# stacks/stacks-project

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 \input{preamble} % OK, start here. % \begin{document} \title{Quot and Hilbert Spaces} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent The purpose of this chapter is to write about Quot and Hilbert functors and to prove that these are algebraic spaces provided certain technical conditions are satisfied. In this chapter we will discuss this in the setting of algebraic space. A reference is Grothendieck's lectures, see \cite{Gr-I}, \cite{Gr-II}, \cite{Gr-III}, \cite{Gr-IV}, \cite{Gr-V}, and \cite{Gr-VI}. Another reference is the paper \cite{olsson-starr}; this paper discusses the more general case of Quot and Hilbert spaces associated to a morphism of algebraic stacks which we will discuss in another chapter, see (insert future reference here). \medskip\noindent In the case of Hilbert spaces there is a more general notion of Hilbert stacks'' which we will discuss in a separate chapter, see (insert future reference here). \medskip\noindent We have intentionally placed this chapter, as well as the chapters Examples of Stacks'', Sheaves on Algebraic Stacks'', Criteria for Representability'', and Artin's Axioms'' before the general development of the theory of algebraic stacks. The reason for this is that starting with the next chapter (see Properties of Stacks, Section \ref{stacks-properties-section-conventions}) we will no longer distinguish between a scheme and the algebraic stack it gives rise to. Thus our language will become more flexible and easier for a human to parse, but also less precise. These first few chapters, including the initial chapter Algebraic Stacks'', lay the groundwork that later allow us to ignore some of the very technical distinctions between different ways of thinking about algebraic stacks. But especially in the chapters Artin's Axioms'' and Criteria of Representability'' we need to be very precise about what objects exactly we are working with, as we are trying to show that certain constructions produce algebraic stacks or algebraic spaces. \medskip\noindent Unfortunately, this means that some of the notation, conventions and terminology is awkward and may seem backwards to the more experienced reader. We hope the reader will forgive us! \section{Conventions} \label{section-conventions} \noindent The standing assumption is that all schemes are contained in a big fppf site $\Sch_{fppf}$. And all rings $A$ considered have the property that $\Spec(A)$ is (isomorphic) to an object of this big site. \medskip\noindent Let $S$ be a scheme and let $X$ be an algebraic space over $S$. In this chapter and the following we will write $X \times_S X$ for the product of $X$ with itself (in the category of algebraic spaces over $S$), instead of $X \times X$. \section{The Hom functor} \label{section-hom} \noindent In this section we study the functor of homomorphisms defined below. \begin{situation} \label{situation-hom} Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_X$-modules. For any scheme $T$ over $B$ we will denote $\mathcal{F}_T$ and $\mathcal{G}_T$ the base changes of $\mathcal{F}$ and $\mathcal{G}$ to $T$, in other words, the pullbacks via the projection morphism $X_T = X \times_B T \to X$. We consider the functor \begin{equation} \label{equation-hom} \mathit{Hom}(\mathcal{F}, \mathcal{G}) : (\Sch/B)^{opp} \longrightarrow \textit{Sets},\quad T \longrightarrow \Hom_{\mathcal{O}_{X_T}}(\mathcal{F}_T, \mathcal{G}_T) \end{equation} \end{situation} \noindent In Situation \ref{situation-hom} we sometimes think of $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ as a functor $(\Sch/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\mathit{Hom}(\mathcal{F}, \mathcal{G}) \to B$. Namely, if $T$ is a scheme over $S$, then an element of $\mathit{Hom}(\mathcal{F}, \mathcal{G})(T)$ consists of a pair $(h, u)$, where $h$ is a morphism $h : T \to B$ and $u : \mathcal{F}_T \to \mathcal{G}_T$ is an $\mathcal{O}_{X_T}$-module map where $X_T = T \times_{h, B} X$ and $\mathcal{F}_T$ and $\mathcal{G}_T$ are the pullbacks to $X_T$. In particular, when we say that $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is an algebraic space, we mean that the corresponding functor $(\Sch/S)^{opp} \to \textit{Sets}$ is an algebraic space. \begin{lemma} \label{lemma-hom-sheaf} In Situation \ref{situation-hom} the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ satisfies the sheaf property for the fpqc topology. \end{lemma} \begin{proof} Let $\{T_i \to T\}_{i \in I}$ be an fpqc covering of schemes over $B$. Set $X_i = X_{T_i} = X \times_S T_i$ and $\mathcal{F}_i = u_{T_i}$ and $\mathcal{G}_i = \mathcal{G}_{T_i}$. Note that $\{X_i \to X_T\}_{i \in I}$ is an fpqc covering of $X_T$, see Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-fpqc}. Thus a family of maps $u_i : \mathcal{F}_i \to \mathcal{G}_i$ such that $u_i$ and $u_j$ restrict to the same map on $X_{T_i \times_T T_j}$ comes from a unique map $u : \mathcal{F}_T \to \mathcal{G}_T$ by descent (Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}). \end{proof} \noindent Sanity check: $\mathit{Hom}$ sheaf plays the same role among algebraic spaces over $S$. \begin{lemma} \label{lemma-extend-hom-to-spaces} In Situation \ref{situation-hom}. Let $T$ be an algebraic space over $S$. We have $$\Mor_{\Sh((\Sch/S)_{fppf})}(T, \mathit{Hom}(\mathcal{F}, \mathcal{G})) = \{(h, u) \mid h : T \to B, u : \mathcal{F}_T \to \mathcal{G}_T\}$$ where $\mathcal{F}_T, \mathcal{G}_T$ denote the pullbacks of $\mathcal{F}$ and $\mathcal{G}$ to the algebraic space $X \times_{B, h} T$. \end{lemma} \begin{proof} Choose a scheme $U$ and a surjective \'etale morphism $p : U \to T$. Let $R = U \times_T U$ with projections $t, s : R \to U$. \medskip\noindent Let $v : T \to \mathit{Hom}(\mathcal{F}, \mathcal{G})$ be a natural transformation. Then $v(p)$ corresponds to a pair $(h_U, u_U)$ over $U$. As $v$ is a transformation of functors we see that the pullbacks of $(h_U, u_U)$ by $s$ and $t$ agree. Since $T = U/R$ (Spaces, Lemma \ref{spaces-lemma-space-presentation}), we obtain a morphism $h : T \to B$ such that $h_U = h \circ p$. Then $\mathcal{F}_U$ is the pullback of $\mathcal{F}_T$ to $X_U$ and similarly for $\mathcal{G}_U$. Hence $u_U$ descends to a $\mathcal{O}_{X_T}$-module map $u : \mathcal{F}_T \to \mathcal{G}_T$ by Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}. \medskip\noindent Conversely, let $(h, u)$ be a pair over $T$. Then we get a natural transformation $v : T \to \mathit{Hom}(\mathcal{F}, \mathcal{G})$ by sending a morphism $a : T' \to T$ where $T'$ is a scheme to $(h \circ a, a^*u)$. We omit the verification that the construction of this and the previous paragraph are mutually inverse. \end{proof} \begin{remark} \label{remark-hom-base-change} In Situation \ref{situation-hom} let $B' \to B$ be a morphism of algebraic spaces over $S$. Set $X' = X \times_B B'$ and denote $\mathcal{F}'$, $\mathcal{G}'$ the pullback of $\mathcal{F}$, $\mathcal{G}$ to $X'$. Then we obtain a functor $\mathit{Hom}(\mathcal{F}', \mathcal{G}') : (\Sch/B')^{opp} \to \textit{Sets}$ associated to the base change $f' : X' \to B'$. For a scheme $T$ over $B'$ it is clear that we have $$\mathit{Hom}(\mathcal{F}', \mathcal{G}')(T) = \mathit{Hom}(\mathcal{F}, \mathcal{G})(T)$$ where on the right hand side we think of $T$ as a scheme over $B$ via the composition $T \to B' \to B$. This trivial remark will occasionally be useful to change the base algebraic space. \end{remark} \begin{lemma} \label{lemma-hom-sheaf-in-X} In Situation \ref{situation-hom} let $\{X_i \to X\}_{i \in I}$ be an fppf covering and for each $i, j \in I$ let $\{X_{ijk} \to X_i \times_X X_j\}$ be an fppf covering. Denote $\mathcal{F}_i$, resp.\ $\mathcal{F}_{ijk}$ the pullback of $\mathcal{F}$ to $X_i$, resp.\ $X_{ijk}$. Similarly define $\mathcal{G}_i$ and $\mathcal{G}_{ijk}$. For every scheme $T$ over $B$ the diagram $$\xymatrix{ \mathit{Hom}(\mathcal{F}, \mathcal{G})(T) \ar[r] & \prod\nolimits_i \mathit{Hom}(\mathcal{F}_i, \mathcal{G}_i)(T) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod\nolimits_{i, j, k} \mathit{Hom}(\mathcal{F}_{ijk}, \mathcal{G}_{ijk})(T) }$$ presents the first arrow as the equalizer of the other two. \end{lemma} \begin{proof} Let $u_i : \mathcal{F}_{i, T} \to \mathcal{G}_{i, T}$ be an element in the equalizer of $\text{pr}_0^*$ and $\text{pr}_1^*$. Since the base change of an fppf covering is an fppf covering (Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-fppf}) we see that $\{X_{i, T} \to X_T\}_{i \in I}$ and $\{X_{ijk, T} \to X_{i, T} \times_{X_T} X_{j, T}\}$ are fppf coverings. Applying Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} we first conclude that $u_i$ and $u_j$ restrict to the same morphism over $X_{i, T} \times_{X_T} X_{j, T}$, whereupon a second application shows that there is a unique morphism $u : \mathcal{F}_T \to \mathcal{G}_T$ restricting to $u_i$ for each $i$. This finishes the proof. \end{proof} \begin{lemma} \label{lemma-hom-limits} In Situation \ref{situation-hom}. If $\mathcal{F}$ is of finite presentation and $f$ is quasi-compact and quasi-separated, then $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is limit preserving. \end{lemma} \begin{proof} Let $T = \lim_{i \in I} T_i$ be a directed limit of affine $B$-schemes. We have to show that $$\mathit{Hom}(\mathcal{F}, \mathcal{G})(T) = \colim \mathit{Hom}(\mathcal{F}, \mathcal{G})(T_i)$$ Pick $0 \in I$. We may replace $B$ by $T_0$, $X$ by $X_{T_0}$, $\mathcal{F}$ by $\mathcal{F}_{T_0}$, $\mathcal{G}$ by $\mathcal{G}_{T_0}$, and $I$ by $\{i \in I \mid i \geq 0\}$. See Remark \ref{remark-hom-base-change}. Thus we may assume $B = \Spec(R)$ is affine. \medskip\noindent When $B$ is affine, then $X$ is quasi-compact and quasi-separated. Choose a surjective \'etale morphism $U \to X$ where $U$ is an affine scheme (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Since $X$ is quasi-separated, the scheme $U \times_X U$ is quasi-compact and we may choose a surjective \'etale morphism $V \to U \times_X U$ where $V$ is an affine scheme. Applying Lemma \ref{lemma-hom-sheaf-in-X} we see that $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is the equalizer of two maps between $$\mathit{Hom}(\mathcal{F}|_U, \mathcal{G}|_U) \quad\text{and}\quad \mathit{Hom}(\mathcal{F}|_V, \mathcal{G}|_V)$$ This reduces us to the case that $X$ is affine. \medskip\noindent In the affine case the statement of the lemma reduces to the following problem: Given a ring map $R \to A$, two $A$-modules $M$, $N$ and a directed system of $R$-algebras $C = \colim C_i$. When is it true that the map $$\colim \Hom_{A \otimes_R C_i}(M \otimes_R C_i, N \otimes_R C_i) \longrightarrow \Hom_{A \otimes_R C}(M \otimes_R C, N \otimes_R C)$$ is bijective? By Algebra, Lemma \ref{algebra-lemma-module-map-property-in-colimit} this holds if $M \otimes_R C$ is of finite presentation over $A \otimes_R C$, i.e., when $M$ is of finite presentation over $A$. \end{proof} \begin{lemma} \label{lemma-hom-closed} Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : X' \to X$ be a closed immersion of algebraic spaces over $B$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module and let $\mathcal{G}'$ be a quasi-coherent $\mathcal{O}_{X'}$-module. Then $$\mathit{Hom}(\mathcal{F}, i_*\mathcal{G}') = \mathit{Hom}(i^*\mathcal{F}, \mathcal{G}')$$ as functors on $(\Sch/B)$. \end{lemma} \begin{proof} Let $g : T \to B$ be a morphism where $T$ is a scheme. Denote $i_T : X'_T \to X_T$ the base change of $i$. Denote $h : X_T \to X$ and $h' : X'_T \to X'$ the projections. Observe that $(h')^*i^*\mathcal{F} = i_T^*h^*\mathcal{F}$. As a closed immersion is affine (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-affine}) we have $h^*i_*\mathcal{G} = i_{T, *}(h')^*\mathcal{G}$ by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-affine-base-change}. Thus we have \begin{align*} \mathit{Hom}(\mathcal{F}, i_*\mathcal{G}')(T) & = \Hom_{\mathcal{O}_{X_T}}(h^*\mathcal{F}, h^*i_*\mathcal{G}') \\ & = \Hom_{\mathcal{O}_{X_T}}(h^*\mathcal{F}, i_{T, *}(h')^*\mathcal{G}) \\ & = \Hom_{\mathcal{O}_{X'_T}}(i_T^*h^*\mathcal{F}, (h')^*\mathcal{G}) \\ & = \Hom_{\mathcal{O}_{X'_T}}((h')^*i^*\mathcal{F}, (h')^*\mathcal{G}) \\ & = \mathit{Hom}(i^*\mathcal{F}, \mathcal{G}')(T) \end{align*} as desired. The middle equality follows from the adjointness of the functors $i_{T, *}$ and $i_T^*$. \end{proof} \begin{lemma} \label{lemma-cohomology-perfect-complex} Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $K$ be a pseudo-coherent object of $D(\mathcal{O}_B)$. \begin{enumerate} \item If for all $g : T \to B$ in $(\Sch/B)$ the cohomology sheaf $H^{-1}(Lg^*K)$ is zero, then the functor $$(\Sch/B)^{opp} \longrightarrow \textit{Sets},\quad (g : T \to B) \longmapsto H^0(T, H^0(Lg^*K))$$ is an algebraic space affine and of finite presentation over $B$. \item If for all $g : T \to B$ in $(\Sch/B)$ the cohomology sheaves $H^i(Lg^*K)$ are zero for $i < 0$, then $K$ is perfect with tor amplitude in $[0, b]$ for some $b \geq 0$ and the functor $$(\Sch/B)^{opp} \longrightarrow \textit{Sets},\quad (g : T \to B) \longmapsto H^0(T, Lg^*K)$$ is an algebraic space affine and of finite presentation over $B$. \end{enumerate} \end{lemma} \begin{proof} Under the assumptions of (2) we have $H^0(T, Lg^*K) = H^0(T, H^0(Lg^*K))$. Let us prove that the rule $T \mapsto H^0(T, H^0(Lg^*K))$ satisfies the sheaf property for the fppf topology. To do this assume we have an fppf covering $\{h_i : T_i \to T\}$ of a scheme $g : T \to B$ over $B$. Set $g_i = g \circ h_i$. Note that since $h_i$ is flat, we have $Lh_i^* = h_i^*$ and $h_i^*$ commutes with taking cohomology. Hence $$H^0(T_i, H^0(Lg_i^*K)) = H^0(T_i, H^0(h_i^*Lg^*K)) = H^0(T, h_i^*H^0(Lg^*K))$$ Similarly for the pullback to $T_i \times_T T_j$. Since $Lg^*K$ is a pseudo-coherent complex on $T$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pseudo-coherent-pullback}) the cohomology sheaf $\mathcal{F} = H^0(Lg^*K)$ is quasi-coherent (Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-pseudo-coherent}). Hence by Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} we see that $$H^0(T, \mathcal{F}) = \Ker( \prod H^0(T_i, h_i^*\mathcal{F}) \to \prod H^0(T_i \times_T T_j, (T_i \times_T T_j \to T)^*\mathcal{F}))$$ In this way we see that the rules in (1) and (2) satisfy the sheaf property for fppf coverings. This means we may apply Bootstrap, Lemma \ref{bootstrap-lemma-locally-algebraic-space-finite-type} to see it suffices to prove the representability \'etale locally on $B$. Moreover, we may check whether the end result is affine and of finite presentation \'etale locally on $B$, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-affine-local} and \ref{spaces-morphisms-lemma-finite-presentation-local}. Hence we may assume that $B$ is an affine scheme. \medskip\noindent Assume $B = \Spec(A)$ is an affine scheme. By the results of Derived Categories of Spaces, Lemmas \ref{spaces-perfect-lemma-pseudo-coherent}, \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}, and \ref{spaces-perfect-lemma-descend-pseudo-coherent} we deduce that in the rest of the proof we may think of $K$ as a perfect object of the derived category of complexes of modules on $B$ in the Zariski topology. By Derived Categories of Schemes, Lemmas \ref{perfect-lemma-pseudo-coherent}, \ref{perfect-lemma-affine-compare-bounded}, and \ref{perfect-lemma-pseudo-coherent-affine} we can find a pseudo-coherent complex $M^\bullet$ of $A$-modules such that $K$ is the corresponding object of $D(\mathcal{O}_B)$. Our assumption on pullbacks implies that $M^\bullet \otimes^\mathbf{L}_A \kappa(\mathfrak p)$ has vanishing $H^{-1}$ for all primes $\mathfrak p \subset A$. By More on Algebra, Lemma \ref{more-algebra-lemma-cut-complex-in-two} we can write $$M^\bullet = \tau_{\geq 0}M^\bullet \oplus \tau_{\leq - 1}M^\bullet$$ with $\tau_{\geq 0}M^\bullet$ perfect with Tor amplitude in $[0, b]$ for some $b \geq 0$ (here we also have used More on Algebra, Lemmas \ref{more-algebra-lemma-glue-perfect} and \ref{more-algebra-lemma-glue-tor-amplitude}). Note that in case (2) we also see that $\tau_{\leq - 1}M^\bullet = 0$ in $D(A)$ whence $M^\bullet$ and $K$ are perfect with tor amplitude in $[0, b]$. For any $B$-scheme $g : T \to B$ we have $$H^0(T, H^0(Lg^*K)) = H^0(T, H^0(Lg^*\tau_{\geq 0}K))$$ (by the dual of Derived Categories, Lemma \ref{derived-lemma-negative-vanishing}) hence we may replace $K$ by $\tau_{\geq 0}K$ and correspondingly $M^\bullet$ by $\tau_{\geq 0}M^\bullet$. In other words, we may assume $M^\bullet$ has tor amplitude in $[0, b]$. \medskip\noindent Assume $M^\bullet$ has tor amplitude in $[0, b]$. We may assume $M^\bullet$ is a bounded above complex of finite free $A$-modules (by our definition of pseudo-coherent complexes, see More on Algebra, Definition \ref{more-algebra-definition-pseudo-coherent} and the discussion following the definition). By More on Algebra, Lemma \ref{more-algebra-lemma-last-one-flat} we see that $M = \Coker(M^{- 1} \to M^0)$ is flat. By Algebra, Lemma \ref{algebra-lemma-finite-projective} we see that $M$ is finite locally free. Hence $M^\bullet$ is quasi-isomorphic to $$M \to M^1 \to M^2 \to \ldots \to M^d \to 0 \ldots$$ Note that this is a K-flat complex (Cohomology, Lemma \ref{cohomology-lemma-bounded-flat-K-flat}), hence derived pullback of $K$ via a morphism $T \to B$ is computed by the complex $$g^*\widetilde{M} \to g^*\widetilde{M^1} \to \ldots$$ Thus it suffices to show that the functor $$(g : T \to B) \longmapsto \Ker( \Gamma(T,g^*\widetilde{M}) \to \Gamma(T, g^*(\widetilde{M^1}) )$$ is representable by an affine scheme of finite presentation over $B$. \medskip\noindent We may still replace $B$ by the members of an affine open covering in order to prove this last statement. Hence we may assume that $M$ is finite free (recall that $M^1$ is finite free to begin with). Write $M = A^{\oplus n}$ and $M^1 = A^{\oplus m}$. Let the map $M \to M^1$ be given by the $m \times n$ matrix $(a_{ij})$ with coefficients in $A$. Then $\widetilde{M} = \mathcal{O}_B^{\oplus n}$ and $\widetilde{M^1} = \mathcal{O}_B^{\oplus m}$. Thus the functor above is equal to the functor $$(g : T \to B) \longmapsto \{(f_1, \ldots, f_n) \in \Gamma(T, \mathcal{O}_T) \mid \sum g^\sharp(a_{ij}f_i) = 0, j = 1, \ldots, m\}$$ Clearly this is representable by the affine scheme $$\Spec\left(A[x_1, \ldots, x_n]/(\sum a_{ij}x_i; j = 1, \ldots, m)\right)$$ and the lemma has been proved. \end{proof} \noindent The functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is representable in a number of situations. All of our results will be based on the following basic case. The proof of this lemma as given below is in some sense the natural generalization to the proof of \cite[III, Cor 7.7.8]{EGA}. \begin{lemma} \label{lemma-noetherian-hom} In Situation \ref{situation-hom} assume that \begin{enumerate} \item $B$ is a Noetherian algebraic space, \item $f$ is locally of finite type and quasi-separated, \item $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module, and \item $\mathcal{G}$ is a finite type $\mathcal{O}_X$-module, flat over $B$, with support proper over $B$. \end{enumerate} Then the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is an algebraic space affine and of finite presentation over $B$. \end{lemma} \begin{proof} We may replace $X$ by a quasi-compact open neighbourhood of the support of $\mathcal{G}$, hence we may assume $X$ is Noetherian. In this case $X$ and $f$ are quasi-compact and quasi-separated. Choose an approximation $P \to \mathcal{F}$ by a perfect complex $P$ of the triple $(X, \mathcal{F}, 0)$, see Derived Categories of Spaces, Definition \ref{spaces-perfect-definition-approximation-holds} and Theorem \ref{spaces-perfect-theorem-approximation}). Then the induced map $$\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) \longrightarrow \Hom_{D(\mathcal{O}_X)}(P, \mathcal{G})$$ is an isomorphism because $P \to \mathcal{F}$ induces an isomorphism $H^0(P) \to \mathcal{F}$ and $H^i(P) = 0$ for $i > 0$. Moreover, for any morphism $g : T \to B$ denote $h : X_T = T \times_B X \to X$ the projection and set $P_T = Lh^*P$. Then it is equally true that $$\Hom_{\mathcal{O}_{X_T}}(\mathcal{F}_T, \mathcal{G}_T) \longrightarrow \Hom_{D(\mathcal{O}_{X_T})}(P_T, \mathcal{G}_T)$$ is an isomorphism, as $P_T = Lh^*P \to Lh^*\mathcal{F} \to \mathcal{F}_T$ induces an isomorphism $H^0(P_T) \to \mathcal{F}_T$ (because $h^*$ is right exact and $H^i(P) = 0$ for $i > 0$). Thus it suffices to prove the result for the functor $$T \longmapsto \Hom_{D(\mathcal{O}_{X_T})}(P_T, \mathcal{G}_T).$$ By the Leray spectral sequence (see Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}) we have $$\Hom_{D(\mathcal{O}_{X_T})}(P_T, \mathcal{G}_T) = H^0(X_T, R\SheafHom(P_T, \mathcal{G}_T)) = H^0(T, Rf_{T, *}R\SheafHom(P_T, \mathcal{G}_T))$$ where $f_T : X_T \to T$ is the base change of $f$. By Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-base-change-RHom} we have $$Rf_{T, *}R\SheafHom(P_T, \mathcal{G}_T) = Lg^*Rf_*R\SheafHom(P, \mathcal{G}).$$ By Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-compute-ext-perfect} the object $K = Rf_*R\SheafHom(P, \mathcal{G})$ of $D(\mathcal{O}_B)$ is perfect. This means we can apply Lemma \ref{lemma-cohomology-perfect-complex} as long as we can prove that the cohomology sheaf $H^i(Lg^*K)$ is $0$ for all $i < 0$ and $g : T \to B$ as above. This is clear from the last displayed formula as the cohomology sheaves of $Rf_{T, *}R\SheafHom(P_T, \mathcal{G}_T)$ are zero in negative degrees due to the fact that $R\SheafHom(P_T, \mathcal{G}_T)$ has vanishing cohomology sheaves in negative degrees as $P_T$ is perfect with vanishing cohomology sheaves in positive degrees. \end{proof} \noindent Here is a cheap consequence of Lemma \ref{lemma-noetherian-hom}. \begin{proposition} \label{proposition-hom} In Situation \ref{situation-hom} assume that \begin{enumerate} \item $f$ is of finite presentation, and \item $\mathcal{G}$ is a finitely presented $\mathcal{O}_X$-module, flat over $B$, with support proper over $B$. \end{enumerate} Then the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is an algebraic space affine over $B$. If $\mathcal{F}$ is of finite presentation, then $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is of finite presentation over $B$. \end{proposition} \begin{proof} By Lemma \ref{lemma-hom-sheaf} the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ satisfies the sheaf property for fppf coverings. This mean we may\footnote{We omit the verification of the set theoretical condition (3) of the referenced lemma.} apply Bootstrap, Lemma \ref{bootstrap-lemma-locally-algebraic-space} to check the representability \'etale locally on $B$. Moreover, we may check whether the end result is affine or of finite presentation \'etale locally on $B$, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-affine-local} and \ref{spaces-morphisms-lemma-finite-presentation-local}. Hence we may assume that $B$ is an affine scheme. \medskip\noindent Assume $B$ is an affine scheme. As $f$ is of finite presentation, it follows $X$ is quasi-compact and quasi-separated. Thus we can write $\mathcal{F} = \colim \mathcal{F}_i$ as a filtered colimit of $\mathcal{O}_X$-modules of finite presentation (Limits of Spaces, Lemma \ref{spaces-limits-lemma-colimit-finitely-presented}). It is clear that $$\mathit{Hom}(\mathcal{F}, \mathcal{G}) = \lim \mathit{Hom}(\mathcal{F}_i, \mathcal{G})$$ Hence if we can show that each $\mathit{Hom}(\mathcal{F}_i, \mathcal{G})$ is representable by an affine scheme, then we see that the same thing holds for $\mathit{Hom}(\mathcal{F}, \mathcal{G})$. Use the material in Limits, Section \ref{limits-section-limits} and Limits of Spaces, Section \ref{spaces-limits-section-limits}. Thus we may assume that $\mathcal{F}$ is of finite presentation. \medskip\noindent Say $B = \Spec(R)$. Write $R = \colim R_i$ with each $R_i$ a finite type $\mathbf{Z}$-algebra. Set $B_i = \Spec(R_i)$. By the results of Limits of Spaces, Lemmas \ref{spaces-limits-lemma-descend-finite-presentation} and \ref{spaces-limits-lemma-descend-modules-finite-presentation} we can find an $i$, a morphism of algebraic spaces $X_i \to B_i$, and finitely presented $\mathcal{O}_{X_i}$-modules $\mathcal{F}_i$ and $\mathcal{G}_i$ such that the base change of $(X_i, \mathcal{F}_i, \mathcal{G}_i)$ to $B$ recovers $(X, \mathcal{F}, \mathcal{G})$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-flat} we may, after increasing $i$, assume that $\mathcal{G}_i$ is flat over $B_i$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-eventually-proper-support} we may similarly assume the scheme theoretic support of $\mathcal{G}_i$ is proper over $B_i$. At this point we can apply Lemma \ref{lemma-noetherian-hom} to see that $H_i = \mathit{Hom}(\mathcal{F}_i, \mathcal{G}_i)$ is an algebraic space affine of finite presentation over $B_i$. Pulling back to $B$ (using Remark \ref{remark-hom-base-change}) we see that $H_i \times_{B_i} B = \mathit{Hom}(\mathcal{F}, \mathcal{G})$ and we win. \end{proof} \section{The Isom functor} \label{section-isom} \noindent In Situation \ref{situation-hom} we can consider the subfunctor $$\mathit{Isom}(\mathcal{F}, \mathcal{G}) \subset \mathit{Hom}(\mathcal{F}, \mathcal{G})$$ whose value on a scheme $T$ over $B$ is the set of {\it invertible} $\mathcal{O}_{X_T}$-homomorphisms $u : \mathcal{F}_T \to \mathcal{G}_T$. \medskip\noindent We sometimes think of $\mathit{Isom}(\mathcal{F}, \mathcal{G})$ as a functor $(\Sch/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\mathit{Isom}(\mathcal{F}, \mathcal{G}) \to B$. Namely, if $T$ is a scheme over $S$, then an element of $\mathit{Isom}(\mathcal{F}, \mathcal{G})(T)$ consists of a pair $(h, u)$, where $h$ is a morphism $h : T \to B$ and $u : \mathcal{F}_T \to \mathcal{G}_T$ is an $\mathcal{O}_{X_T}$-module isomorphism where $X_T = T \times_{h, B} X$ and $\mathcal{F}_T$ and $\mathcal{G}_T$ are the pullbacks to $X_T$. In particular, when we say that $\mathit{Isom}(\mathcal{F}, \mathcal{G})$ is an algebraic space, we mean that the corresponding functor $(\Sch/S)^{opp} \to \textit{Sets}$ is an algebraic space. \begin{lemma} \label{lemma-isom-sheaf} In Situation \ref{situation-hom} the functor $\mathit{Isom}(\mathcal{F}, \mathcal{G})$ satisfies the sheaf property for the fpqc topology. \end{lemma} \begin{proof} We have already seen that $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ satisfies the sheaf property. Hence it remains to show the following: Given an fpqc covering $\{T_i \to T\}_{i \in I}$ of schemes over $B$ and an $\mathcal{O}_{X_T}$-linear map $u : \mathcal{F}_T \to \mathcal{G}_T$ such that $u_{T_i}$ is an isomorphism for all $i$, then $u$ is an isomorphism. Since $\{X_i \to X_T\}_{i \in I}$ is an fpqc covering of $X_T$, see Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-fpqc}, this follows from Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}. \end{proof} \noindent Sanity check: $\mathit{Isom}$ sheaf plays the same role among algebraic spaces over $S$. \begin{lemma} \label{lemma-extend-isom-to-spaces} In Situation \ref{situation-hom}. Let $T$ be an algebraic space over $S$. We have $$\Mor_{\Sh((\Sch/S)_{fppf})}(T, \mathit{Isom}(\mathcal{F}, \mathcal{G})) = \{(h, u) \mid h : T \to B, u : \mathcal{F}_T \to \mathcal{G}_T\text{ isomorphism}\}$$ where $\mathcal{F}_T, \mathcal{G}_T$ denote the pullbacks of $\mathcal{F}$ and $\mathcal{G}$ to the algebraic space $X \times_{B, h} T$. \end{lemma} \begin{proof} Observe that the left and right hand side of the equality are subsets of the left and right hand side of the equality in Lemma \ref{lemma-extend-hom-to-spaces}. We omit the verification that these subsets correspond under the identification given in the proof of that lemma. \end{proof} \begin{proposition} \label{proposition-isom} In Situation \ref{situation-hom} assume that \begin{enumerate} \item $f$ is of finite presentation, and \item $\mathcal{F}$ and $\mathcal{G}$ are finitely presented $\mathcal{O}_X$-modules, flat over $B$, with support proper over $B$. \end{enumerate} Then the functor $\mathit{Isom}(\mathcal{F}, \mathcal{G})$ is an algebraic space affine of finite presentation over $B$. \end{proposition} \begin{proof} We will use the abbreviations $H = \mathit{Hom}(\mathcal{F}, \mathcal{G})$, $I = \mathit{Hom}(\mathcal{F}, \mathcal{F})$, $H' = \mathit{Hom}(\mathcal{G}, \mathcal{F})$, and $I' = \mathit{Hom}(\mathcal{G}, \mathcal{G})$. By Proposition \ref{proposition-hom} the functors $H$, $I$, $H'$, $I'$ are algebraic spaces and the morphisms $H \to B$, $I \to B$, $H' \to B$, and $I' \to B$ are affine and of finite presentation. The composition of maps gives a morphism $$c : H' \times_B H \longrightarrow I \times_B I',\quad (u', u) \longmapsto (u \circ u', u' \circ u)$$ of algebraic spaces over $B$. Since $I \times_B I' \to B$ is separated, the section $\sigma : B \to I \times_B I'$ corresponding to $(\text{id}_\mathcal{F}, \text{id}_\mathcal{G})$ is a closed immersion (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-section-immersion}). Moreover, $\sigma$ is of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-permanence}). Hence $$\mathit{Isom}(\mathcal{F}, \mathcal{G}) = (H' \times_B H) \times_{c, I \times_B I', \sigma} B$$ is an algebraic space affine of finite presentation over $B$ as well. Some details omitted. \end{proof} \section{The stack of coherent sheaves} \label{section-stack-coherent-sheaves} \noindent 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 \cite[Theorem 2.1.1]{lieblich_remarks} which treats the case of the stack of coherent sheaves on an Artin stack over a base. \begin{situation} \label{situation-coherent} 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 $\textit{Coh}_{X/B}$ the category whose objects are triples $(T, g, \mathcal{F})$ where \begin{enumerate} \item $T$ is a scheme over $S$, \item $g : T \to B$ is a morphism over $S$, and setting $X_T = T \times_{g, B} X$ \item $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_{X_T}$-module of finite presentation, flat over $T$, with support proper over $T$. \end{enumerate} A morphism $(T, g, \mathcal{F}) \to (T', g', \mathcal{F}')$ is given by a pair $(h, \varphi)$ where \begin{enumerate} \item $h : T \to T'$ is a morphism of schemes over $B$ (i.e., $g' \circ h = g$), and \item $\varphi : (h')^*\mathcal{F}' \to \mathcal{F}$ is an isomorphism of $\mathcal{O}_{X_T}$-modules where $h' : X_T \to X_{T'}$ is the base change of $h$. \end{enumerate} \end{situation} \noindent Thus $\textit{Coh}_{X/B}$ is a category and the rule $$p : \textit{Coh}_{X/B} \longrightarrow (\Sch/S)_{fppf}, \quad (T, g, \mathcal{F}) \longmapsto T$$ is a functor. For a scheme $T$ over $S$ we denote $\textit{Coh}_{X/B, T}$ the fibre category of $p$ over $T$. These fibre categories are groupoids. \begin{lemma} \label{lemma-coherent-fibred-in-groupoids} In Situation \ref{situation-coherent} the functor $p : \textit{Coh}_{X/B} \longrightarrow (\Sch/S)_{fppf}$ is fibred in groupoids. \end{lemma} \begin{proof} We show that $p$ is fibred in groupoids by checking conditions (1) and (2) of Categories, Definition \ref{categories-definition-fibred-groupoids}. Given an object $(T', g', \mathcal{F}')$ of $\textit{Coh}_{X/B}$ 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 $\textit{Coh}_{X/B}$ 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 $\textit{Coh}_{X/B}$ 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). \end{proof} \begin{lemma} \label{lemma-coherent-diagonal} In Situation \ref{situation-coherent}. Denote $\mathcal{X} = \textit{Coh}_{X/B}$. Then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces. \end{lemma} \begin{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 \ref{algebraic-lemma-representable-diagonal}. 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 \ref{proposition-isom}. \end{proof} \begin{lemma} \label{lemma-coherent-stack} In Situation \ref{situation-coherent} the functor $p : \textit{Coh}_{X/B} \longrightarrow (\Sch/S)_{fppf}$ is a stack in groupoids. \end{lemma} \begin{proof} To prove that $\textit{Coh}_{X/B}$ 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 \ref{lemma-coherent-diagonal}, see Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-diagonal}. 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 $\textit{Coh}_{X/B}$. 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 \ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}. 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 $\textit{Coh}_{X/B}$ 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 \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} 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 \ref{spaces-morphisms-lemma-flat-permanence} we see that $\mathcal{F}$ is flat over $T$ and Descent on Spaces, Lemma \ref{spaces-descent-lemma-finite-presentation-descends} guarantees that $\mathcal{Q}$ is of finite presentation as an $\mathcal{O}_{X_T}$-module. Finally, by Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-proper} we see that the scheme theoretic support of $\mathcal{F}$ is proper over $T$ as we've assume 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 \ref{spaces-morphisms-lemma-flat-pullback-support}). In this way and we obtain our desired object over $T$. \end{proof} \begin{remark} \label{remark-coherent-base-change} In Situation \ref{situation-coherent} the rule $(T, g, \mathcal{F}) \mapsto (T, g)$ defines a $1$-morphism $$\textit{Coh}_{X/B} \longrightarrow \mathcal{S}_B$$ of stacks in groupoids (see Lemma \ref{lemma-coherent-stack}, Algebraic Stacks, Section \ref{algebraic-section-split}, and Examples of Stacks, Section \ref{examples-stacks-section-stack-associated-to-sheaf}). 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 $\textit{Coh}_{X'/B'} \to (\Sch/S)_{fppf}$ associated to the base change $f' : X' \to B'$. In this situation the diagram $$\vcenter{ \xymatrix{ \textit{Coh}_{X'/B'} \ar[r] \ar[d] & \textit{Coh}_{X/B} \ar[d] \\ \mathcal{S}_{B'} \ar[r] & \mathcal{S}_B } } \quad \begin{matrix} \text{or in} \\ \text{another} \\ \text{notation} \end{matrix} \quad \vcenter{ \xymatrix{ \textit{Coh}_{X'/B'} \ar[r] \ar[d] & \textit{Coh}_{X/B} \ar[d] \\ \Sch/B' \ar[r] & \Sch/B } }$$ is $2$-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space. \end{remark} \begin{lemma} \label{lemma-coherent-limits} In Situation \ref{situation-coherent} assume that $B \to S$ is locally of finite presentation. Then $p : \textit{Coh}_{X/B} \to (\Sch/S)_{fppf}$ is limit preserving (Artin's Axioms, Definition \ref{artin-definition-limit-preserving}). \end{lemma} \begin{proof} Write $B(T)$ for the discrete category whose objects are the $S$-morphisms $T \to B$. Let $T = \lim T_i$ be a filtered limit of affine schemes over $S$. Assigning to an object $(T, h, \mathcal{F})$ of $\textit{Coh}_{X/B, T}$ the object $h$ of $B(T)$ gives us a commutative diagram of fibre categories $$\xymatrix{ \colim \textit{Coh}_{X/B, T_i} \ar[r] \ar[d] & \textit{Coh}_{X/B, T} \ar[d] \\ \colim B(T_i) \ar[r] & B(T) }$$ We have to show the top horizontal arrow is an equivalence. Since we have assume that $B$ is locally of finite presentation over $S$ we see from Limits of Spaces, Remark \ref{spaces-limits-remark-limit-preserving} that the bottom horizontal arrow is an equivalence. This means that we may assume $T = \lim T_i$ 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 $X_T = \colim X_i$ 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 \ref{spaces-limits-lemma-descend-modules-finite-presentation} we see that $$\colim \textit{FP}(X_i) = \textit{FP}(X_T).$$ where $\textit{FP}(W)$ is short hand for the category of finitely presented $\mathcal{O}_W$-modules. The results of Limits of Spaces, Lemmas \ref{spaces-limits-lemma-descend-flat} and \ref{spaces-limits-lemma-eventually-proper-support} 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. \end{proof} \begin{lemma} \label{lemma-coherent-RS-star} In Situation \ref{situation-coherent}. 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 \ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$\textit{Coh}_{X/B, Y'} \longrightarrow \textit{Coh}_{X/B, Y} \times_{\textit{Coh}_{X/B, Z}} \textit{Coh}_{X/B, Z'}$$ is an equivalence. \end{lemma} \begin{proof} Observe that the corresponding map $$B(Y') \longrightarrow B(Y) \times_{B(Z)} B(Z')$$ is a bijection, see Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-pushout-along-thickening-schemes}. Thus using the commutative diagram $$\xymatrix{ \textit{Coh}_{X/B, Y'} \ar[r] \ar[d] & \textit{Coh}_{X/B, Y} \times_{\textit{Coh}_{X/B, Z}} \textit{Coh}_{X/B, Z'} \ar[d] \\ B(Y') \ar[r] & B(Y) \times_{B(Z)} B(Z') }$$ we see that we may assume that $Y'$ is a scheme over $B'$. By Remark \ref{remark-coherent-base-change} 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 \ref{spaces-pushouts-lemma-space-over-pushout-flat-modules}. \end{proof} \begin{lemma} \label{lemma-coherent-over-first-order-thickening} 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 \begin{enumerate} \item $\mathcal{F}'$ is a quasi-coherent $\mathcal{O}_{X'}$-module of finite presentation, \item $\mathcal{F}'$ is an $\mathcal{O}_{X'}$-module of finite presentation, \item $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module of finite presentation, \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, \end{enumerate} \end{lemma} \begin{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 \ref{defos-lemma-deform-fp-module-ringed-topoi}. \end{proof} \begin{lemma} \label{lemma-coherent-tangent-space} In Situation \ref{situation-coherent} 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 $x_0 = (\Spec(k), g_0, \mathcal{G}_0)$ be an object of $\mathcal{X} = \textit{Coh}_{X/B}$ over $k$. Then the spaces $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}_{x_0}(\mathcal{F}_{\mathcal{X}, k, x_0})$ (Artin's Axioms, Section \ref{artin-section-tangent-spaces}) are finite dimensional. \end{lemma} \begin{proof} Observe that by Lemma \ref{lemma-coherent-RS-star} our stack in groupoids $\mathcal{X}$ satisfies property (RS*) defined in Artin's Axioms, Section \ref{artin-section-inf}. In particular $\mathcal{X}$ satisfies (RS). Hence all associated predeformation categories are deformation categories (Artin's Axioms, Lemma \ref{artin-lemma-deformation-category}) and the statement makes sense. \medskip\noindent In this paragraph we show that we can reduce to the case $B = \Spec(k)$. Set $X_0 = \Spec(k) \times_{g_0, B} X$ and denote $\mathcal{X}_0 = \textit{Coh}_{X_0/k}$. In Remark \ref{remark-coherent-base-change} we have seen that $\mathcal{X}_0$ is the $2$-fibre product of $\mathcal{X}$ with $\Spec(k)$ over $B$ as categories fibred in groupoids over $(\Sch/S)_{fppf}$. Thus by Artin's Axioms, Lemma \ref{artin-lemma-fibre-product-tangent-spaces} we reduce to proving that $B$, $\Spec(k)$, and $\mathcal{X}_0$ have finite dimensional tangent spaces and infinitesimal automorphism spaces. The tangent space of $B$ and $\Spec(k)$ are finite dimensional by Artin's Axioms, Lemma \ref{artin-lemma-finite-dimension} and of course these have vanishing $\text{Inf}$. Thus it suffices to deal with $\mathcal{X}_0$. \medskip\noindent Let $k[\epsilon]$ be the dual numbers over $k$. Let $\Spec(k[\epsilon]) \to B$ be the composition of $g_0 : \Spec(k) \to B$ and the morphism $\Spec(k[\epsilon]) \to \Spec(k)$ coming from the inclusion $k \to k[\epsilon]$. Set $X_0 = \Spec(k) \times_B X$ and $X_\epsilon = \Spec(k[\epsilon]) \times_B X$. Observe that $X_\epsilon$ is a first order thickening of $X_0$ flat over the first order thickening $\Spec(k) \to \Spec(k[\epsilon])$. Unwinding the definitions and using Lemma \ref{lemma-coherent-over-first-order-thickening} 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 \ref{defos-lemma-flat-ringed-topoi} we conclude that $$T\mathcal{F}_{\mathcal{X}_0, k, x_0} = \text{Ext}^1_{\mathcal{O}_{X_0}}(\mathcal{G}_0, \mathcal{G}_0)$$ Here we have used the identification $\epsilon k[\epsilon] \cong k$ of $k[\epsilon]$-modules. Using Deformation Theory, Lemma \ref{defos-lemma-flat-ringed-topoi} once more we see that $$\text{Inf}_{x_0}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \text{Ext}^0_{\mathcal{O}_{X_0}}(\mathcal{G}_0, \mathcal{G}_0)$$ These spaces are finite dimensional over $k$ as $\mathcal{G}_0$ has support proper over $\Spec(k)$. Namely, $X_0$ is of finite presentation over $\Spec(k)$, 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 \ref{spaces-perfect-lemma-compute-ext} to conclude the desired finiteness. \end{proof} \begin{lemma} \label{lemma-coherent-existence} In Situation \ref{situation-coherent} assume that $S$ is a locally Noetherian scheme and that $f : X \to B$ is separated. Let $\mathcal{X} = \textit{Coh}_{X/B}$. Then the functor Artin's Axioms, Equation (\ref{artin-equation-approximation}) is an equivalence. \end{lemma} \begin{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 \ref{artin-lemma-effective}, it suffices to prove this for those objects lying over a given morphism $\Spec(A) \to B$. \medskip\noindent Set $X_A = \Spec(A) \times_B X$ and $X_n = \Spec(A/\mathfrak m^n) \times_B X$. By Grothendieck's existence theorem (More on Morphisms of Spaces, Theorem \ref{spaces-more-morphisms-theorem-grothendieck-existence}) we see that the category of coherent modules $\mathcal{F}$ on $X_A$ with support proper over $\Spec(A)$ is equivalent to the category of systems $(\mathcal{F}_n)$ of coherent modules $\mathcal{F}_n$ on $X_n$ with support proper over $\Spec(A/\mathfrak m^n)$. 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 \ref{spaces-more-morphisms-remark-reformulate-existence-theorem}. 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 \ref{spaces-more-morphisms-lemma-flatness-over-Noetherian-ring}. \end{proof} \begin{lemma} \label{lemma-coherent-defo-thy} In Situation \ref{situation-coherent} assume that $S$ is a locally Noetherian scheme, $S = B$, and $f : X \to B$ is flat. Let $\mathcal{X} = \textit{Coh}_{X/B}$. Then we have openness of versality for $\mathcal{X}$ (see Artin's Axioms, Definition \ref{artin-definition-openness-versality}). \end{lemma} \begin{proof}[First proof] This proof is based on the criterion of Artin's Axioms, Lemma \ref{artin-lemma-dual-openness}. 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 \ref{morphisms-lemma-point-finite-type}) and $U = \Spec(A)$ with $U \to S$ mapping into an affine open $\Spec(\Lambda)$ of $S$. Let $\mathcal{F}$ be the coherent module on $X_A = \Spec(A) \times_S X$ flat over $A$ corresponding to the given object $x$. \medskip\noindent According to Deformation Theory, Lemma \ref{defos-lemma-flat-ringed-topoi} we have an isomorphism of functors $$T_x(M) = \text{Ext}^1_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)$$ and given any surjection $A' \to A$ of $\Lambda$-algebras with square zero kernel $I$ we have an obstruction class $$\xi_{A'} \in \text{Ext}^2_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A I)$$ This uses that for any $A' \to A$ as above the base change $X_{A'} = \Spec(A') \times_B X$ 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 \ref{defos-lemma-functorial-ringed-topoi}. Apply Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-compute-ext} to the computation of the Ext groups $\text{Ext}^i_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)$ for $i \leq m$ with $m = 2$. We find a perfect object $K \in D(A)$ and functorial isomorphisms $$H^i(K \otimes_A^\mathbf{L} M) \longrightarrow \text{Ext}^i_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)$$ 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 \ref{artin-situation-dual}. Finally, condition (iv) of Artin's Axioms, Lemma \ref{artin-lemma-dual-obstruction} holds by Deformation Theory, Lemma \ref{defos-lemma-verify-iv-ringed-topoi}. Thus Artin's Axioms, Lemma \ref{artin-lemma-dual-openness} does indeed apply and the lemma is proved. \end{proof} \begin{proof}[Second proof] This proof is based on Artin's Axioms, Lemma \ref{artin-lemma-get-openness-obstruction-theory}. Conditions (1), (2), and (3) of that lemma correspond to Lemmas \ref{lemma-coherent-diagonal}, \ref{lemma-coherent-RS-star}, and \ref{lemma-coherent-limits}. \medskip\noindent We have constructed an obstruction theory in the chapter on deformation theory. Namely, given an $S$-algebra $A$ and an object $x$ of $\textit{Coh}_{X/B}$ over $\Spec(A)$ given by $\mathcal{F}$ on $X_A$ we set $\mathcal{O}_x(M) = \text{Ext}^2_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)$ and if $A' \to A$ is a surjection with kernel $I$, then as obstruction element we take the element $$o_x(A') = o(\mathcal{F}, \mathcal{F} \otimes_A I, 1) \in \mathcal{O}_x(I) = \text{Ext}^2_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A I)$$ of Deformation Theory, Lemma \ref{defos-lemma-flat-ringed-topoi}. All properties of an obstruction theory as defined in Artin's Axioms, Definition \ref{artin-definition-obstruction-theory} 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 \ref{artin-lemma-get-openness-obstruction-theory} it suffices to check functoriality of obstruction classes for a fixed $A$ which follows from Deformation Theory, Lemma \ref{defos-lemma-functorial-ringed-topoi}. Deformation Theory, Lemma \ref{defos-lemma-flat-ringed-topoi} also tells us that $T_x(M) = \text{Ext}^1_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)$ for any $A$-module $M$. \medskip\noindent 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 \ref{spaces-perfect-lemma-compute-ext} to the computation of the Ext groups $\text{Ext}^i_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)$ for $i \leq m$ with $m = 2$. We find a perfect object $K \in D(A)$ and functorial isomorphisms $$H^i(K \otimes_A^\mathbf{L} M) \longrightarrow \text{Ext}^i_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)$$ 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 \ref{more-algebra-lemma-pseudo-coherent-tensor}. \end{proof} \begin{theorem}[Algebraicity of stack coherent sheaves; flat case] \label{theorem-coherent-algebraic} Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation, separated, and flat\footnote{This assumption is not necessary. See Section \ref{section-not-flat}.}. Then $\textit{Coh}_{X/B}$ is an algebraic stack over $S$. \end{theorem} \begin{proof} Set $\mathcal{X} = \textit{Coh}_{X/B}$. We have seen that $\mathcal{X}$ is a stack in groupoids over $(\Sch/S)_{fppf}$ with diagonal representable by algebraic spaces (Lemmas \ref{lemma-coherent-stack} and \ref{lemma-coherent-diagonal}). Hence it suffices to find a scheme $W$ and a surjective and smooth morphism $W \to \mathcal{X}$. \medskip\noindent Let $B'$ be a scheme and let $B' \to B$ be a surjective \'etale morphism. Set $X' = B' \times_B X$ and denote $f' : X' \to B'$ the projection. Then $\mathcal{X}' = \textit{Coh}_{X'/B'}$ 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 \ref{remark-coherent-base-change}). By the material in Algebraic Stacks, Section \ref{algebraic-section-representable-properties} the morphism $\mathcal{X}' \to \mathcal{X}$ is surjective and \'etale. Hence it suffices to prove the result for $\mathcal{X}'$. In other words, we may assume $B$ is a scheme. \medskip\noindent Assume $B$ is a scheme. In this case we may replace $S$ by $B$, see Algebraic Stacks, Section \ref{algebraic-section-change-base-scheme}. Thus we may assume $S = B$. \medskip\noindent Assume $S = B$. Choose an affine open covering $S = \bigcup U_i$. Denote $\mathcal{X}_i$ the restriction of $\mathcal{X}$ to $(\Sch/U_i)_{fppf}$. 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. \medskip\noindent Assume $S = B$ is affine, say $S = \Spec(\Lambda)$. Write $\Lambda = \colim \Lambda_i$ as a filtered colimit with each $\Lambda_i$ of finite type over $\mathbf{Z}$. For some $i$ we can find a morphism of algebraic spaces $X_i \to \Spec(\Lambda_i)$ which is of finite presentation and flat and whose base change to $\Lambda$ is $X$. See Limits of Spaces, Lemmas \ref{spaces-limits-lemma-descend-finite-presentation} and \ref{spaces-limits-lemma-descend-flat}. If we show that $\textit{Coh}_{X_i/\Spec(\Lambda_i)}$ is an algebraic stack, then it follows by base change (Remark \ref{remark-coherent-base-change} and Algebraic Stacks, Section \ref{algebraic-section-change-base-scheme}) that $\mathcal{X}$ is an algebraic stack. Thus we may assume that $\Lambda$ is a finite type $\mathbf{Z}$-algebra. \medskip\noindent Assume $S = B = \Spec(\Lambda)$ 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 \ref{artin-lemma-diagonal-representable} to conclude that $\mathcal{X}$ is an algebraic stack. Note that $\Lambda$ is a G-ring, see More on Algebra, Proposition \ref{more-algebra-proposition-ubiquity-G-ring}. Hence all local rings of $S$ are G-rings. Thus (5) holds. By Lemma \ref{lemma-coherent-defo-thy} 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 \ref{artin-section-axioms}. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas \ref{lemma-coherent-stack}, \ref{lemma-coherent-limits}, \ref{lemma-coherent-RS-star}, \ref{lemma-coherent-tangent-space}. Condition (3) follows from Lemma \ref{lemma-coherent-existence}. Finally, condition (1) is Lemma \ref{lemma-coherent-diagonal}. This finishes the proof of the theorem. \end{proof} \section{The stack of coherent sheaves in the non-flat case} \label{section-not-flat} \noindent In Theorem \ref{theorem-coherent-algebraic} the assumption that $f : X \to B$ is flat is not necessary. In this section we give a different proof which avoids the flatness assumption and avoids checking openness of versality by using the results in Flatness on Spaces, Section \ref{spaces-flat-section-existence} and Artin's Axioms, Section \ref{artin-section-strong-formal-effectiveness}. \medskip\noindent For a different approach to this problem the reader may wish to consult \cite{ArtinI} and follow the method discussed in the papers \cite{olsson-starr}, \cite{lieblich_remarks}, \cite{olsson_proper}, \cite{Hall-Rydh}, \cite{Hall-Rydh-Hilbert}, \cite{rydh_representability}. Some of these papers deal with the more general case of the stack of coherent sheaves on an algebraic stack over an algebraic stack and others deal with similar problems in the case of Hilbert stacks or Quot functors. Our strategy will be to show algebraicity of some cases of Hilbert stacks and Quot functors as a consequence of the algebraicity of the stack of coherent sheaves. \begin{theorem}[Algebraicity of stack coherent sheaves; general case] \label{theorem-coherent-algebraic-general} Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation and separated. Then $\textit{Coh}_{X/B}$ is an algebraic stack over $S$. \end{theorem} \begin{proof} Only the last step of the proof is different from the proof in the flat case, but we repeat all the arguments here to make sure everything works. \medskip\noindent Set $\mathcal{X} = \textit{Coh}_{X/B}$. We have seen that $\mathcal{X}$ is a stack in groupoids over $(\Sch/S)_{fppf}$ with diagonal representable by algebraic spaces (Lemmas \ref{lemma-coherent-stack} and \ref{lemma-coherent-diagonal}). Hence it suffices to find a scheme $W$ and a surjective and smooth morphism $W \to \mathcal{X}$. \medskip\noindent Let $B'$ be a scheme and let $B' \to B$ be a surjective \'etale morphism. Set $X' = B' \times_B X$ and denote $f' : X' \to B'$ the projection. Then $\mathcal{X}' = \textit{Coh}_{X'/B'}$ 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 \ref{remark-coherent-base-change}). By the material in Algebraic Stacks, Section \ref{algebraic-section-representable-properties} the morphism $\mathcal{X}' \to \mathcal{X}$ is surjective and \'etale. Hence it suffices to prove the result for $\mathcal{X}'$. In other words, we may assume $B$ is a scheme. \medskip\noindent Assume $B$ is a scheme. In this case we may replace $S$ by $B$, see Algebraic Stacks, Section \ref{algebraic-section-change-base-scheme}. Thus we may assume $S = B$. \medskip\noindent Assume $S = B$. Choose an affine open covering $S = \bigcup U_i$. Denote $\mathcal{X}_i$ the restriction of $\mathcal{X}$ to $(\Sch/U_i)_{fppf}$. 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. \medskip\noindent Assume $S = B$ is affine, say $S = \Spec(\Lambda)$. Write $\Lambda = \colim \Lambda_i$ as a filtered colimit with each $\Lambda_i$ of finite type over $\mathbf{Z}$. For some $i$ we can find a morphism of algebraic spaces $X_i \to \Spec(\Lambda_i)$ which is of finite presentation and whose base change to $\Lambda$ is $X$. See Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-finite-presentation}. If we show that $\textit{Coh}_{X_i/\Spec(\Lambda_i)}$ is an algebraic stack, then it follows by base change (Remark \ref{remark-coherent-base-change} and Algebraic Stacks, Section \ref{algebraic-section-change-base-scheme}) that $\mathcal{X}$ is an algebraic stack. Thus we may assume that $\Lambda$ is a finite type $\mathbf{Z}$-algebra. \medskip\noindent Assume $S = B = \Spec(\Lambda)$ 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 \ref{artin-lemma-diagonal-representable} to conclude that $\mathcal{X}$ is an algebraic stack. Note that $\Lambda$ is a G-ring, see More on Algebra, Proposition \ref{more-algebra-proposition-ubiquity-G-ring}. Hence all local rings of $S$ are G-rings. Thus (5) holds. To check (2) we have to verify axioms [-1], [0], [1], [2], and [3] of Artin's Axioms, Section \ref{artin-section-axioms}. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas \ref{lemma-coherent-stack}, \ref{lemma-coherent-limits}, \ref{lemma-coherent-RS-star}, \ref{lemma-coherent-tangent-space}. Condition (3) is Lemma \ref{lemma-coherent-existence}. Condition (1) is Lemma \ref{lemma-coherent-diagonal}. \medskip\noindent It remains to show condition (4) which is openness of versality. To see this we will use Artin's Axioms, Lemma \ref{artin-lemma-SGE-implies-openness-versality}. We have already seen that $\mathcal{X}$ has diagonal representable by algebraic spaces, has (RS*), and is limit preserving (see lemmas used above). Hence we only need to see that $\mathcal{X}$ satisfies the strong formal effectiveness formulated in Artin's Axioms, Lemma \ref{artin-lemma-SGE-implies-openness-versality}. This is Flatness on Spaces, Theorem \ref{spaces-flat-theorem-existence} and the proof is complete. \end{proof} \section{The functor of quotients} \label{section-functor-quotients} \noindent In this section we discuss some generalities regarding the functor $Q_{\mathcal{F}/X/B}$ defined below. The notation $\text{Quot}_{\mathcal{F}/X/B}$ is reserved for a subfunctor of $\text{Q}_{\mathcal{F}/X/B}$. We urge the reader to skip this section on a first reading. \begin{situation} \label{situation-q} Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. For any scheme $T$ over $B$ we will denote $X_T$ the base change of $X$ to $T$ and $\mathcal{F}_T$ the pullback of $\mathcal{F}$ via the projection morphism $X_T = X \times_B T \to X$. Given such a $T$ we set $$\text{Q}_{\mathcal{F}/X/B}(T) = \left\{ \begin{matrix} \text{quotients }\mathcal{F}_T \to \mathcal{Q}\text{ where } \mathcal{Q}\text{ is a}\\ \text{quasi-coherent } \mathcal{O}_{X_T}\text{-module flat over }T \end{matrix} \right\}$$ We identify quotients if they have the same kernel. Suppose that $T' \to T$ is a morphism of schemes over $B$ and $\mathcal{F}_T \to \mathcal{Q}$ is an element of $\text{Q}_{\mathcal{F}/X/B}(T)$. Then the pullback $\mathcal{Q}' = (X_{T'} \to X_T)^*\mathcal{Q}$ is a quasi-coherent $\mathcal{O}_{X_{T'}}$-module flat over $T'$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-module-flat}. Thus we obtain a functor \begin{equation} \label{equation-q} \text{Q}_{\mathcal{F}/X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets} \end{equation} This is the {\it functor of quotients of $\mathcal{F}/X/B$}. We define a subfunctor \begin{equation} \label{equation-q-fp} \text{Q}^{fp}_{\mathcal{F}/X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets} \end{equation} which assigns to $T$ the subset of $\text{Q}_{\mathcal{F}/X/B}(T)$ consisting of those quotients $\mathcal{F}_T \to \mathcal{Q}$ such that $\mathcal{Q}$ is of finite presentation as an $\mathcal{O}_{X_T}$-module. This is a subfunctor by Properties of Spaces, Section \ref{spaces-properties-section-properties-modules}. \end{situation} \noindent In Situation \ref{situation-q} we sometimes think of $\text{Q}_{\mathcal{F}/X/B}$ as a functor $(\Sch/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\text{Q}_{\mathcal{F}/X/S} \to B$. Namely, if $T$ is a scheme over $S$, then an element of $\text{Q}_{\mathcal{F}/X/B}(T)$ is a pair $(h, \mathcal{Q})$ where $h$ a morphism $h : T \to B$ and $\mathcal{Q}$ is a $T$-flat quotient $\mathcal{F}_T \to \mathcal{Q}$ of finite presentation on $X_T = X \times_{B, h} T$. In particular, when we say that $\text{Q}_{\mathcal{F}/X/S}$ is an algebraic space, we mean that the corresponding functor $(\Sch/S)^{opp} \to \textit{Sets}$ is an algebraic space. Similar remarks apply to $\text{Q}^{fp}_{\mathcal{F}/X/B}$. \begin{remark} \label{remark-q-base-change} In Situation \ref{situation-q} let $B' \to B$ be a morphism of algebraic spaces over $S$. Set $X' = X \times_B B'$ and denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X'$. Thus we have the functor $Q_{\mathcal{F}'/X'/B'}$ on the category of schemes over $B'$. For a scheme $T$ over $B'$ it is clear that we have $$Q_{\mathcal{F}'/X'/B'}(T) = Q_{\mathcal{F}/X/B}(T)$$ where on the right hand side we think of $T$ as a scheme over $B$ via the composition $T \to B' \to B$. Similar remarks apply to $\text{Q}^{fp}_{\mathcal{F}/X/B}$. These trivial remarks will occasionally be useful to change the base algebraic space. \end{remark} \begin{remark} \label{remark-q-sheaf} Let $S$ be a scheme, $X$ an algebraic space over $S$, and $\mathcal{F}$ a quasi-coherent $\mathcal{O}_X$-module. Suppose that $\{f_i : X_i \to X\}_{i \in I}$ is an fpqc covering and for each $i, j \in I$ we are given an fpqc covering $\{X_{ijk} \to X_i \times_X X_j\}$. In this situation we have a bijection $$\left\{ \begin{matrix} \text{quotients }\mathcal{F} \to \mathcal{Q}\text{ where } \\ \mathcal{Q}\text{ is a quasi-coherent }\\ \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{families of quotients }f_i^*\mathcal{F} \to \mathcal{Q}_i \text{ where } \\ \mathcal{Q}_i\text{ is quasi-coherent and } \mathcal{Q}_i\text{ and }\mathcal{Q}_j\\ \text{ restrict to the same quotient on }X_{ijk} \end{matrix} \right\}$$ Namely, let $(f_i^*\mathcal{F} \to \mathcal{Q}_i)_{i \in I}$ be an element of the right hand side. Then since $\{X_{ijk} \to X_i \times_X X_j\}$ is an fpqc covering we see that the pullbacks of $\mathcal{Q}_i$ and $\mathcal{Q}_j$ restrict to the same quotient of the pullback of $\mathcal{F}$ to $X_i \times_X X_j$ (by fully faithfulness in Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}). Hence we obtain a descent datum for quasi-coherent modules with respect to $\{X_i \to X\}_{i \in I}$. By Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} we find a map of quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F} \to \mathcal{Q}$ whose restriction to $X_i$ recovers the given maps $f_i^*\mathcal{F} \to \mathcal{Q}_i$. Since the family of morphisms $\{X_i \to X\}$ is jointly surjective and flat, for every point $x \in |X|$ there exists an $i$ and a point $x_i \in |X_i|$ mapping to $x$. Note that the induced map on local rings $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X_i, \overline{x_i}}$ is faithfully flat, see Morphisms of Spaces, Section \ref{spaces-morphisms-section-flat}. Thus we see that $\mathcal{F} \to \mathcal{Q}$ is surjective. \end{remark} \begin{lemma} \label{lemma-q-sheaf} In Situation \ref{situation-q}. The functors $\text{Q}_{\mathcal{F}/X/B}$ and $\text{Q}^{fp}_{\mathcal{F}/X/B}$ satisfy the sheaf property for the fpqc topology. \end{lemma} \begin{proof} Let $\{T_i \to T\}_{i \in I}$ be an fpqc covering of schemes over $S$. Set $X_i = X_{T_i} = X \times_S T_i$ and $\mathcal{F}_i = \mathcal{F}_{T_i}$. Note that $\{X_i \to X_T\}_{i \in I}$ is an fpqc covering of $X_T$ (Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-fpqc}) and that $X_{T_i \times_T T_{i'}} = X_i \times_{X_T} X_{i'}$. Suppose that $\mathcal{F}_i \to \mathcal{Q}_i$ is a collection of elements of $\text{Q}_{\mathcal{F}/X/B}(T_i)$ such that $\mathcal{Q}_i$ and $\mathcal{Q}_{i'}$ restrict to the same element of $\text{Q}_{\mathcal{F}/X/B}(T_i \times_T T_{i'})$. By Remark \ref{remark-q-sheaf} we obtain a surjective map of quasi-coherent $\mathcal{O}_{X_T}$-modules $\mathcal{F}_T \to \mathcal{Q}$ whose restriction to $X_i$ recovers the given quotients. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-permanence} we see that $\mathcal{Q}$ is flat over $T$. Finally, in the case of $\text{Q}^{fp}_{\mathcal{F}/X/B}$, i.e., if $\mathcal{Q}_i$ are of finite presentation, then Descent on Spaces, Lemma \ref{spaces-descent-lemma-finite-presentation-descends} guarantees that $\mathcal{Q}$ is of finite presentation as an $\mathcal{O}_{X_T}$-module. \end{proof} \noindent Sanity check: $\text{Q}_{\mathcal{F}/X/B}$, $\text{Q}^{fp}_{\mathcal{F}/X/B}$ play the same role among algebraic spaces over $S$. \begin{lemma} \label{lemma-extend-q-to-spaces} In Situation \ref{situation-q}. Let $T$ be an algebraic space over $S$. We have $$\Mor_{\Sh((\Sch/S)_{fppf})}(T, \text{Q}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}\\ \mathcal{Q}\text{ is quasi-coherent and flat over }T \end{matrix} \right\}$$ where $\mathcal{F}_T$ denotes the pullback of $\mathcal{F}$ to the algebraic space $X \times_{B, h} T$. Similarly, we have $$\Mor_{\Sh((\Sch/S)_{fppf})}(T, \text{Q}^{fp}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}\\ \mathcal{Q}\text{ is of finite presentation and flat over }T \end{matrix} \right\}$$ \end{lemma} \begin{proof} Choose a scheme $U$ and a surjective \'etale morphism $p : U \to T$. Let $R = U \times_T U$ with projections $t, s : R \to U$. \medskip\noindent Let $v : T \to \text{Q}_{\mathcal{F}/X/B}$ be a natural transformation. Then $v(p)$ corresponds to a pair $(h_U, \mathcal{F}_U \to \mathcal{Q}_U)$ over $U$. As $v$ is a transformation of functors we see that the pullbacks of $(h_U, \mathcal{F}_U \to \mathcal{Q}_U)$ by $s$ and $t$ agree. Since $T = U/R$ (Spaces, Lemma \ref{spaces-lemma-space-presentation}), we obtain a morphism $h : T \to B$ such that $h_U = h \circ p$. By Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} the quotient $\mathcal{Q}_U$ descends to a quotient $\mathcal{F}_T \to \mathcal{Q}$ over $X_T$. Since $U \to T$ is surjective and flat, it follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-permanence} that $\mathcal{Q}$ is flat over $T$. \medskip\noindent Conversely, let $(h, \mathcal{F}_T \to \mathcal{Q})$ be a pair over $T$. Then we get a natural transformation $v : T \to \text{Q}_{\mathcal{F}/X/B}$ by sending a morphism $a : T' \to T$ where $T'$ is a scheme to $(h \circ a, \mathcal{F}_{T'} \to a^*\mathcal{Q})$. We omit the verification that the construction of this and the previous paragraph are mutually inverse. \medskip\noindent In the case of $\text{Q}^{fp}_{\mathcal{F}/X/B}$ we add: given a morphism $h : T \to B$, a quasi-coherent sheaf on $X_T$ is of finite presentation as an $\mathcal{O}_{X_T}$-module if and only if the pullback to $X_U$ is of finite presentation as an $\mathcal{O}_{X_U}$-module. This follows from the fact that $X_U \to X_T$ is surjective and \'etale and Descent on Spaces, Lemma \ref{spaces-descent-lemma-finite-presentation-descends}. \end{proof} \begin{lemma} \label{lemma-q-sheaf-in-X} In Situation \ref{situation-q} let $\{X_i \to X\}_{i \in I}$ be an fpqc covering and for each $i, j \in I$ let $\{X_{ijk} \to X_i \times_X X_j\}$ be an fpqc covering. Denote $\mathcal{F}_i$, resp.\ $\mathcal{F}_{ijk}$ the pullback of $\mathcal{F}$ to $X_i$, resp.\ $X_{ijk}$. For every scheme $T$ over $B$ the diagram $$\xymatrix{ Q_{\mathcal{F}/X/B}(T) \ar[r] & \prod\nolimits_i Q_{\mathcal{F}_i/X_i/B}(T) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod\nolimits_{i, j, k} Q_{\mathcal{F}_{ijk}/X_{ijk}/B}(T) }$$ presents the first arrow as the equalizer of the other two. The same is true for the functor $\text{Q}^{fp}_{\mathcal{F}/X/B}$. \end{lemma} \begin{proof} Let $\mathcal{F}_{i, T} \to \mathcal{Q}_i$ be an element in the equalizer of $\text{pr}_0^*$ and $\text{pr}_1^*$. By Remark \ref{remark-q-sheaf} we obtain a surjection $\mathcal{F}_T \to \mathcal{Q}$ of quasi-coherent $\mathcal{O}_{X_T}$-modules whose restriction to $X_{i, T}$ recovers $\mathcal{F}_i \to \mathcal{Q}_i$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-permanence} we see that $\mathcal{Q}$ is flat over $T$ as desired. In the case of the functor $\text{Q}^{fp}_{\mathcal{F}/X/B}$, i.e., if $\mathcal{Q}_i$ is of finite presentation, then $\mathcal{Q}$ is of finite presentation too by Descent on Spaces, Lemma \ref{spaces-descent-lemma-finite-presentation-descends}. \end{proof} \begin{lemma} \label{lemma-q-limit-preserving} In Situation \ref{situation-q} assume also that (a) $f$ is quasi-compact and quasi-separated and (b) $\mathcal{F}$ is of finite presentation. Then the functor $\text{Q}^{fp}_{\mathcal{F}/X/B}$ is limit preserving in the following sense: If $T = \lim T_i$ is a directed limit of affine schemes over $B$, then $\text{Q}^{fp}_{\mathcal{F}/X/B}(T) = \colim \text{Q}^{fp}_{\mathcal{F}/X/B}(T_i)$. \end{lemma} \begin{proof} Let $T = \lim T_i$ be as in the statement of the lemma. Choose $i_0 \in I$ and replace $I$ by $\{i \in I \mid i \geq i_0\}$. We may set $B = S = T_{i_0}$ and we may replace $X$ by $X_{T_0}$ and $\mathcal{F}$ by the pullback to $X_{T_0}$. Then $X_T = \lim X_{T_i}$, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-directed-inverse-system-has-limit}. Let $\mathcal{F}_T \to \mathcal{Q}$ be an element of $\text{Q}^{fp}_{\mathcal{F}/X/B}(T)$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-modules-finite-presentation} there exists an $i$ and a map $\mathcal{F}_{T_i} \to \mathcal{Q}_i$ of $\mathcal{O}_{X_{T_i}}$-modules of finite presentation whose pullback to $X_T$ is the given quotient map. \medskip\noindent We still have to check that, after possibly increasing $i$, the map $\mathcal{F}_{T_i} \to \mathcal{Q}_i$ is surjective and $\mathcal{Q}_i$ is flat over $T_i$. To do this, choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). We may check surjectivity and flatness over $T_i$ after pulling back to the \'etale cover $U_{T_i} \to X_{T_i}$ (by definition). This reduces us to the case where $X = \Spec(B_0)$ is an affine scheme of finite presentation over $B = S = T_0 = \Spec(A_0)$. Writing $T_i = \Spec(A_i)$, then $T = \Spec(A)$ with $A = \colim A_i$ we have reached the following algebra problem. Let $M_i \to N_i$ be a map of finitely presented $B_0 \otimes_{A_0} A_i$-modules such that $M_i \otimes_{A_i} A \to N_i \otimes_{A_i} A$ is surjective and $N_i \otimes_{A_i} A$ is flat over $A$. Show that for some $i' \geq i$ $M_i \otimes_{A_i} A_{i'} \to N_i \otimes_{A_i} A_{i'}$ is surjective and $N_i \otimes_{A_i} A_{i'}$ is flat over $A$. The first follows from Algebra, Lemma \ref{algebra-lemma-module-map-property-in-colimit} and the second from Algebra, Lemma \ref{algebra-lemma-flat-finite-presentation-limit-flat}. \end{proof} \begin{lemma} \label{lemma-q-RS-star} In Situation \ref{situation-q}. Let $$\xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' }$$ be a pushout in the category of schemes over $B$ where $Z \to Z'$ is a thickening and $Z \to Y$ is affine, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. Then the natural map $$Q_{\mathcal{F}/X/B}(Y') \longrightarrow Q_{\mathcal{F}/X/B}(Y) \times_{Q_{\mathcal{F}/X/B}(Z)} Q_{\mathcal{F}/X/B}(Z')$$ is bijective. If $X \to B$ is locally of finite presentation, then the same thing is true for $Q^{fp}_{\mathcal{F}/X/B}$. \end{lemma} \begin{proof} Let us construct an inverse map. Namely, suppose we have $\mathcal{F}_Y \to \mathcal{A}$, $\mathcal{F}_{Z'} \to \mathcal{B}'$, and an isomorphism $\mathcal{A}|_{X_Z} \to \mathcal{B}'|_{X_Z}$ compatible with the given surjections. Then we apply Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-space-over-pushout-flat-modules} to get a quasi-coherent module $\mathcal{A}'$ on $X_{Y'}$ flat over $Y'$. Since this sheaf is constructed as a fibre product (see proof of cited lemma) there is a canonical map $\mathcal{F}_{Y'} \to \mathcal{A}'$. That this map is surjective can be seen because it factors as $$\begin{matrix} \mathcal{F}_{Y'} \\ \downarrow \\ (X_Y \to X_{Y'})_*\mathcal{F}_Y \times_{(X_Z \to X_{Y'})_*\mathcal{F}_Z} (X_{Z'} \to X_{Y'})_*\mathcal{F}_{Z'} \\ \downarrow \\ \mathcal{A}' = (X_Y \to X_{Y'})_*\mathcal{A} \times_{(X_Z \to X_{Y'})_*\mathcal{A}|_{X_Z}} (X_{Z'} \to X_{Y'})_*\mathcal{B}' \end{matrix}$$ and the first arrow is surjective by More on Algebra, Lemma \ref{more-algebra-lemma-module-over-fibre-product-bis} and the second by More on Algebra, Lemma \ref{more-algebra-lemma-surjection-module-over-fibre-product}. \medskip\noindent In the case of $Q^{fp}_{\mathcal{F}/X/B}$ all we have to show is that the construction above produces a finitely presented module. This is explained in More on Algebra, Remark \ref{more-algebra-remark-relative-modules-over-fibre-product} in the commutative algebra setting. The current case of modules over algebraic spaces follows from this by \'etale localization. \end{proof} \begin{remark}[Obstructions for quotients] \label{remark-q-obs} In Situation \ref{situation-q} {\bf assume} that $\mathcal{F}$ is flat over $B$. Let $T \subset T'$ be an first order thickening of schemes over $B$ with ideal sheaf $\mathcal{J}$. Then $X_T \subset X_{T'}$ is a first order thickening of algebraic spaces whose ideal sheaf $\mathcal{I}$ is a quotient of $f_T^*\mathcal{J}$. We will think of sheaves on $X_{T'}$, resp.\ $T'$ as sheaves on $X_T$, resp.\ $T$ using the fundamental equivalence described in More on Morphisms of Spaces, Section \ref{spaces-more-morphisms-section-thickenings}. Let $$0 \to \mathcal{K} \to \mathcal{F}_T \to \mathcal{Q} \to 0$$ define an element $x$ of $Q_{\mathcal{F}/X/B}(T)$. Since $\mathcal{F}_{T'}$ is flat over $T'$ we have a short exact sequence $$0 \to f_T^*\mathcal{J} \otimes_{\mathcal{O}_{X_T}} \mathcal{F}_T \xrightarrow{i} \mathcal{F}_{T'} \xrightarrow{\pi} \mathcal{F}_T \to 0$$ and we have $f_T^*\mathcal{J} \otimes_{\mathcal{O}_{X_T}} \mathcal{F}_T = \mathcal{I} \otimes_{\mathcal{O}_{X_T}} \mathcal{F}_T$, see Deformation Theory, Lemma \ref{defos-lemma-deform-module-ringed-topoi}. Let us use the abbreviation $f_T^*\mathcal{J} \otimes_{\mathcal{O}_{X_T}} \mathcal{G} = \mathcal{G} \otimes_{\mathcal{O}_T} \mathcal{J}$ for an $\mathcal{O}_{X_T}$-module $\mathcal{G}$. Since $\mathcal{Q}$ is flat over $T$, we obtain a short exact sequence $$0 \to \mathcal{K} \otimes_{\mathcal{O}_T} \mathcal{J} \to \mathcal{F}_T \otimes_{\mathcal{O}_T} \mathcal{J} \to \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J} \to \to 0$$ Combining the above we obtain an canonical extension $$0 \to \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J} \to \pi^{-1}(\mathcal{K})/i(\mathcal{K} \otimes_{\mathcal{O}_T} \mathcal{J}) \to \mathcal{K} \to 0$$ of $\mathcal{O}_{X_T}$-modules. This defines a canonical class $$o_x(T') \in \text{Ext}^1_{\mathcal{O}_{X_T}}(\mathcal{K}, \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J})$$ If $o_x(T')$ is zero, then we obtain a splitting of the short exact sequence defining it, in other words, we obtain a $\mathcal{O}_{X_{T'}}$-submodule $\mathcal{K}' \subset \pi^{-1}(\mathcal{K})$ sitting in a short exact sequence $0 \to \mathcal{K} \otimes_{\mathcal{O}_T} \mathcal{J} \to \mathcal{K}' \to \mathcal{K} \to 0$. Then it follows from the lemma reference above that $\mathcal{Q}' = \mathcal{F}_{T'}/\mathcal{K}'$ is a lift of $x$ to an element of $Q_{\mathcal{F}/X/B}(T')$. Conversely, the reader sees that the existence of a lift implies that $o_x(T')$ is zero. Moreover, if $x \in Q_{\mathcal{F}/X/B}^{fp}(T)$, then automatically $x' \in Q_{\mathcal{F}/X/B}^{fp}(T')$ by Deformation Theory, Lemma \ref{defos-lemma-deform-fp-module-ringed-topoi}. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi). \end{remark} \begin{remark}[Deformations of quotients] \label{remark-q-defos} In Situation \ref{situation-q} {\bf assume} that $\mathcal{F}$ is flat over $B$. We continue the discussion of Remark \ref{remark-q-obs}. Assume $o_x(T') = 0$. Then we claim that the set of lifts $x' \in Q_{\mathcal{F}/X/B}(T')$ is a principal homogeneous space under the group $$\Hom_{\mathcal{O}_{X_T}}(\mathcal{K}, \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J})$$ Namely, given any $\mathcal{F}_{T'} \to \mathcal{Q}'$ flat over $T'$ lifting the quotient $\mathcal{Q}$ we obtain a commutative diagram with exact rows and columns $$\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & \mathcal{K} \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{F}_T \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{Q} \otimes \mathcal{J} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}' \ar[r] \ar[d] & \mathcal{F}_{T'} \ar[r] \ar[d] & \mathcal{Q}' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[d] \ar[r] & \mathcal{F}_T \ar[d] \ar[r] & \mathcal{Q} \ar[d] \ar[r] & 0 \\ & 0 & 0 & 0 }$$ (to see this use the observations made in the previous remark). Given a map $\varphi : \mathcal{K} \to \mathcal{Q} \otimes \mathcal{J}$ we can consider the subsheaf $\mathcal{K}'_\varphi \subset \mathcal{F}_{T'}$ consisting of those local sections $s$ whose image in $\mathcal{F}_T$ is a local section $k$ of $\mathcal{K}$ and whose image in $\mathcal{Q}'$ is the local section $\varphi(k)$ of $\mathcal{Q} \otimes \mathcal{J}$. Then set $\mathcal{Q}'_\varphi = \mathcal{F}_{T'}/\mathcal{K}'_\varphi$. Conversely, any second lift of $x$ corresponds to one the qotients constructed in this manner. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi). \end{remark} \section{The Quot functor} \label{section-quot} \noindent In this section we prove the Quot functor is an algebraic space. \begin{situation} \label{situation-quot} 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. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. For any scheme $T$ over $B$ we will denote $X_T$ the base change of $X$ to $T$ and $\mathcal{F}_T$ the pullback of $\mathcal{F}$ via the projection morphism $X_T = X \times_S T \to X$. Given such a $T$ we set $$\text{Quot}_{\mathcal{F}/X/B}(T) = \left\{ \begin{matrix} \text{quotients }\mathcal{F}_T \to \mathcal{Q}\text{ where } \mathcal{Q}\text{ is a quasi-coherent }\\ \mathcal{O}_{X_T}\text{-module of finite presentation, flat over }T\\ \text{with support proper over }T \end{matrix} \right\}$$ By Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-base-change-module-support-proper-over-base} this is a subfunctor of the functor $Q^{fp}_{\mathcal{F}/X/B}$ we discussed in Section \ref{section-functor-quotients}. Thus we obtain a functor \begin{equation} \label{equation-quot} \text{Quot}_{\mathcal{F}/X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets} \end{equation} This is the {\it Quot functor} associated to $\mathcal{F}/X/B$. \end{situation} \noindent In Situation \ref{situation-quot} we sometimes think of $\text{Quot}_{\mathcal{F}/X/B}$ as a functor $(\Sch/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\text{Quot}_{\mathcal{F}/X/B} \to B$. Namely, if $T$ is a scheme over $S$, then an element of $\text{Quot}_{\mathcal{F}/X/B}(T)$ is a pair $(h, \mathcal{Q})$ where $h$ is a morphism $h : T \to B$ and $Q$ is a finitely presented, $T$-flat quotient $\mathcal{F}_T \to \mathcal{Q}$ on $X_T = X \times_{B, h} T$ with support proper over $T$. In particular, when we say that $\text{Quot}_{\mathcal{F}/X/B}$ is an algebraic space, we mean that the corresponding functor $(\Sch/S)^{opp} \to \textit{Sets}$ is an algebraic space. \begin{lemma} \label{lemma-quot-sheaf} In Situation \ref{situation-quot}. The functor $\text{Quot}_{\mathcal{F}/X/B}$ satisfies the sheaf property for the fpqc topology. \end{lemma} \begin{proof} In Lemma \ref{lemma-q-sheaf} we have seen that the functor $\text{Q}^{fp}_{\mathcal{F}/X/S}$ is a sheaf. Recall that for a scheme $T$ over $S$ the subset $\text{Quot}_{\mathcal{F}/X/S}(T) \subset \text{Q}_{\mathcal{F}/X/S}(T)$ picks out those quotients whose support is proper over $T$. This defines a subsheaf by the result of Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-proper} combined with Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-pullback-support} which shows that taking scheme theoretic support commutes with flat base change. \end{proof} \noindent Sanity check: $\text{Quot}_{\mathcal{F}/X/B}$ plays the same role among algebraic spaces over $S$. \begin{lemma} \label{lemma-extend-quot-to-spaces} In Situation \ref{situation-quot}. Let $T$ be an algebraic space over $S$. We have $$\Mor_{\Sh((\Sch/S)_{fppf})}(T, \text{Quot}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}\\ \mathcal{Q}\text{ is of finite presentation and}\\ \text{flat over }T\text{ with support proper over }T \end{matrix} \right\}$$ where $\mathcal{F}_T$ denotes the pullback of $\mathcal{F}$ to the algebraic space $X \times_{B, h} T$. \end{lemma} \begin{proof} Observe that the left and right hand side of the equality are subsets of the left and right hand side of the second equality in Lemma \ref{lemma-extend-q-to-spaces}. To see that these subsets correspond under the identification given in the proof of that lemma it suffices to show: given $h : T \to B$, a surjective \'etale morphism $U \to T$, a finite type quasi-coherent $\mathcal{O}_{X_T}$-module $\mathcal{Q}$ the following are equivalent \begin{enumerate} \item the scheme theoretic support of $\mathcal{Q}$ is proper over $T$, and \item the scheme theoretic support of $(X_U \to X_T)^*\mathcal{Q}$ is proper over $U$. \end{enumerate} This follows from Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-proper} combined with Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-pullback-support} which shows that taking scheme theoretic support commutes with flat base change. \end{proof} \begin{proposition} \label{proposition-quot} Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. If $f$ is of finite presentation and separated, then $\text{Quot}_{\mathcal{F}/X/B}$ is an algebraic space. If $\mathcal{F}$ is of finite presentation, then $\text{Quot}_{\mathcal{F}/X/B} \to B$ is locally of finite presentation. \end{proposition} \begin{proof} By Lemma \ref{lemma-quot-sheaf} we have that $\text{Quot}_{\mathcal{F}/X/B}$ is a sheaf in the fppf topology. Let $\textit{Quot}_{\mathcal{F}/X/B}$ be the stack in groupoids corresponding to $\text{Quot}_{\mathcal{F}/X/S}$, see Algebraic Stacks, Section \ref{algebraic-section-split}. By Algebraic Stacks, Proposition \ref{algebraic-proposition-algebraic-stack-no-automorphisms} it suffices to show that $\textit{Quot}_{\mathcal{F}/X/B}$ is an algebraic stack. Consider the $1$-morphism of stacks in groupoids $$\textit{Quot}_{\mathcal{F}/X/S} \longrightarrow \textit{Coh}_{X/B}$$ on $(\Sch/S)_{fppf}$ which associates to the quotient $\mathcal{F}_T \to \mathcal{Q}$ the coherent sheaf $\mathcal{Q}$. By Theorem \ref{theorem-coherent-algebraic-general} we know that $\textit{Coh}_{X/B}$ is an algebraic stack. By Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-morphism-to-algebraic} it suffices to show that this $1$-morphism is representable by algebraic spaces. \medskip\noindent Let $T$ be a scheme over $S$ and let the object $(h, \mathcal{G})$ of $\textit{Coh}_{X/B}$ over $T$ correspond to a $1$-morphism $\xi : (\Sch/T)_{fppf} \to \textit{Coh}_{X/B}$. The $2$-fibre product $$\mathcal{Z} = (\Sch/T)_{fppf} \times_{\xi, \textit{Coh}_{X/B}} \textit{Quot}_{\mathcal{F}/X/S}$$ is a stack in setoids, see Stacks, Lemma \ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids}. The corresponding sheaf of sets (i.e., functor, see Stacks, Lemmas \ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids} and \ref{stacks-lemma-when-stack-in-sets}) assigns to a scheme $T'/T$ the set of surjections $u : \mathcal{F}_{T'} \to \mathcal{G}_{T'}$ of quasi-coherent modules on $X_{T'}$. Thus we see that $\mathcal{Z}$ is representable by an open subspace (by Flatness on Spaces, Lemma \ref{spaces-flat-lemma-F-surj-open}) of the algebraic space $\mathit{Hom}(\mathcal{F}_T, \mathcal{G})$ from Proposition \ref{proposition-hom}. \end{proof} \begin{remark}[Quot via Artin's axioms] \label{remark-quot-via-artins-axioms} Let $S$ be a Noetherian scheme all of whose local rings are G-rings. Let $X$ be an algebraic space over $S$ whose structure morphism $f : X \to S$ is of finite presentation and separated. Let $\mathcal{F}$ be a finitely presented quasi-coherent sheaf on $X$ flat over $S$. In this remark we sketch how one can use Artin's axioms to prove that $\text{Quot}_{\mathcal{F}/X/S}$ is an algebraic space locally of finite presentation over $S$ and avoid using the algebraicity of the stack of coherent sheaves as was done in the proof of Proposition \ref{proposition-quot}. \medskip\noindent We check the conditions listed in Artin's Axioms, Proposition \ref{artin-proposition-spaces-diagonal-representable}. Representability of the diagonal of $\text{Quot}_{\mathcal{F}/X/S}$ can be seen as follows: suppose we have two quotients $\mathcal{F}_T \to \mathcal{Q}_i$, $i = 1, 2$. Denote $\mathcal{K}_1$ the kernel of the first one. Then we have to show that the locus of $T$ over which $u : \mathcal{K}_1 \to \mathcal{Q}_2$ becomes zero is representable. This follows for example from Flatness on Spaces, Lemma \ref{spaces-flat-lemma-F-zero-closed-proper} or from a discussion of the $\mathit{Hom}$ sheaf earlier in this chapter. Axioms [0] (sheaf), [1] (limits), [2] (Rim-Schlessinger) follow from Lemmas \ref{lemma-quot-sheaf}, \ref{lemma-q-limit-preserving}, and \ref{lemma-q-RS-star} (plus some extra work to deal with the properness condition). Axiom [3] (finite dimensionality of tangent spaces) follows from the description of the infinitesimal deformations in Remark \ref{remark-q-defos} and finiteness of cohomology of coherent sheaves on proper algebraic spaces over fields (Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-pushforward-coherent}). Axiom [4] (effectiveness of formal objects) follows from Grothendieck's existence theorem (More on Morphisms of Spaces, Theorem \ref{spaces-more-morphisms-theorem-grothendieck-existence}). As usual, the trickiest to verify is axiom [5] (openness of versality). One can for example use the obstruction theory described in Remark \ref{remark-q-obs} and the description of deformations in Remark \ref{remark-q-defos} to do this using the criterion in Artin's Axioms, Lemma \ref{artin-lemma-get-openness-obstruction-theory}. Please compare with the second proof of Lemma \ref{lemma-coherent-defo-thy}. \end{remark} \section{The Hilbert functor} \label{section-hilb} \noindent In this section we prove the Hilb functor is an algebraic space. \begin{situation} \label{situation-hilb} 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. For any scheme $T$ over $B$ we will denote $X_T$ the base change of $X$ to $T$. Given such a $T$ we set $$\text{Hilb}_{X/B}(T) = \left\{ \begin{matrix} \text{closed subspaces }Z \subset X_T\text{ such that }Z \to T\\ \text{is of finite presentation, flat, and proper} \end{matrix} \right\}$$ Since base change preserves the required properties (Spaces, Lemma \ref{spaces-lemma-base-change-immersions} and Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-finite-presentation}, \ref{spaces-morphisms-lemma-base-change-flat}, and \ref{spaces-morphisms-lemma-base-change-proper}) we obtain a functor \begin{equation} \label{equation-hilb} \text{Hilb}_{X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets} \end{equation} This is the {\it Hilbert functor} associated to $X/B$. \end{situation} \noindent In Situation \ref{situation-hilb} we sometimes think of $\text{Hilb}_{X/B}$ as a functor $(\Sch/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\text{Hilb}_{X/S} \to B$. Namely, if $T$ is a scheme over $S$, then an element of $\text{Hilb}_{X/B}(T)$ is a pair $(h, Z)$ where $h$ is a morphism $h : T \to B$ and $Z \subset X_T = X \times_{B, h} T$ is a closed subscheme, flat, proper, and of finite presentation over $T$. In particular, when we say that $\text{Hilb}_{X/B}$ is an algebraic space, we mean that the corresponding functor $(\Sch/S)^{opp} \to \textit{Sets}$ is an algebraic space. \medskip\noindent Of course the Hilbert functor is just a special case of the Quot functor. \begin{lemma} \label{lemma-hilb-is-quot} In Situation \ref{situation-hilb} we have $\text{Hilb}_{X/B} = \text{Quot}_{\mathcal{O}_X/X/B}$. \end{lemma} \begin{proof} Let $T$ be a scheme over $B$. Given an element $Z \in \text{Hilb}_{X/B}(T)$ we can consider the quotient $\mathcal{O}_{X_T} \to i_*\mathcal{O}_Z$ where $i : Z \to X_T$ is the inclusion morphism. Note that $i_*\mathcal{O}_Z$ is quasi-coherent. Since $Z \to T$ and $X_T \to T$ are of finite presentation, we see that $i$ is of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-permanence}), hence $i_*\mathcal{O}_Z$ is an $\mathcal{O}_{X_T}$-module of finite presentation (Descent on Spaces, Lemma \ref{spaces-descent-lemma-finite-finitely-presented-module}). Since $Z \to T$ is proper we see that $i_*\mathcal{O}_Z$ has support proper over $T$ (as defined in Derived Categories of Spaces, Section \ref{spaces-perfect-section-proper-over-base}). Since $\mathcal{O}_Z$ is flat over $T$ and $i$ is affine, we see that $i_*\mathcal{O}_Z$ is flat over $T$ (small argument omitted). Hence $\mathcal{O}_{X_T} \to i_*\mathcal{O}_Z$ is an element of $\text{Quot}_{\mathcal{O}_X/X/B}(T)$. \medskip\noindent Conversely, given an element $\mathcal{O}_{X_T} \to \mathcal{Q}$ of $\text{Quot}_{\mathcal{O}_X/X/B}(T)$, we can consider the closed immersion $i : Z \to X_T$ corresponding to the quasi-coherent ideal sheaf $\mathcal{I} = \Ker(\mathcal{O}_{X_T} \to \mathcal{Q})$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}). By construction of $Z$ we see that $\mathcal{Q} = i_*\mathcal{O}_Z$. Then we can read the arguments given above backwards to see that $Z$ defines an element of $\text{Hilb}_{X/B}(T)$. For example, $\mathcal{I}$ is quasi-coherent of finite type (Modules on Sites, Lemma \ref{sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation}) hence $i : Z \to X_T$ is of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}) hence $Z \to T$ is of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-finite-presentation}). Properness of $Z \to T$ follows from the discussion in Derived Categories of Spaces, Section \ref{spaces-perfect-section-proper-over-base}. Flatness of $Z \to T$ follows from flatness of $\mathcal{Q}$ over $T$. \medskip\noindent We omit the (immediate) verification that the two constructions given above are mutually inverse. \end{proof} \noindent Sanity check: $\mathit{Hilb}_{X/B}$ sheaf plays the same role among algebraic spaces over $S$. \begin{lemma} \label{lemma-extend-hilb-to-spaces} In Situation \ref{situation-hilb}. Let $T$ be an algebraic space over $S$. We have $$\Mor_{\Sh((\Sch/S)_{fppf})}(T, \mathit{Hilb}_{X/B}) = \left\{ \begin{matrix} (h, Z)\text{ where }h : T \to B,\ Z \subset X_T \\ \text{finite presentation, flat, proper over }T \end{matrix} \right\}$$ where $X_T = X \times_{B, h} T$. \end{lemma} \begin{proof} By Lemma \ref{lemma-hilb-is-quot} we have $\text{Hilb}_{X/B} = \text{Quot}_{\mathcal{O}_X/X/B}$. Thus we can apply Lemma \ref{lemma-extend-quot-to-spaces} to see that the left hand side is bijective with the set of surjections $\mathcal{O}_{X_T} \to \mathcal{Q}$ which are finitely presented, flat over $T$, and have support proper over $T$. Arguing exactly as in the proof of Lemma \ref{lemma-hilb-is-quot} we see that such quotients correspond exactly to the closed immersions $Z \to X_T$ such that $Z \to T$ is proper, flat, and of finite presentation. \end{proof} \begin{proposition} \label{proposition-hilb} Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. If $f$ is of finite presentation and separated, then $\text{Hilb}_{X/B}$ is an algebraic space locally of finite presentation over $B$. \end{proposition} \begin{proof} Immediate consequence of Lemma \ref{lemma-hilb-is-quot} and Proposition \ref{proposition-quot}. \end{proof} \section{The Picard stack} \label{section-picard-stack} \noindent The Picard stack for a morphism of algebraic spaces was introduced in Examples of Stacks, Section \ref{examples-stacks-section-picard-stack}. We will deduce it is an open substack of the stack of coherent sheaves (in good cases) from the following lemma. \begin{lemma} \label{lemma-picard-stack-open-in-coh} Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$ which is flat, of finite presentation, and proper. Then natural map $$\textit{Pic}_{X/B} \longrightarrow \textit{Coh}_{X/B}$$ is representable by open immersions. \end{lemma} \begin{proof} Observe that the map simply sends a triple $(T, g, \mathcal{L})$ as in Examples of Stacks, Section \ref{examples-stacks-section-picard-stack} to the same triple $(T, g, \mathcal{L})$ but where now we view this as a triple of the kind described in Situation \ref{situation-coherent}. This works because the invertible $\mathcal{O}_{X_T}$-module $\mathcal{L}$ is certainly a finitely presented $\mathcal{O}_{X_T}$-module, it is flat over $T$ because $X_T \to T$ is flat, and the support is proper over $T$ as $X_T \to T$ is proper (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-flat} and \ref{spaces-morphisms-lemma-base-change-proper}). Thus the statement makes sense. \medskip\noindent Having said this, it is clear that the content of the lemma is the following: given an object $(T, g, \mathcal{F})$ of $\textit{Coh}_{X/B}$ there is an open subscheme $U \subset T$ such that for a morphism of schemes $T' \to T$ the following are equivalent \begin{enumerate} \item[(a)] $T' \to T$ factors through $U$, \item[(b)] the pullback $\mathcal{F}_{T'}$ of $\mathcal{F}$ by $X_{T'} \to X_T$ is invertible. \end{enumerate} Let $W \subset |X_T|$ be the set of points $x \in |X_T|$ such that $\mathcal{F}$ is locally free in a neighbourhood of $x$. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-finite-free-open}. $W$ is open and formation of $W$ commutes with arbitrary base change. Clearly, if $T' \to T$ satisfies (b), then $|X_{T'}| \to |X_T|$ maps into $W$. Hence we may replace $T$ by the open $T \setminus f_T(|X_T| \setminus W)$ in order to construct $U$. After doing so we reach the situation where $\mathcal{F}$ is finite locally free. In this case we get a disjoint union decomposition $X_T = X_0 \amalg X_1 \amalg X_2 \amalg \ldots$ into open and closed subspaces such that the restriction of $\mathcal{F}$ is locally free of rank $i$ on $X_i$. Then clearly $$U = T \setminus f_T(|X_0| \cup |X_2| \cup |X_3| \cup \ldots )$$ works. (Note that if we assume that $T$ is quasi-compact, then $X_T$ is quasi-compact hence only a finite number of $X_i$ are nonempty and so $U$ is indeed open.) \end{proof} \begin{proposition} \label{proposition-pic} Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. If $f$ is flat, of finite presentation, and proper, then $\textit{Pic}_{X/B}$ is an algebraic stack. \end{proposition} \begin{proof} Immediate consequence of Lemma \ref{lemma-picard-stack-open-in-coh}, Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-morphism-to-algebraic} and either Theorem \ref{theorem-coherent-algebraic} or Theorem \ref{theorem-coherent-algebraic-general} \end{proof} \section{The Picard functor} \label{section-picard-functor} \noindent In this section we revisit the Picard functor discussed in Picard Schemes of Curves, Section \ref{pic-section-picard-functor}. The discussion will be more general as we want to study the Picard functor of a morphism of algebraic spaces as in the section on the Picard stack, see Section \ref{section-picard-stack}. \medskip\noindent Let $S$ be a scheme and let $X$ be an algebraic space over $S$. An invertible sheaf on $X$ is an invertible $\mathcal{O}_X$-module on $X_\etale$, see Modules on Sites, Definition \ref{sites-modules-definition-invertible-sheaf}. The group of isomorphism classes of invertible modules is denoted $\text{Pic}(X)$, see Modules on Sites, Definition \ref{sites-modules-definition-pic}. Given a morphism $f : X \to Y$ of algebraic spaces over $S$ pullback defines a group homomorphism $\text{Pic}(Y) \to \text{Pic}(X)$. The assignment $X \leadsto \text{Pic}(X)$ is a contravariant functor from the category of schemes to the category of abelian groups. This functor is not representable, but it turns out that a relative variant of this construction sometimes is representable. \begin{situation} \label{situation-pic} Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. We define $$\text{Pic}_{X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets}$$ as the fppf sheafification of the functor which to a scheme $T$ over $B$ associates the group $\text{Pic}(X_T)$. \end{situation} \noindent In Situation \ref{situation-pic} we sometimes think of $\text{Pic}_{X/B}$ as a functor $(\Sch/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\text{Pic}_{X/B} \to B$. In this point of view, we define $\text{Pic}_{X/B}$ to be the fppf sheafification of the functor $$T/S \longmapsto \{(h, \mathcal{L}) \mid h : T \to B,\ \mathcal{L} \in \text{Pic}(X \times_{B, h} T)\}$$ In particular, when we say that $\text{Pic}_{X/B}$ is an algebraic space, we mean that the corresponding functor $(\Sch/S)^{opp} \to \textit{Sets}$ is an algebraic space. \medskip\noindent An often used remark is that if $T$ is a scheme over $B$, then $\text{Pic}_{X_T/T}$ is the restriction of $\text{Pic}_{X/B}$ to $(\Sch/T)_{fppf}$. \begin{lemma} \label{lemma-pic-over-pic} In Situation \ref{situation-pic} the functor $\text{Pic}_{X/B}$ is the sheafification of the functor $T \mapsto \Ob(\textit{Pic}_{X/B, T})/\cong$. \end{lemma} \begin{proof} Since the fibre category $\textit{Pic}_{X/B, T}$ of the Picard stack $\textit{Pic}_{X/B}$ over $T$ is the category of invertible sheaves on $X_T$ (see Section \ref{section-picard-stack} and Examples of Stacks, Section \ref{examples-stacks-section-picard-stack}) this is immediate from the definitions. \end{proof} \noindent It turns out to be nontrivial to see what the value of $\text{Pic}_{X/B}$ is on schemes $T$ over $B$. Here is a lemma that helps with this task. \begin{lemma} \label{lemma-flat-geometrically-connected-fibres} In Situation \ref{situation-pic}. If $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all schemes $T$ over $B$, then $$0 \to \text{Pic}(T) \to \text{Pic}(X_T) \to \text{Pic}_{X/B}(T)$$ is an exact sequence for all $T$. \end{lemma} \begin{proof} We may replace $B$ by $T$ and $X$ by $X_T$ and assume that $B = T$ to simplify the notation. Let $\mathcal{N}$ be an invertible $\mathcal{O}_B$-module. If $f^*\mathcal{N} \cong \mathcal{O}_X$, then we see that $f_*f^*\mathcal{N} \cong f_*\mathcal{O}_X \cong \mathcal{O}_B$ by assumption. Since $\mathcal{N}$ is locally trivial, we see that the canonical map $\mathcal{N} \to f_*f^*\mathcal{N}$ is locally an isomorphism (because $\mathcal{O}_B \to f_*f^*\mathcal{O}_B$ is an isomorphism by assumption). Hence we conclude that $\mathcal{N} \to f_*f^*\mathcal{N} \to \mathcal{O}_B$ is an isomorphism and we see that $\mathcal{N}$ is trivial. This proves the first arrow is injective. \medskip\noindent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module which is in the kernel of $\text{Pic}(X) \to \text{Pic}_{X/B}(B)$. Then there exists an fppf covering $\{B_i \to B\}$ such that $\mathcal{L}$ pulls back to the trivial invertible sheaf on $X_{B_i}$. Choose a trivializing section $s_i$. Then $\text{pr}_0^*s_i$ and $\text{pr}_1^*s_j$ are both trivialising sections of $\mathcal{L}$ over $X_{B_i \times_B B_j}$ and hence differ by a multiplicative unit $$f_{ij} \in \Gamma(X_{S_i \times_B B_j}, \mathcal{O}_{X_{B_i \times_B B_j}}^*) = \Gamma(B_i \times_B B_j, \mathcal{O}_{B_i \times_N B_j}^*)$$ (equality by our assumption on pushforward of structure sheaves). Of course these elements satisfy the cocycle condition on $B_i \times_B B_j \times_B B_k$, hence they define a descent datum on invertible sheaves for the fppf covering $\{B_i \to B\}$. By Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent} there is an invertible $\mathcal{O}_B$-module $\mathcal{N}$ with trivializations over $B_i$ whose associated descent datum is $\{f_{ij}\}$. (The proposition applies because $B$ is a scheme by the replacement performed at the start of the proof.) Then $f^*\mathcal{N} \cong \mathcal{L}$ as the functor from descent data to modules is fully faithful. \end{proof} \begin{lemma} \label{lemma-flat-geometrically-connected-fibres-with-section} In Situation \ref{situation-pic} let $\sigma : B \to X$ be a section. Assume that $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all $T$ over $B$. Then $$0 \to \text{Pic}(T) \to \text{Pic}(X_T) \to \text{Pic}_{X/B}(T) \to 0$$ is a split exact sequence with splitting given by $\sigma_T^* : \text{Pic}(X_T) \to \text{Pic}(T)$. \end{lemma} \begin{proof} Denote $K(T) = \Ker(\sigma_T^* : \text{Pic}(X_T) \to \text{Pic}(T))$. Since $\sigma$ is a section of $f$ we see that $\text{Pic}(X_T)$ is the direct sum of $\text{Pic}(T)$ and $K(T)$. Thus by Lemma \ref{lemma-flat-geometrically-connected-fibres} we see that $K(T) \subset \text{Pic}_{X/B}(T)$ for all $T$. Moreover, it is clear from the construction that $\text{Pic}_{X/B}$ is the sheafification of the presheaf $K$. To finish the proof it suffices to show that $K$ satisfies the sheaf condition for fppf coverings which we do in the next paragraph. \medskip\noindent Let $\{T_i \to T\}$ be an fppf covering. Let $\mathcal{L}_i$ be elements of $K(T_i)$ which map to the same elements of $K(T_i \times_T T_j)$ for all $i$ and $j$. Choose an isomorphism $\alpha_i : \mathcal{O}_{T_i} \to \sigma_{T_i}^*\mathcal{L}_i$ for all $i$. Choose an isomorphism $$\varphi_{ij} : \mathcal{L}_i|_{X_{T_i \times_T T_j}} \longrightarrow \mathcal{L}_j|_{X_{T_i \times_T T_j}}$$ If the map $$\alpha_j|_{T_i \times_T T_j} \circ \sigma_{T_i \times_T T_j}^*\varphi_{ij} \circ \alpha_i|_{T_i \times_T T_j} : \mathcal{O}_{T_i \times_T T_j} \to \mathcal{O}_{T_i \times_T T_j}$$ is not equal to multiplication by $1$ but some $u_{ij}$, then we can scale $\varphi_{ij}$ by $u_{ij}^{-1}$ to correct this. Having done this, consider the self map $$\varphi_{ki}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{jk}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{ij}|_{X_{T_i \times_T T_j \times_T T_k}} \quad\text{on}\quad \mathcal{L}_i|_{X_{T_i \times_T T_j \times_T T_k}}$$ which is given by multiplication by some section $f_{ijk}$ of the structure sheaf of $X_{T_i \times_T T_j \times_T T_k}$. By our choice of $\varphi_{ij}$ we see that the pullback of this map by $\sigma$ is equal to multiplication by $1$. By our assumption on functions on $X$, we see that $f_{ijk} = 1$. Thus we obtain a descent datum for the fppf covering $\{X_{T_i} \to X\}$. By Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} there is an invertible $\mathcal{O}_{X_T}$-module $\mathcal{L}$ and an isomorphism $\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}$ whose pullback to $X_{T_i}$ recovers $(\mathcal{L}_i, \alpha_i)$ (small detail omitted). Thus $\mathcal{L}$ defines an object of $K(T)$ as desired. \end{proof} \noindent In Situation \ref{situation-pic} let $\sigma : B \to X$ be a section. We denote $\textit{Pic}_{X/B, \sigma}$ the category defined as follows: \begin{enumerate} \item An object is a quadruple $(T, h, \mathcal{L}, \alpha)$, where $(T, h, \mathcal{L})$ is an object of $\textit{Pic}_{X/B}$ over $T$ and $\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}$ is an isomorphism. \item A morphism $(g, \varphi) : (T, h, \mathcal{L}, \alpha) \to (T', h', \mathcal{L}', \alpha')$ is given by a morphism of schemes $g : T \to T'$ with $h = h' \circ g$ and an isomorphism $\varphi : (g')^*\mathcal{L}' \to \mathcal{L}$ such that $\sigma_T^*\varphi \circ g^*\alpha' = \alpha$. Here $g' : X_{T'} \to X_T$ is the base change of $g$. \end{enumerate} There is a natural faithful forgetful functor $$\textit{Pic}_{X/B, \sigma} \longrightarrow \textit{Pic}_{X/B}$$ In this way we view $\textit{Pic}_{X/B, \sigma}$ as a category over $(\Sch/S)_{fppf}$. \begin{lemma} \label{lemma-pic-with-section-stack} In Situation \ref{situation-pic} let $\sigma : B \to X$ be a section. Then $\textit{Pic}_{X/B, \sigma}$ as defined above is a stack in groupoids over $(\Sch/S)_{fppf}$. \end{lemma} \begin{proof} We already know that $\textit{Pic}_{X/B}$ is a stack in groupoids over $(\Sch/S)_{fppf}$ by Examples of Stacks, Lemma \ref{examples-stacks-lemma-picard-stack}. Let us show descent for objects for $\textit{Pic}_{X/B, \sigma}$. Let $\{T_i \to T\}$ be an fppf covering and let $\xi_i = (T_i, h_i, \mathcal{L}_i, \alpha_i)$ be an object of $\textit{Pic}_{X/B, \sigma}$ lying over $T_i$, and let $\varphi_{ij} : \text{pr}_0^*\xi_i \to \text{pr}_1^*\xi_j$ be a descent datum. Applying the result for $\textit{Pic}_{X/B}$ we see that we may assume we have an object $(T, h, \mathcal{L})$ of $\textit{Pic}_{X/B}$ over $T$ which pulls back to $\xi_i$ for all $i$. Then we get $$\alpha_i : \mathcal{O}_{T_i} \to \sigma_{T_i}^*\mathcal{L}_i = (T_i \to T)^*\sigma_T^*\mathcal{L}$$ Since the maps $\varphi_{ij}$ are compatible with the $\alpha_i$ we see that $\alpha_i$ and $\alpha_j$ pullback to the same map on $T_i \times_T T_j$. By descent of quasi-coherent sheaves (Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}, we see that the $\alpha_i$ are the restriction of a single map $\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}$ as desired. We omit the proof of descent for morphisms. \end{proof} \begin{lemma} \label{lemma-compare-pic-with-section} In Situation \ref{situation-pic} let $\sigma : B \to X$ be a section. The morphism $\textit{Pic}_{X/B, \sigma} \to \textit{Pic}_{X/B}$ is representable, surjective, and smooth. \end{lemma} \begin{proof} Let $T$ be a scheme and let $(\Sch/T)_{fppf} \to \textit{Pic}_{X/B}$ be given by the object $\xi = (T, h, \mathcal{L})$ of $\textit{Pic}_{X/B}$ over $T$. We have to show that $$(\Sch/T)_{fppf} \times_{\xi, \textit{Pic}_{X/B}} \textit{Pic}_{X/B, \sigma}$$ is representable by a scheme $V$ and that the corresponding morphism $V \to T$ is surjective and smooth. See Algebraic Stacks, Sections \ref{algebraic-section-representable-morphism}, \ref{algebraic-section-morphisms-representable-by-algebraic-spaces}, and \ref{algebraic-section-representable-properties}. The forgetful functor $\textit{Pic}_{X/B, \sigma} \to \textit{Pic}_{X/B}$ is faithful on fibre categories and for $T'/T$ the set of isomorphism classes is the set of isomorphisms $$\alpha' : \mathcal{O}_{T'} \longrightarrow (T' \to T)^*\sigma_T^*\mathcal{L}$$ See Algebraic Stacks, Lemma \ref{algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids}. We know this functor is representable by an affine scheme $U$ of finite presentation over $T$ by Proposition \ref{proposition-isom} (applied to $\text{id} : T \to T$ and $\mathcal{O}_T$ and $\sigma^*\mathcal{L}$). Working Zariski locally on $T$ we may assume that $\sigma_T^*\mathcal{L}$ is isomorphic to $\mathcal{O}_T$ and then we see that our functor is representable by $\mathbf{G}_m \times T$ over $T$. Hence $U \to T$ Zariski locally on $T$ looks like the projection $\mathbf{G}_m \times T \to T$ which is indeed smooth and surjective. \end{proof} \begin{lemma} \label{lemma-flat-geometrically-connected-fibres-with-section-functor-stack} In Situation \ref{situation-pic} let $\sigma : B \to X$ be a section. If $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all $T$ over $B$, then $\textit{Pic}_{X/B, \sigma} \to (\Sch/S)_{fppf}$ is fibred in setoids with set of isomorphism classes over $T$ given by $$\coprod\nolimits_{h : T \to B} \Ker(\sigma_T^* : \text{Pic}(X \times_{B, h} T) \to \text{Pic}(T))$$ \end{lemma} \begin{proof} If $\xi = (T, h, \mathcal{L}, \alpha)$ is an object of $\textit{Pic}_{X/B, \sigma}$ over $T$, then an automorphism $\varphi$ of $\xi$ is given by multiplication with an invertible global section $u$ of the structure sheaf of $X_T$ such that moreover $\sigma_T^*u = 1$. Then $u = 1$ by our assumption that $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism. Hence $\textit{Pic}_{X/B, \sigma}$ is fibred in setoids over $(\Sch/S)_{fppf}$. Given $T$ and $h : T \to B$ the set of isomorphism classes of pairs $(\mathcal{L}, \alpha)$ is the same as the set of isomorphism classes of $\mathcal{L}$ with $\sigma_T^*\mathcal{L} \cong \mathcal{O}_T$ (isomorphism not specified). This is clear because any two choices of $\alpha$ differ by a global unit on $T$ and this is the same thing as a global unit on $X_T$. \end{proof} \begin{proposition} \label{proposition-pic-functor} Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume that \begin{enumerate} \item $f$ is flat, of finite presentation, and proper, and \item $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all schemes $T$ over $B$. \end{enumerate} Then $\text{Pic}_{X/B}$ is an algebraic space. \end{proposition} \noindent In the situation of the proposition the algebraic stack $\textit{Pic}_{X/B}$ is a gerbe over the algebraic space $\text{Pic}_{X/B}$. After developing the general theory of gerbes, this provides a shorter proof of the proposition (but using more general theory). \begin{proof} There exists a surjective, flat, finitely presented morphism $B' \to B$ of algebraic spaces such that the base change $X' = X \times_B B'$ over $B'$ has a section: namely, we can take $B' = X$. Observe that $\text{Pic}_{X'/B'} = B' \times_B \text{Pic}_{X/B}$. Hence $\text{Pic}_{X'/B'} \to \text{Pic}_{X/B}$ is representable by algebraic spaces, surjective, flat, and finitely presented. Hence, if we can show that $\text{Pic}_{X'/B'}$ is an algebraic space, then it follows that $\text{Pic}_{X/B}$ is an algebraic space by Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}. In this way we reduce to the case described in the next paragraph. \medskip\noindent In addition to the assumptions of the proposition, assume that we have a section $\sigma : B \to X$. By Proposition \ref{proposition-pic} we see that $\textit{Pic}_{X/B}$ is an algebraic stack. By Lemma \ref{lemma-compare-pic-with-section} and Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-morphism-to-algebraic} we see that $\textit{Pic}_{X/B, \sigma}$ is an algebraic stack. By Lemma \ref{lemma-flat-geometrically-connected-fibres-with-section-functor-stack} and Algebraic Stacks, Lemma \ref{algebraic-lemma-characterize-representable-by-space} we see that $T \mapsto \Ker(\sigma_T^* : \text{Pic}(X_T) \to \text{Pic}(T))$ is an algebraic space. By Lemma \ref{lemma-flat-geometrically-connected-fibres-with-section} this functor is the same as $\text{Pic}_{X/B}$. \end{proof} \begin{lemma} \label{lemma-diagonal-pic} With assumptions and notation as in Proposition \ref{proposition-pic-functor}. Then the diagonal $\text{Pic}_{X/B} \to \text{Pic}_{X/B} \times_B \text{Pic}_{X/B}$ is representable by immersions. In other words, $\text{Pic}_{X/B} \to B$ is locally separated. \end{lemma} \begin{proof} Let $T$ be a scheme over $B$ and let $s, t \in \text{Pic}_{X/B}(T)$. We want to show that there exists a locally closed subscheme $Z \subset T$ such that $s|_Z = t|_Z$ and such that a morphism $T' \to T$ factors through $Z$ if and only if $s|_{T'} = t|_{T'}$. \medskip\noindent We first reduce the general problem to the case where $s$ and $t$ come from invertible modules on $X_T$. We suggest the reader skip this step. Choose an fppf covering $\{T_i \to T\}_{i \in I}$ such that $s|_{T_i}$ and $t|_{T_i}$ come from $\text{Pic}(X_{T_i})$ for all $i$. Suppose that we can show the result for all the pairs $s|_{T_i}, t|_{T_i}$. Then we obtain locally closed subschemes $Z_i \subset T_i$ with the desired universal property. It follows that $Z_i$ and $Z_j$ have the same scheme theoretic inverse image in $T_i \times_T T_j$. This determines a descend datum on $Z_i/T_i$. Since $Z_i \to T_i$ is locally quasi-finite, it follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} that we obtain a locally quasi-finite morphism $Z \to T$ recovering $Z_i \to T_i$ by base change. Then $Z \to T$ is an immersion by Descent, Lemma \ref{descent-lemma-descending-fppf-property-immersion}. Finally, because $\text{Pic}_{X/B}$ is an fppf sheaf, we conclude that $s|_Z = t|_Z$ and that $Z$ satisfies the universal property mentioned above. \medskip\noindent Assume $s$ and $t$ come from invertible modules $\mathcal{V}$, $\mathcal{W}$ on $X_T$. Set $\mathcal{L} = \mathcal{V} \otimes \mathcal{W}^{\otimes -1}$ We are looking for a locally closed subscheme $Z$ of $T$ such that $T' \to T$ factors through $Z$ if and only if $\mathcal{L}_{X_{T'}}$ is the pullback of an invertible sheaf on $T'$, see Lemma \ref{lemma-flat-geometrically-connected-fibres}. Hence the existence of $Z$ follows from More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-diagonal-picard-flat-proper}. \end{proof} \section{Relative morphisms} \label{section-relative-morphisms} \noindent We continue the discussion from Criteria for Representability, Section \ref{criteria-section-relative-morphisms}. In that section, starting with a scheme $S$ and morphisms of algebraic spaces $Z \to B$ and $X \to B$ over $S$ we constructed a functor $$\mathit{Mor}_B(Z, X) : (\Sch/B)^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \{f : Z_T \to X_T\}$$ We sometimes think of $\mathit{Mor}_B(Z, X)$ as a functor $(\Sch/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\mathit{Mor}_B(Z, X) \to B$. Namely, if $T$ is a scheme over $S$, then an element of $\mathit{Mor}_B(Z, X)(T)$ is a pair $(f, h)$ where $h$ is a morphism $h : T \to B$ and $f : Z \times_{B, h} T \to X \times_{B, h} T$ is a morphism of algebraic spaces over $T$. In particular, when we say that $\mathit{Mor}_B(Z, X)$ is an algebraic space, we mean that the corresponding functor $(\Sch/S)^{opp} \to \textit{Sets}$ is an algebraic space. \begin{lemma} \label{lemma-Mor-into-Hilb} Let $S$ be a scheme. Consider morphisms of algebraic spaces $Z \to B$ and $X \to B$ over $S$. If $X \to B$ is separated and $Z \to B$ is of finite presentation, flat, and proper, then there is a natural injective transformation of functors $$\mathit{Mor}_B(Z, X) \longrightarrow \text{Hilb}_{Z \times_B X/B}$$ which maps a morphism $f : Z_T \to X_T$ to its graph. \end{lemma} \begin{proof} Given a scheme $T$ over $B$ and a morphism $f_T : Z_T \to X_T$ over $T$, the graph of $f$ is the morphism $\Gamma_f = (\text{id}, f) : Z_T \to Z_T \times_T X_T = (Z \times_B X)_T$. Recall that being separated, flat, proper, or finite presentation are properties of morphisms of algebraic spaces which are stable under base change (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-separated}, \ref{spaces-morphisms-lemma-base-change-flat}, \ref{spaces-morphisms-lemma-base-change-proper}, and \ref{spaces-morphisms-lemma-base-change-finite-presentation}). Hence $\Gamma_f$ is a closed immersion by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-semi-diagonal}. Moreover, $\Gamma_f(Z_T)$ is flat, proper, and of finite presentation over $T$. Thus $\Gamma_f(Z_T)$ defines an element of $\text{Hilb}_{Z \times_B X/B}(T)$. To show the transformation is injective it suffices to show that two morphisms with the same graph are the same. This is true because if $Y \subset (Z \times_B X)_T$ is the graph of a morphism $f$, then we can recover $f$ by using the inverse of $\text{pr}_1|_Y : Y \to Z_T$ composed with $\text{pr}_2|_Y$. \end{proof} \begin{lemma} \label{lemma-Mor-into-Hilb-open} Assumption and notation as in Lemma \ref{lemma-Mor-into-Hilb}. The transformation $\mathit{Mor}_B(Z, X) \longrightarrow \text{Hilb}_{Z \times_B X/B}$ is representable by open immersions. \end{lemma} \begin{proof} Let $T$ be a scheme over $B$ and let $Y \subset (Z \times_B X)_T$ be an element of $\text{Hilb}_{Z \times_B X/B}(T)$. Then we see that $Y$ is the graph of a morphism $Z_T \to X_T$ over $T$ if and only if $k = \text{pr}_1|_Y : Y \to Z_T$ is an isomorphism. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-where-isomorphism} there exists an open subscheme $V \subset T$ such that for any morphism of schemes $T' \to T$ we have $k_{T'} : Y_{T'} \to Z_{T'}$ is an isomorphism if and only if $T' \to T$ factors through $V$. This proves the lemma. \end{proof} \begin{proposition} \label{proposition-Mor} Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of algebraic spaces over $S$. Assume $X \to B$ is of finite presentation and separated and $Z \to B$ is of finite presentation, flat, and proper. Then $\mathit{Mor}_B(Z, X)$ is an algebraic space locally of finite presentation over $B$. \end{proposition} \begin{proof} Immediate consequence of Lemma \ref{lemma-Mor-into-Hilb-open} and Proposition \ref{proposition-hilb}. \end{proof} \section{The stack of algebraic spaces} \label{section-stack-of-spaces} \noindent This section continuous the discussion started in Examples of Stacks, Sections \ref{examples-stacks-section-stack-of-spaces}, \ref{examples-stacks-section-stack-of-finite-type-spaces}, and \ref{examples-stacks-section-stack-in-groupoids-of-finite-type-spaces}. Working over $\mathbf{Z}$, the discussion therein shows that we have a stack in groupoids $$p'_{ft} : \textit{Spaces}'_{ft} \longrightarrow \Sch_{fppf}$$ parametrizing (nonflat) families of finite type algebraic spaces. More precisely, an object\footnote{We always perform a replacement as in Examples of Stacks, Lemma \ref{examples-stacks-lemma-stack-ft-spaces}.} of $\textit{Spaces}'_{ft}$ is a finite type morphism $X \to S$ from an algebraic space $X$ to a scheme $S$ and a morphism $(X' \to S') \to (X \to S)$ 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 $$\xymatrix{ X' \ar[d] \ar[r]_f & X \ar[d] \\ S' \ar[r]^g & S }$$ inducing an isomorphism $X' \to S' \times_S X$, in other words, the diagram is cartesian in the category of algebraic spaces. The functor $p'_{ft}$ sends $(X \to S)$ to $S$ and sends $(f, g)$ to $g$. We define a full subcategory $$\textit{Spaces}'_{fp, flat, proper} \subset \textit{Spaces}'_{ft}$$ consisting of objects $X \to S$ of $\textit{Spaces}'_{ft}$ such that $X \to S$ is of finite presentation, flat, and proper. We denote $$p'_{fp, flat, proper} : \textit{Spaces}'_{fp, flat, proper} \longrightarrow \Sch_{fppf}$$ the restriction of the functor $p'_{ft}$ to the indicated subcategory. We first review the results already obtained in the references listed above, and then we start adding further results. \begin{lemma} \label{lemma-spaces-fibred-in-groupoids} The category $\textit{Spaces}'_{ft}$ is fibred in groupoids over $\Sch_{fppf}$. The same is true for $\textit{Spaces}'_{fp, flat, proper}$. \end{lemma} \begin{proof} We have seen this in Examples of Stacks, Section \ref{examples-stacks-section-stack-in-groupoids-of-finite-type-spaces} for the case of $\textit{Spaces}'_{ft}$ 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 \ref{categories-definition-fibred-groupoids}. \medskip\noindent Condition (1). Let $X \to S$ be an object of $\textit{Spaces}'_{ft}$ 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 \ref{spaces-morphisms-lemma-base-change-finite-type}. 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 \ref{spaces-morphisms-lemma-base-change-finite-presentation}, \ref{spaces-morphisms-lemma-base-change-flat}, and \ref{spaces-morphisms-lemma-base-change-proper}. \medskip\noindent Condition (2). Consider morphisms $(f, g) : (X' \to S') \to (X \to S)$ and $(a, b) : (Y \to T) \to (X \to S)$ of $\textit{Spaces}'_{ft}$. 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 $\textit{Spaces}'_{ft}$ 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 $\textit{Spaces}'_{ft}$ satisfying (1). \end{proof} \begin{lemma} \label{lemma-spaces-diagonal} The diagonal $$\Delta : \textit{Spaces}'_{fp, flat, proper} \longrightarrow \textit{Spaces}'_{fp, flat, proper} \times \textit{Spaces}'_{fp, flat, proper}$$ is representable by algebraic spaces. \end{lemma} \begin{proof} We will use criterion (2) of Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-diagonal}. 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 $$\mathit{Isom}_S(X, Y) : (\Sch/S)_{fppf} \longrightarrow \textit{Sets}, \quad T \longmapsto \{f : X_T \to Y_T \text{ isomorphism}\}$$ is an algebraic space. An elementary argument shows that $\mathit{Isom}_S(X, Y)$ sits in a fibre product $$\xymatrix{ \mathit{Isom}_S(X, Y) \ar[r] \ar[d] & S \ar[d]_{(\text{id}, \text{id})} \\ \mathit{Mor}_S(X, Y) \times \mathit{Mor}_S(Y, X) \ar[r] & \mathit{Mor}_S(X, X) \times \mathit{Mor}_S(Y, Y) }$$ The bottom arrow sends $(\varphi, \psi)$ to $(\psi \circ \varphi, \varphi \circ \psi)$. By Proposition \ref{proposition-Mor} 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. \end{proof} \begin{lemma} \label{lemma-spaces-stack} The category $\textit{Spaces}'_{ft}$ is a stack in groupoids over $\Sch_{fppf}$. The same is true for $\textit{Spaces}'_{fp, flat, proper}$. \end{lemma} \begin{proof} The reason this lemma holds is the slogan: any fppf descent datum for algebraic spaces is effective, see Bootstrap, Section \ref{bootstrap-section-applications}. More precisely, the lemma for $\textit{Spaces}'_{ft}$ follows from Examples of Stacks, Lemma \ref{examples-stacks-lemma-stack-of-finite-type-spaces} as we saw in Examples of Stacks, Section \ref{examples-stacks-section-stack-in-groupoids-of-finite-type-spaces}. However, let us review the proof. We need to check conditions (1), (2), and (3) of Stacks, Definition \ref{stacks-definition-stack-in-groupoids}. \medskip\noindent Property (1) we have seen in Lemma \ref{lemma-spaces-fibred-in-groupoids}. \medskip\noindent Property (2) follows from Lemma \ref{lemma-spaces-diagonal} in the case of $\textit{Spaces}'_{fp, flat, proper}$. In the case of $\textit{Spaces}'_{ft}$ it follows from Examples of Stacks, Lemma \ref{examples-stacks-lemma-pre-stack-of-spaces} (and this is really the correct'' reference). \medskip\noindent Condition (3) for $\textit{Spaces}'_{ft}$ is checked as follows. Suppose given \begin{enumerate} \item an fppf covering $\{U_i \to U\}_{i \in I}$ in $\Sch_{fppf}$, \item for each $i \in I$ an algebraic space $X_i$ of finite type over $U_i$, and \item 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 $U_i \times_U U_j \times_U U_k$. \end{enumerate} 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 $\varphi_{ij}$. First, note that by Sites, Lemma \ref{sites-lemma-glue-sheaves} there exists a sheaf $X$ on $(\Sch/U)_{fppf}$ recovering the $X_i$ and the $\varphi_{ij}$. Then by Bootstrap, Lemma \ref{bootstrap-lemma-locally-algebraic-space-finite-type} we see that $X$ is an algebraic space. By Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-finite-type} we see that $X \to U$ is of finite type. In the case of $\textit{Spaces}'_{fp, flat, proper}$ one additionally uses Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-finite-presentation}, \ref{spaces-descent-lemma-descending-property-flat}, and \ref{spaces-descent-lemma-descending-property-proper} in the last step. \end{proof} \begin{remark} \label{remark-spaces-base-change} Let $B$ be an algebraic space over $\Spec(\mathbf{Z})$. Let $B\textit{-Spaces}'_{ft}$ be the category consisting of pairs $(X \to S, h : S \to B)$ where $X \to S$ is an object of $\textit{Spaces}'_{ft}$ and $h : S \to B$ is a morphism. A morphism $(X' \to S', h') \to (X \to S, h)$ in $B\textit{-Spaces}'_{ft}$ is a morphism $(f, g)$ in $\textit{Spaces}'_{ft}$ such that $h \circ g = h'$. In this situation the diagram $$\xymatrix{ B\textit{-Spaces}'_{ft} \ar[r] \ar[d] & \textit{Spaces}'_{ft} \ar[d] \\ (\Sch/B)_{fppf} \ar[r] & \Sch_{fppf} }$$ is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\textit{Spaces}'_{ft}$ to the case of families over a given base algebraic space. Of course, a similar construction works for $B\textit{-Spaces}'_{fp, flat, proper}$ \end{remark} \begin{lemma} \label{lemma-spaces-limits} The stack $p'_{fp, flat, proper} : \textit{Spaces}'_{fp, flat, proper} \to \Sch_{fppf}$ is limit preserving (Artin's Axioms, Definition \ref{artin-definition-limit-preserving}). \end{lemma} \begin{proof} Let $T = \lim T_i$ be the limits of a directed inverse system of affine schemes. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-finite-presentation} 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 \ref{spaces-limits-lemma-descend-flat} and \ref{spaces-limits-lemma-eventually-proper}. \end{proof}