The Stacks project

98.5 The stack of coherent sheaves

In this section we prove that the stack of coherent sheaves on $X/B$ is algebraic under suitable hypotheses. This is a special case of [Theorem 2.1.1, lieblich_remarks] which treats the case of the stack of coherent sheaves on an Artin stack over a base.

Situation 98.5.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation. We denote $\mathcal{C}\! \mathit{oh}_{X/B}$ the category whose objects are triples $(T, g, \mathcal{F})$ where

  1. $T$ is a scheme over $S$,

  2. $g : T \to B$ is a morphism over $S$, and setting $X_ T = T \times _{g, B} X$

  3. $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_{X_ T}$-module of finite presentation, flat over $T$, with support proper over $T$.

A morphism $(T, g, \mathcal{F}) \to (T', g', \mathcal{F}')$ is given by a pair $(h, \varphi )$ where

  1. $h : T \to T'$ is a morphism of schemes over $B$ (i.e., $g' \circ h = g$), and

  2. $\varphi : (h')^*\mathcal{F}' \to \mathcal{F}$ is an isomorphism of $\mathcal{O}_{X_ T}$-modules where $h' : X_ T \to X_{T'}$ is the base change of $h$.

Thus $\mathcal{C}\! \mathit{oh}_{X/B}$ is a category and the rule

\[ p : \mathcal{C}\! \mathit{oh}_{X/B} \longrightarrow (\mathit{Sch}/S)_{fppf}, \quad (T, g, \mathcal{F}) \longmapsto T \]

is a functor. For a scheme $T$ over $S$ we denote $\mathcal{C}\! \mathit{oh}_{X/B, T}$ the fibre category of $p$ over $T$. These fibre categories are groupoids.

Lemma 98.5.2. In Situation 98.5.1 the functor $p : \mathcal{C}\! \mathit{oh}_{X/B} \longrightarrow (\mathit{Sch}/S)_{fppf}$ is fibred in groupoids.

Proof. We show that $p$ is fibred in groupoids by checking conditions (1) and (2) of Categories, Definition 4.35.1. Given an object $(T', g', \mathcal{F}')$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ and a morphism $h : T \to T'$ of schemes over $S$ we can set $g = h \circ g'$ and $\mathcal{F} = (h')^*\mathcal{F}'$ where $h' : X_ T \to X_{T'}$ is the base change of $h$. Then it is clear that we obtain a morphism $(T, g, \mathcal{F}) \to (T', g', \mathcal{F}')$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ lying over $h$. This proves (1). For (2) suppose we are given morphisms

\[ (h_1, \varphi _1) : (T_1, g_1, \mathcal{F}_1) \to (T, g, \mathcal{F}) \quad \text{and}\quad (h_2, \varphi _2) : (T_2, g_2, \mathcal{F}_2) \to (T, g, \mathcal{F}) \]

of $\mathcal{C}\! \mathit{oh}_{X/B}$ and a morphism $h : T_1 \to T_2$ such that $h_2 \circ h = h_1$. Then we can let $\varphi $ be the composition

\[ (h')^*\mathcal{F}_2 \xrightarrow {(h')^*\varphi _2^{-1}} (h')^*(h_2)^*\mathcal{F} = (h_1)^*\mathcal{F} \xrightarrow {\varphi _1} \mathcal{F}_1 \]

to obtain the morphism $(h, \varphi ) : (T_1, g_1, \mathcal{F}_1) \to (T_2, g_2, \mathcal{F}_2)$ that witnesses the truth of condition (2). $\square$

Lemma 98.5.3. In Situation 98.5.1. Denote $\mathcal{X} = \mathcal{C}\! \mathit{oh}_{X/B}$. Then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces.

Proof. Consider two objects $x = (T, g, \mathcal{F})$ and $y = (T, h, \mathcal{G})$ of $\mathcal{X}$ over a scheme $T$. We have to show that $\mathit{Isom}_\mathcal {X}(x, y)$ is an algebraic space over $T$, see Algebraic Stacks, Lemma 93.10.11. If for $a : T' \to T$ the restrictions $x|_{T'}$ and $y|_{T'}$ are isomorphic in the fibre category $\mathcal{X}_{T'}$, then $g \circ a = h \circ a$. Hence there is a transformation of presheaves

\[ \mathit{Isom}_\mathcal {X}(x, y) \longrightarrow \text{Equalizer}(g, h) \]

Since the diagonal of $B$ is representable (by schemes) this equalizer is a scheme. Thus we may replace $T$ by this equalizer and the sheaves $\mathcal{F}$ and $\mathcal{G}$ by their pullbacks. Thus we may assume $g = h$. In this case we have $\mathit{Isom}_\mathcal {X}(x, y) = \mathit{Isom}(\mathcal{F}, \mathcal{G})$ and the result follows from Proposition 98.4.3. $\square$

Lemma 98.5.4. In Situation 98.5.1 the functor $p : \mathcal{C}\! \mathit{oh}_{X/B} \longrightarrow (\mathit{Sch}/S)_{fppf}$ is a stack in groupoids.

Proof. To prove that $\mathcal{C}\! \mathit{oh}_{X/B}$ is a stack in groupoids, we have to show that the presheaves $\mathit{Isom}$ are sheaves and that descent data are effective. The statement on $\mathit{Isom}$ follows from Lemma 98.5.3, see Algebraic Stacks, Lemma 93.10.11. Let us prove the statement on descent data. Suppose that $\{ a_ i : T_ i \to T\} $ is an fppf covering of schemes over $S$. Let $(\xi _ i, \varphi _{ij})$ be a descent datum for $\{ T_ i \to T\} $ with values in $\mathcal{C}\! \mathit{oh}_{X/B}$. For each $i$ we can write $\xi _ i = (T_ i, g_ i, \mathcal{F}_ i)$. Denote $\text{pr}_0 : T_ i \times _ T T_ j \to T_ i$ and $\text{pr}_1 : T_ i \times _ T T_ j \to T_ j$ the projections. The condition that $\xi _ i|_{T_ i \times _ T T_ j} = \xi _ j|_{T_ i \times _ T T_ j}$ implies in particular that $g_ i \circ \text{pr}_0 = g_ j \circ \text{pr}_1$. Thus there exists a unique morphism $g : T \to B$ such that $g_ i = g \circ a_ i$, see Descent on Spaces, Lemma 73.7.2. Denote $X_ T = T \times _{g, B} X$. Set $X_ i = X_{T_ i} = T_ i \times _{g_ i, B} X = T_ i \times _{a_ i, T} X_ T$ and

\[ X_{ij} = X_{T_ i} \times _{X_ T} X_{T_ j} = X_ i \times _{X_ T} X_ j \]

with projections $\text{pr}_ i$ and $\text{pr}_ j$ to $X_ i$ and $X_ j$. Observe that the pullback of $(T_ i, g_ i, \mathcal{F}_ i)$ by $\text{pr}_0 : T_ i \times _ T T_ j \to T_ i$ is given by $(T_ i \times _ T T_ j, g_ i \circ \text{pr}_0, \text{pr}_ i^*\mathcal{F}_ i)$. Hence a descent datum for $\{ T_ i \to T\} $ in $\mathcal{C}\! \mathit{oh}_{X/B}$ is given by the objects $(T_ i, g \circ a_ i, \mathcal{F}_ i)$ and for each pair $i, j$ an isomorphism of $\mathcal{O}_{X_{ij}}$-modules

\[ \varphi _{ij} : \text{pr}_ i^*\mathcal{F}_ i \longrightarrow \text{pr}_ j^*\mathcal{F}_ j \]

satisfying the cocycle condition over (the pullback of $X$ to) $T_ i \times _ T T_ j \times _ T T_ k$. Ok, and now we simply use that $\{ X_ i \to X_ T\} $ is an fppf covering so that we can view $(\mathcal{F}_ i, \varphi _{ij})$ as a descent datum for this covering. By Descent on Spaces, Proposition 73.4.1 this descent datum is effective and we obtain a quasi-coherent sheaf $\mathcal{F}$ over $X_ T$ restricting to $\mathcal{F}_ i$ on $X_ i$. By Morphisms of Spaces, Lemma 66.31.5 we see that $\mathcal{F}$ is flat over $T$ and Descent on Spaces, Lemma 73.6.2 guarantees that $\mathcal{Q}$ is of finite presentation as an $\mathcal{O}_{X_ T}$-module. Finally, by Descent on Spaces, Lemma 73.11.19 we see that the scheme theoretic support of $\mathcal{F}$ is proper over $T$ as we've assumed the scheme theoretic support of $\mathcal{F}_ i$ is proper over $T_ i$ (note that taking scheme theoretic support commutes with flat base change by Morphisms of Spaces, Lemma 66.30.10). In this way we obtain our desired object over $T$. $\square$

Remark 98.5.5. In Situation 98.5.1 the rule $(T, g, \mathcal{F}) \mapsto (T, g)$ defines a $1$-morphism

\[ \mathcal{C}\! \mathit{oh}_{X/B} \longrightarrow \mathcal{S}_ B \]

of stacks in groupoids (see Lemma 98.5.4, Algebraic Stacks, Section 93.7, and Examples of Stacks, Section 94.10). Let $B' \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{S}_{B'} \to \mathcal{S}_ B$ be the associated $1$-morphism of stacks fibred in sets. Set $X' = X \times _ B B'$. We obtain a stack in groupoids $\mathcal{C}\! \mathit{oh}_{X'/B'} \to (\mathit{Sch}/S)_{fppf}$ associated to the base change $f' : X' \to B'$. In this situation the diagram

\[ \vcenter { \xymatrix{ \mathcal{C}\! \mathit{oh}_{X'/B'} \ar[r] \ar[d] & \mathcal{C}\! \mathit{oh}_{X/B} \ar[d] \\ \mathcal{S}_{B'} \ar[r] & \mathcal{S}_ B } } \quad \begin{matrix} \text{or in} \\ \text{another} \\ \text{notation} \end{matrix} \quad \vcenter { \xymatrix{ \mathcal{C}\! \mathit{oh}_{X'/B'} \ar[r] \ar[d] & \mathcal{C}\! \mathit{oh}_{X/B} \ar[d] \\ \mathit{Sch}/B' \ar[r] & \mathit{Sch}/B } } \]

is $2$-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space.

Lemma 98.5.6. In Situation 98.5.1 assume that $B \to S$ is locally of finite presentation. Then $p : \mathcal{C}\! \mathit{oh}_{X/B} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving (Artin's Axioms, Definition 97.11.1).

Proof. Write $B(T)$ for the discrete category whose objects are the $S$-morphisms $T \to B$. Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a filtered limit of affine schemes over $S$. Assigning to an object $(T, h, \mathcal{F})$ of $\mathcal{C}\! \mathit{oh}_{X/B, T}$ the object $h$ of $B(T)$ gives us a commutative diagram of fibre categories

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathcal{C}\! \mathit{oh}_{X/B, T_ i} \ar[r] \ar[d] & \mathcal{C}\! \mathit{oh}_{X/B, T} \ar[d] \\ \mathop{\mathrm{colim}}\nolimits B(T_ i) \ar[r] & B(T) } \]

We have to show the top horizontal arrow is an equivalence. Since we have assumed that $B$ is locally of finite presentation over $S$ we see from Limits of Spaces, Remark 69.3.11 that the bottom horizontal arrow is an equivalence. This means that we may assume $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a filtered limit of affine schemes over $B$. Denote $g_ i : T_ i \to B$ and $g : T \to B$ the corresponding morphisms. Set $X_ i = T_ i \times _{g_ i, B} X$ and $X_ T = T \times _{g, B} X$. Observe that $X_ T = \mathop{\mathrm{colim}}\nolimits X_ i$ and that the algebraic spaces $X_ i$ and $X_ T$ are quasi-separated and quasi-compact (as they are of finite presentation over the affines $T_ i$ and $T$). By Limits of Spaces, Lemma 69.7.2 we see that

\[ \mathop{\mathrm{colim}}\nolimits \textit{FP}(X_ i) = \textit{FP}(X_ T). \]

where $\textit{FP}(W)$ is short hand for the category of finitely presented $\mathcal{O}_ W$-modules. The results of Limits of Spaces, Lemmas 69.6.12 and 69.12.3 tell us the same thing is true if we replace $\textit{FP}(X_ i)$ and $\textit{FP}(X_ T)$ by the full subcategory of objects flat over $T_ i$ and $T$ with scheme theoretic support proper over $T_ i$ and $T$. This proves the lemma. $\square$

Lemma 98.5.7. In Situation 98.5.1. Let

\[ \xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' } \]

be a pushout in the category of schemes over $S$ where $Z \to Z'$ is a thickening and $Z \to Y$ is affine, see More on Morphisms, Lemma 37.14.3. Then the functor on fibre categories

\[ \mathcal{C}\! \mathit{oh}_{X/B, Y'} \longrightarrow \mathcal{C}\! \mathit{oh}_{X/B, Y} \times _{\mathcal{C}\! \mathit{oh}_{X/B, Z}} \mathcal{C}\! \mathit{oh}_{X/B, Z'} \]

is an equivalence.

Proof. Observe that the corresponding map

\[ B(Y') \longrightarrow B(Y) \times _{B(Z)} B(Z') \]

is a bijection, see Pushouts of Spaces, Lemma 80.6.1. Thus using the commutative diagram

\[ \xymatrix{ \mathcal{C}\! \mathit{oh}_{X/B, Y'} \ar[r] \ar[d] & \mathcal{C}\! \mathit{oh}_{X/B, Y} \times _{\mathcal{C}\! \mathit{oh}_{X/B, Z}} \mathcal{C}\! \mathit{oh}_{X/B, Z'} \ar[d] \\ B(Y') \ar[r] & B(Y) \times _{B(Z)} B(Z') } \]

we see that we may assume that $Y'$ is a scheme over $B'$. By Remark 98.5.5 we may replace $B$ by $Y'$ and $X$ by $X \times _ B Y'$. Thus we may assume $B = Y'$. In this case the statement follows from Pushouts of Spaces, Lemma 80.6.6. $\square$

Lemma 98.5.8. Let

\[ \xymatrix{ X \ar[d] \ar[r]_ i & X' \ar[d] \\ T \ar[r] & T' } \]

be a cartesian square of algebraic spaces where $T \to T'$ is a first order thickening. Let $\mathcal{F}'$ be an $\mathcal{O}_{X'}$-module flat over $T'$. Set $\mathcal{F} = i^*\mathcal{F}'$. The following are equivalent

  1. $\mathcal{F}'$ is a quasi-coherent $\mathcal{O}_{X'}$-module of finite presentation,

  2. $\mathcal{F}'$ is an $\mathcal{O}_{X'}$-module of finite presentation,

  3. $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module of finite presentation,

  4. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation,

Proof. Recall that a finitely presented module is quasi-coherent hence the equivalence of (1) and (2) and (3) and (4). The equivalence of (2) and (4) is a special case of Deformation Theory, Lemma 90.11.3. $\square$

Lemma 98.5.9. In Situation 98.5.1 assume that $S$ is a locally Noetherian scheme and $B \to S$ is locally of finite presentation. Let $k$ be a finite type field over $S$ and let $x_0 = (\mathop{\mathrm{Spec}}(k), g_0, \mathcal{G}_0)$ be an object of $\mathcal{X} = \mathcal{C}\! \mathit{oh}_{X/B}$ over $k$. Then the spaces $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ (Artin's Axioms, Section 97.8) are finite dimensional.

Proof. Observe that by Lemma 98.5.7 our stack in groupoids $\mathcal{X}$ satisfies property (RS*) defined in Artin's Axioms, Section 97.21. In particular $\mathcal{X}$ satisfies (RS). Hence all associated predeformation categories are deformation categories (Artin's Axioms, Lemma 97.6.1) and the statement makes sense.

In this paragraph we show that we can reduce to the case $B = \mathop{\mathrm{Spec}}(k)$. Set $X_0 = \mathop{\mathrm{Spec}}(k) \times _{g_0, B} X$ and denote $\mathcal{X}_0 = \mathcal{C}\! \mathit{oh}_{X_0/k}$. In Remark 98.5.5 we have seen that $\mathcal{X}_0$ is the $2$-fibre product of $\mathcal{X}$ with $\mathop{\mathrm{Spec}}(k)$ over $B$ as categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Thus by Artin's Axioms, Lemma 97.8.2 we reduce to proving that $B$, $\mathop{\mathrm{Spec}}(k)$, and $\mathcal{X}_0$ have finite dimensional tangent spaces and infinitesimal automorphism spaces. The tangent space of $B$ and $\mathop{\mathrm{Spec}}(k)$ are finite dimensional by Artin's Axioms, Lemma 97.8.1 and of course these have vanishing $\text{Inf}$. Thus it suffices to deal with $\mathcal{X}_0$.

Let $k[\epsilon ]$ be the dual numbers over $k$. Let $\mathop{\mathrm{Spec}}(k[\epsilon ]) \to B$ be the composition of $g_0 : \mathop{\mathrm{Spec}}(k) \to B$ and the morphism $\mathop{\mathrm{Spec}}(k[\epsilon ]) \to \mathop{\mathrm{Spec}}(k)$ coming from the inclusion $k \to k[\epsilon ]$. Set $X_0 = \mathop{\mathrm{Spec}}(k) \times _ B X$ and $X_\epsilon = \mathop{\mathrm{Spec}}(k[\epsilon ]) \times _ B X$. Observe that $X_\epsilon $ is a first order thickening of $X_0$ flat over the first order thickening $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k[\epsilon ])$. Unwinding the definitions and using Lemma 98.5.8 we see that $T\mathcal{F}_{\mathcal{X}_0, k, x_0}$ is the set of lifts of $\mathcal{G}_0$ to a flat module on $X_\epsilon $. By Deformation Theory, Lemma 90.12.1 we conclude that

\[ T\mathcal{F}_{\mathcal{X}_0, k, x_0} = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_{X_0}}(\mathcal{G}_0, \mathcal{G}_0) \]

Here we have used the identification $\epsilon k[\epsilon ] \cong k$ of $k[\epsilon ]$-modules. Using Deformation Theory, Lemma 90.12.1 once more we see that

\[ \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_{X_0}}(\mathcal{G}_0, \mathcal{G}_0) \]

These spaces are finite dimensional over $k$ as $\mathcal{G}_0$ has support proper over $\mathop{\mathrm{Spec}}(k)$. Namely, $X_0$ is of finite presentation over $\mathop{\mathrm{Spec}}(k)$, hence Noetherian. Since $\mathcal{G}_0$ is of finite presentation it is a coherent $\mathcal{O}_{X_0}$-module. Thus we may apply Derived Categories of Spaces, Lemma 74.8.4 to conclude the desired finiteness. $\square$

Lemma 98.5.10. In Situation 98.5.1 assume that $S$ is a locally Noetherian scheme and that $f : X \to B$ is separated. Let $\mathcal{X} = \mathcal{C}\! \mathit{oh}_{X/B}$. Then the functor Artin's Axioms, Equation (97.9.3.1) is an equivalence.

Proof. Let $A$ be an $S$-algebra which is a complete local Noetherian ring with maximal ideal $\mathfrak m$ whose residue field $k$ is of finite type over $S$. We have to show that the category of objects over $A$ is equivalent to the category of formal objects over $A$. Since we know this holds for the category $\mathcal{S}_ B$ fibred in sets associated to $B$ by Artin's Axioms, Lemma 97.9.5, it suffices to prove this for those objects lying over a given morphism $\mathop{\mathrm{Spec}}(A) \to B$.

Set $X_ A = \mathop{\mathrm{Spec}}(A) \times _ B X$ and $X_ n = \mathop{\mathrm{Spec}}(A/\mathfrak m^ n) \times _ B X$. By Grothendieck's existence theorem (More on Morphisms of Spaces, Theorem 75.42.11) we see that the category of coherent modules $\mathcal{F}$ on $X_ A$ with support proper over $\mathop{\mathrm{Spec}}(A)$ is equivalent to the category of systems $(\mathcal{F}_ n)$ of coherent modules $\mathcal{F}_ n$ on $X_ n$ with support proper over $\mathop{\mathrm{Spec}}(A/\mathfrak m^ n)$. The equivalence sends $\mathcal{F}$ to the system $(\mathcal{F} \otimes _ A A/\mathfrak m^ n)$. See discussion in More on Morphisms of Spaces, Remark 75.42.12. To finish the proof of the lemma, it suffices to show that $\mathcal{F}$ is flat over $A$ if and only if all $\mathcal{F} \otimes _ A A/\mathfrak m^ n$ are flat over $A/\mathfrak m^ n$. This follows from More on Morphisms of Spaces, Lemma 75.24.3. $\square$

Lemma 98.5.11. In Situation 98.5.1 assume that $S$ is a locally Noetherian scheme, $S = B$, and $f : X \to B$ is flat. Let $\mathcal{X} = \mathcal{C}\! \mathit{oh}_{X/B}$. Then we have openness of versality for $\mathcal{X}$ (see Artin's Axioms, Definition 97.13.1).

First proof. This proof is based on the criterion of Artin's Axioms, Lemma 97.24.4. Let $U \to S$ be of finite type morphism of schemes, $x$ an object of $\mathcal{X}$ over $U$ and $u_0 \in U$ a finite type point such that $x$ is versal at $u_0$. After shrinking $U$ we may assume that $u_0$ is a closed point (Morphisms, Lemma 29.16.1) and $U = \mathop{\mathrm{Spec}}(A)$ with $U \to S$ mapping into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$ of $S$. Let $\mathcal{F}$ be the coherent module on $X_ A = \mathop{\mathrm{Spec}}(A) \times _ S X$ flat over $A$ corresponding to the given object $x$.

According to Deformation Theory, Lemma 90.12.1 we have an isomorphism of functors

\[ T_ x(M) = \mathop{\mathrm{Ext}}\nolimits ^1_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M) \]

and given any surjection $A' \to A$ of $\Lambda $-algebras with square zero kernel $I$ we have an obstruction class

\[ \xi _{A'} \in \mathop{\mathrm{Ext}}\nolimits ^2_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A I) \]

This uses that for any $A' \to A$ as above the base change $X_{A'} = \mathop{\mathrm{Spec}}(A') \times _ B X$ is flat over $A'$. Moreover, the construction of the obstruction class is functorial in the surjection $A' \to A$ (for fixed $A$) by Deformation Theory, Lemma 90.12.3. Apply Derived Categories of Spaces, Lemma 74.23.3 to the computation of the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ for $i \leq m$ with $m = 2$. We find a perfect object $K \in D(A)$ and functorial isomorphisms

\[ H^ i(K \otimes _ A^\mathbf {L} M) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M) \]

for $i \leq m$ compatible with boundary maps. This object $K$, together with the displayed identifications above gives us a datum as in Artin's Axioms, Situation 97.24.2. Finally, condition (iv) of Artin's Axioms, Lemma 97.24.3 holds by Deformation Theory, Lemma 90.12.5. Thus Artin's Axioms, Lemma 97.24.4 does indeed apply and the lemma is proved. $\square$

Second proof. This proof is based on Artin's Axioms, Lemma 97.22.2. Conditions (1), (2), and (3) of that lemma correspond to Lemmas 98.5.3, 98.5.7, and 98.5.6.

We have constructed an obstruction theory in the chapter on deformation theory. Namely, given an $S$-algebra $A$ and an object $x$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ over $\mathop{\mathrm{Spec}}(A)$ given by $\mathcal{F}$ on $X_ A$ we set $\mathcal{O}_ x(M) = \mathop{\mathrm{Ext}}\nolimits ^2_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ and if $A' \to A$ is a surjection with kernel $I$, then as obstruction element we take the element

\[ o_ x(A') = o(\mathcal{F}, \mathcal{F} \otimes _ A I, 1) \in \mathcal{O}_ x(I) = \mathop{\mathrm{Ext}}\nolimits ^2_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A I) \]

of Deformation Theory, Lemma 90.12.1. All properties of an obstruction theory as defined in Artin's Axioms, Definition 97.22.1 follow from this lemma except for functoriality of obstruction classes as formulated in condition (ii) of the definition. But as stated in the footnote to assumption (4) of Artin's Axioms, Lemma 97.22.2 it suffices to check functoriality of obstruction classes for a fixed $A$ which follows from Deformation Theory, Lemma 90.12.3. Deformation Theory, Lemma 90.12.1 also tells us that $T_ x(M) = \mathop{\mathrm{Ext}}\nolimits ^1_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ for any $A$-module $M$.

To finish the proof it suffices to show that $T_ x(\prod M_ n) = \prod T_ x(M_ n)$ and $\mathcal{O}_ x(\prod M_ n) = \prod \mathcal{O}_ x(M)$. Apply Derived Categories of Spaces, Lemma 74.23.3 to the computation of the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ for $i \leq m$ with $m = 2$. We find a perfect object $K \in D(A)$ and functorial isomorphisms

\[ H^ i(K \otimes _ A^\mathbf {L} M) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M) \]

for $i = 1, 2$. A straightforward argument shows that

\[ H^ i(K \otimes _ A^\mathbf {L} \prod M_ n) = \prod H^ i(K \otimes _ A^\mathbf {L} M_ n) \]

whenever $K$ is a pseudo-coherent object of $D(A)$. In fact, this property (for all $i$) characterizes pseudo-coherent complexes, see More on Algebra, Lemma 15.65.5. $\square$

Proof. Set $\mathcal{X} = \mathcal{C}\! \mathit{oh}_{X/B}$. We have seen that $\mathcal{X}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ with diagonal representable by algebraic spaces (Lemmas 98.5.4 and 98.5.3). Hence it suffices to find a scheme $W$ and a surjective and smooth morphism $W \to \mathcal{X}$.

Let $B'$ be a scheme and let $B' \to B$ be a surjective ├ętale morphism. Set $X' = B' \times _ B X$ and denote $f' : X' \to B'$ the projection. Then $\mathcal{X}' = \mathcal{C}\! \mathit{oh}_{X'/B'}$ is equal to the $2$-fibre product of $\mathcal{X}$ with the category fibred in sets associated to $B'$ over the category fibred in sets associated to $B$ (Remark 98.5.5). By the material in Algebraic Stacks, Section 93.10 the morphism $\mathcal{X}' \to \mathcal{X}$ is surjective and ├ętale. Hence it suffices to prove the result for $\mathcal{X}'$. In other words, we may assume $B$ is a scheme.

Assume $B$ is a scheme. In this case we may replace $S$ by $B$, see Algebraic Stacks, Section 93.19. Thus we may assume $S = B$.

Assume $S = B$. Choose an affine open covering $S = \bigcup U_ i$. Denote $\mathcal{X}_ i$ the restriction of $\mathcal{X}$ to $(\mathit{Sch}/U_ i)_{fppf}$. If we can find schemes $W_ i$ over $U_ i$ and surjective smooth morphisms $W_ i \to \mathcal{X}_ i$, then we set $W = \coprod W_ i$ and we obtain a surjective smooth morphism $W \to \mathcal{X}$. Thus we may assume $S = B$ is affine.

Assume $S = B$ is affine, say $S = \mathop{\mathrm{Spec}}(\Lambda )$. Write $\Lambda = \mathop{\mathrm{colim}}\nolimits \Lambda _ i$ as a filtered colimit with each $\Lambda _ i$ of finite type over $\mathbf{Z}$. For some $i$ we can find a morphism of algebraic spaces $X_ i \to \mathop{\mathrm{Spec}}(\Lambda _ i)$ which is of finite presentation, separated, and flat and whose base change to $\Lambda $ is $X$. See Limits of Spaces, Lemmas 69.7.1, 69.6.9, and 69.6.12. If we show that $\mathcal{C}\! \mathit{oh}_{X_ i/\mathop{\mathrm{Spec}}(\Lambda _ i)}$ is an algebraic stack, then it follows by base change (Remark 98.5.5 and Algebraic Stacks, Section 93.19) that $\mathcal{X}$ is an algebraic stack. Thus we may assume that $\Lambda $ is a finite type $\mathbf{Z}$-algebra.

Assume $S = B = \mathop{\mathrm{Spec}}(\Lambda )$ is affine of finite type over $\mathbf{Z}$. In this case we will verify conditions (1), (2), (3), (4), and (5) of Artin's Axioms, Lemma 97.17.1 to conclude that $\mathcal{X}$ is an algebraic stack. Note that $\Lambda $ is a G-ring, see More on Algebra, Proposition 15.50.12. Hence all local rings of $S$ are G-rings. Thus (5) holds. By Lemma 98.5.11 we have that $\mathcal{X}$ satisfies openness of versality, hence (4) holds. To check (2) we have to verify axioms [-1], [0], [1], [2], and [3] of Artin's Axioms, Section 97.14. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas 98.5.4, 98.5.6, 98.5.7, 98.5.9. Condition (3) follows from Lemma 98.5.10. Finally, condition (1) is Lemma 98.5.3. This finishes the proof of the theorem. $\square$

[1] This assumption is not necessary. See Section 98.6.

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