}^{-1}\mathcal{O}_Y & \longrightarrow \mathcal{O}_X \\ f_{small}^{-1}\mathcal{O}_Y & \longrightarrow 0\end{aligned}\end{gathered}$$ then there exists a first order thickening $X \subset X''$ with $\mathcal{O}_{X''} = \mathcal{A}$ and applying More on Morphisms of Spaces, Lemma Nilpotent thickenings once more, we obtain a morphism $(f, f'') : (X \subset X'') \to (Y \subset Y')$ with all the desired properties. Thus part (1) follows from Lemma Choices in a square-zero extension of ringed topoi and part (2) from part (2) of Lemma Compatibility of the square-zero extension maps on ringed topoi. (Note that $\mathrm{NL}_{X/Y}$ as defined for a morphism of algebraic spaces in More on Morphisms of Spaces, Section The naive cotangent complex agrees with $\mathrm{NL}_{X/Y}$ as used in Section Nilpotent thickenings and sheaves on ringed sites.) $\square$

Lemma. Comparing square-zero extensions through the quasi-coherator

In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_X(\mathcal{F}) \to DQ_X(\mathcal{G})$ (Derived Categories of Spaces, Section Derived tensor products and Tor amplitude) is an isomorphism. Then the functor $F$ is an equivalence of categories.

Proof. Recall that $\mathrm{NL}_{X/B}$ is an object of $D_\mathrm{QCoh}(\mathcal{O}_X)$, see More on Morphisms of Spaces, Lemma Quasi-coherent complexes and coherent sheaves. Hence our assumption implies the maps $$\operatorname{Ext}^i_X(\mathrm{NL}_{X/B}, \mathcal{F}) \longrightarrow \operatorname{Ext}^i_X(\mathrm{NL}_{X/B}, \mathcal{G})$$ are isomorphisms for all $i$. This implies our functor is fully faithful by Lemma Compatibility of the square-zero extension maps on ringed topoi. On the other hand, the functor is essentially surjective by Lemma Choices in a square-zero extension of ringed topoi because we have the solutions $\mathcal{O}_X \oplus \mathcal{F}$ and $\mathcal{O}_X \oplus \mathcal{G}$ in both categories. $\square$

Lemma. Comparing square-zero extensions over a thickening

In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_X(\mathcal{F}) \to DQ_X(\mathcal{G})$ (Derived Categories of Spaces, Section Derived tensor products and Tor amplitude) is an isomorphism. Then the functor $FT$ is an equivalence of categories.

Proof. A solution of (Sheaves on ringed sites) for $\mathcal{F}$ in particular gives an extension of $f^{-1}\mathcal{O}_{B'}$-algebras $$0 \to \mathcal{F} \to \mathcal{O}' \to \mathcal{O}_X \to 0$$ where $\mathcal{F}$ is an ideal of square zero. Similarly for $\mathcal{G}$. Moreover, given such an extension, we obtain a map $c_{\mathcal{O}'} : f^{-1}\mathcal{J} \to \mathcal{F}$. Thus we are looking at the full subcategory of such extensions of $f^{-1}\mathcal{O}_{B'}$-algebras with $c = c_{\mathcal{O}'}$. Clearly, if $\mathcal{O}'' = F(\mathcal{O}')$ where $F$ is the equivalence of Lemma Comparing square-zero extensions through the quasi-coherator (applied to $X \to B'$ this time), then $c_{\mathcal{O}''}$ is the composition of $c_{\mathcal{O}'}$ and the map $\mathcal{F} \to \mathcal{G}$. This proves the lemma. $\square$

A.12.10 The schematic locus in dimension at most one

The following arguments adapt the human Stacks Project treatment in spaces-over-fields.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. The schematic locus in local dimension at most one

Separated algebraic spaces are schemes in codimension 1.

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. If $X$ is separated, locally Noetherian, and the dimension of the local ring of $X$ at $x$ is $\leq 1$ (Properties of Spaces, Definition Dimension, codimension and local algebra), then there exists an open subspace of $X$ containing $x$ which is a scheme.

Proof. (Please see the remark below for a different approach avoiding the material on finite groupoids.) We can replace $X$ by an quasi-compact neighbourhood of $x$, hence we may assume $X$ is quasi-compact, separated, and Noetherian. There exists a scheme $U$ and a finite surjective morphism $U \to X$, see Limits of Spaces, Proposition Finite algebras. Let $R = U \times_X U$. Then $j : R \to U \times_S U$ is an equivalence relation and we obtain a groupoid scheme $(U, R, s, t, c)$ over $S$ with $s, t$ finite and $U$ Noetherian and separated. Let $\{u_1, \ldots, u_n\} \subset U$ be the set of points mapping to $x$. Then $\dim(\mathcal{O}_{U, u_i}) \leq 1$ by Decent Spaces, Lemma Dimension, codimension and finite algebras.

By More on Groupoids, Lemma An invariant affine neighbourhood of points in local dimension at most one there exists an $R$-invariant affine open $W \subset U$ containing the orbit $\{u_1, \ldots, u_n\}$. Since $U \to X$ is finite surjective the continuous map $|U| \to |X|$ is closed surjective, hence submersive by Topology, Lemma The geometric construction (uncovered prerequisite). Thus $f(W)$ is open and there is an open subspace $X' \subset X$ with $f : W \to X'$ a surjective finite morphism. Then $X'$ is an affine scheme by Cohomology of Spaces, Lemma Affine neighbourhoods and Noetherian rings and the proof is finished. $\square$

A.12.11 Ample bundles on reducible and nonreduced one-dimensional schemes

The following arguments adapt the human Stacks Project treatment in varieties.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Lemma. An element with prescribed negative valuations

Let $A$ be a domain with fraction field $K$. Let $B_1, \ldots, B_r \subset K$ be Noetherian $1$-dimensional semi-local domains whose fraction fields are $K$. If $A \otimes B_i \to K$ are surjective for $i = 1, \ldots, r$, then there exists an $x \in A$ such that $x^{-1}$ is in the Jacobson radical of $B_i$ for $i = 1, \ldots, r$.

Proof. Let $B_i'$ be the integral closure of $B_i$ in $K$. Suppose we find a nonzero $x \in A$ such that $x^{-1}$ is in the Jacobson radical of $B'_i$ for $i = 1, \ldots, r$. Then by Lemma Derived tensor products, Tor amplitude and dimension and codimension, after replacing $x$ by a power we get $x^{-1} \in B_i$. Since $\operatorname{Spec}(B'_i) \to \operatorname{Spec}(B_i)$ is surjective we see that $x^{-1}$ is then also in the Jacobson radical of $B_i$. Thus we may assume that each $B_i$ is a semi-local Dedekind domain.

If $B_i$ is not local, then remove $B_i$ from the list and add back the finite collection of local rings $(B_i)_\mathfrak m$. Thus we may assume that $B_i$ is a discrete valuation ring for $i = 1, \ldots, r$.

Let $v_i : K \to \mathbf{Z}$, $i = 1, \ldots, r$ be the corresponding discrete valuations (see Algebra, Lemma Characterizations of Dedekind domains). We are looking for a nonzero $x \in A$ with $v_i(x) < 0$ for $i = 1, \ldots, r$. We will prove this by induction on $r$.

If $r = 1$ and the result is wrong, then $A \subset B$ and the map $A \otimes B \to K$ is not surjective, contradiction.

If $r > 1$, then by induction we can find a nonzero $x \in A$ such that $v_i(x) < 0$ for $i = 1, \ldots, r - 1$. If $v_r(x) < 0$ then we are done, so we may assume $v_r(x) \geq 0$. By the base case we can find $y \in A$ nonzero such that $v_r(y) < 0$. After replacing $x$ by a power we may assume that $v_i(x) < v_i(y)$ for $i = 1, \ldots, r - 1$. Then $x + y$ is the element we are looking for. $\square$

Lemma. A globally generated line bundle cutting out an affine open

Let $X$ be an integral separated scheme. Let $U \subset X$ be a nonempty affine open such that $X \setminus U$ is a finite set of points $x_1, \ldots, x_r$ with $\mathcal{O}_{X, x_i}$ Noetherian of dimension $1$. Then there exists a globally generated invertible $\mathcal{O}_X$-module $\mathcal{L}$ and a section $s$ such that $U = X_s$.

Proof. Say $U = \operatorname{Spec}(A)$ and let $K$ be the function field of $X$. Write $B_i = \mathcal{O}_{X, x_i}$ and $\mathfrak m_i = \mathfrak m_{x_i}$. Since $x_i \not \in U$ we see that the open $U \times_X \operatorname{Spec}(B_i)$ of $\operatorname{Spec}(B_i)$ has only one point, i.e., $U \times_X \operatorname{Spec}(B_i) = \operatorname{Spec}(K)$. Since $X$ is separated, we find that $\operatorname{Spec}(K)$ is a closed subscheme of $U \times \operatorname{Spec}(B_i)$, i.e., the map $A \otimes B_i \to K$ is a surjection. By Lemma An element with prescribed negative valuations we can find a nonzero $f \in A$ such that $f^{-1} \in \mathfrak m_i$ for $i = 1, \ldots, r$. Pick opens $x_i \in U_i \subset X$ such that $f^{-1} \in \mathcal{O}(U_i)$. Then $$\mathcal{U} : X = U \cup \bigcup U_i$$ is an open covering of $X$. Consider the $2$-cocycle with values in $\mathcal{O}_X^*$ given by $f$ on $U \cap U_i$ and by $1$ on $U_i \cap U_j$. This defines a line bundle $\mathcal{L}$ with two sections:

1. a section $s$ defined by $1$ on $U$ and $f^{-1}$ on $U_i$ is as in the statement of the lemma, and

2. a section $t$ defined by $f$ on $U$ and $1$ on $U_i$.

Note that $X_t \supset U_1 \cup \ldots \cup U_r$. Hence $s, t$ generate $\mathcal{L}$ and the lemma is proved. $\square$

Lemma. Line bundles and ampleness

Let $X$ be a quasi-compact scheme. If for every $x \in X$ there exists a pair $(\mathcal{L}, s)$ consisting of a globally generated invertible sheaf $\mathcal{L}$ and a global section $s$ such that $x \in X_s$ and $X_s$ is affine, then $X$ has an ample invertible sheaf.

Proof. Since $X$ is quasi-compact we can find a finite collection $(\mathcal{L}_i, s_i)$, $i = 1, \ldots, n$ of pairs such that $\mathcal{L}_i$ is globally generated, $X_{s_i}$ is affine and $X = \bigcup X_{s_i}$. Again because $X$ is quasi-compact we can find, for each $i$, a finite collection of sections $t_{i, j}$ of $\mathcal{L}_i$, $j = 1, \ldots, m_i$ such that $X = \bigcup X_{t_{i, j}}$. Set $t_{i, 0} = s_i$. Consider the invertible sheaf $$\mathcal{L} = \mathcal{L}_1 \otimes_{\mathcal{O}_X} \ldots \otimes_{\mathcal{O}_X} \mathcal{L}_n$$ and the global sections $$\tau_J = t_{1, j_1} \otimes \ldots \otimes t_{n, j_n}$$ By Properties, Lemma Affine neighbourhoods the open $X_{\tau_J}$ is affine as soon as $j_i = 0$ for some $i$. It is a simple matter to see that these opens cover $X$. Hence $\mathcal{L}$ is ample by definition. $\square$

Lemma. Dimension, codimension and line bundles and ampleness

Let $X$ be a Noetherian integral separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf.

Proof. Choose an affine open covering $X = U_1 \cup \ldots \cup U_n$. Since $X$ is Noetherian, each of the sets $X \setminus U_i$ is finite. Thus by Lemma A globally generated line bundle cutting out an affine open we can find a pair $(\mathcal{L}_i, s_i)$ consisting of a globally generated invertible sheaf $\mathcal{L}_i$ and a global section $s_i$ such that $U_i = X_{s_i}$. We conclude that $X$ has an ample invertible sheaf by Lemma Line bundles and ampleness. $\square$

Lemma. Line bundles and ampleness

Let $X$ be a scheme. Let $Z_1, \ldots, Z_n \subset X$ be closed subschemes. Let $\mathcal{L}_i$ be an invertible sheaf on $Z_i$. Assume that

1. $X$ is reduced,

2. $X = \bigcup Z_i$ set theoretically, and

3. $Z_i \cap Z_j$ is a discrete topological space for $i \not = j$.

Then there exists an invertible sheaf $\mathcal{L}$ on $X$ whose restriction to $Z_i$ is $\mathcal{L}_i$. Moreover, if we are given sections $s_i \in \Gamma(Z_i, \mathcal{L}_i)$ which are nonvanishing at the points of $Z_i \cap Z_j$, then we can choose $\mathcal{L}$ such that there exists a $s \in \Gamma(X, \mathcal{L})$ with $s|_{Z_i} = s_i$ for all $i$.

Proof. The existence of $\mathcal{L}$ can be deduced from Lemma Finite algebras but we will also give a direct proof and we will use the direct proof to see the statement about sections is true. Set $T = \bigcup_{i \not = j} Z_i \cap Z_j$. As $X$ is reduced we have $$X \setminus T = \bigcup (Z_i \setminus T)$$ as schemes. Assumption (3) implies $T$ is a discrete subset of $X$. Thus for each $t \in T$ we can find an open $U_t \subset X$ with $t \in U_t$ but $t' \not \in U_t$ for $t' \in T$, $t' \not = t$. By shrinking $U_t$ if necessary, we may assume that there exist isomorphisms $\varphi_{t, i} : \mathcal{L}_i|_{U_t \cap Z_i} \to \mathcal{O}_{U_t \cap Z_i}$. Furthermore, for each $i$ choose an open covering $$Z_i \setminus T = \bigcup\nolimits_j U_{ij}$$ such that there exist isomorphisms $\varphi_{i, j} : \mathcal{L}_i|_{U_{ij}} \cong \mathcal{O}_{U_{ij}}$. Observe that $$\mathcal{U} : X = \bigcup U_t \cup \bigcup U_{ij}$$ is an open covering of $X$. We claim that we can use the isomorphisms $\varphi_{t, i}$ and $\varphi_{i, j}$ to define a $2$-cocycle with values in $\mathcal{O}_X^*$ for this covering that defines $\mathcal{L}$ as in the statement of the lemma.

Namely, if $i \not = i'$, then $U_{i, j} \cap U_{i', j'} = \emptyset$ and there is nothing to do. For $U_{i, j} \cap U_{i, j'}$ we have $\mathcal{O}_X(U_{i, j} \cap U_{i, j'}) = \mathcal{O}_{Z_i}(U_{i, j} \cap U_{i, j'})$ by the first remark of the proof. Thus the transition function for $\mathcal{L}_i$ (more precisely $\varphi_{i, j} \circ \varphi_{i, j'}^{-1}$) defines the value of our cocycle on this intersection. For $U_t \cap U_{i, j}$ we can do the same thing. Finally, for $t \not = t'$ we have $$U_t \cap U_{t'} = \coprod (U_t \cap U_{t'}) \cap Z_i$$ and moreover the intersection $U_t \cap U_{t'} \cap Z_i$ is contained in $Z_i \setminus T$. Hence by the same reasoning as before we see that $$\mathcal{O}_X(U_t \cap U_{t'}) = \prod \mathcal{O}_{Z_i}(U_t \cap U_{t'} \cap Z_i)$$ and we can use the transition functions for $\mathcal{L}_i$ (more precisely $\varphi_{t, i} \circ \varphi_{t', i}^{-1}$) to define the value of our cocycle on $U_t \cap U_{t'}$. This finishes the proof of existence of $\mathcal{L}$.

Given sections $s_i$ as in the last assertion of the lemma, in the argument above, we choose $U_t$ such that $s_i|_{U_t \cap Z_i}$ is nonvanishing and we choose $\varphi_{t, i}$ such that $\varphi_{t, i}(s_i|_{U_t \cap Z_i}) = 1$. Then using $1$ over $U_t$ and $\varphi_{i, j}(s_i|_{U_{i, j}})$ over $U_{i, j}$ will define a section of $\mathcal{L}$ which restricts to $s_i$ over $Z_i$. $\square$

Remark. Conductor ideals on one-dimensional schemes

Let $A$ be a reduced ring. Let $I, J$ be ideals of $A$ such that $V(I) \cup V(J) = \operatorname{Spec}(A)$. Set $B = A/J$. Then $I \to IB$ is an isomorphism of $A$-modules. Namely, we have $IB = I + J/J = I/(I \cap J)$ and $I \cap J$ is zero because $A$ is reduced and $\operatorname{Spec}(A) = V(I) \cup V(J) = V(I \cap J)$. Thus for any projective $A$-module $P$ we also have $IP = I(P/JP)$.

Lemma. Dimension, codimension and line bundles and ampleness

Let $X$ be a Noetherian reduced separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf.

Proof. Let $Z_i$, $i = 1, \ldots, n$ be the irreducible components of $X$. We view these as reduced closed subschemes of $X$. By Lemma Dimension, codimension and line bundles and ampleness there exist ample invertible sheaves $\mathcal{L}_i$ on $Z_i$. Set $T = \bigcup_{i \not = j} Z_i \cap Z_j$. As $X$ is Noetherian of dimension $1$, the set $T$ is finite and consists of closed points of $X$. For each $i$ we may, possibly after replacing $\mathcal{L}_i$ by a power, choose $s_i \in \Gamma(Z_i, \mathcal{L}_i)$ such that $(Z_i)_{s_i}$ is affine and contains $T \cap Z_i$, see Properties, Lemma Line bundles, ampleness and affine neighbourhoods.

By Lemma Line bundles and ampleness we can find an invertible sheaf $\mathcal{L}$ on $X$ and $s \in \Gamma(X, \mathcal{L})$ such that $(\mathcal{L}, s)|_{Z_i} = (\mathcal{L}_i, s_i)$. Observe that $X_s$ contains $T$ and is set theoretically equal to the affine closed subschemes $(Z_i)_{s_i}$. Thus it is affine by Limits, Lemma Affine neighbourhoods. To finish the proof, it suffices to find for every $x \in X$, $x \not \in T$ an integer $m > 0$ and a section $t \in \Gamma(X, \mathcal{L}^{\otimes m})$ such that $X_t$ is affine and $x \in X_t$. Since $x \not \in T$ we see that $x \in Z_i$ for some unique $i$, say $i = 1$. Let $Z \subset X$ be the reduced closed subscheme whose underlying topological space is $Z_2 \cup \ldots \cup Z_n$. Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal sheaf of $Z$. Denote that $\mathcal{I}_1 \subset \mathcal{O}_{Z_1}$ the inverse image of this ideal sheaf under the inclusion morphism $Z_1 \to X$. Observe that $$\Gamma(X, \mathcal{I}\mathcal{L}^{\otimes m}) = \Gamma(Z_1, \mathcal{I}_1 \mathcal{L}_1^{\otimes m})$$ see Remark Conductor ideals on one-dimensional schemes. Thus it suffices to find $m > 0$ and $t \in \Gamma(Z_1, \mathcal{I}_1 \mathcal{L}_1^{\otimes m})$ with $x \in (Z_1)_t$ affine. Since $\mathcal{L}_1$ is ample and since $x$ is not in $Z_1 \cap T = V(\mathcal{I}_1)$ we can find a section $t_1 \in \Gamma(Z_1, \mathcal{I}_1 \mathcal{L}_1^{\otimes m_1})$ with $x \in (Z_1)_{t_1}$, see Properties, Proposition Criteria for line bundles and ampleness. Since $\mathcal{L}_1$ is ample we can find a section $t_2 \in \Gamma(Z_1, \mathcal{L}_1^{\otimes m_2})$ with $x \in (Z_1)_{t_2}$ and $(Z_1)_{t_2}$ affine, see Properties, Definition Ample invertible sheaves. Set $m = m_1 + m_2$ and $t = t_1 t_2$. Then $t \in \Gamma(Z_1, \mathcal{I}_1 \mathcal{L}_1^{\otimes m})$ with $x \in (Z_1)_t$ by construction and $(Z_1)_t$ is affine by Properties, Lemma Affine neighbourhoods. $\square$

Lemma. Lifting derived Hom, Ext and dimension and codimension

Let $i : Z \to X$ be a closed immersion of schemes. If the underlying topological space of $X$ is Noetherian and $\dim(X) \leq 1$, then $\operatorname{Pic}(X) \to \operatorname{Pic}(Z)$ is surjective.

Proof. Consider the short exact sequence $$0 \to (1 + \mathcal{I}) \cap \mathcal{O}_X^* \to \mathcal{O}^*_X \to i_*\mathcal{O}^*_Z \to 0$$ of sheaves of abelian groups on $X$ where $\mathcal{I}$ is the quasi-coherent sheaf of ideals corresponding to $Z$. Since $\dim(X) \leq 1$ we see that $H^2(X, \mathcal{F}) = 0$ for any abelian sheaf $\mathcal{F}$, see Cohomology, Proposition Vanishing and Noetherian rings. Hence the map $H^1(X, \mathcal{O}^*_X) \to H^1(X, i_*\mathcal{O}_Z^*)$ is surjective. By Cohomology, Lemma Sheaf cohomology and diagonals and separation we have $H^1(X, i_*\mathcal{O}_Z^*) = H^1(Z, \mathcal{O}_Z^*)$. This proves the lemma by Cohomology, Lemma Line bundles and ampleness. $\square$

Proposition. Dimension, codimension and line bundles and ampleness

Let $X$ be a Noetherian separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf.

Proof. Let $Z \subset X$ be the reduction of $X$. By Lemma Dimension, codimension and line bundles and ampleness the scheme $Z$ has an ample invertible sheaf. Thus by Lemma Lifting derived Hom, Ext and dimension and codimension there exists an invertible $\mathcal{O}_X$-module $\mathcal{L}$ on $X$ whose restriction to $Z$ is ample. Then $\mathcal{L}$ is ample by an application of Cohomology of Schemes, Lemma Line bundles and ampleness (uncovered prerequisite). $\square$

Lemma. Projective, locally free modules and dimension and codimension

Let $X$ be a separated finite type scheme over a field $k$. If $\dim(X) \leq 1$ then $X$ is H-quasi-projective over $k$.

Proof. By Proposition Dimension, codimension and line bundles and ampleness the scheme $X$ has an ample invertible sheaf $\mathcal{L}$. By Morphisms, Lemma Projective, locally free modules and finite algebras (uncovered prerequisite) we see that $X$ is isomorphic to a locally closed subscheme of $\mathbf{P}^n_k$ over $\operatorname{Spec}(k)$. This is the definition of being H-quasi-projective over $k$, see Morphisms, Definition Projective and locally free modules. $\square$

Lemma. Projectivity of proper field schemes of dimension at most one

Let $X$ be a proper scheme over a field $k$. If $\dim(X) \leq 1$ then $X$ is H-projective over $k$.

Proof. By Lemma Projective, locally free modules and dimension and codimension we see that $X$ is a locally closed subscheme of $\mathbf{P}^n_k$ for some field $k$. Since $X$ is proper over $k$ it follows that $X$ is a closed subscheme of $\mathbf{P}^n_k$ (Morphisms, Lemma Proper morphisms (uncovered prerequisite)). $\square$

A.12.12 Artin checks for polarized schemes and all curve spaces

The following arguments adapt the human Stacks Project treatment in quot.tex. Source correspondence and the exact lower prerequisites are retained in the accompanying record.

Situation. The groupoid of polarized proper schemes

We define a category $\mathcal{P}\!ol$ as follows. Objects are pairs $(X \to S, \mathcal{L})$ where

1. $X \to S$ is a morphism of schemes which is proper, flat, and of finite presentation, and

2. $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module which is relatively ample on $X/S$ (Morphisms, Definition Line bundles and ampleness).

A morphism $(X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ between objects is given by a triple $(f, g, \varphi)$ where $f : X' \to X$ and $g : S' \to S$ are morphisms of schemes which fit into a commutative diagram $$\begin{gathered}\begin{matrix}X' & X \\ S' & S\end{matrix} \\[6pt] \begin{aligned}X' & \longrightarrow S' \\ X' & \xrightarrow{f} X \\ X & \longrightarrow S \\ S' & \xrightarrow{g} S\end{aligned}\end{gathered}$$ inducing an isomorphism $X' \to S' \times_S X$, in other words, the diagram is cartesian, and $\varphi : f^*\mathcal{L} \to \mathcal{L}'$ is an isomorphism. Composition is defined in the obvious manner (see Examples of Stacks, Sections The geometric construction and Quasi-coherent complexes and coherent sheaves). The forgetful functor $$p : \mathcal{P}\!ol \longrightarrow \mathrm{Sch}_{fppf},\quad (X \to S, \mathcal{L}) \longmapsto S$$ is how we view $\mathcal{P}\!ol$ as a category over $\mathrm{Sch}_{fppf}$ (see Section Conventions for the moduli groupoids for notation).

Lemma. The groupoid of proper flat spaces

The category $\mathcal{S}\!paces'_{ft}$ is fibred in groupoids over $\mathrm{Sch}_{fppf}$. The same is true for $\mathcal{S}\!paces'_{fp, flat, proper}$.

Proof. We have seen this in Examples of Stacks, Section Groupoids, equivalence relations and finite algebras for the case of $\mathcal{S}\!paces'_{ft}$ and this easily implies the result for the other case. However, let us also prove this directly by checking conditions (1) and (2) of Categories, Definition Groupoids and equivalence relations.

Condition (1). Let $X \to S$ be an object of $\mathcal{S}\!paces'_{ft}$ and let $S' \to S$ be a morphism of schemes. Then we set $X' = S' \times_S X$. Note that $X' \to S'$ is of finite type by Morphisms of Spaces, Lemma Base change for finite algebras. to obtain a morphism $(X' \to S') \to (X \to S)$ lying over $S' \to S$. Argue similarly for the other case using Morphisms of Spaces, Lemmas Base change for finite presentation and finite algebras, Base change for flatness, and Base change for proper morphisms.

Condition (2). Consider morphisms $(f, g) : (X' \to S') \to (X \to S)$ and $(a, b) : (Y \to T) \to (X \to S)$ of $\mathcal{S}\!paces'_{ft}$. Given a morphism $h : T \to S'$ with $g \circ h = b$ we have to show there is a unique morphism $(k, h) : (Y \to T) \to (X' \to S')$ of $\mathcal{S}\!paces'_{ft}$ such that $(f, g) \circ (k, h) = (a, b)$. This is clear from the fact that $X' = S' \times_S X$. The same therefore works for any full subcategory of $\mathcal{S}\!paces'_{ft}$ satisfying (1). $\square$

Lemma. The diagonal of the proper-space groupoid

The diagonal $$\Delta : \mathcal{S}\!paces'_{fp, flat, proper} \longrightarrow \mathcal{S}\!paces'_{fp, flat, proper} \times \mathcal{S}\!paces'_{fp, flat, proper}$$ is representable by algebraic spaces.

Proof. We will use criterion (2) of Algebraic Stacks, Lemma Diagonals and separation (uncovered prerequisite). Let $S$ be a scheme and let $X$ and $Y$ be algebraic spaces of finite presentation over $S$, flat over $S$, and proper over $S$. We have to show that the functor $$\mathit{Isom}_S(X, Y) : (\mathrm{Sch}/S)_{fppf} \longrightarrow \textit{Sets}, \quad T \longmapsto \{f : X_T \to Y_T \text{ isomorphism}\}$$ is an algebraic space. An elementary argument shows that $\mathit{Isom}_S(X, Y)$ sits in a fibre product $$\begin{gathered}\begin{matrix}\mathit{Isom}_S(X, Y) & S \\ \mathit{Mor}_S(X, Y) \times \mathit{Mor}_S(Y, X) & \mathit{Mor}_S(X, X) \times \mathit{Mor}_S(Y, Y)\end{matrix} \\[6pt] \begin{aligned}\mathit{Isom}_S(X, Y) & \longrightarrow S \\ \mathit{Isom}_S(X, Y) & \longrightarrow \mathit{Mor}_S(X, Y) \times \mathit{Mor}_S(Y, X) \\ S & \xrightarrow{(\text{id}, \text{id})} \mathit{Mor}_S(X, X) \times \mathit{Mor}_S(Y, Y) \\ \mathit{Mor}_S(X, Y) \times \mathit{Mor}_S(Y, X) & \longrightarrow \mathit{Mor}_S(X, X) \times \mathit{Mor}_S(Y, Y)\end{aligned}\end{gathered}$$ The bottom arrow sends $(\varphi, \psi)$ to $(\psi \circ \varphi, \varphi \circ \psi)$. By Proposition Representability of the morphism functor the functors on the bottom row are algebraic spaces over $S$. Hence the result follows from the fact that the category of algebraic spaces over $S$ has fibre products. $\square$

Lemma. Descent of proper flat spaces

The category $\mathcal{S}\!paces'_{ft}$ is a stack in groupoids over $\mathrm{Sch}_{fppf}$. The same is true for $\mathcal{S}\!paces'_{fp, flat, proper}$.

Proof. The reason this lemma holds is the slogan: any fppf descent datum for algebraic spaces is effective, see Bootstrap, Section The geometric construction (uncovered prerequisite). More precisely, the lemma for $\mathcal{S}\!paces'_{ft}$ follows from Examples of Stacks, Lemma Finite algebras (uncovered prerequisite) as we saw in Examples of Stacks, Section Groupoids, equivalence relations and finite algebras. However, let us review the proof. We need to check conditions (1), (2), and (3) of Stacks, Definition Groupoids and equivalence relations.

Property (1) we have seen in Lemma The groupoid of proper flat spaces.

Property (2) follows from Lemma The diagonal of the proper-space groupoid in the case of $\mathcal{S}\!paces'_{fp, flat, proper}$. In the case of $\mathcal{S}\!paces'_{ft}$ it follows from Examples of Stacks, Lemma The geometric construction (uncovered prerequisite) (and this is really the "correct" reference).

Condition (3) for $\mathcal{S}\!paces'_{ft}$ is checked as follows. Suppose given

1. an fppf covering $\{U_i \to U\}_{i \in I}$ in $\mathrm{Sch}_{fppf}$,

2. for each $i \in I$ an algebraic space $X_i$ of finite type over $U_i$, and

3. for each $i, j \in I$ an isomorphism $\varphi_{ij} : X_i \times_U U_j \to U_i \times_U X_j$ of algebraic spaces over $U_i \times_U U_j$ satisfying the cocycle condition over $U_i \times_U U_j \times_U U_k$.

We have to show there exists an algebraic space $X$ of finite type over $U$ and isomorphisms $X_{U_i} \cong X_i$ over $U_i$ recovering the isomorphisms $\varphi_{ij}$. This follows from Bootstrap, Lemma The geometric construction (uncovered prerequisite) part (2). By Descent on Spaces, Lemma Descent of proper morphisms and finite algebras we see that $X \to U$ is of finite type. In the case of $\mathcal{S}\!paces'_{fp, flat, proper}$ one additionally uses Descent on Spaces, Lemma Descent of finite presentation and proper morphisms, Descent of flatness and proper morphisms, and Descent of proper morphisms in the last step. $\square$

Lemma. Limit preservation for proper flat spaces

The stack $p'_{fp, flat, proper} : \mathcal{S}\!paces'_{fp, flat, proper} \to \mathrm{Sch}_{fppf}$ is limit preserving (Artin's Axioms, Definition Limit-preserving deformation groupoids).

Proof. Let $T = \varprojlim T_i$ be the limits of a directed inverse system of affine schemes. By Limits of Spaces, Lemma Descent of finite presentation and finite algebras the category of algebraic spaces of finite presentation over $T$ is the colimit of the categories of algebraic spaces of finite presentation over $T_i$. To finish the proof use that flatness and properness descends through the limit, see Limits of Spaces, Lemmas Descent of flatness and Proper morphisms. $\square$

Lemma. Infinitesimal patching of proper flat spaces

Let $$\begin{gathered}\begin{matrix}T & T' \\ S & S'\end{matrix} \\[6pt] \begin{aligned}T & \longrightarrow T' \\ T & \longrightarrow S \\ T' & \longrightarrow S' \\ S & \longrightarrow S'\end{aligned}\end{gathered}$$ be a pushout in the category of schemes where $T \to T'$ is a thickening and $T \to S$ is affine, see More on Morphisms, Lemma Pushouts along a thickening of algebraic spaces. Then the functor on fibre categories $$\begin{matrix} \mathcal{S}\!paces'_{fp, flat, proper, S'} \\ \downarrow \\ \mathcal{S}\!paces'_{fp, flat, proper, S} \times_{\mathcal{S}\!paces'_{fp, flat, proper, T}} \mathcal{S}\!paces'_{fp, flat, proper, T'} \end{matrix}$$ is an equivalence.

Proof. The functor is an equivalence if we drop "proper" from the list of conditions and replace "of finite presentation" by "locally of finite presentation", see Pushouts of Spaces, Lemma Patching flat algebraic spaces. Thus it suffices to show that given a morphism $X' \to S'$ of an algebraic space to $S'$ which is flat and locally of finite presentation, then $X' \to S'$ is proper if and only if $S \times_{S'} X' \to S$ and $T' \times_{S'} X' \to T'$ are proper. One implication follows from the fact that properness is preserved under base change (Morphisms of Spaces, Lemma Base change for proper morphisms) and the other from the fact that properness of $S \times_{S'} X' \to S$ implies properness of $X' \to S'$ by More on Morphisms of Spaces, Lemma Proper morphisms and nilpotent thickenings. $\square$

Lemma. Finite tangent spaces of proper flat spaces

Let $k$ be a field and let $x = (X \to \operatorname{Spec}(k))$ be an object of $\mathcal{X} = \mathcal{S}\!paces'_{fp, flat, proper}$ over $\operatorname{Spec}(k)$.

1. If $k$ is of finite type over $\mathbf{Z}$, then the vector spaces $T\mathcal{F}_{\mathcal{X}, k, x}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x})$ (see Artin's Axioms, Section Tangent spaces of deformation groupoids) are finite dimensional, and

2. in general the vector spaces $T_x(k)$ and $\text{Inf}_x(k)$ (see Artin's Axioms, Section Infinitesimal automorphisms) are finite dimensional.

Proof. The discussion in Artin's Axioms, Section Tangent spaces of deformation groupoids only applies to fields of finite type over the base scheme $\operatorname{Spec}(\mathbf{Z})$. Our stack satisfies (RS*) by Lemma Infinitesimal patching of proper flat spaces and we may apply Artin's Axioms, Lemma Lifting properties under infinitesimal patching to get the vector spaces $T_x(k)$ and $\text{Inf}_x(k)$ mentioned in (2). Moreover, in the finite type case these spaces agree with the ones mentioned in (1) by Artin's Axioms, Remark Comparing deformation categories. With this out of the way we can start the proof. Observe that the first order thickening $\operatorname{Spec}(k) \to \operatorname{Spec}(k[\epsilon]) = \operatorname{Spec}(k[k])$ has conormal module $k$. Hence the formula in Deformation Theory, Lemma Deformations of algebraic spaces describing infinitesimal deformations of $X$ and infinitesimal automorphisms of $X$ become $$T_x(k) = \operatorname{Ext}^1_{\mathcal{O}_X}(\mathrm{NL}_{X/k}, \mathcal{O}_X) \quad\text{and}\quad \text{Inf}_x(k) = \operatorname{Ext}^0_{\mathcal{O}_X}(\mathrm{NL}_{X/k}, \mathcal{O}_X)$$ By More on Morphisms of Spaces, Lemma Finite presentation from a finite-type flat family and the fact that $X$ is Noetherian, we see that $\mathrm{NL}_{X/k}$ has coherent cohomology sheaves zero except in degrees $0$ and $-1$. By Derived Categories of Spaces, Lemma Derived Hom, Ext and finite algebras the displayed $\operatorname{Ext}$-groups are finite $k$-vector spaces and the proof is complete. $\square$

Lemma. Openness of versality for proper flat spaces

The stack in groupoids $\mathcal{X} = \mathcal{S}\!paces'_{fp, flat, proper}$ satisfies openness of versality over $\operatorname{Spec}(\mathbf{Z})$. Similarly, after base change (Remark Base change of the proper-space groupoid) openness of versality holds over any Noetherian base scheme $S$.

Proof. For the "usual" proof of this fact, please see the discussion in the remark following this proof. We will prove this using Artin's Axioms, Lemma Strong effectivity implies openness of versality. We have already seen that $\mathcal{X}$ has diagonal representable by algebraic spaces, has (RS*), and is limit preserving, see Lemmas The diagonal of the proper-space groupoid, Infinitesimal patching of proper flat spaces, and Limit preservation for proper flat spaces. Hence we only need to see that $\mathcal{X}$ satisfies the strong formal effectiveness formulated in Artin's Axioms, Lemma Strong effectivity implies openness of versality.

Let $(R_n)$ be an inverse system of rings such that $R_n \to R_m$ is surjective with square zero kernel for all $n \geq m$. Let $X_n \to \operatorname{Spec}(R_n)$ be a finitely presented, flat, proper morphism where $X_n$ is an algebraic space and let $X_{n + 1} \to X_n$ be a morphism over $\operatorname{Spec}(R_{n + 1})$ inducing an isomorphism $X_n = X_{n + 1} \times_{\operatorname{Spec}(R_{n + 1})} \operatorname{Spec}(R_n)$. We have to find a flat, proper, finitely presented morphism $X \to \operatorname{Spec}(\varprojlim R_n)$ whose source is an algebraic space such that $X_n$ is the base change of $X$ for all $n$.

Let $I_n = \operatorname{Ker}(R_n \to R_1)$. We may think of $(X_1 \subset X_n) \to (\operatorname{Spec}(R_1) \subset \operatorname{Spec}(R_n))$ as a morphism of first order thickenings. (Please read some of the material on thickenings of algebraic spaces in More on Morphisms of Spaces, Section Nilpotent thickenings before continuing.) The structure sheaf of $X_n$ is an extension $$0 \to \mathcal{O}_{X_1} \otimes_{R_1} I_n \to \mathcal{O}_{X_n} \to \mathcal{O}_{X_1} \to 0$$ over $0 \to I_n \to R_n \to R_1$, see More on Morphisms of Spaces, Lemma Square-zero deformations and their comparison maps. Let's consider the extension $$0 \to \varprojlim \mathcal{O}_{X_1} \otimes_{R_1} I_n \to \varprojlim \mathcal{O}_{X_n} \to \mathcal{O}_{X_1} \to 0$$ over $0 \to \varprojlim I_n \to \varprojlim R_n \to R_1 \to 0$. The displayed sequence is exact as the $R^1\varprojlim$ of the system of kernels is zero by Derived Categories of Spaces, Lemma Quasi-coherent complexes and coherent sheaves. Observe that the map $$\mathcal{O}_{X_1} \otimes_{R_1} \varprojlim I_n \longrightarrow \varprojlim \mathcal{O}_{X_1} \otimes_{R_1} I_n$$ induces an isomorphism upon applying the functor $DQ_X$, see Derived Categories of Spaces, Lemma Pullback of derived quasi-coherent complexes. Hence we obtain a unique extension $$0 \to \mathcal{O}_{X_1} \otimes_{R_1} \varprojlim I_n \to \mathcal{O}' \to \mathcal{O}_{X_1} \to 0$$ over $0 \to \varprojlim I_n \to \varprojlim R_n \to R_1 \to 0$ by the equivalence of categories of Deformation Theory, Lemma Comparing square-zero extensions over a thickening. The sheaf $\mathcal{O}'$ determines a first order thickening of algebraic spaces $X_1 \subset X$ over $\operatorname{Spec}(R_1) \subset \operatorname{Spec}(\varprojlim R_n)$ by More on Morphisms of Spaces, Lemma Nilpotent thickenings. Observe that $X \to \operatorname{Spec}(\varprojlim R_n)$ is flat by the already used More on Morphisms of Spaces, Lemma Square-zero deformations and their comparison maps. By More on Morphisms of Spaces, Lemma Properties preserved under a square-zero deformation we see that $X \to \operatorname{Spec}(\varprojlim R_n)$ is proper and of finite presentation. This finishes the proof. $\square$

Lemma. Base change of polarized proper schemes

The category $\mathcal{P}\!ol$ is fibred in groupoids over $\mathcal{S}\!paces'_{fp, flat, proper}$. The category $\mathcal{P}\!ol$ is fibred in groupoids over $\mathrm{Sch}_{fppf}$.

Proof. We check conditions (1) and (2) of Categories, Definition Groupoids and equivalence relations.

Condition (1). Let $(X \to S, \mathcal{L})$ be an object of $\mathcal{P}\!ol$ and let $(X' \to S') \to (X \to S)$ be a morphism of $\mathcal{S}\!paces'_{fp, flat, proper}$. Then we let $\mathcal{L}'$ be the pullback of $\mathcal{L}$ to $X'$. Observe that $X, S, S'$ are schemes, hence $X'$ is a scheme as well (as the fibre product of schemes). Then $\mathcal{L}'$ is ample on $X'/S'$ by Morphisms, Lemma Base change for line bundles and ampleness (uncovered prerequisite). In this way we obtain a morphism $(X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ lying over $(X' \to S') \to (X \to S)$.

Condition (2). Consider morphisms $(f, g, \varphi) : (X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ and $(a, b, \psi) : (Y \to T, \mathcal{N}) \to (X \to S, \mathcal{L})$ of $\mathcal{P}\!ol$. Given a morphism $(k, h) : (Y \to T) \to (X' \to S')$ of $\mathcal{S}\!paces'_{fp, flat, proper}$ with $(f, g) \circ (k, h) = (a, b)$ we have to show there is a unique morphism $(k, h, \chi) : (Y \to T, \mathcal{N}) \to (X' \to S', \mathcal{L}')$ of $\mathcal{P}\!ol$ such that $(f, g, \varphi) \circ (k, h, \chi) = (a, b, \psi)$. We can just take $$\chi = \psi \circ (k^*\varphi)^{-1}$$ This proves condition (2). A composition of functors defining fibred categories defines a fibred category, see Categories, Lemma The geometric construction (uncovered prerequisite). This we see that $\mathcal{P}\!ol$ is fibred in groupoids over $\mathrm{Sch}_{fppf}$ (strictly speaking we should check the fibre categories are groupoids and apply Categories, Lemma Groupoids and equivalence relations (uncovered prerequisite)). $\square$

Lemma. Descent of polarized proper schemes

The category $\mathcal{P}\!ol$ is a stack in groupoids over $\mathcal{S}\!paces'_{fp, flat, proper}$ (endowed with the inherited topology, see Stacks, Definition The geometric construction). The category $\mathcal{P}\!ol$ is a stack in groupoids over $\mathrm{Sch}_{fppf}$.

Proof. We prove $\mathcal{P}\!ol$ is a stack in groupoids over $\mathcal{S}\!paces'_{fp, flat, proper}$ by checking conditions (1), (2), and (3) of Stacks, Definition Groupoids and equivalence relations. We have already seen (1) in Lemma Base change of polarized proper schemes.

A covering of $\mathcal{S}\!paces'_{fp, flat, proper}$ comes about in the following manner: Let $X \to S$ be an object of $\mathcal{S}\!paces'_{fp, flat, proper}$. Suppose that $\{S_i \to S\}_{i \in I}$ is a covering of $\mathrm{Sch}_{fppf}$. Set $X_i = S_i \times_S X$. Then $\{(X_i \to S_i) \to (X \to S)\}_{i \in I}$ is a covering of $\mathcal{S}\!paces'_{fp, flat, proper}$ and every covering of $\mathcal{S}\!paces'_{fp, flat, proper}$ is isomorphic to one of these. Set $S_{ij} = S_i \times_S S_j$ and $X_{ij} = S_{ij} \times_S X$ so that $(X_{ij} \to S_{ij}) = (X_i \to S_i) \times_{(X \to S)} (X_j \to S_j)$. Next, suppose that $\mathcal{L}, \mathcal{N}$ are ample invertible sheaves on $X/S$ so that $(X \to S, \mathcal{L})$ and $(X \to S, \mathcal{N})$ are two objects of $\mathcal{P}\!ol$ over the object $(X \to S)$. To check descent for morphisms, we assume we have morphisms $(\text{id}, \text{id}, \varphi_i)$ from $(X_i \to S_i, \mathcal{L}|_{X_i})$ to $(X_i \to S_i, \mathcal{N}|_{X_i})$ whose base changes to morphisms from $(X_{ij} \to S_{ij}, \mathcal{L}|_{X_{ij}})$ to $(X_{ij} \to S_{ij}, \mathcal{N}|_{X_{ij}})$ agree. Then $\varphi_i : \mathcal{L}|_{X_i} \to \mathcal{N}|_{X_i}$ are isomorphisms of invertible modules over $X_i$ such that $\varphi_i$ and $\varphi_j$ restrict to the same isomorphisms over $X_{ij}$. By descent for quasi-coherent sheaves (Descent on Spaces, Proposition Quasi-coherent complexes and coherent sheaves) we obtain a unique isomorphism $\varphi : \mathcal{L} \to \mathcal{N}$ whose restriction to $X_i$ recovers $\varphi_i$.

Decent for objects is proved in exactly the same manner. Namely, suppose that $\{(X_i \to S_i) \to (X \to S)\}_{i \in I}$ is a covering of $\mathcal{S}\!paces'_{fp, flat, proper}$ as above. Suppose we have objects $(X_i \to S_i, \mathcal{L}_i)$ of $\mathcal{P}\!ol$ lying over $(X_i \to S_i)$ and a descent datum $$(\text{id}, \text{id}, \varphi_{ij}) : (X_{ij} \to S_{ij}, \mathcal{L}_i|_{X_{ij}}) \to (X_{ij} \to S_{ij}, \mathcal{L}_j|_{X_{ij}})$$ satisfying the obvious cocycle condition over $(X_{ijk} \to S_{ijk})$ for every triple of indices. Then by descent for quasi-coherent sheaves (Descent on Spaces, Proposition Quasi-coherent complexes and coherent sheaves) we obtain a unique invertible $\mathcal{O}_X$-module $\mathcal{L}$ and isomorphisms $\mathcal{L}|_{X_i} \to \mathcal{L}_i$ recovering the descent datum $\varphi_{ij}$. To show that $(X \to S, \mathcal{L})$ is an object of $\mathcal{P}\!ol$ we have to prove that $\mathcal{L}$ is ample. This follows from Descent on Spaces, Lemma Descent of proper morphisms and line bundles and ampleness.

Since we already have seen that $\mathcal{S}\!paces'_{fp, flat, proper}$ is a stack in groupoids over $\mathrm{Sch}_{fppf}$ (Lemma Descent of proper flat spaces) it now follows formally that $\mathcal{P}\!ol$ is a stack in groupoids over $\mathrm{Sch}_{fppf}$. See Stacks, Lemma The geometric construction (uncovered prerequisite). $\square$

Lemma. The diagonal of the polarized moduli stack

The diagonal $$\Delta : \mathcal{P}\!ol \longrightarrow \mathcal{P}\!ol \times \mathcal{P}\!ol$$ is representable by algebraic spaces.

Proof. This is a formal consequence of Lemmas The forgetful map from polarized schemes to proper spaces and The diagonal of the proper-space groupoid. See Criteria for Representability, Lemma Diagonals and separation (uncovered prerequisite). $\square$

Lemma. Limit preservation for polarized proper schemes

The stack in groupoids $\mathcal{P}\!ol$ is limit preserving (Artin's Axioms, Definition Limit-preserving deformation groupoids).

Proof. Let $I$ be a directed set and let $(A_i, \varphi_{ii'})$ be a system of rings over $I$. Set $S = \operatorname{Spec}(A)$ and $S_i = \operatorname{Spec}(A_i)$. We have to show that on fibre categories we have $$\mathcal{P}\!ol_S = \mathop{\operatorname{colim}} \mathcal{P}\!ol_{S_i}$$ We know that the category of schemes of finite presentation over $S$ is the colimit of the category of schemes of finite presentation over $S_i$, see Limits, Lemma Descent of finite presentation and finite algebras. Moreover, given $X_i \to S_i$ of finite presentation, with limit $X \to S$, then the category of invertible $\mathcal{O}_X$-modules $\mathcal{L}$ is the colimit of the categories of invertible $\mathcal{O}_{X_i}$-modules $\mathcal{L}_i$, see Limits, Lemma Descent of finite presentation and modules and Descent of finite locally free and invertible modules. If $X \to S$ is proper and flat, then for sufficiently large $i$ the morphism $X_i \to S_i$ is proper and flat too, see Limits, Lemmas Proper morphisms and Descent of finite presentation and flatness. Finally, if $\mathcal{L}$ is ample on $X$ then $\mathcal{L}_i$ is ample on $X_i$ for $i$ sufficiently large, see Limits, Lemma Filtered limits and line bundles and ampleness. Putting everything together finishes the proof. $\square$

Lemma. Infinitesimal patching of polarized proper schemes

In Situation The groupoid of flat sheaves with proper support. Let $$\begin{gathered}\begin{matrix}T & T' \\ S & S'\end{matrix} \\[6pt] \begin{aligned}T & \longrightarrow T' \\ T & \longrightarrow S \\ T' & \longrightarrow S' \\ S & \longrightarrow S'\end{aligned}\end{gathered}$$ be a pushout in the category of schemes where $T \to T'$ is a thickening and $T \to S$ is affine, see More on Morphisms, Lemma Pushouts along a thickening of algebraic spaces. Then the functor on fibre categories $$\mathcal{P}\!ol_{S'} \longrightarrow \mathcal{P}\!ol_S \times_{\mathcal{P}\!ol_T} \mathcal{P}\!ol_{T'}$$ is an equivalence.

Proof. By More on Morphisms, Lemma Flatness and groupoids and equivalence relations there is an equivalence $$\textit{flat-lfp}_{S'} \longrightarrow \textit{flat-lfp}_S \times_{\textit{flat-lfp}_T} \textit{flat-lfp}_{T'}$$ where $\textit{flat-lfp}_S$ signifies the category of schemes flat and locally of finite presentation over $S$. Let $X'/S'$ on the left hand side correspond to the triple $(X/S, Y'/T', \varphi)$ on the right hand side. Set $Y = T \times_{T'} Y'$ which is isomorphic with $T \times_S X$ via $\varphi$. Then More on Morphisms, Lemma Flatness and modules shows that we have an equivalence $$\textit{QCoh-flat}_{X'/S'} \longrightarrow \textit{QCoh-flat}_{X/S} \times_{\textit{QCoh-flat}_{Y/T}} \textit{QCoh-flat}_{Y'/T'}$$ where $\textit{QCoh-flat}_{X/S}$ signifies the category of quasi-coherent $\mathcal{O}_X$-modules flat over $S$. Since $X \to S$, $Y \to T$, $X' \to S'$, $Y' \to T'$ are flat, this will in particular apply to invertible modules to give an equivalence of categories $$\textit{Pic}(X') \longrightarrow \textit{Pic}(X) \times_{\textit{Pic}(Y)} \textit{Pic}(Y')$$ where $\textit{Pic}(X)$ signifies the category of invertible $\mathcal{O}_X$-modules. There is a small point here: one has to show that if an object $\mathcal{F}'$ of $\textit{QCoh-flat}_{X'/S'}$ pulls back to invertible modules on $X$ and $Y'$, then $\mathcal{F}'$ is an invertible $\mathcal{O}_{X'}$-module. It follows from the cited lemma that $\mathcal{F}'$ is an $\mathcal{O}_{X'}$-module of finite presentation. By More on Morphisms, Lemma Projective, locally free modules and flatness it suffices to check the restriction of $\mathcal{F}'$ to fibres of $X' \to S'$ is invertible. But the fibres of $X' \to S'$ are the same as the fibres of $X \to S$ and hence these restrictions are invertible.

Having said the above we obtain an equivalence of categories if we drop the assumption (for the category of objects over $S$) that $X \to S$ be proper and the assumption that $\mathcal{L}$ be ample. Now it is clear that if $X' \to S'$ is proper, then $X \to S$ and $Y' \to T'$ are proper (Morphisms, Lemma Base change for proper morphisms (uncovered prerequisite)). Conversely, if $X \to S$ and $Y' \to T'$ are proper, then $X' \to S'$ is proper by More on Morphisms, Lemma Proper morphisms and nilpotent thickenings. Similarly, if $\mathcal{L}'$ is ample on $X'/S'$, then $\mathcal{L}'|_X$ is ample on $X/S$ and $\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$ (Morphisms, Lemma Base change for line bundles and ampleness (uncovered prerequisite)). Finally, if $\mathcal{L}'|_X$ is ample on $X/S$ and $\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$, then $\mathcal{L}'$ is ample on $X'/S'$ by More on Morphisms, Lemma Proper morphisms and line bundles and ampleness. $\square$

Lemma. Strong effectivity of polarized proper schemes (Strong formal effectiveness for polarized schemes)

Grothendieck's algebraization theorem continues to hold in the non-Noetherian setting if one assumes flatness and finite presentation.

Let $(R_n)$ be an inverse system of rings with surjective transition maps whose kernels are locally nilpotent. Set $R = \varprojlim R_n$. Set $S_n = \operatorname{Spec}(R_n)$ and $S = \operatorname{Spec}(R)$. Consider a commutative diagram $$\begin{gathered}\begin{matrix}X_1 & X_2 & X_3 & \ldots \\ S_1 & S_2 & S_3 & \ldots\end{matrix} \\[6pt] \begin{aligned}X_1 & \xrightarrow{i_1} X_2 \\ X_1 & \longrightarrow S_1 \\ X_2 & \xrightarrow{i_2} X_3 \\ X_2 & \longrightarrow S_2 \\ X_3 & \longrightarrow \ldots \\ X_3 & \longrightarrow S_3 \\ S_1 & \longrightarrow S_2 \\ S_2 & \longrightarrow S_3 \\ S_3 & \longrightarrow \ldots\end{aligned}\end{gathered}$$ of schemes with cartesian squares. Suppose given $(\mathcal{L}_n, \varphi_n)$ where each $\mathcal{L}_n$ is an invertible sheaf on $X_n$ and $\varphi_n : i_n^*\mathcal{L}_{n + 1} \to \mathcal{L}_n$ is an isomorphism. If

1. $X_n \to S_n$ is proper, flat, of finite presentation, and

2. $\mathcal{L}_1$ is ample on $X_1$

then there exists a morphism of schemes $X \to S$ proper, flat, and of finite presentation and an ample invertible $\mathcal{O}_X$-module $\mathcal{L}$ and isomorphisms $X_n \cong X \times_S S_n$ and $\mathcal{L}_n \cong \mathcal{L}|_{X_n}$ compatible with the morphisms $i_n$ and $\varphi_n$.

Proof. Choose $d_0$ for $X_1 \to S_1$ and $\mathcal{L}_1$ as in More on Morphisms, Lemma Uniform section algebras under nilpotent deformation. For any $n \geq 1$ set $$A_n = R_n \oplus \bigoplus\nolimits_{d \geq d_0} H^0(X_n, \mathcal{L}_n^{\otimes d})$$ By the lemma each $A_n$ is a finitely presented graded $R_n$-algebra whose homogeneous parts $(A_n)_d$ are finite projective $R_n$-modules such that $X_n = \text{Proj}(A_n)$ and $\mathcal{L}_n = \mathcal{O}_{\text{Proj}(A_n)}(1)$. The lemma also guarantees that the maps $$A_1 \leftarrow A_2 \leftarrow A_3 \leftarrow \ldots$$ induce isomorphisms $A_n = A_m \otimes_{R_m} R_n$ for $n \leq m$. We set $$B = \bigoplus\nolimits_{d \geq 0} B_d \quad\text{with}\quad B_d = \varprojlim_n (A_n)_d$$ By More on Algebra, Lemma Compatible finite projectives over an inverse limit we see that $B_d$ is a finite projective $R$-module and that $B \otimes_R R_n = A_n$. Thus the scheme $$X = \text{Proj}(B) \quad\text{and}\quad \mathcal{L} = \mathcal{O}_X(1)$$ is flat over $S$ and $\mathcal{L}$ is a quasi-coherent $\mathcal{O}_X$-module flat over $S$, see Divisors, Lemma Flatness. Because formation of Proj commutes with base change (Constructions, Lemma Base change for the geometric construction (uncovered prerequisite)) we obtain canonical isomorphisms $$X \times_S S_n = X_n \quad\text{and}\quad \mathcal{L}|_{X_n} \cong \mathcal{L}_n$$ compatible with the transition maps of the system. Thus we may think of $X_1 \subset X$ as a closed subscheme. Below we will show that $B$ is of finite presentation over $R$. By Divisors, Lemmas Proper morphisms and Finite presentation and finite algebras this implies that $X \to S$ is of finite presentation and proper and that $\mathcal{L} = \mathcal{O}_X(1)$ is of finite presentation as an $\mathcal{O}_X$-module. Since the restriction of $\mathcal{L}$ to the base change $X_1 \to S_1$ is invertible, we see from More on Morphisms, Lemma Projective, locally free modules and finite algebras that $\mathcal{L}$ is invertible on an open neighbourhood of $X_1$ in $X$. Since $X \to S$ is closed and since $\operatorname{Ker}(R \to R_1)$ is contained in the Jacobson radical (More on Algebra, Lemma Henselian pairs from locally nilpotent inverse systems) we see that any open neighbourhood of $X_1$ in $X$ is equal to $X$. Thus $\mathcal{L}$ is invertible. Finally, the set of points in $S$ where $\mathcal{L}$ is ample on the fibre is open in $S$ (More on Morphisms, Lemma Line bundles and ampleness) and contains $S_1$ hence equals $S$. Thus $X \to S$ and $\mathcal{L}$ have all the properties required of them in the statement of the lemma.

We prove the claim above. Choose a presentation $A_1 = R_1[X_1, \ldots, X_s]/(F_1, \ldots, F_t)$ where $X_i$ are variables having degrees $d_i$ and $F_j$ are homogeneous polynomials in $X_i$ of degree $e_j$. Then we can choose a map $$\Psi : R[X_1, \ldots, X_s] \longrightarrow B$$ lifting the map $R_1[X_1, \ldots, X_s] \to A_1$. Since each $B_d$ is finite projective over $R$ we conclude from Nakayama's lemma (Algebra, Lemma Nakayama's lemma using again that $\operatorname{Ker}(R \to R_1)$ is contained in the Jacobson radical of $R$) that $\Psi$ is surjective in every degree. Each polynomial degree piece is finite free, and $B_d$ is finite projective. Thus the degreewise surjection splits, so reduction identifies its kernel with the kernel of the reduced presentation. This projectivity, rather than right exactness alone, lets us find homogeneous $G_1, \ldots, G_t \in \operatorname{Ker}(\Psi)$ mapping to $F_1, \ldots, F_t$ in $R_1[X_1, \ldots, X_s]$. Observe that $\operatorname{Ker}(\Psi)_d$ is a finite projective $R$-module for all $d \geq 0$ as the kernel of the surjection $R[X_1, \ldots, X_s]_d \to B_d$ of finite projective $R$-modules. We conclude from Nakayama's lemma once more that the finite degreewise quotient of $\operatorname{Ker}(\Psi)$ by the ideal generated by $G_1, \ldots, G_t$ is zero after reduction and hence zero. Therefore $\operatorname{Ker}(\Psi)$ is generated by these homogeneous relations. $\square$

Lemma. Formal effectivity of polarized proper schemes

Consider the stack $\mathcal{P}\!ol$ over the base scheme $\operatorname{Spec}(\mathbf{Z})$. Then every formal object is effective.

Proof. For definitions of the notions in the lemma, please see Artin's Axioms, Section Formal objects and their markings. From the definitions we see the lemma follows immediately from the more general Lemma Strong effectivity of polarized proper schemes. $\square$

Lemma. Openness of versality for polarized proper schemes

The stack in groupoids $\mathcal{P}\!ol$ satisfies openness of versality over $\operatorname{Spec}(\mathbf{Z})$. Similarly, after base change (Remark Base change of the polarized moduli stack) openness of versality holds over any Noetherian base scheme $S$.

Proof. This follows from Artin's Axioms, Lemma Strong effectivity implies openness of versality and Lemmas The diagonal of the polarized moduli stack, Infinitesimal patching of polarized proper schemes, Limit preservation for polarized proper schemes, and Strong effectivity of polarized proper schemes. For the "usual" proof of this fact, please see the discussion in the remark following this proof. $\square$

Theorem. Algebraicity of the full polarized proper-scheme stack (Algebraicity of the stack of polarized schemes)

The stack $\mathcal{P}\!ol$ (Situation The groupoid of polarized proper schemes) is algebraic. In fact, for any algebraic space $B$ the stack $B\textit{-Polarized}$ (Remark Base change of the polarized moduli stack) is algebraic.

Proof. The absolute case follows from Artin's Axioms, Lemma Representability of a stack diagonal and Lemmas The diagonal of the polarized moduli stack, Infinitesimal patching of polarized proper schemes, Limit preservation for polarized proper schemes, Formal effectivity of polarized proper schemes, and Openness of versality for polarized proper schemes. The case over $B$ follows from this, the description of $B\textit{-Polarized}$ as a $2$-fibre product in Remark Base change of the polarized moduli stack, and the fact that algebraic stacks have $2$-fibre products, see Algebraic Stacks, Lemma Tensor products and direct sums (uncovered prerequisite). $\square$

Situation. The groupoid of proper flat spaces of dimension at most one

We define a category $\mathcal{C}\!urves$ as follows:

1. Objects are families of curves. More precisely, an object is a morphism $f : X \to S$ where the base $S$ is a scheme, the total space $X$ is an algebraic space, and $f$ is flat, proper, of finite presentation, and has relative dimension $\leq 1$ (Morphisms of Spaces, Definition Relative dimension).

2. A morphism $(X' \to S') \to (X \to S)$ between objects is given by a pair $(f, g)$ where $f : X' \to X$ is a morphism of algebraic spaces and $g : S' \to S$ is a morphism of schemes which fit into a commutative diagram $$\begin{gathered}\begin{matrix}X' & X \\ S' & S\end{matrix} \\[6pt] \begin{aligned}X' & \longrightarrow S' \\ X' & \xrightarrow{f} X \\ X & \longrightarrow S \\ S' & \xrightarrow{g} S\end{aligned}\end{gathered}$$ inducing an isomorphism $X' \to S' \times_S X$, in other words, the diagram is cartesian.

The forgetful functor $$p : \mathcal{C}\!urves \longrightarrow \mathrm{Sch}_{fppf},\quad (X \to S) \longmapsto S$$ is how we view $\mathcal{C}\!urves$ as a category over $\mathrm{Sch}_{fppf}$ (see Section Conventions for the moduli groupoids for notation).

Lemma. The groupoid of proper flat curve spaces

The category $\mathcal{C}\!urves$ is fibred in groupoids over $\mathrm{Sch}_{fppf}$.

Proof. Using the embedding (The inclusion of curve spaces in proper spaces), the description of the image, and the corresponding fact for $\mathcal{S}\!paces'_{fp, flat, proper}$ (Lemma The groupoid of proper flat spaces) this reduces to the following statement: Given a morphism $$\begin{gathered}\begin{matrix}X' & X \\ S' & S\end{matrix} \\[6pt] \begin{aligned}X' & \longrightarrow X \\ X' & \longrightarrow S' \\ X & \longrightarrow S \\ S' & \longrightarrow S\end{aligned}\end{gathered}$$ in $\mathcal{S}\!paces'_{fp, flat, proper}$ (recall that this implies in particular the diagram is cartesian) if $X \to S$ has relative dimension $\leq 1$, then $X' \to S'$ has relative dimension $\leq 1$. This follows from Morphisms of Spaces, Lemma Base change for dimension and codimension. $\square$

Lemma. Descent of proper flat curve spaces

The category $\mathcal{C}\!urves$ is a stack in groupoids over $\mathrm{Sch}_{fppf}$.

Proof. Using the embedding (The inclusion of curve spaces in proper spaces), the description of the image, and the corresponding fact for $\mathcal{S}\!paces'_{fp, flat, proper}$ (Lemma Descent of proper flat spaces) this reduces to the following statement: Given an object $X \to S$ of $\mathcal{S}\!paces'_{fp, flat, proper}$ and an fppf covering $\{S_i \to S\}_{i \in I}$ the following are equivalent:

1. $X \to S$ has relative dimension $\leq 1$, and

2. for each $i$ the base change $X_i \to S_i$ has relative dimension $\leq 1$.

This follows from Morphisms of Spaces, Lemma Base change for dimension and codimension. $\square$

Lemma. The diagonal of the proper curve-space stack

The diagonal $$\Delta : \mathcal{C}\!urves \longrightarrow \mathcal{C}\!urves \times \mathcal{C}\!urves$$ is representable by algebraic spaces.

Proof. This is immediate from the fully faithful embedding (The inclusion of curve spaces in proper spaces) and the corresponding fact for $\mathcal{S}\!paces'_{fp, flat, proper}$ (Lemma The diagonal of the proper-space groupoid). $\square$

Lemma. Limit preservation for proper flat curve spaces

The stack $\mathcal{C}\!urves \to \mathrm{Sch}_{fppf}$ is limit preserving (Artin's Axioms, Definition Limit-preserving deformation groupoids).

Proof. Using the embedding (The inclusion of curve spaces in proper spaces), the description of the image, and the corresponding fact for $\mathcal{S}\!paces'_{fp, flat, proper}$ (Lemma Limit preservation for proper flat spaces) this reduces to the following statement: Let $T = \varprojlim T_i$ be the limits of a directed inverse system of affine schemes. Let $i \in I$ and let $X_i \to T_i$ be an object of $\mathcal{S}\!paces'_{fp, flat, proper}$ over $T_i$. Assume that $T \times_{T_i} X_i \to T$ has relative dimension $\leq 1$. Then for some $i' \geq i$ the morphism $T_{i'} \times_{T_i} X_i \to T_i$ has relative dimension $\leq 1$. This follows from Limits of Spaces, Lemma Dimension and codimension. $\square$

Lemma. Infinitesimal patching of proper flat curve spaces

Let $$\begin{gathered}\begin{matrix}T & T' \\ S & S'\end{matrix} \\[6pt] \begin{aligned}T & \longrightarrow T' \\ T & \longrightarrow S \\ T' & \longrightarrow S' \\ S & \longrightarrow S'\end{aligned}\end{gathered}$$ be a pushout in the category of schemes where $T \to T'$ is a thickening and $T \to S$ is affine, see More on Morphisms, Lemma Pushouts along a thickening of algebraic spaces. Then the functor on fibre categories $$\mathcal{C}\!urves_{S'} \longrightarrow \mathcal{C}\!urves_S \times_{\mathcal{C}\!urves_T} \mathcal{C}\!urves_{T'}$$ is an equivalence.

Proof. Using the embedding (The inclusion of curve spaces in proper spaces), the description of the image, and the corresponding fact for $\mathcal{S}\!paces'_{fp, flat, proper}$ (Lemma Infinitesimal patching of proper flat spaces) this reduces to the following statement: given a morphism $X' \to S'$ of an algebraic space to $S'$ which is of finite presentation, flat, proper then $X' \to S'$ has relative dimension $\leq 1$ if and only if $S \times_{S'} X' \to S$ and $T' \times_{S'} X' \to T'$ have relative dimension $\leq 1$. One implication follows from the fact that having relative dimension $\leq 1$ is preserved under base change (Morphisms of Spaces, Lemma Base change for dimension and codimension). The other follows from the fact that having relative dimension $\leq 1$ is checked on the fibres and that the fibres of $X' \to S'$ (over points of the scheme $S'$) are the same as the fibres of $S \times_{S'} X' \to S$ since $S \to S'$ is a thickening by More on Morphisms, Lemma Pushouts along a thickening of algebraic spaces. $\square$

Lemma. Finite tangent spaces of proper flat curve spaces

Let $k$ be a field and let $x = (X \to \operatorname{Spec}(k))$ be an object of $\mathcal{X} = \mathcal{C}\!urves$ over $\operatorname{Spec}(k)$.

1. If $k$ is of finite type over $\mathbf{Z}$, then the vector spaces $T\mathcal{F}_{\mathcal{X}, k, x}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x})$ (see Artin's Axioms, Section Tangent spaces of deformation groupoids) are finite dimensional, and

2. in general the vector spaces $T_x(k)$ and $\text{Inf}_x(k)$ (see Artin's Axioms, Section Infinitesimal automorphisms) are finite dimensional.

Proof. This is immediate from the fully faithful embedding (The inclusion of curve spaces in proper spaces) and the corresponding fact for $\mathcal{S}\!paces'_{fp, flat, proper}$ (Lemma Finite tangent spaces of proper flat spaces). $\square$

Lemma. Formal effectivity of proper flat curve spaces

Consider the stack $\mathcal{C}\!urves$ over the base scheme $\operatorname{Spec}(\mathbf{Z})$. Then every formal object is effective.

Proof. For definitions of the notions in the lemma, please see Artin's Axioms, Section Formal objects and their markings. Let $(A, \mathfrak m, \kappa)$ be a Noetherian complete local ring. Let $(X_n \to \operatorname{Spec}(A/\mathfrak m^n))$ be a formal object of $\mathcal{C}\!urves$ over $A$. By More on Morphisms of Spaces, Lemma Algebraization of a proper flat finitely presented family of dimension at most one there exists a projective morphism $X \to \operatorname{Spec}(A)$ and a compatible system of ismomorphisms $X \times_{\operatorname{Spec}(A)} \operatorname{Spec}(A/\mathfrak m^n) \cong X_n$. By More on Morphisms, Lemma Derived tensor products, Tor amplitude and flatness we see that $X \to \operatorname{Spec}(A)$ is flat. By More on Morphisms, Lemma Flatness and dimension and codimension we see that $X \to \operatorname{Spec}(A)$ has relative dimension $\leq 1$. This proves the lemma. $\square$

Lemma. Openness of versality for proper flat curve spaces

The stack in groupoids $\mathcal{X} = \mathcal{C}\!urves$ satisfies openness of versality over $\operatorname{Spec}(\mathbf{Z})$. Similarly, after base change (Remark Base change of the curve-space stack) openness of versality holds over any Noetherian base scheme $S$.

Proof. This is immediate from the fully faithful embedding (The inclusion of curve spaces in proper spaces) and the corresponding fact for $\mathcal{S}\!paces'_{fp, flat, proper}$ (Lemma Openness of versality for proper flat spaces). $\square$

Theorem. Algebraicity of the full stack of proper flat curve spaces (Algebraicity of the stack of curves)

Source credit: See the original source citation dJHS (Proposition 3.3, page 8) and the original source citation Smyth (Appendix B by Jack Hall, Theorem B.1).

The stack $\mathcal{C}\!urves$ (Situation The groupoid of proper flat spaces of dimension at most one) is algebraic. In fact, for any algebraic space $B$ the stack $B\text{-}\mathcal{C}\!urves$ (Remark Base change of the curve-space stack) is algebraic.

Proof. The absolute case follows from Artin's Axioms, Lemma Representability of a stack diagonal and Lemmas The diagonal of the proper curve-space stack, Infinitesimal patching of proper flat curve spaces, Limit preservation for proper flat curve spaces, Formal effectivity of proper flat curve spaces, and Openness of versality for proper flat curve spaces. The case over $B$ follows from this, the description of $B\text{-}\mathcal{C}\!urves$ as a $2$-fibre product in Remark Base change of the curve-space stack, and the fact that algebraic stacks have $2$-fibre products, see Algebraic Stacks, Lemma Tensor products and direct sums (uncovered prerequisite). $\square$

A.12.13 Complete dévissage and flatness dimension strata

These complete native scheme arguments support the incorporated algebraic-space flattening proof. They do not stand in for the remaining one-step dévissage and finite-generic-rank representability arguments; those precise labels are recorded next. In the complete-dévissage proposition, $\mathcal F$ is a finite-type quasi-coherent module as required by the section and the actual consumer; its printed statement has omitted that quantification. In the dimension-stratum proof, after $T$-base change all references to relative flatness are over $T$.

Proposition. Complete dévissage at a point

Let $\mathcal F$ be a finite-type quasi-coherent module on $X$.

Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $x \in X$ be a point with image $s \in S$. There exists a commutative diagram $$\begin{gathered}\begin{matrix}(X, x) & (X', x') \\ (S, s) & (S', s')\end{matrix} \\[6pt] \begin{aligned}(X, x) & \longrightarrow (S, s) \\ (X', x') & \xrightarrow{g} (X, x) \\ (X', x') & \longrightarrow (S', s') \\ (S', s') & \longrightarrow (S, s)\end{aligned}\end{gathered}$$ of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that $g^*\mathcal{F}/X'/S'$ has a complete dévissage at $x$.

Proof. We prove this by induction on the integer $d = \dim_x(\text{Supp}(\mathcal{F}_s))$. By Lemma Purity and dévissage there exists a diagram $$\begin{gathered}\begin{matrix}(X, x) & (X', x') \\ (S, s) & (S', s')\end{matrix} \\[6pt] \begin{aligned}(X, x) & \longrightarrow (S, s) \\ (X', x') & \xrightarrow{g} (X, x) \\ (X', x') & \longrightarrow (S', s') \\ (S', s') & \longrightarrow (S, s)\end{aligned}\end{gathered}$$ of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that $g^*\mathcal{F}/X'/S'$ has a one step dévissage at $x'$. The local nature of the problem implies that we may replace $(X, x) \to (S, s)$ by $(X', x') \to (S', s')$. Thus after doing so we may assume that there exists a one step dévissage $(Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)$ of $\mathcal{F}/X/S$ at $x$.

We apply Lemma A generic free presentation in dévissage to find a map $$\alpha_1 : \mathcal{O}_{Y_1}^{\oplus r_1} \longrightarrow \pi_{1, *}\mathcal{G}_1$$ which induces an isomorphism of vector spaces over $\kappa(\xi_1)$ where $\xi_1 \in Y_1$ is the unique generic point of the fibre of $Y_1$ over $s$. Moreover $\dim_{y_1}(\text{Supp}(\operatorname{Coker}(\alpha_1)_s)) < d$. It may happen that the stalk of $\operatorname{Coker}(\alpha_1)_s$ at $y_1$ is zero. In this case we may shrink $Y_1$ by Lemma Restriction of a dévissage to a neighbourhood (the indicated step) and assume that $\operatorname{Coker}(\alpha_1) = 0$ so we obtain a complete dévissage of length zero.

Assume now that the stalk of $\operatorname{Coker}(\alpha_1)_s$ at $y_1$ is not zero. In this case, by induction, there exists a commutative diagram

$$\begin{gathered}\begin{matrix}(Y_1, y_1) & (Y'_1, y'_1) \\ (S, s) & (S', s')\end{matrix} \\[6pt] \begin{aligned}(Y_1, y_1) & \longrightarrow (S, s) \\ (Y'_1, y'_1) & \xrightarrow{h} (Y_1, y_1) \\ (Y'_1, y'_1) & \longrightarrow (S', s') \\ (S', s') & \longrightarrow (S, s)\end{aligned}\end{gathered}$$ of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that $h^*\operatorname{Coker}(\alpha_1)/Y'_1/S'$ has a complete dévissage $$(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 2, \ldots, n}$$ at $y'_1$. (In particular $i_2 : Z_2 \to Y'_1$ is a closed immersion into $Y'_2$.) At this point we apply Lemma Étale morphisms to $S, X, \mathcal{F}, x, s$, the system $(Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)$ and diagram (the displayed identity). We obtain a diagram $$\begin{gathered}\begin{matrix}\phantom{X} & \phantom{X} & (X'', x'') & (Z''_1, z''_1) \\ (X, x) & (Z_1, z_1) & (S'', s'') & (Y''_1, y''_1) \\ (S, s) & (Y_1, y_1)\end{matrix} \\[6pt] \begin{aligned}(X'', x'') & \longrightarrow (X, x) \\ (X'', x'') & \longrightarrow (S'', s'') \\ (Z''_1, z''_1) & \longrightarrow (X'', x'') \\ (Z''_1, z''_1) & \longrightarrow (Z_1, z_1) \\ (Z''_1, z''_1) & \longrightarrow (Y''_1, y''_1) \\ (X, x) & \longrightarrow (S, s) \\ (Z_1, z_1) & \longrightarrow (X, x) \\ (Z_1, z_1) & \longrightarrow (Y_1, y_1) \\ (S'', s'') & \longrightarrow (S, s) \\ (Y''_1, y''_1) & \longrightarrow (Y_1, y_1) \\ (Y''_1, y''_1) & \longrightarrow (S'', s'') \\ (Y_1, y_1) & \longrightarrow (S, s)\end{aligned}\end{gathered}$$ with all the properties as listed in the referenced lemma. In particular $Y''_1 \subset Y'_1 \times_{S'} S''$. Set $X_1 = Y'_1 \times_{S'} S''$ and let $\mathcal{F}_1$ denote the pullback of $\operatorname{Coker}(\alpha_1)$. By Lemma Base change for complete rings and formal power series the system

$$(Z_k \times_{S'} S'', Y_k \times_{S'} S'', i''_k, \pi''_k, \mathcal{G}''_k, \alpha''_k, z''_k, y''_k)_{k = 2, \ldots, n}$$ is a complete dévissage of $\mathcal{F}_1$ to $X_1$. Again, the nature of the problem allows us to replace $(X, x) \to (S, s)$ by $(X'', x'') \to (S'', s'')$. In this we see that we may assume:

1. There exists a one step dévissage $(Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)$ of $\mathcal{F}/X/S$ at $x$,

2. there exists an $\alpha_1 : \mathcal{O}_{Y_1}^{\oplus r_1} \to \pi_{1, *}\mathcal{G}_1$ such that $\alpha \otimes \kappa(\xi_1)$ is an isomorphism,

3. $Y_1 \subset X_1$ is open, $y_1 = x_1$, and $\mathcal{F}_1|_{Y_1} \cong \operatorname{Coker}(\alpha_1)$, and

4. there exists a complete dévissage $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 2, \ldots, n}$ of $\mathcal{F}_1/X_1/S$ at $x_1$.

To finish the proof all we have to do is shrink the one step dévissage and the complete dévissage such that they fit together to a complete dévissage. (We suggest the reader do this on their own using Lemmas Restriction of a dévissage to a neighbourhood and Restriction of a complete dévissage instead of reading the proof that follows.) Since $Y_1 \subset X_1$ is an open neighbourhood of $x_1$ we may apply Lemma Restriction of a complete dévissage (the indicated step) to find a standard shrinking $S', X'_1, Z'_2, Y'_2, \ldots, Y'_n$ of the datum (d) so that $X'_1 \subset Y_1$. Note that $X'_1$ is also a standard open of the affine scheme $Y_1$. Next, we shrink the datum (a) as follows: first we shrink the base $S$ to $S'$, see Lemma Restriction of a dévissage to a neighbourhood (the indicated step) and then we shrink the result to $S''$, $X''$, $Z''_1$, $Y''_1$ using Lemma Restriction of a dévissage to a neighbourhood (the indicated step) such that eventually $Y''_1 = X'_1 \times_S S''$ and $S'' \subset S'$. Then we see that $$Z''_1, Y''_1, Z'_2 \times_{S'} S'', Y'_2 \times_{S'} S'', \ldots, Y'_n \times_{S'} S''$$ gives the complete dévissage we were looking for. $\square$

Lemma. Complete dévissage around a fibre

Let $X \to S$ be a finite type morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$ be a point. There exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and étale morphisms $h_i : Y_i \to X_{S'}$, $i = 1, \ldots, n$ such that for each $i$ there exists a complete dévissage of $\mathcal{F}_i/Y_i/S'$ over $s'$, where $\mathcal{F}_i$ is the pullback of $\mathcal{F}$ to $Y_i$ and such that $X_s = (X_{S'})_{s'} \subset \bigcup h_i(Y_i)$.

Proof. For every point $x \in X_s$ we can find a diagram $$\begin{gathered}\begin{matrix}(X, x) & (X', x') \\ (S, s) & (S', s')\end{matrix} \\[6pt] \begin{aligned}(X, x) & \longrightarrow (S, s) \\ (X', x') & \xrightarrow{g} (X, x) \\ (X', x') & \longrightarrow (S', s') \\ (S', s') & \longrightarrow (S, s)\end{aligned}\end{gathered}$$ of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that $g^*\mathcal{F}/X'/S'$ has a complete dévissage at $x'$. As $X \to S$ is of finite type the fibre $X_s$ is quasi-compact, and since each $g : X' \to X$ as above is open we can cover $X_s$ by a finite union of $g(X'_{s'})$. Thus we can find a finite family of such diagrams $$\begin{gathered}\begin{matrix}(X, x) & (X'_i, x'_i) \\ (S, s) & (S'_i, s'_i)\end{matrix} \\[6pt] \begin{aligned}(X, x) & \longrightarrow (S, s) \\ (X'_i, x'_i) & \xrightarrow{g_i} (X, x) \\ (X'_i, x'_i) & \longrightarrow (S'_i, s'_i) \\ (S'_i, s'_i) & \longrightarrow (S, s)\end{aligned}\end{gathered} \quad i = 1, \ldots, n$$ such that $X_s = \bigcup g_i(X'_i)$. Set $S' = S'_1 \times_S \ldots \times_S S'_n$ and let $Y_i = X_i \times_{S'_i} S'$ be the base change of $X'_i$ to $S'$. By Lemma Base change for complete rings and formal power series we see that the pullback of $\mathcal{F}$ to $Y_i$ has a complete dévissage over $s$ and we win. $\square$

Lemma. Representing a flatness dimension stratum

In Situation The universal flatness dimension stratum. Let $s \in S$ let $d \geq 0$. Assume

1. there exists a complete dévissage of $\mathcal{F}/X/S$ over some point $s \in S$,

2. $X$ is of finite presentation over $S$,

3. $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, and

4. $\mathcal{F}$ is flat in dimensions $\geq d + 1$ over $S$.

Then after possibly replacing $S$ by an open neighbourhood of $s$ the functor $F_d$ (the displayed identity) is representable by a monomorphism $Z_d \to S$ of finite presentation.

Proof. A preliminary remark is that $X$, $S$ are affine schemes and that it suffices to prove $F_d$ is representable by a monomorphism of finite presentation $Z_d \to S$ on the category of affine schemes over $S$. (Of course we do not require $Z_d$ to be affine.) Hence throughout the proof of the lemma we work in the category of affine schemes over $S$.

Let $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k)_{k = 1, \ldots, n}$ be a complete dévissage of $\mathcal{F}/X/S$ over $s$, see Definition Complete dévissage. We will use induction on the length $n$ of the dévissage. Recall that $Y_k \to S$ is smooth with geometrically irreducible fibres, see Definition Purity and dévissage. Let $d_k$ be the relative dimension of $Y_k$ over $S$. Recall that $i_{k, *}\mathcal{G}_k = \operatorname{Coker}(\alpha_k)$ and that $i_k$ is a closed immersion. By the definitions referenced above we have $d_1 = \dim(\text{Supp}(\mathcal{F}_s))$ and $$d_k = \dim(\text{Supp}(\operatorname{Coker}(\alpha_{k - 1})_s)) = \dim(\text{Supp}(\mathcal{G}_{k, s}))$$ for $k = 2, \ldots, n$. It follows that $d_1 > d_2 > \ldots > d_n \geq 0$ because $\alpha_k$ is an isomorphism in the generic point of $(Y_k)_s$.

Note that $i_1$ is a closed immersion and $\mathcal{F} = i_{1, *}\mathcal{G}_1$. Hence for any morphism of schemes $T \to S$ with $T$ affine, we have $\mathcal{F}_T = i_{1, T, *}\mathcal{G}_{1, T}$ and $i_{1, T}$ is still a closed immersion of schemes over $T$. Thus $\mathcal{F}_T$ is flat in dimensions $\geq d$ over $T$ if and only if $\mathcal{G}_{1, T}$ is flat in dimensions $\geq d$ over $T$. Because $\pi_1 : Z_1 \to Y_1$ is finite we see in the same manner that $\mathcal{G}_{1, T}$ is flat in dimensions $\geq d$ over $T$ if and only if $\pi_{1, T, *}\mathcal{G}_{1, T}$ is flat in dimensions $\geq d$ over $T$. The same arguments work for "flat in dimensions $\geq d + 1$" and we conclude in particular that $\pi_{1, *}\mathcal{G}_1$ is flat over $S$ in dimensions $\geq d + 1$ by our assumption on $\mathcal{F}$.

Suppose that $d_1 > d$. It follows from the discussion above that in particular $\pi_{1, *}\mathcal{G}_1$ is flat over $S$ at the generic point of $(Y_1)_s$. By Lemma Finite presentation we may replace $S$ by an affine neighbourhood of $s$ and assume that $\alpha_1$ is $S$-universally injective. Because $\alpha_1$ is $S$-universally injective, for any morphism $T \to S$ with $T$ affine, we have a short exact sequence $$0 \to \mathcal{O}_{Y_{1, T}}^{\oplus r_1} \to \pi_{1, T, *}\mathcal{G}_{1, T} \to \operatorname{Coker}(\alpha_1)_T \to 0$$ and still the first arrow is $T$-universally injective. Hence the set of points of $(Y_1)_T$ where $\pi_{1, T, *}\mathcal{G}_{1, T}$ is flat over $T$ is the same as the set of points of $(Y_1)_T$ where $\operatorname{Coker}(\alpha_1)_T$ is flat over $T$. In this way the question reduces to the sheaf $\operatorname{Coker}(\alpha_1)$ which has a complete dévissage of length $n - 1$ and we win by induction.

If $d_1 < d$ then $F_d$ is represented by $S$ and we win.

The last case is the case $d_1 = d$. This case follows from a combination of Lemma Comparison for purity and dévissage and Lemma Projective and locally free modules. $\square$

Lemma. Recovering a monomorphism from a locally nilpotent tower

Let $A = \varprojlim A_n$ be a limit of a system of rings whose transition maps are surjective and with locally nilpotent kernels. Let $S = \operatorname{Spec}(A)$. Let $T \to S$ be a monomorphism which is locally of finite type. If $\operatorname{Spec}(A_n) \to S$ factors through $T$ for all $n$, then $T = S$.

Proof. Set $S_n = \operatorname{Spec}(A_n)$. Let $T_0 \subset T$ be the common image of the factorizations $S_n \to T$. Then $T_0$ is quasi-compact. Let $T' \subset T$ be a quasi-compact open containing $T_0$. Then $S_n \to T$ factors through $T'$. If we can show that $T' = S$, then $T' = T = S$. Hence we may assume $T$ is quasi-compact.

Assume $T$ is quasi-compact. In this case $T \to S$ is separated and quasi-finite (Morphisms, Lemma Finite algebras (uncovered prerequisite)). Using Zariski's Main Theorem (in the form of More on Morphisms, Lemma Diagonals, separation and finite algebras) we choose a factorization $T \to W \to S$ with $W \to S$ finite and $T \to W$ an open immersion. Write $W = \operatorname{Spec}(B)$. The (unique) factorizations $S_n \to T$ may be viewed as morphisms into $W$ and we obtain $$A \longrightarrow B \longrightarrow \varprojlim A_n = A$$ Consider the morphism $h : S = \operatorname{Spec}(A) \to \operatorname{Spec}(B) = W$ coming from the arrow on the right. Then $$T \times_{W, h} S$$ is an open subscheme of $S$ containing the image of $S_n \to S$ for all $n$. To finish the proof it suffices to show that any open $U \subset S$ containing the image of $S_n \to S$ for some $n \geq 1$ is equal to $S$. This is true because $(A, \operatorname{Ker}(A \to A_n))$ is a henselian pair (More on Algebra, Lemma Henselian pairs from locally nilpotent inverse systems) and hence every closed point of $S$ is contained in the image of $S_n \to S$. $\square$

Additional proofs of the supporting constructions

These statements and full proofs supply local support for the preceding arguments. They reuse the same Stacks Project edition. A remaining genuine prerequisite is marked explicitly rather than treated as proved.

Descent of algebraic spaces through inverse limits

Lemma. Descent of finite locally free and invertible modules

With notation and assumptions as in Lemma Descent of finite presentation and finite algebras. Then

1. any finite locally free $\mathcal{O}_X$-module is the pullback of a finite locally free $\mathcal{O}_{X_i}$-module for some $i$,

2. any invertible $\mathcal{O}_X$-module is the pullback of an invertible $\mathcal{O}_{X_i}$-module for some $i$.

Proof. Proof of (2). Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Since invertible modules are of finite presentation we can find an $i$ and modules $\mathcal{L}_i$ and $\mathcal{N}_i$ of finite presentation over $X_i$ such that $f_i^*\mathcal{L}_i \cong \mathcal{L}$ and $f_i^*\mathcal{N}_i \cong \mathcal{L}^{\otimes -1}$, see Lemma Descent of finite presentation and modules. Since pullback commutes with tensor product we see that $f_i^*(\mathcal{L}_i \otimes_{\mathcal{O}_{X_i}} \mathcal{N}_i)$ is isomorphic to $\mathcal{O}_X$. Since the tensor product of finitely presented modules is finitely presented, the same lemma implies that $f_{i'i}^*\mathcal{L}_i \otimes_{\mathcal{O}_{X_{i'}}} f_{i'i}^*\mathcal{N}_i$ is isomorphic to $\mathcal{O}_{X_{i'}}$ for some $i' \geq i$. It follows that $f_{i'i}^*\mathcal{L}_i$ is invertible, as follows. On an affine etale chart, let $M$ and $N$ denote the two finitely presented modules with $M\otimes N\cong R$. At each local ring their residue-vector-space dimensions have product one. Nakayama makes both modules cyclic there, so they have the form $R/J$ and $R/K$. Their tensor product is $R/(J+K)\cong R$, whence $J=K=0$. Finite presentation spreads these generators to a neighbourhood; the same tensor argument makes the modules free of rank one on that neighbourhood. Thus the descended module is invertible and the proof of (2) is complete.

Proof of (1). A finite locally free module $E$ has a finitely presented dual $E^\vee$, with evaluation and coevaluation maps satisfying the two triangle identities. Descend the two modules by Lemma Descent of finite presentation and modules, then their two maps by its full faithfulness, and finally the two identities by equality at a later stage in the same colimit category. The resulting module $E_i$ is dualizable. To check that it is finite locally free, work on an affine etale chart and write its coevaluation as a finite sum $\sum_j n_j\otimes m_j$. The triangle identity says $m=\sum_j m_j\operatorname{ev}(m\otimes n_j)$ for every $m$. Hence the maps $R^{\oplus r}\to E_i$, given by the $m_j$, and $E_i\to R^{\oplus r}$, given by these evaluations, compose to the identity. The module is therefore finite projective. A finite projective module is finite locally free: over each local ring choose a residue-field basis, split the resulting surjection from a finite free module, and apply Nakayama to its finite projective kernel; the chosen basis and its inverse maps then spread to a neighbourhood. This proves (1) over arbitrary bases, with the original module as its pullback. $\square$

Lemma. Descent of finite presentation and finite algebras

Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces over $S$. Assume

1. the morphisms $f_{ii'} : X_i \to X_{i'}$ are affine,

2. the spaces $X_i$ are quasi-compact and quasi-separated.

Let $X = \varprojlim_i X_i$. Then the category of algebraic spaces of finite presentation over $X$ is the colimit over $I$ of the categories of algebraic spaces of finite presentation over $X_i$.

Proof. Pick $0 \in I$. Choose a surjective étale morphism $U_0 \to X_0$ where $U_0$ is an affine scheme (Properties of Spaces, Lemma Affine neighbourhoods). Set $U_i = X_i \times_{X_0} U_0$. Set $R_0 = U_0 \times_{X_0} U_0$ and $R_i = R_0 \times_{X_0} X_i$. Denote $s_i, t_i : R_i \to U_i$ and $s, t : R \to U$ the two projections. In the proof of Lemma Descent of algebraic spaces we have seen that there exists a presentation $X = U/R$ with $U = \varprojlim U_i$ and $R = \varprojlim R_i$. Note that $U_i$ and $U$ are affine and that $R_i$ and $R$ are quasi-compact and separated (as $X_i$ is quasi-separated). Let $Y$ be an algebraic space over $S$ and let $Y \to X$ be a morphism of finite presentation. Set $V = U \times_X Y$. This is an algebraic space of finite presentation over $U$. Choose an affine scheme $W$ and a surjective étale morphism $W \to V$. Then $W \to Y$ is surjective étale as well. Set $R' = W \times_Y W$ so that $Y = W/R'$ (see Spaces, Section The geometric construction). Note that $W$ is a scheme of finite presentation over $U$ and that $R'$ is a scheme of finite presentation over $R$ (details omitted). By Limits, Lemma Descent of finite presentation and finite algebras we can find an index $i$ and a morphism of schemes $W_i \to U_i$ of finite presentation whose base change to $U$ gives $W \to U$. Similarly we can find, after possibly increasing $i$, a scheme $R'_i$ of finite presentation over $R_i$ whose base change to $R$ is $R'$. The projection morphisms $s', t' : R' \to W$ are morphisms over the projection morphisms $s, t : R \to U$. Hence we can view $s'$, resp. $t'$ as a morphism between schemes of finite presentation over $U$ (with structure morphism $R' \to U$ given by $R' \to R$ followed by $s$, resp. $t$). Hence we can apply Limits, Lemma Descent of finite presentation and finite algebras again to see that, after possibly increasing $i$, there exist morphisms $s'_i, t'_i : R'_i \to W_i$, whose base change to $U$ is $S', t'$. By Limits, Lemmas Descent of étale morphisms and Descent of finite-presentation descent we may assume that $s'_i, t'_i$ are étale and that $j'_i : R'_i \to W_i \times_{X_i} W_i$ is a monomorphism (here we view $j'_i$ as a morphism of schemes of finite presentation over $U_i$ via one of the projections -- it doesn't matter which one). Setting $Y_i = W_i/R'_i$ (see Spaces, Theorem The geometric construction (uncovered prerequisite)) we obtain an algebraic space of finite presentation over $X_i$ whose base change to $X$ is isomorphic to $Y$.