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 \input{preamble} % OK, start here. % \begin{document} \title{Limits of Algebraic Spaces} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we put material related to limits of algebraic spaces. A first topic is the characterization of algebraic spaces $F$ locally of finite presentation over the base $S$ as limit preserving functors. We continue with a study of limits of inverse systems over directed sets (Categories, Definition \ref{categories-definition-directed-set}) with affine transition maps. We discuss absolute Noetherian approximation for quasi-compact and quasi-separated algebraic spaces following \cite{CLO}. Another approach is due to David Rydh (see \cite{rydh_approx}) whose results also cover absolute Noetherian approximation for certain algebraic stacks. \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{Morphisms of finite presentation} \label{section-finite-presentation} \noindent In this section we generalize Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} to morphisms of algebraic spaces. The motivation for the following definition comes from the proposition just cited. \begin{definition} \label{definition-locally-finite-presentation} Let $S$ be a scheme. \begin{enumerate} \item A functor $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ is said to be {\it limit preserving} or {\it locally of finite presentation} if for every affine scheme $T$ over $S$ which is a limit $T = \lim T_i$ of a directed inverse system of affine schemes $T_i$ over $S$, we have $$F(T) = \colim F(T_i).$$ We sometimes say that $F$ is {\it locally of finite presentation over $S$}. \item Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. A transformation of functors $a : F \to G$ is {\it limit preserving} or {\it locally of finite presentation} if for every scheme $T$ over $S$ and every $y \in G(T)$ the functor $$F_y : (\Sch/T)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad T'/T \longmapsto \{x \in F(T') \mid a(x) = y|_{T'}\}$$ is locally of finite presentation over $T$\footnote{The characterization (2) in Lemma \ref{lemma-characterize-relative-limit-preserving} may be easier to parse.}. We sometimes say that $F$ is {\it relatively limit preserving} over $G$. \end{enumerate} \end{definition} \noindent The functor $F_y$ is in some sense the fiber of $a : F \to G$ over $y$, except that it is a presheaf on the big fppf site of $T$. A formula for this functor is: \begin{equation} \label{equation-fibre-map-functors} F_y = F|_{(\Sch/T)_{fppf}} {\times}_{G|_{(\Sch/T)_{fppf}}} * \end{equation} Here $*$ is the final object in the category of (pre)sheaves on $(\Sch/T)_{fppf}$ (see Sites, Example \ref{sites-example-singleton-sheaf}) and the map $* \to G|_{(\Sch/T)_{fppf}}$ is given by $y$. Note that if $j : (\Sch/T)_{fppf} \to (\Sch/S)_{fppf}$ is the localization functor, then the formula above becomes $F_y = j^{-1}F \times_{j^{-1}G} *$ and $j_!F_y$ is just the fiber product $F \times_{G, y} T$. (See Sites, Section \ref{sites-section-localize}, for information on localization, and especially Sites, Remark \ref{sites-remark-localize-presheaves} for information on $j_!$ for presheaves.) \medskip\noindent At this point we temporarily have two definitions of what it means for a morphism $X \to Y$ of algebraic spaces over $S$ to be locally of finite presentation. Namely, one by Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-locally-finite-presentation} and one using that $X \to Y$ is a transformation of functors so that Definition \ref{definition-locally-finite-presentation} applies (we will use the terminology limit preserving'' for this notion as much as possible). We will show in Proposition \ref{proposition-characterize-locally-finite-presentation} that these two definitions agree. \begin{lemma} \label{lemma-characterize-relative-limit-preserving} Let $S$ be a scheme. Let $a : F \to G$ be a transformation of functors $(\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. The following are equivalent \begin{enumerate} \item $a : F \to G$ is limit preserving, and \item for every every affine scheme $T$ over $S$ which is a limit $T = \lim T_i$ of a directed inverse system of affine schemes $T_i$ over $S$ the diagram of sets $$\xymatrix{ \colim_i F(T_i) \ar[r] \ar[d]_a & F(T) \ar[d]^a \\ \colim_i G(T_i) \ar[r] & G(T) }$$ is a fibre product diagram. \end{enumerate} \end{lemma} \begin{proof} Assume (1). Consider $T = \lim_{i \in I} T_i$ as in (2). Let $(y, x_T)$ be an element of the fibre product $\colim_i G(T_i) \times_{G(T)} F(T)$. Then $y$ comes from $y_i \in G(T_i)$ for some $i$. Consider the functor $F_{y_i}$ on $(\Sch/T_i)_{fppf}$ as in Definition \ref{definition-locally-finite-presentation}. We see that $x_T \in F_{y_i}(T)$. Moreover $T = \lim_{i' \geq i} T_{i'}$ is a directed system of affine schemes over $T_i$. Hence (1) implies that $x_T$ the image of a unique element $x$ of $\colim_{i' \geq i} F_{y_i}(T_{i'})$. Thus $x$ is the unique element of $\colim F(T_i)$ which maps to the pair $(y, x_T)$. This proves that (2) holds. \medskip\noindent Assume (2). Let $T$ be a scheme and $y_T \in G(T)$. We have to show that $F_{y_T}$ is limit preserving. Let $T' = \lim_{i \in I} T'_i$ be an affine scheme over $T$ which is the directed limit of affine scheme $T'_i$ over $T$. Let $x_{T'} \in F_{y_T}$. Pick $i \in I$ which is possible as $I$ is a directed set. Denote $y_i \in F(T'_i)$ the image of $y_{T'}$. Then we see that $(y_i, x_{T'})$ is an element of the fibre product $\colim_i G(T'_i) \times_{G(T')} F(T')$. Hence by (2) we get a unique element $x$ of $\colim_i F(T'_i)$ mapping to $(y_i, x_{T'})$. It is clear that $x$ defines an element of $\colim_i F_y(T'_i)$ mapping to $x_{T'}$ and we win. \end{proof} \begin{lemma} \label{lemma-composition-locally-finite-presentation} Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : G \to H$ be transformations of functors. If $a$ and $b$ are limit preserving, then $$b \circ a : F \longrightarrow H$$ is limit preserving. \end{lemma} \begin{proof} Let $T = \lim_{i \in I} T_i$ as in characterization (2) of Lemma \ref{lemma-characterize-relative-limit-preserving}. Consider the diagram $$\xymatrix{ \colim_i F(T_i) \ar[r] \ar[d]_a & F(T) \ar[d]^a \\ \colim_i G(T_i) \ar[r] \ar[d]_b & G(T) \ar[d]^b \\ \colim_i H(T_i) \ar[r] & H(T) }$$ By assumption the two squares are fibre product squares. Hence the outer rectangle is a fibre product diagram too which proves the lemma. \end{proof} \begin{lemma} \label{lemma-base-change-locally-finite-presentation} Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : H \to G$ be transformations of functors. Consider the fibre product diagram $$\xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G }$$ If $a$ is limit preserving, then the base change $a'$ is limit preserving. \end{lemma} \begin{proof} Omitted. Hint: This is formal. \end{proof} \begin{lemma} \label{lemma-limit-fppf-topology} Let $T$ be an affine scheme which is written as a limit $T = \lim_{i \in I} T_i$ of a directed inverse system of affine schemes. \begin{enumerate} \item Let $\mathcal{V} = \{V_j \to T\}_{j = 1, \ldots, m}$ be a standard fppf covering of $T$, see Topologies, Definition \ref{topologies-definition-standard-fppf}. Then there exists an index $i$ and a standard fppf covering $\mathcal{V}_i = \{V_{i, j} \to T_i\}_{j = 1, \ldots, m}$ whose base change $T \times_{T_i} \mathcal{V}_i$ to $T$ is isomorphic to $\mathcal{V}$. \item Let $\mathcal{V}_i$, $\mathcal{V}'_i$ be a pair of standard fppf coverings of $T_i$. If $f : T \times_{T_i} \mathcal{V} \to T \times_{T_i} \mathcal{V}'_i$ is a morphism of coverings of $T$, then there exists an index $i' \geq i$ and a morphism $f_{i'} : T_{i'} \times_{T_i} \mathcal{V} \to T_{i'} \times_{T_i} \mathcal{V}'_i$ whose base change to $T$ is $f$. \item If $f, g : \mathcal{V} \to \mathcal{V}'_i$ are morphisms of standard fppf coverings of $T_i$ whose base changes $f_T, g_T$ to $T$ are equal then there exists an index $i' \geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$. \end{enumerate} In other words, the category of standard fppf coverings of $T$ is the colimit over $I$ of the categories of standard fppf coverings of $T_i$ \end{lemma} \begin{proof} By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} the category of schemes of finite presentation over $T$ is the colimit over $I$ of the categories of finite presentation over $T_i$. By Limits, Lemmas \ref{limits-lemma-descend-affine-finite-presentation} and \ref{limits-lemma-descend-flat-finite-presentation} the same is true for category of schemes which are affine, flat and of finite presentation over $T$. To finish the proof of the lemma it suffices to show that if $\{V_{j, i} \to T_i\}_{j = 1, \ldots, m}$ is a finite family of flat finitely presented morphisms with $V_{j, i}$ affine, and the base change $\coprod_j T \times_{T_i} V_{j, i} \to T$ is surjective, then for some $i' \geq i$ the morphism $\coprod T_{i'} \times_{T_i} V_{j, i} \to T_{i'}$ is surjective. Denote $W_{i'} \subset T_{i'}$, resp.\ $W \subset T$ the image. Of course $W = T$ by assumption. Since the morphisms are flat and of finite presentation we see that $W_i$ is a quasi-compact open of $T_i$, see Morphisms, Lemma \ref{morphisms-lemma-fppf-open}. Moreover, $W = T \times_{T_i} W_i$ (formation of image commutes with base change). Hence by Limits, Lemma \ref{limits-lemma-descend-opens} we conclude that $W_{i'} = T_{i'}$ for some large enough $i'$ and we win. \end{proof} \begin{lemma} \label{lemma-sheafify-finite-presentation} Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. If $F$ is limit preserving then its sheafification $F^\#$ is limit preserving. \end{lemma} \begin{proof} Assume $F$ is limit preserving. It suffices to show that $F^+$ is limit preserving, since $F^\# = (F^+)^+$, see Sites, Theorem \ref{sites-theorem-plus}. Let $T$ be an affine scheme over $S$, and let $T = \lim T_i$ be written as the directed limit of an inverse system of affine $S$ schemes. Recall that $F^+(T)$ is the colimit of $\check H^0(\mathcal{V}, F)$ where the limit is over all coverings of $T$ in $(\Sch/S)_{fppf}$. Any fppf covering of an affine scheme can be refined by a standard fppf covering, see Topologies, Lemma \ref{topologies-lemma-fppf-affine}. Hence we can write $$F^+(T) = \colim_{\mathcal{V}\text{ standard covering }T} \check H^0(\mathcal{V}, F).$$ By Lemma \ref{lemma-limit-fppf-topology} we may rewrite this as $$\colim_{i \in I} \colim_{\mathcal{V}_i\text{ standard covering }T_i} \check H^0(T \times_{T_i}\mathcal{V}_i, F).$$ (The order of the colimits is irrelevant by Categories, Lemma \ref{categories-lemma-colimits-commute}.) Given a standard fppf covering $\mathcal{V}_i = \{V_j \to T_i\}_{j = 1, \ldots, m}$ of $T_i$ we see that $$T \times_{T_i} V_j = \lim_{i' \geq i} T_{i'} \times_T V_j$$ by Limits, Lemma \ref{limits-lemma-scheme-over-limit}, and similarly $$T \times_{T_i} (V_j \times_{T_i} V_{j'}) = \lim_{i' \geq i} T_{i'} \times_T (V_j \times_{T_i} V_{j'}).$$ As the presheaf $F$ is limit preserving this means that $$\check H^0(T \times_{T_i}\mathcal{V}_i, F) = \colim_{i' \geq i} \check H^0(T_{i'} \times_{T_i}\mathcal{V}_i, F).$$ Hence the colimit expression for $F^+(T)$ above collapses to $$\colim_{i \in I} \colim_{\mathcal{V}\text{ standard covering }T_i} \check H^0(\mathcal{V}, F). = \colim_{i \in I} F^+(T_i).$$ In other words $F^+(T) = \colim_i F^+(T_i)$ and hence the lemma holds. \end{proof} \begin{lemma} \label{lemma-sheaf-finite-presentation} Let $S$ be a scheme. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item $F$ is a sheaf, and \item there exists an fppf covering $\{U_j \to S\}_{j \in J}$ such that $F|_{(\Sch/U_j)_{fppf}}$ is limit preserving. \end{enumerate} Then $F$ is limit preserving. \end{lemma} \begin{proof} Let $T$ be an affine scheme over $S$. Let $I$ be a directed set, and let $T_i$ be an inverse system of affine schemes over $S$ such that $T = \lim T_i$. We have to show that the canonical map $\colim F(T_i) \to F(T)$ is bijective. \medskip\noindent Choose some $0 \in I$ and choose a standard fppf covering $\{V_{0, k} \to T_{0}\}_{k = 1, \ldots, m}$ which refines the pullback $\{U_j \times_S T_0 \to T_0\}$ of the given fppf covering of $S$. For each $i \geq 0$ we set $V_{i, k} = T_i \times_{T_0} V_{0, k}$, and we set $V_k = T \times_{T_0} V_{0, k}$. Note that $V_k = \lim_{i \geq 0} V_{i, k}$, see Limits, Lemma \ref{limits-lemma-scheme-over-limit}. \medskip\noindent Suppose that $x, x' \in \colim F(T_i)$ map to the same element of $F(T)$. Say $x, x'$ are given by elements $x_i, x'_i \in F(T_i)$ for some $i \in I$ (we may choose the same $i$ for both as $I$ is directed). By assumption (2) and the fact that $x_i, x'_i$ map to the same element of $F(T)$ this implies that $$x_i|_{V_{i', k}} = x'_i|_{V_{i', k}}$$ for some suitably large $i' \in I$. We can choose the same $i'$ for each $k$ as $k \in \{1, \ldots, m\}$ ranges over a finite set. Since $\{V_{i', k} \to T_{i'}\}$ is an fppf covering and $F$ is a sheaf this implies that $x_i|_{T_{i'}} = x'_i|_{T_{i'}}$ as desired. This proves that the map $\colim F(T_i) \to F(T)$ is injective. \medskip\noindent To show surjectivity we argue in a similar fashion. Let $x \in F(T)$. By assumption (2) for each $k$ we can choose a $i$ such that $x|_{V_k}$ comes from an element $x_{i, k} \in F(V_{i, k})$. As before we may choose a single $i$ which works for all $k$. By the injectivity proved above we see that $$x_{i, k}|_{V_{i', k} \times_{T_{i'}} V_{i', l}} = x_{i, l}|_{V_{i', k} \times_{T_{i'}} V_{i', l}}$$ for some large enough $i'$. Hence by the sheaf condition of $F$ the elements $x_{i, k}|_{V_{i', k}}$ glue to an element $x_{i'} \in F(T_{i'})$ as desired. \end{proof} \begin{lemma} \label{lemma-sheafify-finite-presentation-map} Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be functors. If $a : F \to G$ is a transformation which is limit preserving, then the induced transformation of sheaves $F^\# \to G^\#$ is limit preserving. \end{lemma} \begin{proof} Suppose that $T$ is a scheme and $y \in G^\#(T)$. We have to show the functor $F^\#_y : (\Sch/T)_{fppf}^{opp} \to \textit{Sets}$ constructed from $F^\# \to G^\#$ and $y$ as in Definition \ref{definition-locally-finite-presentation} is limit preserving. By Equation (\ref{equation-fibre-map-functors}) we see that $F^\#_y$ is a sheaf. Choose an fppf covering $\{V_j \to T\}_{j \in J}$ such that $y|_{V_j}$ comes from an element $y_j \in F(V_j)$. Note that the restriction of $F^\#$ to $(\Sch/V_j)_{fppf}$ is just $F^\#_{y_j}$. If we can show that $F^\#_{y_j}$ is limit preserving then Lemma \ref{lemma-sheaf-finite-presentation} guarantees that $F^\#_y$ is limit preserving and we win. This reduces us to the case $y \in G(T)$. \medskip\noindent Let $y \in G(T)$. In this case we claim that $F^\#_y = (F_y)^\#$. This follows from Equation (\ref{equation-fibre-map-functors}). Thus this case follows from Lemma \ref{lemma-sheafify-finite-presentation}. \end{proof} \begin{proposition} \label{proposition-characterize-locally-finite-presentation} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item The morphism $f$ is a morphism of algebraic spaces which is locally of finite presentation, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-locally-finite-presentation}. \item The morphism $f : X \to Y$ is limit preserving as a transformation of functors, see Definition \ref{definition-locally-finite-presentation}. \end{enumerate} \end{proposition} \begin{proof} Assume (1). Let $T$ be a scheme and let $y \in Y(T)$. We have to show that $T \times_Y X$ is limit preserving over $T$ in the sense of Definition \ref{definition-locally-finite-presentation}. Hence we are reduced to proving that if $X$ is an algebraic space which is locally of finite presentation over $S$ as an algebraic space, then it is limit preserving as a functor $X : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. To see this choose a presentation $X = U/R$, see Spaces, Definition \ref{spaces-definition-presentation}. It follows from Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-locally-finite-presentation} that both $U$ and $R$ are schemes which are locally of finite presentation over $S$. Hence by Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} we have $$U(T) = \colim U(T_i), \quad R(T) = \colim R(T_i)$$ whenever $T = \lim_i T_i$ in $(\Sch/S)_{fppf}$. It follows that the presheaf $$(\Sch/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad W \longmapsto U(W)/R(W)$$ is limit preserving. Hence by Lemma \ref{lemma-sheafify-finite-presentation} its sheafification $X = U/R$ is limit preserving too. \medskip\noindent Assume (2). Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Next, choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. By Lemma \ref{lemma-base-change-locally-finite-presentation} the transformation of functors $V \times_Y X \to V$ is limit preserving. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-locally-finite-presentation} the morphism of algebraic spaces $U \to V \times_Y X$ is locally of finite presentation, hence limit preserving as a transformation of functors by the first part of the proof. By Lemma \ref{lemma-composition-locally-finite-presentation} the composition $U \to V \times_Y X \to V$ is limit preserving as a transformation of functors. Hence the morphism of schemes $U \to V$ is locally of finite presentation by Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} (modulo a set theoretic remark, see last paragraph of the proof). This means, by definition, that (1) holds. \medskip\noindent Set theoretic remark. Let $U \to V$ be a morphism of $(\Sch/S)_{fppf}$. In the statement of Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} we characterize $U \to V$ as being locally of finite presentation if for {\it all} directed inverse systems $(T_i, f_{ii'})$ of affine schemes over $V$ we have $U(T) = \colim V(T_i)$, but in the current setting we may only consider affine schemes $T_i$ over $V$ which are (isomorphic to) an object of $(\Sch/S)_{fppf}$. So we have to make sure that there are enough affines in $(\Sch/S)_{fppf}$ to make the proof work. Inspecting the proof of (2) $\Rightarrow$ (1) of Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} we see that the question reduces to the case that $U$ and $V$ are affine. Say $U = \Spec(A)$ and $V = \Spec(B)$. By construction of $(\Sch/S)_{fppf}$ the spectrum of any ring of cardinality $\leq |B|$ is isomorphic to an object of $(\Sch/S)_{fppf}$. Hence it suffices to observe that in the "only if" part of the proof of Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation} only $A$-algebras of cardinality $\leq |B|$ are used. \end{proof} \begin{remark} \label{remark-limit-preserving} Here is an important special case of Proposition \ref{proposition-characterize-locally-finite-presentation}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is locally of finite presentation over $S$ if and only if $X$, as a functor $(\Sch/S)^{opp} \to \textit{Sets}$, is limit preserving. Compare with Limits, Remark \ref{limits-remark-limit-preserving}. In fact, we will see in Lemma \ref{lemma-surjection-is-enough} below that it suffices if the map $$\colim X(T_i) \longrightarrow X(T)$$ is surjective whenever $T = \lim T_i$ is a directed limit of affine schemes over $S$. \end{remark} \begin{lemma} \label{lemma-surjection-is-enough} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If for every directed limit $T = \lim_{i \in I} T_i$ of affine schemes over $S$ the map $$\colim X(T_i) \longrightarrow X(T) \times_{Y(T)} \colim Y(T_i)$$ is surjective, then $f$ is locally of finite presentation. In other words, in Proposition \ref{proposition-characterize-locally-finite-presentation} part (2) it suffices to check surjectivity in the criterion of Lemma \ref{lemma-characterize-relative-limit-preserving}. \end{lemma} \begin{proof} Choose a scheme $V$ and a surjective \'etale morphism $g : V \to Y$. Next, choose a scheme $U$ and a surjective \'etale morphism $h : U \to V \times_Y X$. It suffices to show for $T = \lim T_i$ as in the lemma that the map $$\colim U(T_i) \longrightarrow U(T) \times_{V(T)} \colim V(T_i)$$ is surjective, because then $U \to V$ will be locally of finite presentation by Limits, Lemma \ref{limits-lemma-surjection-is-enough} (modulo a set theoretic remark exactly as in the proof of Proposition \ref{proposition-characterize-locally-finite-presentation}). Thus we take $a : T \to U$ and $b_i : T_i \to V$ which determine the same morphism $T \to V$. Picture $$\xymatrix{ T \ar[d]_a \ar[rr]_{p_i} & & T_i \ar[d]^{b_i} \ar@{..>}[ld] \\ U \ar[r]^-h & X \times_Y V \ar[d] \ar[r] & V \ar[d]^g \\ & X \ar[r]^f & Y }$$ By the assumption of the lemma after increasing $i$ we can find a morphism $c_i : T_i \to X$ such that $h \circ a = (b_i, c_i) \circ p_i : T_i \to V \times_Y X$ and such that $f \circ c_i = g \circ b_i$. Since $h$ is an \'etale morphism of algebraic spaces (and hence locally of finite presentation), we have the surjectivity of $$\colim U(T_i) \longrightarrow U(T) \times_{(X \times_Y V)(T)} \colim (X \times_Y V)(T_i)$$ by Proposition \ref{proposition-characterize-locally-finite-presentation}. Hence after increasing $i$ again we can find the desired morphism $a_i : T_i \to U$ with $a = a_i \circ p_i$ and $b_i = (U \to V) \circ a_i$. \end{proof} \section{Limits of algebraic spaces} \label{section-limits} \noindent The following lemma explains how we think of limits of algebraic spaces in this chapter. We will use (without further mention) that the base change of an affine morphism of algebraic spaces is affine (see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-affine}). \begin{lemma} \label{lemma-directed-inverse-system-has-limit} Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ in the category of algebraic spaces over $S$. If the morphisms $f_{ii'} : X_i \to X_{i'}$ are affine, then the limit $X = \lim_i X_i$ (as an fppf sheaf) is an algebraic space. Moreover, \begin{enumerate} \item each of the morphisms $f_i : X \to X_i$ is affine, \item for any $i \in I$ and any morphism of algebraic spaces $T \to X_i$ we have $$X \times_{X_i} T = \lim_{i' \geq i} X_{i'} \times_{X_i} T.$$ as algebraic spaces over $S$. \end{enumerate} \end{lemma} \begin{proof} Part (2) is a formal consequence of the existence of the limit $X = \lim X_i$ as an algebraic space over $S$. Choose an element $0 \in I$ (this is possible as a directed set is nonempty). Choose a scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $R_0 = U_0 \times_{X_0} U_0$ so that $X_0 = U_0/R_0$. For $i \geq 0$ set $U_i = X_i \times_{X_0} U_0$ and $R_i = X_i \times_{X_0} R_0 = U_i \times_{X_i} U_i$. By Limits, Lemma \ref{limits-lemma-directed-inverse-system-has-limit} we see that $U = \lim_{i \geq 0} U_i$ and $R = \lim_{i \geq 0} R_i$ are schemes. Moreover, the two morphisms $s, t : R \to U$ are the base change of the two projections $R_0 \to U_0$ by the morphism $U \to U_0$, in particular \'etale. The morphism $R \to U \times_S U$ defines an equivalence relation as directed a limit of equivalence relations is an equivalence relation. Hence the morphism $R \to U \times_S U$ is an \'etale equivalence relation. We claim that the natural map \begin{equation} \label{equation-isomorphism-sheaves} U/R \longrightarrow \lim X_i \end{equation} is an isomorphism of fppf sheaves on the category of schemes over $S$. The claim implies $X = \lim X_i$ is an algebraic space by Spaces, Theorem \ref{spaces-theorem-presentation}. \medskip\noindent Let $Z$ be a scheme and let $a : Z \to \lim X_i$ be a morphism. Then $a = (a_i)$ where $a_i : Z \to X_i$. Set $W_0 = Z \times_{a_0, X_0} U_0$. Note that $W_0 = Z \times_{a_i, X_i} U_i$ for all $i \geq 0$ by our choice of $U_i \to X_i$ above. Hence we obtain a morphism $W_0 \to \lim_{i \geq 0} U_i = U$. Since $W_0 \to Z$ is surjective and \'etale, we conclude that (\ref{equation-isomorphism-sheaves}) is a surjective map of sheaves. Finally, suppose that $Z$ is a scheme and that $a, b : Z \to U/R$ are two morphisms which are equalized by (\ref{equation-isomorphism-sheaves}). We have to show that $a = b$. After replacing $Z$ by the members of an fppf covering we may assume there exist morphisms $a', b' : Z \to U$ which give rise to $a$ and $b$. The condition that $a, b$ are equalized by (\ref{equation-isomorphism-sheaves}) means that for each $i \geq 0$ the compositions $a_i', b_i' : Z \to U \to U_i$ are equal as morphisms into $U_i/R_i = X_i$. Hence $(a_i', b_i') : Z \to U_i \times_S U_i$ factors through $R_i$, say by some morphism $c_i : Z \to R_i$. Since $R = \lim_{i \geq 0} R_i$ we see that $c = \lim c_i : Z \to R$ is a morphism which shows that $a, b$ are equal as morphisms of $Z$ into $U/R$. \medskip\noindent Part (1) follows as we have seen above that $U_i \times_{X_i} X = U$ and $U \to U_i$ is affine by construction. \end{proof} \begin{lemma} \label{lemma-space-over-limit} Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces over $S$ with affine transition maps. Let $X = \lim_i X_i$. Let $0 \in I$. Suppose that $T \to X_0$ is a morphism of algebraic spaces. Then $$T \times_{X_0} X = \lim_{i \geq 0} T \times_{X_0} X_i$$ as algebraic spaces over $S$. \end{lemma} \begin{proof} The limit $X$ is an algebraic space by Lemma \ref{lemma-directed-inverse-system-has-limit}. The equality is formal, see Categories, Lemma \ref{categories-lemma-colimits-commute}. \end{proof} \begin{lemma} \label{lemma-directed-inverse-system-closed-immersions} Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{i'i}) \to (Y_i, g_{i'i})$ be a morphism of inverse systems over $I$ of algebraic spaces over $S$. Assume \begin{enumerate} \item the morphisms $f_{i'i} : X_{i'} \to X_i$ are affine, \item the morphisms $g_{i'i} : Y_{i'} \to Y_i$ are affine, \item the morphisms $X_i \to Y_i$ are closed immersions. \end{enumerate} Then $\lim X_i \to \lim Y_i$ is a closed immersion. \end{lemma} \begin{proof} Observe that $\lim X_i$ and $\lim Y_i$ exist by Lemma \ref{lemma-directed-inverse-system-has-limit}. Pick $0 \in I$ and choose an affine scheme $V_0$ and an \'etale morphism $V_0 \to Y_0$. Then the morphisms $V_i = Y_i \times_{Y_0} V_0 \to U_i = X_i \times_{Y_0} V_0$ are closed immersions of affine schemes. Hence the morphism $V = Y \times_{Y_0} V_0 \to U = X \times_{Y_0} V_0$ is a closed immersion because $V = \lim V_i$, $U = \lim U_i$ and because a limit of closed immersions of affine schemes is a closed immersion: a filtered colimit of surjective ring maps is surjective. Since the \'etale morphisms $V \to Y$ form an \'etale covering of $Y$ as we vary our choice of $V_0 \to Y_0$ we see that the lemma is true. \end{proof} \begin{lemma} \label{lemma-directed-inverse-system-reduced} Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{i'i})$ be an inverse systems over $I$ of algebraic spaces over $S$. If $X_i$ is reduced for all $i$, then $X$ is reduced. \end{lemma} \begin{proof} Observe that $\lim X_i$ exists by Lemma \ref{lemma-directed-inverse-system-has-limit}. Pick $0 \in I$ and choose an affine scheme $V_0$ and an \'etale morphism $U_0 \to X_0$. Then the affine schemes $U_i = X_i \times_{X_0} U_0$ are reduced. Hence $U = X \times_{X_0} U_0$ is a reduced affine scheme as a limit of reduced affine schemes: a filtered colimit of reduced rings is reduced. Since the \'etale morphisms $U \to X$ form an \'etale covering of $X$ as we vary our choice of $U_0 \to X_0$ we see that the lemma is true. \end{proof} \begin{lemma} \label{lemma-better-characterize-relative-limit-preserving} Let $S$ be a scheme. Let $X \to Y$ be a morphism of algebraic spaces over $S$. The equivalent conditions (1) and (2) of Proposition \ref{proposition-characterize-locally-finite-presentation} are also equivalent to \begin{enumerate} \item[(3)] for every directed limit $T = \lim T_i$ of quasi-compact and quasi-separated algebraic spaces $T_i$ over $S$ with affine transition morphisms the diagram of sets $$\xymatrix{ \colim_i \Mor(T_i, X) \ar[r] \ar[d] & \Mor(T, X) \ar[d] \\ \colim_i \Mor(T_i, Y) \ar[r] & \Mor(T, Y) }$$ is a fibre product diagram. \end{enumerate} \end{lemma} \begin{proof} It is clear that (3) implies (2). We will assume (2) and prove (3). The proof is rather formal and we encourage the reader to find their own proof. \medskip\noindent Let us first prove that (3) holds when $T_i$ is in addition assumed separated for all $i$. Choose $i \in I$ and choose a surjective \'etale morphism $U_i \to T_i$ where $U_i$ is affine. Using Lemma \ref{lemma-space-over-limit} we see that with $U = U_i \times_{T_i} T$ and $U_{i'} = U_i \times_{T_i} T_{i'}$ we have $U = \lim_{i' \geq i} U_{i'}$. Of course $U$ and $U_{i'}$ are affine (see Lemma \ref{lemma-directed-inverse-system-has-limit}). Since $T_i$ is separated, the fibre product $V_i = U_i \times_{T_i} U_i$ is an affine scheme as well and we obtain affine schemes $V = V_i \times_{T_i} T$ and $V_{i'} = V_i \times_{T_i} T_{i'}$ with $V = \lim_{i' \geq i} V_{i'}$. Observe that $U \to T$ and $U_i \to T_i$ are surjective \'etale and that $V = U \times_T U$ and $V_{i'} = U_{i'} \times_{T_{i'}} U_{i'}$. Note that $\Mor(T, X)$ is the equalizer of the two maps $\Mor(U, X) \to \Mor(V, X)$; this is true for example because $X$ as a sheaf on $(\Sch/S)_{fppf}$ is the coequalizer of the two maps $h_V \to h_u$. Similarly $\Mor(T_{i'}, X)$ is the equalizer of the two maps $\Mor(U_{i'}, X) \to \Mor(V_{i'}, X)$. And of course the same thing is true with $X$ replaced with $Y$. Condition (2) says that the diagrams of in (3) are fibre products in the case of $U = \lim U_i$ and $V = \lim V_i$. It follows formally that the same thing is true for $T = \lim T_i$. \medskip\noindent In the general case, choose an affine scheme $U$, an $i \in I$, and a surjective \'etale morphism $U \to T_i$. Repeating the argument of the previous paragraph we still achieve the proof: the schemes $V_{i'}$, $V$ are no longer affine, but they are still quasi-compact and separated and the result of the preceding paragraph applies. \end{proof} \section{Descending properties} \label{section-descent} \noindent This section is the analogue of Limits, Section \ref{limits-section-descent}. \begin{lemma} \label{lemma-inverse-limit-sets} Let $S$ be a scheme. Let $X = \lim_{i \in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). If each $X_i$ is decent (for example quasi-separated or locally separated) then $|X| = \lim_i |X_i|$ as sets. \end{lemma} \begin{proof} There is a canonical map $|X| \to \lim |X_i|$. Choose $0 \in I$. If $W_0 \subset X_0$ is an open subspace, then we have $f_0^{-1}W_0 = \lim_{i \geq 0} f_{i0}^{-1}W_0$, see Lemma \ref{lemma-directed-inverse-system-has-limit}. Hence, if we can prove the lemma for inverse systems where $X_0$ is quasi-compact, then the lemma follows in general. Thus we may and do assume $X_0$ is quasi-compact. \medskip\noindent Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $U_i = X_i \times_{X_0} U_0$ and $U = X \times_{X_0} U_0$. Set $R_i = U_i \times_{X_i} U_i$ and $R = U \times_X U$. Recall that $U = \lim U_i$ and $R = \lim R_i$, see proof of Lemma \ref{lemma-directed-inverse-system-has-limit}. Recall that $|X| = |U|/|R|$ and $|X_i| = |U_i|/|R_i|$. By Limits, Lemma \ref{limits-lemma-topology-limit} we have $|U| = \lim |U_i|$ and $|R| = \lim |R_i|$. \medskip\noindent Surjectivity of $|X| \to \lim |X_i|$. Let $(x_i) \in \lim |X_i|$. Denote $S_i \subset |U_i|$ the inverse image of $x_i$. This is a finite nonempty set by the definition of decent spaces (Decent Spaces, Definition \ref{decent-spaces-definition-very-reasonable}). Hence $\lim S_i$ is nonempty, see Categories, Lemma \ref{categories-lemma-nonempty-limit}. Let $(u_i) \in \lim S_i \subset \lim |U_i|$. By the above this determines a point $u \in |U|$ which maps to an $x \in |X|$ mapping to the given element $(x_i)$ of $\lim |X_i|$. \medskip\noindent Injectivity of $|X| \to \lim |X_i|$. Suppose that $x, x' \in |X|$ map to the same point of $\lim |X_i|$. Choose lifts $u, u' \in |U|$ and denote $u_i, u'_i \in |U_i|$ the images. For each $i$ let $T_i \subset |R_i|$ be the set of points mapping to $(u_i, u'_i) \in |U_i| \times |U_i|$. This is a finite set by the definition of decent spaces (Decent Spaces, Definition \ref{decent-spaces-definition-very-reasonable}). Moreover $T_i$ is nonempty as we've assumed that $x$ and $x'$ map to the same point of $X_i$. Hence $\lim T_i$ is nonempty, see Categories, Lemma \ref{categories-lemma-nonempty-limit}. As before let $r \in |R| = \lim |R_i|$ be a point corresponding to an element of $\lim T_i$. Then $r$ maps to $(u, u')$ in $|U| \times |U|$ by construction and we see that $x = x'$ in $|X|$ as desired. \medskip\noindent Parenthetical statement: A quasi-separated algebraic space is decent, see Decent Spaces, Section \ref{decent-spaces-section-reasonable-decent} (the key observation to this is Properties of Spaces, Lemma \ref{spaces-properties-lemma-finite-fibres-presentation}). A locally separated algebraic space is decent by Decent Spaces, Lemma \ref{decent-spaces-lemma-locally-separated-decent}. \end{proof} \begin{lemma} \label{lemma-topology-limit} With same notation and assumptions as in Lemma \ref{lemma-inverse-limit-sets} we have $|X| = \lim_i |X_i|$ as topological spaces. \end{lemma} \begin{proof} We will use the criterion of Topology, Lemma \ref{topology-lemma-characterize-limit}. We have seen that $|X| = \lim_i |X_i|$ as sets in Lemma \ref{lemma-inverse-limit-sets}. The maps $f_i : X \to X_i$ are morphisms of algebraic spaces hence determine continuous maps $|X| \to |X_i|$. Thus $f_i^{-1}(U_i)$ is open for each open $U_i \subset |X_i|$. Finally, let $x \in |X|$ and let $x \in V \subset |X|$ be an open neighbourhood. We have to find an $i$ and an open neighbourhood $W_i \subset |X_i|$ of the image $x$ with $f_i^{-1}(W_i) \subset V$. Choose $0 \in I$. Choose a scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $U = X \times_{X_0} U_0$ and $U_i = X_i \times_{X_0} U_0$ for $i \geq 0$. Then $U = \lim_{i \geq 0} U_i$ in the category of schemes by Lemma \ref{lemma-directed-inverse-system-has-limit}. Choose $u \in U$ mapping to $x$. By the result for schemes (Limits, Lemma \ref{limits-lemma-inverse-limit-top}) we can find an $i \geq 0$ and an open neighbourhood $E_i \subset U_i$ of the image of $u$ whose inverse image in $U$ is contained in the inverse image of $V$ in $U$. Then we can set $W_i \subset |X_i|$ equal to the image of $E_i$. This works because $|U_i| \to |X_i|$ is open. \end{proof} \begin{lemma} \label{lemma-limit-nonempty} Let $S$ be a scheme. Let $X = \lim_{i \in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). If each $X_i$ is quasi-compact and nonempty, then $|X|$ is nonempty. \end{lemma} \begin{proof} Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $U_i = X_i \times_{X_0} U_0$ and $U = X \times_{X_0} U_0$. Then each $U_i$ is a nonempty affine scheme. Hence $U = \lim U_i$ is nonempty (Limits, Lemma \ref{limits-lemma-limit-nonempty}) and thus $X$ is nonempty. \end{proof} \begin{lemma} \label{lemma-inverse-limit-irreducibles} Let $S$ be a scheme. Let $X = \lim_{i \in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). Let $x \in |X|$ with images $x_i \in |X_i|$. If each $X_i$ is decent, then $\overline{\{x\}} = \lim_i \overline{\{x_i\}}$ as sets and as algebraic spaces if endowed with reduced induced scheme structure. \end{lemma} \begin{proof} Set $Z = \overline{\{x\}} \subset |X|$ and $Z_i = \overline{\{x_i\}} \subset |X_i|$. Since $|X| \to |X_i|$ is continuous we see that $Z$ maps into $Z_i$ for each $i$. Hence we obtain an injective map $Z \to \lim Z_i$ because $|X| = \lim |X_i|$ as sets (Lemma \ref{lemma-inverse-limit-sets}). Suppose that $x' \in |X|$ is not in $Z$. Then there is an open subset $U \subset |X|$ with $x' \in U$ and $x \not \in U$. Since $|X| = \lim |X_i|$ as topological spaces (Lemma \ref{lemma-topology-limit}) we can write $U = \bigcup_{j \in J} f_j^{-1}(U_j)$ for some subset $J \subset I$ and opens $U_j \subset |X_j|$, see Topology, Lemma \ref{topology-lemma-describe-limits}. Then we see that for some $j \in J$ we have $f_j(x') \in U_j$ and $f_j(x) \not \in U_j$. In other words, we see that $f_j(x') \not \in Z_j$. Thus $Z = \lim Z_i$ as sets. \medskip\noindent Next, endow $Z$ and $Z_i$ with their reduced induced scheme structures, see Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}. The transition morphisms $X_{i'} \to X_i$ induce affine morphisms $Z_{i'} \to Z_i$ and the projections $X \to X_i$ induce compatible morphisms $Z \to Z_i$. Hence we obtain morphisms $Z \to \lim Z_i \to X$ of algebraic spaces. By Lemma \ref{lemma-directed-inverse-system-closed-immersions} we see that $\lim Z_i \to X$ is a closed immersion. By Lemma \ref{lemma-directed-inverse-system-reduced} the algebraic space $\lim Z_i$ is reduced. By the above $Z \to \lim Z_i$ is bijective on points. By uniqueness of the reduced induced closed subscheme structure we find that this morphism is an isomorphism of algebraic spaces. \end{proof} \begin{situation} \label{situation-descent} Let $S$ be a scheme. Let $X = \lim_{i \in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). We assume that $X_i$ is quasi-compact and quasi-separated for all $i \in I$. We also choose an element $0 \in I$. \end{situation} \begin{lemma} \label{lemma-descend-section} Notation and assumptions as in Situation \ref{situation-descent}. Suppose that $\mathcal{F}_0$ is a quasi-coherent sheaf on $X_0$. Set $\mathcal{F}_i = f_{0i}^*\mathcal{F}_0$ for $i \geq 0$ and set $\mathcal{F} = f_0^*\mathcal{F}_0$. Then $$\Gamma(X, \mathcal{F}) = \colim_{i \geq 0} \Gamma(X_i, \mathcal{F}_i)$$ \end{lemma} \begin{proof} Choose a surjective \'etale morphism $U_0 \to X_0$ where $U_0$ is an affine scheme (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Set $U_i = X_i \times_{X_0} U_0$. Set $R_0 = U_0 \times_{X_0} U_0$ and $R_i = R_0 \times_{X_0} X_i$. In the proof of Lemma \ref{lemma-directed-inverse-system-has-limit} we have seen that there exists a presentation $X = U/R$ with $U = \lim U_i$ and $R = \lim R_i$. Note that $U_i$ and $U$ are affine and that $R_i$ and $R$ are quasi-compact and separated (as $X_i$ is quasi-separated). Hence Limits, Lemma \ref{limits-lemma-descend-section} implies that $$\mathcal{F}(U) = \colim \mathcal{F}_i(U_i) \quad\text{and}\quad \mathcal{F}(R) = \colim \mathcal{F}_i(R_i).$$ The lemma follows as $\Gamma(X, \mathcal{F}) = \Ker(\mathcal{F}(U) \to \mathcal{F}(R))$ and similarly $\Gamma(X_i, \mathcal{F}_i) = \Ker(\mathcal{F}_i(U_i) \to \mathcal{F}_i(R_i))$ \end{proof} \begin{lemma} \label{lemma-descend-opens} Notation and assumptions as in Situation \ref{situation-descent}. For any quasi-compact open subspace $U \subset X$ there exists an $i$ and a quasi-compact open $U_i \subset X_i$ whose inverse image in $X$ is $U$. \end{lemma} \begin{proof} Follows formally from the construction of limits in Lemma \ref{lemma-directed-inverse-system-has-limit} and the corresponding result for schemes: Limits, Lemma \ref{limits-lemma-descend-opens}. \end{proof} \noindent The following lemma will be superseded by the stronger Lemma \ref{lemma-descend-isomorphism}. \begin{lemma} \label{lemma-descend-equality} Notation and assumptions as in Situation \ref{situation-descent}. Let $f_0 : Y_0 \to Z_0$ be a morphism of algebraic spaces over $X_0$. Assume (a) $Y_0 \to X_0$ and $Z_0 \to X_0$ are representable, (b) $Y_0$, $Z_0$ quasi-compact and quasi-separated, (c) $f_0$ locally of finite presentation, and (d) $Y_0 \times_{X_0} X \to Z_0 \times_{X_0} X$ an isomorphism. Then there exists an $i \geq 0$ such that $Y_0 \times_{X_0} X_i \to Z_0 \times_{X_0} X_i$ is an isomorphism. \end{lemma} \begin{proof} Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $U_i = U_0 \times_{X_0} X_i$ and $U = U_0 \times_{X_0} X$. Apply Limits, Lemma \ref{limits-lemma-descend-isomorphism} to see that $Y_0 \times_{X_0} U_i \to Z_0 \times_{X_0} U_i$ is an isomorphism of schemes for some $i \geq 0$ (details omitted). As $U_i \to X_i$ is surjective \'etale, it follows that $Y_0 \times_{X_0} X_i \to Z_0 \times_{X_0} X_i$ is an isomorphism (details omitted). \end{proof} \begin{lemma} \label{lemma-descend-separated} Notation and assumptions as in Situation \ref{situation-descent}. If $X$ is separated, then $X_i$ is separated for some $i \in I$. \end{lemma} \begin{proof} Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. For $i \geq 0$ set $U_i = U_0 \times_{X_0} X_i$ and set $U = U_0 \times_{X_0} X$. Note that $U_i$ and $U$ are affine schemes which come equipped with surjective \'etale morphisms $U_i \to X_i$ and $U \to X$. Set $R_i = U_i \times_{X_i} U_i$ and $R = U \times_X U$ with projections $s_i, t_i : R_i \to U_i$ and $s, t : R \to U$. Note that $R_i$ and $R$ are quasi-compact separated schemes (as the algebraic spaces $X_i$ and $X$ are quasi-separated). The maps $s_i : R_i \to U_i$ and $s : R \to U$ are of finite type. By definition $X_i$ is separated if and only if $(t_i, s_i) : R_i \to U_i \times U_i$ is a closed immersion, and since $X$ is separated by assumption, the morphism $(t, s) : R \to U \times U$ is a closed immersion. Since $R \to U$ is of finite type, there exists an $i$ such that the morphism $R \to U_i \times U$ is a closed immersion (Limits, Lemma \ref{limits-lemma-finite-type-eventually-closed}). Fix such an $i \in I$. Apply Limits, Lemma \ref{limits-lemma-descend-closed-immersion-finite-presentation} to the system of morphisms $R_{i'} \to U_i \times U_{i'}$ for $i' \geq i$ (this is permissible as indeed $R_{i'} = R_i \times_{U_i \times U_i} U_i \times U_{i'}$) to see that $R_{i'} \to U_i \times U_{i'}$ is a closed immersion for $i'$ sufficiently large. This implies immediately that $R_{i'} \to U_{i'} \times U_{i'}$ is a closed immersion finishing the proof of the lemma. \end{proof} \begin{lemma} \label{lemma-limit-is-affine} Notation and assumptions as in Situation \ref{situation-descent}. If $X$ is affine, then there exists an $i$ such that $X_i$ is affine. \end{lemma} \begin{proof} Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $U = U_0 \times_{X_0} X$ and $U_i = U_0 \times_{X_0} X_i$ for $i \geq 0$. Since the transition morphisms are affine, the algebraic spaces $U_i$ and $U$ are affine. Thus $U \to X$ is an \'etale morphism of affine schemes. Hence we can write $X = \Spec(A)$, $U = \Spec(B)$ and $$B = A[x_1, \ldots, x_n]/(g_1, \ldots, g_n)$$ such that $\Delta = \det(\partial g_\lambda/\partial x_\mu)$ is invertible in $B$, see Algebra, Lemma \ref{algebra-lemma-etale-standard-smooth}. Set $A_i = \mathcal{O}_{X_i}(X_i)$. We have $A = \colim A_i$ by Lemma \ref{lemma-descend-section}. After increasing $0$ we may assume we have $g_{1, i}, \ldots, g_{n, i} \in A_i[x_1, \ldots, x_n]$ mapping to $g_1, \ldots, g_n$. Set $$B_i = A_i[x_1, \ldots, x_n]/(g_{1, i}, \ldots, g_{n, i})$$ for all $i \geq 0$. Increasing $0$ if necessary we may assume that $\Delta_i = \det(\partial g_{\lambda, i}/\partial x_\mu)$ is invertible in $B_i$ for all $i \geq 0$. Thus $A_i \to B_i$ is an \'etale ring map. After increasing $0$ we may assume also that $\Spec(B_i) \to \Spec(A_i)$ is surjective, see Limits, Lemma \ref{limits-lemma-descend-surjective}. Increasing $0$ yet again we may choose elements $h_{1, i}, \ldots, h_{n, i} \in \mathcal{O}_{U_i}(U_i)$ which map to the classes of $x_1, \ldots, x_n$ in $B = \mathcal{O}_U(U)$ and such that $g_{\lambda, i}(h_{\nu, i}) = 0$ in $\mathcal{O}_{U_i}(U_i)$. Thus we obtain a commutative diagram \begin{equation} \label{equation-to-show-cartesian} \vcenter{ \xymatrix{ X_i \ar[d] & U_i \ar[l] \ar[d] \\ \Spec(A_i) & \Spec(B_i) \ar[l] } } \end{equation} By construction $B_i = B_0 \otimes_{A_0} A_i$ and $B = B_0 \otimes_{A_0} A$. Consider the morphism $$f_0 : U_0 \longrightarrow X_0 \times_{\Spec(A_0)} \Spec(B_0)$$ This is a morphism of quasi-compact and quasi-separated algebraic spaces representable, separated and \'etale over $X_0$. The base change of $f_0$ to $X$ is an isomorphism by our choices. Hence Lemma \ref{lemma-descend-equality} guarantees that there exists an $i$ such that the base change of $f_0$ to $X_i$ is an isomorphism, in other words the diagram (\ref{equation-to-show-cartesian}) is cartesian. Thus Descent, Lemma \ref{descent-lemma-descent-data-sheaves} applied to the fppf covering $\{\Spec(B_i) \to \Spec(A_i)\}$ combined with Descent, Lemma \ref{descent-lemma-affine} give that $X_i \to \Spec(A_i)$ is representable by a scheme affine over $\Spec(A_i)$ as desired. (Of course it then also follows that $X_i = \Spec(A_i)$ but we don't need this.) \end{proof} \begin{lemma} \label{lemma-limit-is-scheme} Notation and assumptions as in Situation \ref{situation-descent}. If $X$ is a scheme, then there exists an $i$ such that $X_i$ is a scheme. \end{lemma} \begin{proof} Choose a finite affine open covering $X = \bigcup W_j$. By Lemma \ref{lemma-descend-opens} we can find an $i \in I$ and open subspaces $W_{j, i} \subset X_i$ whose base change to $X$ is $W_j \to X$. By Lemma \ref{lemma-limit-is-affine} we may assume that each $W_{j, i}$ is an affine scheme. This means that $X_i$ is a scheme (see for example Properties of Spaces, Section \ref{spaces-properties-section-schematic}). \end{proof} \begin{lemma} \label{lemma-finite-type-eventually-closed} Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $B$ with affine transition morphisms. Let $Y \to X$ be a morphism of algebraic spaces over $B$. \begin{enumerate} \item If $Y \to X$ is a closed immersion, $X_i$ quasi-compact, and $Y \to B$ locally of finite type, then $Y \to X_i$ is a closed immersion for $i$ large enough. \item If $Y \to X$ is an immersion, $X_i$ quasi-separated, $Y \to B$ locally of finite type, and $Y$ quasi-compact, then $Y \to X_i$ is an immersion for $i$ large enough. \item If $Y \to X$ is an isomorphism, $X_i$ quasi-compact, $X_i \to B$ locally of finite type, the transition morphisms $X_{i'} \to X_i$ are closed immersions, and $Y \to B$ is locally of finite presentation, then $Y \to X_i$ is an isomorphism for $i$ large enough. \item If $Y \to X$ is a monomorphism, $X_i$ quasi-separated, $Y \to B$ locally of finite type, and $Y$ quasi-compact, then $Y \to X_i$ is a monomorphism for $i$ large enough. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). Choose $0 \in I$. As $X_0$ is quasi-compact, we can choose an affine scheme $W$ and an \'etale morphism $W \to B$ such that the image of $|X_0| \to |B|$ is contained in $|W| \to |B|$. Choose an affine scheme $U_0$ and an \'etale morphism $U_0 \to X_0 \times_B W$ such that $U_0 \to X_0$ is surjective. (This is possible by our choice of $W$ and the fact that $X_0$ is quasi-compact; details omitted.) Let $V \to Y$, resp.\ $U \to X$, resp.\ $U_i \to X_i$ be the base change of $U_0 \to X_0$ (for $i \geq 0$). It suffices to prove that $V \to U_i$ is a closed immersion for $i$ sufficiently large. Thus we reduce to proving the result for $V \to U = \lim U_i$ over $W$. This follows from the case of schemes, which is Limits, Lemma \ref{limits-lemma-finite-type-eventually-closed}. \medskip\noindent Proof of (2). Choose $0 \in I$. Choose a quasi-compact open subspace $X'_0 \subset X_0$ such that $Y \to X_0$ factors through $X'_0$. After replacing $X_i$ by the inverse image of $X'_0$ for $i \geq 0$ we may assume all $X_i'$ are quasi-compact and quasi-separated. Let $U \subset X$ be a quasi-compact open such that $Y \to X$ factors through a closed immersion $Y \to U$ ($U$ exists as $Y$ is quasi-compact). By Lemma \ref{lemma-descend-opens} we may assume that $U = \lim U_i$ with $U_i \subset X_i$ quasi-compact open. By part (1) we see that $Y \to U_i$ is a closed immersion for some $i$. Thus (2) holds. \medskip\noindent Proof of (3). Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $U_i = X_i \times_{X_0} U_0$, $U = X \times_{X_0} U_0 = Y \times_{X_0} U_0$. Then $U = \lim U_i$ is a limit of affine schemes, the transition maps of the system are closed immersions, and $U \to U_0$ is of finite presentation (because $U \to B$ is locally of finite presentation and $U_0 \to B$ is locally of finite type and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-permanence}). Thus we've reduced to the following algebra fact: If $A = \lim A_i$ is a directed colimit of $R$-algebras with surjective transition maps and $A$ of finite presentation over $A_0$, then $A = A_i$ for some $i$. Namely, write $A = A_0/(f_1, \ldots, f_n)$. Pick $i$ such that $f_1, \ldots, f_n$ map to zero under the surjective map $A_0 \to A_i$. \medskip\noindent Proof of (4). Set $Z_i = Y \times_{X_i} Y$. As the transition morphisms $X_{i'} \to X_i$ are affine hence separated, the transition morphisms $Z_{i'} \to Z_i$ are closed immersions, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fibre-product-after-map}. We have $\lim Z_i = Y \times_X Y = Y$ as $Y \to X$ is a monomorphism. Choose $0 \in I$. Since $Y \to X_0$ is locally of finite type (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}) the morphism $Y \to Z_0$ is locally of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}). The morphisms $Z_i \to Z_0$ are locally of finite type (they are closed immersions). Finally, $Z_i = Y \times_{X_i} Y$ is quasi-compact as $X_i$ is quasi-separated and $Y$ is quasi-compact. Thus part (3) applies to $Y = \lim_{i \geq 0} Z_i$ over $Z_0$ and we conclude $Y = Z_i$ for some $i$. This proves (4) and the lemma. \end{proof} \begin{lemma} \label{lemma-eventually-separated} Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \begin{enumerate} \item $Y$ is quasi-separated, \item $X_i$ is quasi-compact and quasi-separated, \item the morphism $X \to Y$ is separated. \end{enumerate} Then $X_i \to Y$ is separated for all $i$ large enough. \end{lemma} \begin{proof} Let $0 \in I$. Choose an affine scheme $W$ and an \'etale morphism $W \to Y$ such that the image of $|W| \to |Y|$ contains the image of $|X_0| \to |Y|$. This is possible as $X_0$ is quasi-compact. It suffices to check that $W \times_Y X_i \to W$ is separated for some $i \geq 0$ because the diagonal of $W \times_Y X_i$ over $W$ is the base change of $X_i \to X_i \times_Y X_i$ by the surjective \'etale morphism $(X_i \times_Y X_i) \times_Y W \to X_i \times_Y X_i$. Since $Y$ is quasi-separated the algebraic spaces $W \times_Y X_i$ are quasi-compact (as well as quasi-separated). Thus we may base change to $W$ and assume $Y$ is an affine scheme. When $Y$ is an affine scheme, we have to show that $X_i$ is a separated algebraic space for $i$ large enough and we are given that $X$ is a separated algebraic space. Thus this case follows from Lemma \ref{lemma-descend-separated}. \end{proof} \begin{lemma} \label{lemma-eventually-affine} Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \begin{enumerate} \item $Y$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item $X \to Y$ affine. \end{enumerate} Then $X_i \to Y$ is affine for $i$ large enough. \end{lemma} \begin{proof} Choose an affine scheme $W$ and a surjective \'etale morphism $W \to Y$. Then $X \times_Y W$ is affine and it suffices to check that $X_i \times_Y W$ is affine for some $i$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-local}). This follows from Lemma \ref{lemma-limit-is-affine}. \end{proof} \begin{lemma} \label{lemma-eventually-finite} Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \begin{enumerate} \item $Y$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item the transition morphisms $X_{i'} \to X_i$ are finite, \item $X_i \to Y$ locally of finite type \item $X \to Y$ integral. \end{enumerate} Then $X_i \to Y$ is finite for $i$ large enough. \end{lemma} \begin{proof} Choose an affine scheme $W$ and a surjective \'etale morphism $W \to Y$. Then $X \times_Y W$ is finite over $W$ and it suffices to check that $X_i \times_Y W$ is finite over $W$ for some $i$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-local}). By Lemma \ref{lemma-limit-is-scheme} this reduces us to the case of schemes. In the case of schemes it follows from Limits, Lemma \ref{limits-lemma-eventually-finite}. \end{proof} \begin{lemma} \label{lemma-eventually-closed-immersion} Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \begin{enumerate} \item $Y$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item the transition morphisms $X_{i'} \to X_i$ are closed immersions, \item $X_i \to Y$ locally of finite type \item $X \to Y$ is a closed immersion. \end{enumerate} Then $X_i \to Y$ is a closed immersion for $i$ large enough. \end{lemma} \begin{proof} Choose an affine scheme $W$ and a surjective \'etale morphism $W \to Y$. Then $X \times_Y W$ is a closed subspace of $W$ and it suffices to check that $X_i \times_Y W$ is a closed subspace $W$ for some $i$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-local}). By Lemma \ref{lemma-limit-is-scheme} this reduces us to the case of schemes. In the case of schemes it follows from Limits, Lemma \ref{limits-lemma-eventually-closed-immersion}. \end{proof} \section{Descending properties of morphisms} \label{section-descent-of-properties} \noindent This section is the analogue of Section \ref{section-descent} for properties of morphisms. We will work in the following situation. \begin{situation} \label{situation-descent-property} Let $S$ be a scheme. Let $B = \lim B_i$ be a limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). Let $0 \in I$ and let $f_0 : X_0 \to Y_0$ be a morphism of algebraic spaces over $B_0$. Assume $B_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated. Let $f_i : X_i \to Y_i$ be the base change of $f_0$ to $B_i$ and let $f : X \to Y$ be the base change of $f_0$ to $B$. \end{situation} \begin{lemma} \label{lemma-descend-etale} With notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is \'etale, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is \'etale for some $i \geq 0$. \end{lemma} \begin{proof} Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to V_0 \times_{Y_0} X_0$. Diagram $$\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }$$ The vertical arrows are surjective and \'etale by construction. We can base change this diagram to $B_i$ or $B$ to get $$\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }$$ Note that $U_i, V_i, U, V$ are affine schemes, the vertical morphisms are surjective \'etale, and the limit of the morphisms $U_i \to V_i$ is $U \to V$. Recall that $X_i \to Y_i$ is \'etale if and only if $U_i \to V_i$ is \'etale and similarly $X \to Y$ is \'etale if and only if $U \to V$ is \'etale (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-local}). Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from Limits, Lemma \ref{limits-lemma-descend-etale}. \end{proof} \begin{lemma} \label{lemma-descend-smooth} With notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is smooth, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is smooth for some $i \geq 0$. \end{lemma} \begin{proof} Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to V_0 \times_{Y_0} X_0$. Diagram $$\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }$$ The vertical arrows are surjective and \'etale by construction. We can base change this diagram to $B_i$ or $B$ to get $$\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }$$ Note that $U_i, V_i, U, V$ are affine schemes, the vertical morphisms are surjective \'etale, and the limit of the morphisms $U_i \to V_i$ is $U \to V$. Recall that $X_i \to Y_i$ is smooth if and only if $U_i \to V_i$ is smooth and similarly $X \to Y$ is smooth if and only if $U \to V$ is smooth (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-smooth}). Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from Limits, Lemma \ref{limits-lemma-descend-smooth}. \end{proof} \begin{lemma} \label{lemma-descend-surjective} With notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is surjective, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is surjective for some $i \geq 0$. \end{lemma} \begin{proof} Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to V_0 \times_{Y_0} X_0$. Diagram $$\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }$$ The vertical arrows are surjective and \'etale by construction. We can base change this diagram to $B_i$ or $B$ to get $$\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }$$ Note that $U_i, V_i, U, V$ are affine schemes, the vertical morphisms are surjective \'etale, the limit of the morphisms $U_i \to V_i$ is $U \to V$, and the morphisms $U_i \to X_i \times_{Y_i} V_i$ and $U \to X \times_Y V$ are surjective (as base changes of $U_0 \to X_0 \times_{Y_0} V_0$). In particular, we see that $X_i \to Y_i$ is surjective if and only if $U_i \to V_i$ is surjective and similarly $X \to Y$ is surjective if and only if $U \to V$ is surjective. Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from the case of schemes (Limits, Lemma \ref{limits-lemma-descend-surjective}). \end{proof} \begin{lemma} \label{lemma-descend-universally-injective} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is universally injective, \item $f_0$ is locally of finite type, \end{enumerate} then $f_i$ is universally injective for some $i \geq 0$. \end{lemma} \begin{proof} Recall that a morphism $X \to Y$ is universally injective if and only if the diagonal $X \to X \times_Y X$ is surjective (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-universally-injective} and Lemma \ref{spaces-morphisms-lemma-universally-injective}). Observe that $X_0 \to X_0 \times_{Y_0} X_0$ is of locally of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}). Hence the lemma follows from Lemma \ref{lemma-descend-surjective} by considering the morphism $X_0 \to X_0 \times_{Y_0} X_0$. \end{proof} \begin{lemma} \label{lemma-descend-affine} Notation and assumptions as in Situation \ref{situation-descent-property}. If $f$ is affine, then $f_i$ is affine for some $i \geq 0$. \end{lemma} \begin{proof} Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Set $V_i = V_0 \times_{Y_0} Y_i$ and $V = V_0 \times_{Y_0} Y$. Since $f$ is affine we see that $V \times_Y X = \lim V_i \times_{Y_i} X_i$ is affine. By Lemma \ref{lemma-limit-is-affine} we see that $V_i \times_{Y_i} X_i$ is affine for some $i \geq 0$. For this $i$ the morphism $f_i$ is affine (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-local}). \end{proof} \begin{lemma} \label{lemma-descend-finite} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is finite, \item $f_0$ is locally of finite type, \end{enumerate} then $f_i$ is finite for some $i \geq 0$. \end{lemma} \begin{proof} Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Set $V_i = V_0 \times_{Y_0} Y_i$ and $V = V_0 \times_{Y_0} Y$. Since $f$ is finite we see that $V \times_Y X = \lim V_i \times_{Y_i} X_i$ is a scheme finite over $V$. By Lemma \ref{lemma-limit-is-affine} we see that $V_i \times_{Y_i} X_i$ is affine for some $i \geq 0$. Increasing $i$ if necessary we find that $V_i \times_{Y_i} X_i \to V_i$ is finite by Limits, Lemma \ref{limits-lemma-descend-finite-finite-presentation}. For this $i$ the morphism $f_i$ is finite (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-local}). \end{proof} \begin{lemma} \label{lemma-descend-closed-immersion} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a closed immersion, \item $f_0$ is locally of finite type, \end{enumerate} then $f_i$ is a closed immersion for some $i \geq 0$. \end{lemma} \begin{proof} Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Set $V_i = V_0 \times_{Y_0} Y_i$ and $V = V_0 \times_{Y_0} Y$. Since $f$ is a closed immersion we see that $V \times_Y X = \lim V_i \times_{Y_i} X_i$ is a closed subscheme of the affine scheme $V$. By Lemma \ref{lemma-limit-is-affine} we see that $V_i \times_{Y_i} X_i$ is affine for some $i \geq 0$. Increasing $i$ if necessary we find that $V_i \times_{Y_i} X_i \to V_i$ is a closed immersion by Limits, Lemma \ref{limits-lemma-descend-closed-immersion-finite-presentation}. For this $i$ the morphism $f_i$ is a closed immersion (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-local}). \end{proof} \begin{lemma} \label{lemma-descend-separated-morphism} Notation and assumptions as in Situation \ref{situation-descent-property}. If $f$ is separated, then $f_i$ is separated for some $i \geq 0$. \end{lemma} \begin{proof} Apply Lemma \ref{lemma-descend-closed-immersion} to the diagonal morphism $\Delta_{X_0/Y_0} : X_0 \to X_0 \times_{Y_0} X_0$. (Diagonal morphisms are locally of finite type and the fibre product $X_0 \times_{Y_0} X_0$ is quasi-compact and quasi-separated. Some details omitted.) \end{proof} \begin{lemma} \label{lemma-descend-isomorphism} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a isomorphism, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is a isomorphism for some $i \geq 0$. \end{lemma} \begin{proof} Being an isomorphism is equivalent to being \'etale, universally injective, and surjective, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-universally-injective-open}. Thus the lemma follows from Lemmas \ref{lemma-descend-etale}, \ref{lemma-descend-surjective}, and \ref{lemma-descend-universally-injective}. \end{proof} \begin{lemma} \label{lemma-descend-monomorphism} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a monomorphism, \item $f_0$ is locally of finite type, \end{enumerate} then $f_i$ is a monomorphism for some $i \geq 0$. \end{lemma} \begin{proof} Recall that a morphism is a monomorphism if and only if the diagonal is an isomorphism. The morphism $X_0 \to X_0 \times_{Y_0} X_0$ is locally of finite presentation by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}. Since $X_0 \times_{Y_0} X_0$ is quasi-compact and quasi-separated we conclude from Lemma \ref{lemma-descend-isomorphism} that $\Delta_i : X_i \to X_i \times_{Y_i} X_i$ is an isomorphism for some $i \geq 0$. For this $i$ the morphism $f_i$ is a monomorphism. \end{proof} \begin{lemma} \label{lemma-descend-flat} Notation and assumptions as in Situation \ref{situation-descent-property}. Let $\mathcal{F}_0$ be a quasi-coherent $\mathcal{O}_{X_0}$-module and denote $\mathcal{F}_i$ the pullback to $X_i$ and $\mathcal{F}$ the pullback to $X$. If \begin{enumerate} \item $\mathcal{F}$ is flat over $Y$, \item $\mathcal{F}_0$ is of finite presentation, and \item $f_0$ is locally of finite presentation, \end{enumerate} then $\mathcal{F}_i$ is flat over $Y_i$ for some $i \geq 0$. In particular, if $f_0$ is locally of finite presentation and $f$ is flat, then $f_i$ is flat for some $i \geq 0$. \end{lemma} \begin{proof} Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to V_0 \times_{Y_0} X_0$. Diagram $$\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }$$ The vertical arrows are surjective and \'etale by construction. We can base change this diagram to $B_i$ or $B$ to get $$\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }$$ Note that $U_i, V_i, U, V$ are affine schemes, the vertical morphisms are surjective \'etale, and the limit of the morphisms $U_i \to V_i$ is $U \to V$. Recall that $\mathcal{F}_i$ is flat over $Y_i$ if and only if $\mathcal{F}_i|_{U_i}$ is flat over $V_i$ and similarly $\mathcal{F}$ is flat over $Y$ if and only if $\mathcal{F}|_U$ is flat over $V$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-flat}). Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from Limits, Lemma \ref{limits-lemma-descend-module-flat-finite-presentation}. \end{proof} \begin{lemma} \label{lemma-eventually-proper} Assumptions and notation as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is proper, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i$ such that $f_i$ is proper. \end{lemma} \begin{proof} Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Set $V_i = Y_i \times_{Y_0} V_0$ and $V = Y \times_{Y_0} V_0$. It suffices to prove that the base change of $f_i$ to $V_i$ is proper, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-proper-local}. Thus we may assume $Y_0$ is affine. \medskip\noindent By Lemma \ref{lemma-descend-separated-morphism} we see that $f_i$ is separated for some $i \geq 0$. Replacing $0$ by $i$ we may assume that $f_0$ is separated. Observe that $f_0$ is quasi-compact. Thus $f_0$ is separated and of finite type. By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow} we can choose a diagram $$\xymatrix{ X_0 \ar[rd] & X_0' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_{Y_0} \ar[dl] \\ & Y_0 & }$$ where $X_0' \to \mathbf{P}^n_{Y_0}$ is an immersion, and $\pi : X_0' \to X_0$ is proper and surjective. Introduce $X' = X_0' \times_{Y_0} Y$ and $X_i' = X_0' \times_{Y_0} Y_i$. By Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-composition-proper} and \ref{spaces-morphisms-lemma-base-change-proper} we see that $X' \to Y$ is proper. Hence $X' \to \mathbf{P}^n_Y$ is a closed immersion (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-closed-permanence}). By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper} it suffices to prove that $X'_i \to Y_i$ is proper for some $i$. By Lemma \ref{lemma-descend-closed-immersion} we find that $X'_i \to \mathbf{P}^n_{Y_i}$ is a closed immersion for $i$ large enough. Then $X'_i \to Y_i$ is proper and we win. \end{proof} \begin{lemma} \label{lemma-eventually-relative-dimension} Assumptions and notation as in Situation \ref{situation-descent-property}. Let $d \geq 0$. If \begin{enumerate} \item $f$ has relative dimension $\leq d$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-dimension}), and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i$ such that $f_i$ has relative dimension $\leq d$. \end{lemma} \begin{proof} Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to V_0 \times_{Y_0} X_0$. Diagram $$\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }$$ The vertical arrows are surjective and \'etale by construction. We can base change this diagram to $B_i$ or $B$ to get $$\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }$$ Note that $U_i, V_i, U, V$ are affine schemes, the vertical morphisms are surjective \'etale, and the limit of the morphisms $U_i \to V_i$ is $U \to V$. In this situation $X_i \to Y_i$ has relative dimension $\leq d$ if and only if $U_i \to V_i$ has relative dimension $\leq d$ (as defined in Morphisms, Definition \ref{morphisms-definition-relative-dimension-d}). To see the equivalence, use that the definition for morphisms of algebraic spaces involves Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-dimension-fibre} which uses \'etale localization. The same is true for $X \to Y$ and $U \to V$. Since $f_0$ is locally of finite type, so is the morphism $U_0 \to V_0$. Hence the lemma follows from the more general Limits, Lemma \ref{limits-lemma-limit-dimension}. \end{proof} \section{Descending relative objects} \label{section-descending-relative} \noindent The following lemma is typical of the type of results in this section. \begin{lemma} \label{lemma-descend-finite-presentation} Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces over $S$. Assume \begin{enumerate} \item the morphisms $f_{ii'} : X_i \to X_{i'}$ are affine, \item the spaces $X_i$ are quasi-compact and quasi-separated. \end{enumerate} Let $X = \lim_i X_i$. Then the category of algebraic spaces of finite presentation over $X$ is the colimit over $I$ of the categories of algebraic spaces of finite presentation over $X_i$. \end{lemma} \begin{proof} Pick $0 \in I$. Choose a surjective \'etale morphism $U_0 \to X_0$ where $U_0$ is an affine scheme (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Set $U_i = X_i \times_{X_0} U_0$. Set $R_0 = U_0 \times_{X_0} U_0$ and $R_i = R_0 \times_{X_0} X_i$. Denote $s_i, t_i : R_i \to U_i$ and $s, t : R \to U$ the two projections. In the proof of Lemma \ref{lemma-directed-inverse-system-has-limit} we have seen that there exists a presentation $X = U/R$ with $U = \lim U_i$ and $R = \lim R_i$. Note that $U_i$ and $U$ are affine and that $R_i$ and $R$ are quasi-compact and separated (as $X_i$ is quasi-separated). Let $Y$ be an algebraic space over $S$ and let $Y \to X$ be a morphism of finite presentation. Set $V = U \times_X Y$. This is an algebraic space of finite presentation over $U$. Choose an affine scheme $W$ and a surjective \'etale morphism $W \to V$. Then $W \to Y$ is surjective \'etale as well. Set $R' = W \times_Y W$ so that $Y = W/R'$ (see Spaces, Section \ref{spaces-section-presentations}). Note that $W$ is a scheme of finite presentation over $U$ and that $R'$ is a scheme of finite presentation over $R$ (details omitted). By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} we can find an index $i$ and a morphism of schemes $W_i \to U_i$ of finite presentation whose base change to $U$ gives $W \to U$. Similarly we can find, after possibly increasing $i$, a scheme $R'_i$ of finite presentation over $R_i$ whose base change to $R$ is $R'$. The projection morphisms $s', t' : R' \to W$ are morphisms over the projection morphisms $s, t : R \to U$. Hence we can view $s'$, resp.\ $t'$ as a morphism between schemes of finite presentation over $U$ (with structure morphism $R' \to U$ given by $R' \to R$ followed by $s$, resp.\ $t$). Hence we can apply Limits, Lemma \ref{limits-lemma-descend-finite-presentation} again to see that, after possibly increasing $i$, there exist morphisms $s'_i, t'_i : R'_i \to W_i$, whose base change to $U$ is $S', t'$. By Limits, Lemmas \ref{limits-lemma-descend-etale} and \ref{limits-lemma-descend-monomorphism} we may assume that $s'_i, t'_i$ are \'etale and that $j'_i : R'_i \to W_i \times_{X_i} W_i$ is a monomorphism (here we view $j'_i$ as a morphism of schemes of finite presentation over $U_i$ via one of the projections -- it doesn't matter which one). Setting $Y_i = W_i/R'_i$ (see Spaces, Theorem \ref{spaces-theorem-presentation}) we obtain an algebraic space of finite presentation over $X_i$ whose base change to $X$ is isomorphic to $Y$. \medskip\noindent This shows that every algebraic space of finite presentation over $X$ comes from an algebraic space of finite presentation over some $X_i$, i.e., it shows that the functor of the lemma is essentially surjective. To show that it is fully faithful, consider an index $0 \in I$ and two algebraic spaces $Y_0, Z_0$ of finite presentation over $X_0$. Set $Y_i = X_i \times_{X_0} Y_0$, $Y = X \times_{X_0} Y_0$, $Z_i = X_i \times_{X_0} Z_0$, and $Z = X \times_{X_0} Z_0$. Let $\alpha : Y \to Z$ be a morphism of algebraic spaces over $X$. Choose a surjective \'etale morphism $V_0 \to Y_0$ where $V_0$ is an affine scheme. Set $V_i = V_0 \times_{Y_0} Y_i$ and $V = V_0 \times_{Y_0} Y$ which are affine schemes endowed with surjective \'etale morphisms to $Y_i$ and $Y$. The composition $V \to Y \to Z \to Z_0$ comes from a (essentially unique) morphism $V_i \to Z_0$ for some $i \geq 0$ by Proposition \ref{proposition-characterize-locally-finite-presentation} (applied to $Z_0 \to X_0$ which is of finite presentation by assumption). After increasing $i$ the two compositions $$V_i \times_{Y_i} V_i \to V_i \to Z_0$$ are equal as this is true in the limit. Hence we obtain a (essentially unique) morphism $Y_i \to Z_0$. Since this is a morphism over $X_0$ it induces a morphism into $Z_i = Z_0 \times_{X_0} X_i$ as desired. \end{proof} \begin{lemma} \label{lemma-descend-modules-finite-presentation} With notation and assumptions as in Lemma \ref{lemma-descend-finite-presentation}. The category of $\mathcal{O}_X$-modules of finite presentation is the colimit over $I$ of the categories $\mathcal{O}_{X_i}$-modules of finite presentation. \end{lemma} \begin{proof} Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $U_i = X_i \times_{X_0} U_0$. Set $R_0 = U_0 \times_{X_0} U_0$ and $R_i = R_0 \times_{X_0} X_i$. Denote $s_i, t_i : R_i \to U_i$ and $s, t : R \to U$ the two projections. In the proof of Lemma \ref{lemma-directed-inverse-system-has-limit} we have seen that there exists a presentation $X = U/R$ with $U = \lim U_i$ and $R = \lim R_i$. Note that $U_i$ and $U$ are affine and that $R_i$ and $R$ are quasi-compact and separated (as $X_i$ is quasi-separated). Moreover, it is also true that $R \times_{s, U, t} R = \colim R_i \times_{s_i, U_i, t_i} R_i$. Thus we know that $\QCoh(\mathcal{O}_U) = \colim \QCoh(\mathcal{O}_{U_i})$, $\QCoh(\mathcal{O}_R) = \colim \QCoh(\mathcal{O}_{R_i})$, and $\QCoh(\mathcal{O}_{R \times_{s, U, t} R}) = \colim \QCoh(\mathcal{O}_{R_i \times_{s_i, U_i, t_i} R_i})$ by Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation}. We have $\QCoh(\mathcal{O}_X) = \QCoh(U, R, s, t, c)$ and $\QCoh(\mathcal{O}_{X_i}) = \QCoh(U_i, R_i, s_i, t_i, c_i)$, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-quasi-coherent}. Thus the result follows formally. \end{proof} \begin{lemma} \label{lemma-descend-invertible-modules} With notation and assumptions as in Lemma \ref{lemma-descend-finite-presentation}. Then any invertible $\mathcal{O}_X$-module is the pullback of an invertible $\mathcal{O}_{X_i}$-module for some $i$. \end{lemma} \begin{proof} Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Since invertible modules are of finite presentation we can find an $i$ and modules $\mathcal{L}_i$ and $\mathcal{N}_i$ of finite presentation over $X_i$ such that $f_i^*\mathcal{L}_i \cong \mathcal{L}$ and $f_i^*\mathcal{N}_i \cong \mathcal{L}^{\otimes -1}$, see Lemma \ref{lemma-descend-modules-finite-presentation}. Since pullback commutes with tensor product we see that $f_i^*(\mathcal{L}_i \otimes_{\mathcal{O}_{X_i}} \mathcal{N}_i)$ is isomorphic to $\mathcal{O}_X$. Since the tensor product of finitely presented modules is finitely presented, the same lemma implies that $f_{i'i}^*\mathcal{L}_i \otimes_{\mathcal{O}_{X_{i'}}} f_{i'i}^*\mathcal{N}_i$ is isomorphic to $\mathcal{O}_{X_{i'}}$ for some $i' \geq i$. It follows that $f_{i'i}^*\mathcal{L}_i$ is invertible (Modules on Sites, Lemma \ref{sites-modules-lemma-invertible}) and the proof is complete. \end{proof} \section{Absolute Noetherian approximation} \label{section-approximation} \noindent The following result is \cite[Theorem 1.2.2]{CLO}. A key ingredient in the proof is Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}. \begin{proposition} \label{proposition-approximate} \begin{reference} Our proof follows closely the proof given in \cite[Theorem 1.2.2]{CLO}. \end{reference} Let $X$ be a quasi-compact and quasi-separated algebraic space over $\Spec(\mathbf{Z})$. There exist a directed set $I$ and an inverse system of algebraic spaces $(X_i, f_{ii'})$ over $I$ such that \begin{enumerate} \item the transition morphisms $f_{ii'}$ are affine \item each $X_i$ is quasi-separated and of finite type over $\mathbf{Z}$, and \item $X = \lim X_i$. \end{enumerate} \end{proposition} \begin{proof} We apply Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated} to get open subspaces $U_p \subset X$, schemes $V_p$, and morphisms $f_p : V_p \to U_p$ with properties as stated. Note that $f_n : V_n \to U_n$ is an \'etale morphism of algebraic spaces whose restriction to the inverse image of $T_n = (V_n)_{red}$ is an isomorphism. Hence $f_n$ is an isomorphism, for example by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-universally-injective-open}. In particular $U_n$ is a quasi-compact and separated scheme. Thus we can write $U_n = \lim U_{n, i}$ as a directed limit of schemes of finite type over $\mathbf{Z}$ with affine transition morphisms, see Limits, Proposition \ref{limits-proposition-approximate}. Thus, applying descending induction on $p$, we see that we have reduced to the problem posed in the following paragraph. \medskip\noindent Here we have $U \subset X$, $U = \lim U_i$, $Z \subset X$, and $f : V \to X$ with the following properties \begin{enumerate} \item $X$ is a quasi-compact and quasi-separated algebraic space, \item $V$ is a quasi-compact and separated scheme, \item $U \subset X$ is a quasi-compact open subspace, \item $(U_i, g_{ii'})$ is a directed inverse system of quasi-separated algebraic spaces of finite type over $\mathbf{Z}$ with affine transition morphisms whose limit is $U$, \item $Z \subset X$ is a closed subspace such that $|X| = |U| \amalg |Z|$, \item $f : V \to X$ is a surjective \'etale morphism such that $f^{-1}(Z) \to Z$ is an isomorphism. \end{enumerate} Problem: Show that the conclusion of the proposition holds for $X$. \medskip\noindent Note that $W = f^{-1}(U) \subset V$ is a quasi-compact open subscheme \'etale over $U$. Hence we may apply Lemmas \ref{lemma-descend-finite-presentation} and \ref{lemma-descend-etale} to find an index $0 \in I$ and an \'etale morphism $W_0 \to U_0$ of finite presentation whose base change to $U$ produces $W$. Setting $W_i = W_0 \times_{U_0} U_i$ we see that $W = \lim_{i \geq 0} W_i$. After increasing $0$ we may assume the $W_i$ are schemes, see Lemma \ref{lemma-limit-is-scheme}. Moreover, $W_i$ is of finite type over $\mathbf{Z}$. \medskip\noindent Apply Limits, Lemma \ref{limits-lemma-approximate} to $W = \lim_{i \geq 0} W_i$ and the inclusion $W \subset V$. Replace $I$ by the directed set $J$ found in that lemma. This allows us to write $V$ as a directed limit $V = \lim V_i$ of finite type schemes over $\mathbf{Z}$ with affine transition maps such that each $V_i$ contains $W_i$ as an open subscheme (compatible with transition morphisms). For each $i$ we can form the push out $$\xymatrix{ W_i \ar[r] \ar[d]_\Delta & V_i \ar[d] \\ W_i \times_{U_i} W_i \ar[r] & R_i }$$ in the category of schemes. Namely, the left vertical and upper horizontal arrows are open immersions of schemes. In other words, we can construct $R_i$ as the glueing of $V_i$ and $W_i \times_{U_i} W_i$ along the common open $W_i$ (see Schemes, Section \ref{schemes-section-glueing-schemes}). Note that the \'etale projection maps $W_i \times_{U_i} W_i \to W_i$ extend to \'etale morphisms $s_i, t_i : R_i \to V_i$. It is clear that the morphism $j_i = (t_i, s_i) : R_i \to V_i \times V_i$ is an \'etale equivalence relation on $V_i$. Note that $W_i \times_{U_i} W_i$ is quasi-compact (as $U_i$ is quasi-separated and $W_i$ quasi-compact) and $V_i$ is quasi-compact, hence $R_i$ is quasi-compact. For $i \geq i'$ the diagram \begin{equation} \label{equation-cartesian} \vcenter{ \xymatrix{ R_i \ar[r] \ar[d]_{s_i} & R_{i'} \ar[d]^{s_{i'}} \\ V_i \ar[r] & V_{i'} } } \end{equation} is cartesian because $$(W_{i'} \times_{U_{i'}} W_{i'}) \times_{U_{i'}} U_i = W_{i'} \times_{U_{i'}} U_i \times_{U_i} U_i \times_{U_{i'}} W_{i'} = W_i \times_{U_i} W_i.$$ Consider the algebraic space $X_i = V_i/R_i$ (see Spaces, Theorem \ref{spaces-theorem-presentation}). As $V_i$ is of finite type over $\mathbf{Z}$ and $R_i$ is quasi-compact we see that $X_i$ is quasi-separated and of finite type over $\mathbf{Z}$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-separated} and Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-surjection-from-quasi-compact} and \ref{spaces-morphisms-lemma-finite-type-local}). As the construction of $R_i$ above is compatible with transition morphisms, we obtain morphisms of algebraic spaces $X_i \to X_{i'}$ for $i \geq i'$. The commutative diagrams $$\xymatrix{ V_i \ar[r] \ar[d] & V_{i'} \ar[d] \\ X_i \ar[r] & X_{i'} }$$ are cartesian as (\ref{equation-cartesian}) is cartesian, see Groupoids, Lemma \ref{groupoids-lemma-criterion-fibre-product}. Since $V_i \to V_{i'}$ is affine, this implies that $X_i \to X_{i'}$ is affine, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-local}. Thus we can form the limit $X' = \lim X_i$ by Lemma \ref{lemma-directed-inverse-system-has-limit}. We claim that $X \cong X'$ which finishes the proof of the proposition. \medskip\noindent Proof of the claim. Set $R = \lim R_i$. By construction the algebraic space $X'$ comes equipped with a surjective \'etale morphism $V \to X'$ such that $$V \times_{X'} V \cong R$$ (use Lemma \ref{lemma-directed-inverse-system-has-limit}). By construction $\lim W_i \times_{U_i} W_i = W \times_U W$ and $V = \lim V_i$ so that $R$ is the union of $W \times_U W$ and $V$ glued along $W$. Property (6) implies the projections $V \times_X V \to V$ are isomorphisms over $f^{-1}(Z) \subset V$. Hence the scheme $V \times_X V$ is the union of the opens $\Delta_{V/X}(V)$ and $W \times_U W$ which intersect along $\Delta_{W/X}(W)$. We conclude that there exists a unique isomorphism $R \cong V \times_X V$ compatible with the projections to $V$. Since $V \to X$ and $V \to X'$ are surjective \'etale we see that $$X = V/ V \times_X V = V/R = V/V \times_{X'} V = X'$$ by Spaces, Lemma \ref{spaces-lemma-space-presentation} and we win. \end{proof} \section{Applications} \label{section-applications} \noindent The following lemma can also be deduced directly from Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated} without passing through absolute Noetherian approximation. \begin{lemma} \label{lemma-colimit-finitely-presented} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Every quasi-coherent $\mathcal{O}_X$-module is a filtered colimit of finitely presented $\mathcal{O}_X$-modules. \end{lemma} \begin{proof} We may view as an algebraic space over $\Spec(\mathbf{Z})$, see Spaces, Definition \ref{spaces-definition-base-change} and Properties of Spaces, Definition \ref{spaces-properties-definition-separated}. Thus we may apply Proposition \ref{proposition-approximate} and write $X = \lim X_i$ with $X_i$ of finite presentation over $\mathbf{Z}$. Thus $X_i$ is a Noetherian algebraic space, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-noetherian}. The morphism $X \to X_i$ is affine, see Lemma \ref{lemma-directed-inverse-system-has-limit}. Conclusion by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-direct-colimit-finite-presentation}. \end{proof} \noindent The rest of this section consists of straightforward applications of Lemma \ref{lemma-colimit-finitely-presented}. \begin{lemma} \label{lemma-directed-colimit-finite-type} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is the directed colimit of its finite type quasi-coherent submodules. \end{lemma} \begin{proof} If $\mathcal{G}, \mathcal{H} \subset \mathcal{F}$ are finite type quasi-coherent $\mathcal{O}_X$-submodules then the image of $\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}$ is another finite type quasi-coherent $\mathcal{O}_X$-submodule which contains both of them. In this way we see that the system is directed. To show that $\mathcal{F}$ is the colimit of this system, write $\mathcal{F} = \colim_i \mathcal{F}_i$ as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma \ref{lemma-colimit-finitely-presented}. Then the images $\mathcal{G}_i = \Im(\mathcal{F}_i \to \mathcal{F})$ are finite type quasi-coherent subsheaves of $\mathcal{F}$. Since $\mathcal{F}$ is the colimit of these the result follows. \end{proof} \begin{lemma} \label{lemma-finite-directed-colimit-surjective-maps} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Then we can write $\mathcal{F} = \lim \mathcal{F}_i$ where each $\mathcal{F}_i$ is an $\mathcal{O}_X$-module of finite presentation and all transition maps $\mathcal{F}_i \to \mathcal{F}_{i'}$ surjective. \end{lemma} \begin{proof} Write $\mathcal{F} = \colim \mathcal{G}_i$ as a filtered colimit of finitely presented $\mathcal{O}_X$-modules (Lemma \ref{lemma-colimit-finitely-presented}). We claim that $\mathcal{G}_i \to \mathcal{F}$ is surjective for some $i$. Namely, choose an \'etale surjection $U \to X$ where $U$ is an affine scheme. Choose finitely many sections $s_k \in \mathcal{F}(U)$ generating $\mathcal{F}|_U$. Since $U$ is affine we see that $s_k$ is in the image of $\mathcal{G}_i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{G}_i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{G}_i$ be the kernel of the map $\mathcal{G}_i \to \mathcal{F}$. Write $\mathcal{K} = \colim \mathcal{K}_a$ as the filtered colimit of its finite type quasi-coherent submodules (Lemma \ref{lemma-directed-colimit-finite-type}). Then $\mathcal{F} = \colim \mathcal{G}_i/\mathcal{K}_a$ is a solution to the problem posed by the lemma. \end{proof} \noindent Let $X$ be an algebraic space. In the following lemma we use the notion of a {\it finitely presented quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{A}$}. This means that for every affine $U = \Spec(R)$ \'etale over $X$ we have $\mathcal{A}|_U = \widetilde{A}$ where $A$ is a (commutative) $R$-algebra which is of finite presentation as an $R$-algebra. \begin{lemma} \label{lemma-algebra-directed-colimit-finite-presentation} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A}$ is a directed colimit of finitely presented quasi-coherent $\mathcal{O}_X$-algebras. \end{lemma} \begin{proof} First we write $\mathcal{A} = \colim_i \mathcal{F}_i$ as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma \ref{lemma-colimit-finitely-presented}. For each $i$ let $\mathcal{B}_i = \text{Sym}(\mathcal{F}_i)$ be the symmetric algebra on $\mathcal{F}_i$ over $\mathcal{O}_X$. Write $\mathcal{I}_i = \Ker(\mathcal{B}_i \to \mathcal{A})$. Write $\mathcal{I}_i = \colim_j \mathcal{F}_{i, j}$ where $\mathcal{F}_{i, j}$ is a finite type quasi-coherent submodule of $\mathcal{I}_i$, see Lemma \ref{lemma-directed-colimit-finite-type}. Set $\mathcal{I}_{i, j} \subset \mathcal{I}_i$ equal to the $\mathcal{B}_i$-ideal generated by $\mathcal{F}_{i, j}$. Set $\mathcal{A}_{i, j} = \mathcal{B}_i/\mathcal{I}_{i, j}$. Then $\mathcal{A}_{i, j}$ is a quasi-coherent finitely presented $\mathcal{O}_X$-algebra. Define $(i, j) \leq (i', j')$ if $i \leq i'$ and the map $\mathcal{B}_i \to \mathcal{B}_{i'}$ maps the ideal $\mathcal{I}_{i, j}$ into the ideal $\mathcal{I}_{i', j'}$. Then it is clear that $\mathcal{A} = \colim_{i, j} \mathcal{A}_{i, j}$. \end{proof} \noindent Let $X$ be an algebraic space. In the following lemma we use the notion of a {\it quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{A}$ of finite type}. This means that for every affine $U = \Spec(R)$ \'etale over $X$ we have $\mathcal{A}|_U = \widetilde{A}$ where $A$ is a (commutative) $R$-algebra which is of finite type as an $R$-algebra. \begin{lemma} \label{lemma-algebra-directed-colimit-finite-type} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A}$ is the directed colimit of its finite type quasi-coherent $\mathcal{O}_X$-subalgebras. \end{lemma} \begin{proof} Omitted. Hint: Compare with the proof of Lemma \ref{lemma-directed-colimit-finite-type}. \end{proof} \noindent Let $X$ be an algebraic space. In the following lemma we use the notion of a {\it finite (resp.\ integral) quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{A}$}. This means that for every affine $U = \Spec(R)$ \'etale over $X$ we have $\mathcal{A}|_U = \widetilde{A}$ where $A$ is a (commutative) $R$-algebra which is finite (resp.\ integral) as an $R$-algebra. \begin{lemma} \label{lemma-finite-algebra-directed-colimit-finite-finitely-presented} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a finite quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A} = \colim \mathcal{A}_i$ is a directed colimit of finite and finitely presented quasi-coherent $\mathcal{O}_X$-algebras with surjective transition maps. \end{lemma} \begin{proof} By Lemma \ref{lemma-finite-directed-colimit-surjective-maps} there exists a finitely presented $\mathcal{O}_X$-module $\mathcal{F}$ and a surjection $\mathcal{F} \to \mathcal{A}$. Using the algebra structure we obtain a surjection $$\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}) \longrightarrow \mathcal{A}$$ Denote $\mathcal{J}$ the kernel. Write $\mathcal{J} = \colim \mathcal{E}_i$ as a filtered colimit of finite type $\mathcal{O}_X$-submodules $\mathcal{E}_i$ (Lemma \ref{lemma-directed-colimit-finite-type}). Set $$\mathcal{A}_i = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})/(\mathcal{E}_i)$$ where $(\mathcal{E}_i)$ indicates the ideal sheaf generated by the image of $\mathcal{E}_i \to \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})$. Then each $\mathcal{A}_i$ is a finitely presented $\mathcal{O}_X$-algebra, the transition maps are surjective, and $\mathcal{A} = \colim \mathcal{A}_i$. To finish the proof we still have to show that $\mathcal{A}_i$ is a finite $\mathcal{O}_X$-algebra for $i$ sufficiently large. To do this we choose an \'etale surjective map $U \to X$ where $U$ is an affine scheme. Take generators $f_1, \ldots, f_m \in \Gamma(U, \mathcal{F})$. As $\mathcal{A}(U)$ is a finite $\mathcal{O}_X(U)$-algebra we see that for each $j$ there exists a monic polynomial $P_j \in \mathcal{O}(U)[T]$ such that $P_j(f_j)$ is zero in $\mathcal{A}(U)$. Since $\mathcal{A} = \colim \mathcal{A}_i$ by construction, we have $P_j(f_j) = 0$ in $\mathcal{A}_i(U)$ for all sufficiently large $i$. For such $i$ the algebras $\mathcal{A}_i$ are finite. \end{proof} \begin{lemma} \label{lemma-integral-algebra-directed-colimit-finite} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be an integral quasi-coherent $\mathcal{O}_X$-algebra. Then \begin{enumerate} \item $\mathcal{A}$ is the directed colimit of its finite quasi-coherent $\mathcal{O}_X$-subalgebras, and \item $\mathcal{A}$ is a directed colimit of finite and finitely presented $\mathcal{O}_X$-algebras. \end{enumerate} \end{lemma} \begin{proof} By Lemma \ref{lemma-algebra-directed-colimit-finite-type} we have $\mathcal{A} = \colim \mathcal{A}_i$ where $\mathcal{A}_i \subset \mathcal{A}$ runs through the quasi-coherent $\mathcal{O}_X$-sub algebras of finite type. Any finite type quasi-coherent $\mathcal{O}_X$-subalgebra of $\mathcal{A}$ is finite (use Algebra, Lemma \ref{algebra-lemma-characterize-finite-in-terms-of-integral} on affine schemes \'etale over $X$). This proves (1). \medskip\noindent To prove (2), write $\mathcal{A} = \colim \mathcal{F}_i$ as a colimit of finitely presented $\mathcal{O}_X$-modules using Lemma \ref{lemma-colimit-finitely-presented}. For each $i$, let $\mathcal{J}_i$ be the kernel of the map $$\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i) \longrightarrow \mathcal{A}$$ For $i' \geq i$ there is an induced map $\mathcal{J}_i \to \mathcal{J}_{i'}$ and we have $\mathcal{A} = \colim \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i$. Moreover, the quasi-coherent $\mathcal{O}_X$-algebras $\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i$ are finite (see above). Write $\mathcal{J}_i = \colim \mathcal{E}_{ik}$ as a colimit of finitely presented $\mathcal{O}_X$-modules. Given $i' \geq i$ and $k$ there exists a $k'$ such that we have a map $\mathcal{E}_{ik} \to \mathcal{E}_{i'k'}$ making $$\xymatrix{ \mathcal{J}_i \ar[r] & \mathcal{J}_{i'} \\ \mathcal{E}_{ik} \ar[u] \ar[r] & \mathcal{E}_{i'k'} \ar[u] }$$ commute. This follows from Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-finite-presentation-quasi-compact-colimit}. This induces a map $$\mathcal{A}_{ik} = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/(\mathcal{E}_{ik}) \longrightarrow \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_{i'})/(\mathcal{E}_{i'k'}) = \mathcal{A}_{i'k'}$$ where $(\mathcal{E}_{ik})$ denotes the ideal generated by $\mathcal{E}_{ik}$. The quasi-coherent $\mathcal{O}_X$-algebras $\mathcal{A}_{ki}$ are of finite presentation and finite for $k$ large enough (see proof of Lemma \ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented}). Finally, we have $$\colim \mathcal{A}_{ik} = \colim \mathcal{A}_i = \mathcal{A}$$ Namely, the first equality was shown in the proof of Lemma \ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented} and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_i$. \end{proof} \begin{lemma} \label{lemma-extend} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G} \subset \mathcal{F}|_U$ be a quasi-coherent $\mathcal{O}_U$-submodule which is of finite type. Then there exists a quasi-coherent submodule $\mathcal{G}' \subset \mathcal{F}$ which is of finite type such that $\mathcal{G}'|_U = \mathcal{G}$. \end{lemma} \begin{proof} Denote $j : U \to X$ the inclusion morphism. As $X$ is quasi-separated and $U$ quasi-compact, the morphism $j$ is quasi-compact. Hence $j_*\mathcal{G} \subset j_*\mathcal{F}|_U$ are quasi-coherent modules on $X$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}). Let $\mathcal{H} = \Ker(j_*\mathcal{G} \oplus \mathcal{F} \to j_*\mathcal{F}|_U)$. Then $\mathcal{H}|_U = \mathcal{G}$. By Lemma \ref{lemma-directed-colimit-finite-type} we can find a finite type quasi-coherent submodule $\mathcal{H}' \subset \mathcal{H}$ such that $\mathcal{H}'|_U = \mathcal{H}|_U = \mathcal{G}$. Set $\mathcal{G}' = \Im(\mathcal{H}' \to \mathcal{F})$ to conclude. \end{proof} \section{Relative approximation} \label{section-relative-approximation} \noindent The title of this section refers to the following result. \begin{lemma} \label{lemma-relative-approximation} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that \begin{enumerate} \item $X$ is quasi-compact and quasi-separated, and \item $Y$ is quasi-separated. \end{enumerate} Then $X = \lim X_i$ is a limit of a directed inverse system of algebraic spaces $X_i$ of finite presentation over $Y$ with affine transition morphisms over $Y$. \end{lemma} \begin{proof} Since $|f|(|X|)$ is quasi-compact we may replace $Y$ by a quasi-compact open subspace whose set of points contains $|f|(|X|)$. Hence we may assume $Y$ is quasi-compact as well. Write $X = \lim X_a$ and $Y = \lim Y_b$ as in Proposition \ref{proposition-approximate}, i.e., with $X_a$ and $Y_b$ of finite type over $\mathbf{Z}$ and with affine transition morphisms. By Proposition \ref{proposition-characterize-locally-finite-presentation} we find that for each $b$ there exists an $a$ and a morphism $f_{a, b} : X_a \to Y_b$ making the diagram $$\xymatrix{ X \ar[d] \ar[r] & Y \ar[d] \\ X_a \ar[r] & Y_b }$$ commute. Moreover the same proposition implies that, given a second triple $(a', b', f_{a', b'})$, there exists an $a'' \geq a'$ such that the compositions $X_{a''} \to X_a \to X_b$ and $X_{a''} \to X_{a'} \to X_{b'} \to X_b$ are equal. Consider the set of triples $(a, b, f_{a, b})$ endowed with the preordering $$(a, b, f_{a, b}) \geq (a', b', f_{a', b'}) \Leftrightarrow a \geq a',\ b' \geq b,\text{ and } f_{a', b'} \circ h_{a, a'} = g_{b', b} \circ f_{a, b}$$ where $h_{a, a'} : X_a \to X_{a'}$ and $g_{b', b} : Y_{b'} \to Y_b$ are the transition morphisms. The remarks above show that this system is directed. It follows formally from the equalities $X = \lim X_a$ and $Y = \lim Y_b$ that $$X = \lim_{(a, b, f_{a, b})} X_a \times_{f_{a, b}, Y_b} Y.$$ where the limit is over our directed system above. The transition morphisms $X_a \times_{Y_b} Y \to X_{a'} \times_{Y_{b'}} Y$ are affine as the composition $$X_a \times_{Y_b} Y \to X_a \times_{Y_{b'}} Y \to X_{a'} \times_{Y_{b'}} Y$$ where the first morphism is a closed immersion (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fibre-product-after-map}) and the second is a base change of an affine morphism (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-affine}) and the composition of affine morphisms is affine (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-affine}). The morphisms $f_{a, b}$ are of finite presentation (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation} and \ref{spaces-morphisms-lemma-finite-presentation-permanence}) and hence the base changes $X_a \times_{f_{a, b}, S_b} S \to S$ are of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-finite-presentation}). \end{proof} \section{Finite type closed in finite presentation} \label{section-finite-type-closed-in-finite-presentation} \noindent This section is the analogue of Limits, Section \ref{limits-section-finite-type-closed-in-finite-presentation}. \begin{lemma} \label{lemma-affine-morphism-is-limit} Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. If $Y$ quasi-compact and quasi-separated, then $X$ is a directed limit $X = \lim X_i$ with each $X_i$ affine and of finite presentation over $Y$. \end{lemma} \begin{proof} Consider the quasi-coherent $\mathcal{O}_Y$-module $\mathcal{A} = f_*\mathcal{O}_X$. By Lemma \ref{lemma-algebra-directed-colimit-finite-presentation} we can write $\mathcal{A} = \colim \mathcal{A}_i$ as a directed colimit of finitely presented $\mathcal{O}_Y$-algebras $\mathcal{A}_i$. Set $X_i = \underline{\Spec}_Y(\mathcal{A}_i)$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-spec}. By construction $X_i \to Y$ is affine and of finite presentation and $X = \lim X_i$. \end{proof} \begin{lemma} \label{lemma-integral-limit-finite-and-finite-presentation} Let $S$ be a scheme. Let $f : X \to Y$ be an integral morphism of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \lim X_i$ where $X_i$ are finite and of finite presentation over $Y$. \end{lemma} \begin{proof} Consider the finite quasi-coherent $\mathcal{O}_Y$-module $\mathcal{A} = f_*\mathcal{O}_X$. By Lemma \ref{lemma-integral-algebra-directed-colimit-finite} we can write $\mathcal{A} = \colim \mathcal{A}_i$ as a directed colimit of finite and finitely presented $\mathcal{O}_Y$-algebras $\mathcal{A}_i$. Set $X_i = \underline{\Spec}_Y(\mathcal{A}_i)$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-spec}. By construction $X_i \to Y$ is finite and of finite presentation and $X = \lim X_i$. \end{proof} \begin{lemma} \label{lemma-finite-in-finite-and-finite-presentation} Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \lim X_i$ where the transition maps are closed immersions and the objects $X_i$ are finite and of finite presentation over $Y$. \end{lemma} \begin{proof} Consider the finite quasi-coherent $\mathcal{O}_Y$-module $\mathcal{A} = f_*\mathcal{O}_X$. By Lemma \ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented} we can write $\mathcal{A} = \colim \mathcal{A}_i$ as a directed colimit of finite and finitely presented $\mathcal{O}_Y$-algebras $\mathcal{A}_i$ with surjective transition maps. Set $X_i = \underline{\Spec}_Y(\mathcal{A}_i)$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-spec}. By construction $X_i \to Y$ is finite and of finite presentation, the transition maps are closed immersions, and $X = \lim X_i$. \end{proof} \begin{lemma} \label{lemma-closed-is-limit-closed-and-finite-presentation} \begin{slogan} Closed immersions of qcqs algebraic spaces can be approximated by finitely presented closed immersions. \end{slogan} Let $S$ be a scheme. Let $f : X \to Y$ be a closed immersion of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \lim X_i$ where the transition maps are closed immersions and the morphisms $X_i \to Y$ are closed immersions of finite presentation. \end{lemma} \begin{proof} Let $\mathcal{I} \subset \mathcal{O}_Y$ be the quasi-coherent sheaf of ideals defining $X$ as a closed subspace of $Y$. By Lemma \ref{lemma-directed-colimit-finite-type} we can write $\mathcal{I} = \colim \mathcal{I}_i$ as the filtered colimit of its finite type quasi-coherent submodules. Let $X_i$ be the closed subspace of $X$ cut out by $\mathcal{I}_i$. Then $X_i \to Y$ is a closed immersion of finite presentation, and $X = \lim X_i$. Some details omitted. \end{proof} \begin{lemma} \label{lemma-quasi-affine-closed-in-finite-presentation} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is locally of finite type and quasi-affine, and \item $Y$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a morphism of finite presentation $f' : X' \to Y$ and a closed immersion $X \to X'$ over $Y$. \end{lemma} \begin{proof} By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-characterize-quasi-affine} we can find a factorization $X \to Z \to Y$ where $X \to Z$ is a quasi-compact open immersion and $Z \to Y$ is affine. Write $Z = \lim Z_i$ with $Z_i$ affine and of finite presentation over $Y$ (Lemma \ref{lemma-affine-morphism-is-limit}). For some $0 \in I$ we can find a quasi-compact open $U_0 \subset Z_0$ such that $X$ is isomorphic to the inverse image of $U_0$ in $Z$ (Lemma \ref{lemma-descend-opens}). Let $U_i$ be the inverse image of $U_0$ in $Z_i$, so $U = \lim U_i$. By Lemma \ref{lemma-finite-type-eventually-closed} we see that $X \to U_i$ is a closed immersion for some $i$ large enough. Setting $X' = U_i$ finishes the proof. \end{proof} \begin{lemma} \label{lemma-finite-type-closed-in-finite-presentation} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume: \begin{enumerate} \item $f$ is of locally of finite type. \item $X$ is quasi-compact and quasi-separated, and \item $Y$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a morphism of finite presentation $f' : X' \to Y$ and a closed immersion $X \to X'$ of algebraic spaces over $Y$. \end{lemma} \begin{proof} By Proposition \ref{proposition-approximate} we can write $X = \lim_i X_i$ with $X_i$ quasi-separated of finite type over $\mathbf{Z}$ and with transition morphisms $f_{ii'} : X_i \to X_{i'}$ affine. Consider the commutative diagram $$\xymatrix{ X \ar[r] \ar[rd] & X_{i, Y} \ar[r] \ar[d] & X_i \ar[d] \\ & Y \ar[r] & \Spec(\mathbf{Z}) }$$ Note that $X_i$ is of finite presentation over $\Spec(\mathbf{Z})$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}. Hence the base change $X_{i, Y} \to Y$ is of finite presentation by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-finite-presentation}. Observe that $\lim X_{i, Y} = X \times Y$ and that $X \to X \times Y$ is a monomorphism. By Lemma \ref{lemma-finite-type-eventually-closed} we see that $X \to X_{i, Y}$ is a monomorphism for $i$ large enough. Fix such an $i$. Note that $X \to X_{i, Y}$ is locally of finite type (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}) and a monomorphism, hence separated and locally quasi-finite (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}). Hence $X \to X_{i, Y}$ is representable. Hence $X \to X_{i, Y}$ is quasi-affine because we can use the principle Spaces, Lemma \ref{spaces-lemma-representable-transformations-property-implication} and the result for morphisms of schemes More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-separated-quasi-affine}. Thus Lemma \ref{lemma-quasi-affine-closed-in-finite-presentation} gives a factorization $X \to X' \to X_{i, Y}$ with $X \to X'$ a closed immersion and $X' \to X_{i, Y}$ of finite presentation. Finally, $X' \to Y$ is of finite presentation as a composition of morphisms of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-finite-presentation}). \end{proof} \begin{proposition} \label{proposition-separated-closed-in-finite-presentation} Let $S$ be a scheme. $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is of finite type and separated, and \item $Y$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a separated morphism of finite presentation $f' : X' \to Y$ and a closed immersion $X \to X'$ over $Y$. \end{proposition} \begin{proof} By Lemma \ref{lemma-finite-type-closed-in-finite-presentation} there is a closed immersion $X \to Z$ with $Z/Y$ of finite presentation. Let $\mathcal{I} \subset \mathcal{O}_Z$ be the quasi-coherent sheaf of ideals defining $X$ as a closed subscheme of $Y$. By Lemma \ref{lemma-directed-colimit-finite-type} we can write $\mathcal{I}$ as a directed colimit $\mathcal{I} = \colim_{a \in A} \mathcal{I}_a$ of its quasi-coherent sheaves of ideals of finite type. Let $X_a \subset Z$ be the closed subspace defined by $\mathcal{I}_a$. These form an inverse system indexed by $A$. The transition morphisms $X_a \to X_{a'}$ are affine because they are closed immersions. Each $X_a$ is quasi-compact and quasi-separated since it is a closed subspace of $Z$ and $Z$ is quasi-compact and quasi-separated by our assumptions. We have $X = \lim_a X_a$ as follows directly from the fact that $\mathcal{I} = \colim_{a \in A} \mathcal{I}_a$. Each of the morphisms $X_a \to Z$ is of finite presentation, see Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-finite-presentation}. Hence the morphisms $X_a \to Y$ are of finite presentation. Thus it suffices to show that $X_a \to Y$ is separated for some $a \in A$. This follows from Lemma \ref{lemma-eventually-separated} as we have assumed that $X \to Y$ is separated. \end{proof} \section{Approximating proper morphisms} \label{section-approximate-proper} \begin{lemma} \label{lemma-proper-limit-of-proper-finite-presentation} Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ quasi-compact and quasi-separated. Then $X = \lim X_i$ is a directed limit of algebraic spaces $X_i$ proper and of finite presentation over $Y$ and with transition morphisms and morphisms $X \to X_i$ closed immersions. \end{lemma} \begin{proof} By Proposition \ref{proposition-separated-closed-in-finite-presentation} we can find a closed immersion $X \to X'$ with $X'$ separated and of finite presentation over $Y$. By Lemma \ref{lemma-closed-is-limit-closed-and-finite-presentation} we can write $X = \lim X_i$ with $X_i \to X'$ a closed immersion of finite presentation. We claim that for all $i$ large enough the morphism $X_i \to Y$ is proper which finishes the proof. \medskip\noindent To prove this we may assume that $Y$ is an affine scheme, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-proper-local}. Next, we use the weak version of Chow's lemma, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}, to find a diagram $$\xymatrix{ X' \ar[rd] & X'' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_Y \ar[dl] \\ & Y & }$$ where $X'' \to \mathbf{P}^n_Y$ is an immersion, and $\pi : X'' \to X'$ is proper and surjective. Denote $X'_i \subset X''$, resp.\ $\pi^{-1}(X)$ the scheme theoretic inverse image of $X_i \subset X'$, resp.\ $X \subset X'$. Then $\lim X'_i = \pi^{-1}(X)$. Since $\pi^{-1}(X) \to Y$ is proper (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-composition-proper}), we see that $\pi^{-1}(X) \to \mathbf{P}^n_Y$ is a closed immersion (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-universally-closed-permanence} and \ref{spaces-morphisms-lemma-immersion-when-closed}). Hence for $i$ large enough we find that $X'_i \to \mathbf{P}^n_Y$ is a closed immersion by Lemma \ref{lemma-eventually-closed-immersion}. Thus $X'_i$ is proper over $Y$. For such $i$ the morphism $X_i \to Y$ is proper by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper}. \end{proof} \begin{lemma} \label{lemma-proper-limit-of-proper-finite-presentation-noetherian} Let $f : X \to Y$ be a proper morphism of algebraic spaces over $\mathbf{Z}$ with $Y$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_i : X_i \to Y_i)$ of morphisms of algebraic spaces over $I$, such that the transition morphisms $X_i \to X_{i'}$ and $Y_i \to Y_{i'}$ are affine, such that $f_i$ is proper and of finite presentation, such that $Y_i$ is of finite presentation over $\mathbf{Z}$, and such that $(X \to Y) = \lim (X_i \to Y_i)$. \end{lemma} \begin{proof} By Lemma \ref{lemma-proper-limit-of-proper-finite-presentation} we can write $X = \lim_{k \in K} X_k$ with $X_k \to Y$ proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition \ref{proposition-approximate}) we can write $Y = \lim_{j \in J} Y_j$ with $Y_j$ of finite presentation over $\mathbf{Z}$. For each $k$ there exists a $j$ and a morphism $X_{k, j} \to Y_j$ of finite presentation with $X_k \cong Y \times_{Y_j} X_{k, j}$ as algebraic spaces over $Y$, see Lemma \ref{lemma-descend-finite-presentation}. After increasing $j$ we may assume $X_{k, j} \to Y_j$ is proper, see Lemma \ref{lemma-eventually-proper}. The set $I$ will be consist of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \to Y_j$. For every $k' \geq k$ we can find a $j' \geq j$ and a morphism $X_{j', k'} \to X_{j, k}$ over $Y_{j'} \to Y_j$ whose base change to $Y$ gives the morphism $X_{k'} \to X_k$ (follows again from Lemma \ref{lemma-descend-finite-presentation}). These morphisms form the transition morphisms of the system. Some details omitted. \end{proof} \noindent Recall the scheme theoretic support of a finite type quasi-coherent module, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-support}. \begin{lemma} \label{lemma-eventually-proper-support} Assumptions and notation as in Situation \ref{situation-descent-property}. Let $\mathcal{F}_0$ be a quasi-coherent $\mathcal{O}_{X_0}$-module. Denote $\mathcal{F}$ and $\mathcal{F}_i$ the pullbacks of $\mathcal{F}_0$ to $X$ and $X_i$. Assume \begin{enumerate} \item $f_0$ is locally of finite type, \item $\mathcal{F}_0$ is of finite type, \item the scheme theoretic support of $\mathcal{F}$ is proper over $Y$. \end{enumerate} Then the scheme theoretic support of $\mathcal{F}_i$ is proper over $Y_i$ for some $i$. \end{lemma} \begin{proof} We may replace $X_0$ by the scheme theoretic support of $\mathcal{F}_0$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-support-finite-type} this guarantees that $X_i$ is the support of $\mathcal{F}_i$ and $X$ is the support of $\mathcal{F}$. Then, if $Z \subset X$ denotes the scheme theoretic support of $\mathcal{F}$, we see that $Z \to X$ is a universal homeomorphism. We conclude that $X \to Y$ is proper as this is true for $Z \to Y$ by assumption, see Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper}. By Lemma \ref{lemma-eventually-proper} we see that $X_i \to Y$ is proper for some $i$. Then it follows that the scheme theoretic support $Z_i$ of $\mathcal{F}_i$ is proper over $Y$ by Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-closed-immersion-proper} and \ref{spaces-morphisms-lemma-composition-proper}. \end{proof} \section{Embedding into affine space} \label{section-embedding} \noindent Some technical lemmas to be used in the proof of Chow's lemma later. \begin{lemma} \label{lemma-embedding-into-affine-over-ls-qs} Let $S$ be a scheme. Let $f : U \to X$ be a morphism of algebraic spaces over $S$. Assume $U$ is an affine scheme, $f$ is locally of finite type, and $X$ quasi-separated and locally separated. Then there exists an immersion $U \to \mathbf{A}^n_X$ over $X$. \end{lemma} \begin{proof} Say $U = \Spec(A)$. Write $A = \colim A_i$ as a filtered colimit of finite type $\mathbf{Z}$-subalgebras. For each $i$ the morphism $U \to U_i = \Spec(A_i)$ induces a morphism $$U \longrightarrow X \times U_i$$ over $X$. In the limit the morphism $U \to X \times U$ is an immersion as $X$ is locally separated, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-semi-diagonal}. By Lemma \ref{lemma-finite-type-eventually-closed} we see that $U \to X \times U_i$ is an immersion for some $i$. Since $U_i$ is isomorphic to a closed subscheme of $\mathbf{A}^n_{\mathbf{Z}}$ the lemma follows. \end{proof} \begin{remark} \label{remark-cannot-embed-in-general} We have seen in Examples, Section \ref{examples-section-embedding-affines} that Lemma \ref{lemma-embedding-into-affine-over-ls-qs} does not hold if we drop the assumption that $X$ be locally separated. This raises the question: Does Lemma \ref{lemma-embedding-into-affine-over-ls-qs} hold if we drop the assumption that $X$ be quasi-separated? If you know the answer, please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}. \end{remark} \begin{lemma} \label{lemma-embedding-into-affine-over-qs} Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Assume $X$ Noetherian and $f$ of finite presentation. Then there exists a dense open $V \subset Y$ and an immersion $V \to \mathbf{A}^n_X$. \end{lemma} \begin{proof} The assumptions imply that $Y$ is Noetherian (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-noetherian}). Then $Y$ is quasi-separated, hence has a dense open subscheme (Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}). Thus we may assume that $Y$ is a Noetherian scheme. By removing intersections of irreducible components of $Y$ (use Topology, Lemma \ref{topology-lemma-Noetherian} and Properties, Lemma \ref{properties-lemma-Noetherian-topology}) we may assume that $Y$ is a disjoint union of irreducible Noetherian schemes. Since there is an immersion $$\mathbf{A}^n_X \amalg \mathbf{A}^m_X \longrightarrow \mathbf{A}^{\max(n, m) + 1}_X$$ (details omitted) we see that it suffices to prove the result in case $Y$ is irreducible. \medskip\noindent Assume $Y$ is an irreducible scheme. Let $T \subset |X|$ be the closure of the image of $f : Y \to X$. Note that since $|Y|$ and $|X|$ are sober topological spaces (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-separated-sober}) $T$ is irreducible with a unique generic point $\xi$ which is the image of the generic point $\eta$ of $Y$. Let $\mathcal{I} \subset X$ be a quasi-coherent sheaf of ideals cutting out the reduced induced space structure on $T$ (Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}). Since $\mathcal{O}_{Y, \eta}$ is an Artinian local ring we see that for some $n > 0$ we have $f^{-1}\mathcal{I}^n \mathcal{O}_{Y, \eta} = 0$. As $f^{-1}\mathcal{I}\mathcal{O}_Y$ is a finite type quasi-coherent ideal we conclude that $f^{-1}\mathcal{I}^n\mathcal{O}_V = 0$ for some nonempty open $V \subset Y$. Let $Z \subset X$ be the closed subspace cut out by $\mathcal{I}^n$. By construction $V \to Y \to X$ factors through $Z$. Because $\mathbf{A}^n_Z \to \mathbf{A}^n_X$ is an immersion, we may replace $X$ by $Z$ and $Y$ by $V$. Hence we reach the situation where $Y$ and $X$ are irreducible and $Y \to X$ maps the generic point of $Y$ onto the generic point of $X$. \medskip\noindent Assume $Y$ and $X$ are irreducible, $Y$ is a scheme, and $Y \to X$ maps the generic point of $Y$ onto the generic point of $X$. By Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme} $X$ has a dense open subscheme $U \subset X$. Choose a nonempty affine open $V \subset Y$ whose image in $X$ is contained in $U$. By Morphisms, Lemma \ref{morphisms-lemma-quasi-affine-finite-type-over-S} we may factor $V \to U$ as $V \to \mathbf{A}^n_U \to U$. Composing with $\mathbf{A}^n_U \to \mathbf{A}^n_X$ we obtain the desired immersion. \end{proof} \section{Sections with support in a closed subset} \label{section-sections-with-support-in-closed} \noindent This section is the analogue of Properties, Section \ref{properties-section-sections-with-support-in-closed}. \begin{lemma} \label{lemma-quasi-coherent-finite-type-ideals} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $U \subset X$ be an open subspace. The following are equivalent: \begin{enumerate} \item $U \to X$ is quasi-compact, \item $U$ is quasi-compact, and \item there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ such that $|X| \setminus |U| = |V(\mathcal{I})|$. \end{enumerate} \end{lemma} \begin{proof} Let $W$ be an affine scheme and let $\varphi : W \to X$ be a surjective \'etale morphism, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. If (1) holds, then $\varphi^{-1}(U) \to W$ is quasi-compact, hence $\varphi^{-1}(U)$ is quasi-compact, hence $U$ is quasi-compact (as $|\varphi^{-1}(U)| \to |U|$ is surjective). If (2) holds, then $\varphi^{-1}(U)$ is quasi-compact because $\varphi$ is quasi-compact since $X$ is quasi-separated (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence}). Hence $\varphi^{-1}(U) \to W$ is a quasi-compact morphism of schemes by Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals}. It follows that $U \to X$ is quasi-compact by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-local}. Thus (1) and (2) are equivalent. \medskip\noindent Assume (1) and (2). By Properties of Spaces, Lemma \ref{spaces-properties-lemma-reduced-closed-subspace} there exists a unique quasi-coherent sheaf of ideals $\mathcal{J}$ cutting out the reduced induced closed subspace structure on $|X| \setminus |U|$. Note that $\mathcal{J}|_U = \mathcal{O}_U$ which is an $\mathcal{O}_U$-modules of finite type. As $U$ is quasi-compact it follows from Lemma \ref{lemma-directed-colimit-finite-type} that there exists a quasi-coherent subsheaf $\mathcal{I} \subset \mathcal{J}$ which is of finite type and has the property that $\mathcal{I}|_U = \mathcal{J}|_U$. Then $|X| \setminus |U| = |V(\mathcal{I})|$ and we obtain (3). Conversely, if $\mathcal{I}$ is as in (3), then $\varphi^{-1}(U) \subset W$ is a quasi-compact open by the lemma for schemes (Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals}) applied to $\varphi^{-1}\mathcal{I}$ on $W$. Thus (2) holds. \end{proof} \begin{lemma} \label{lemma-sections-annihilated-by-ideal} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider the sheaf of $\mathcal{O}_X$-modules $\mathcal{F}'$ which associates to every object $U$ of $X_\etale$ the module $$\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}s = 0\}$$ Assume $\mathcal{I}$ is of finite type. Then \begin{enumerate} \item $\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules, \item for affine $U$ in $X_\etale$ we have $\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}(U)s = 0\}$, and \item $\mathcal{F}'_x = \{s \in \mathcal{F}_x \mid \mathcal{I}_x s = 0\}$. \end{enumerate} \end{lemma} \begin{proof} It is clear that the rule defining $\mathcal{F}'$ gives a subsheaf of $\mathcal{F}$. Hence we may work \'etale locally on $X$ to verify the other statements. Thus the lemma reduces to the case of schemes which is Properties, Lemma \ref{properties-lemma-sections-annihilated-by-ideal}. \end{proof} \begin{definition} \label{definition-subsheaf-sections-annihilated-by-ideal} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma \ref{lemma-sections-annihilated-by-ideal} above is called the {\it subsheaf of sections annihilated by $\mathcal{I}$}. \end{definition} \begin{lemma} \label{lemma-push-sections-annihilated-by-ideal} Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_Y$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections annihilated by $f^{-1}\mathcal{I}\mathcal{O}_X$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections annihilated by $\mathcal{I}$. \end{lemma} \begin{proof} Omitted. Hint: The assumption that $f$ is quasi-compact and quasi-separated implies that $f_*\mathcal{F}$ is quasi-coherent (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}) so that Lemma \ref{lemma-sections-annihilated-by-ideal} applies to $\mathcal{I}$ and $f_*\mathcal{F}$. \end{proof} \noindent Next we come to the sheaf of sections supported in a closed subset. Again this isn't always a quasi-coherent sheaf, but if the complement of the closed is retrocompact'' in the given algebraic space, then it is. \begin{lemma} \label{lemma-sections-supported-on-closed-subset} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset and let $U \subset X$ be the open subspace such that $T \amalg |U| = |X|$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider the sheaf of $\mathcal{O}_X$-modules $\mathcal{F}'$ which associates to every object $\varphi : W \to X$ of $X_\etale$ the module $$\mathcal{F}'(W) = \{s \in \mathcal{F}(W) \mid \text{the support of }s\text{ is contained in }|\varphi|^{-1}(T)\}$$ If $U \to X$ is quasi-compact, then \begin{enumerate} \item for $W$ affine there exist a finitely generated ideal $I \subset \mathcal{O}_X(W)$ such that $|\varphi|^{-1}(T) = V(I)$, \item for $W$ and $I$ as in (1) we have $\mathcal{F}'(W) = \{x \in \mathcal{F}(W) \mid I^nx = 0 \text{ for some } n\}$, \item $\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules. \end{enumerate} \end{lemma} \begin{proof} It is clear that the rule defining $\mathcal{F}'$ gives a subsheaf of $\mathcal{F}$. Hence we may work \'etale locally on $X$ to verify the other statements. Thus the lemma reduces to the case of schemes which is Properties, Lemma \ref{properties-lemma-sections-supported-on-closed-subset}. \end{proof} \begin{definition} \label{definition-subsheaf-sections-supported-on-closed} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset whose complement corresponds to an open subspace $U \subset X$ with quasi-compact inclusion morphism $U \to X$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The quasi-coherent subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma \ref{lemma-sections-supported-on-closed-subset} above is called the {\it subsheaf of sections supported on $T$}. \end{definition} \begin{lemma} \label{lemma-push-sections-supported-on-closed-subset} Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $T \subset |Y|$ be a closed subset. Assume $|Y| \setminus T$ corresponds to an open subspace $V \subset Y$ such that $V \to Y$ is quasi-compact. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections supported on $|f|^{-1}T$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections supported on $T$. \end{lemma} \begin{proof} Omitted. Hints: $|X| \setminus |f|^{-1}T$ is the support of the open subspace $U = f^{-1}V \subset X$. Since $V \to Y$ is quasi-compact, so is $U \to X$ (by base change). The assumption that $f$ is quasi-compact and quasi-separated implies that $f_*\mathcal{F}$ is quasi-coherent. Hence Lemma \ref{lemma-sections-supported-on-closed-subset} applies to $T$ and $f_*\mathcal{F}$ as well as to $|f|^{-1}T$ and $\mathcal{F}$. The equality of the given quasi-coherent modules is immediate from the definitions. \end{proof} \section{Characterizing affine spaces} \label{section-affine} \noindent This section is the analogue of Limits, Section \ref{limits-section-affine}. \begin{lemma} \label{lemma-affine} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is surjective and finite, and assume that $X$ is affine. Then $Y$ is affine. \end{lemma} \begin{proof} We may and do view $f : X \to Y$ as a morphism of algebraic space over $\Spec(\mathbf{Z})$ (see Spaces, Definition \ref{spaces-definition-base-change}). Note that a finite morphism is affine and universally closed, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-universally-closed}. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-universally-closed-separated} we see that $Y$ is a separated algebraic space. As $f$ is surjective and $X$ is quasi-compact we see that $Y$ is quasi-compact. \medskip\noindent By Lemma \ref{lemma-finite-in-finite-and-finite-presentation} we can write $X = \lim X_a$ with each $X_a \to Y$ finite and of finite presentation. By Lemma