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 \input{preamble} % OK, start here. % \begin{document} \title{Limits of Schemes} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we put material related to limits of schemes. We mostly study limits of inverse systems over directed sets (Categories, Definition \ref{categories-definition-directed-set}) with affine transition maps. We discuss absolute Noetherian approximation. We characterize schemes locally of finite presentation over a base as those whose associated functor of points is limit preserving. As an application of absolute Noetherian approximation we prove that the image of an affine under an integral morphism is affine. Moreover, we prove some very general variants of Chow's lemma. A basic reference is \cite{EGA}. \section{Directed limits of schemes with affine transition maps} \label{section-limits} \noindent In this section we construct the limit. \begin{lemma} \label{lemma-directed-inverse-system-affine-schemes-has-limit} Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. If all the schemes $S_i$ are affine, then the limit $S = \lim_i S_i$ exists in the category of schemes. In fact $S$ is affine and $S = \Spec(\colim_i R_i)$ with $R_i = \Gamma(S_i, \mathcal{O})$. \end{lemma} \begin{proof} Just define $S = \Spec(\colim_i R_i)$. It follows from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} that $S$ is the limit even in the category of locally ringed spaces. \end{proof} \begin{lemma} \label{lemma-directed-inverse-system-has-limit} Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. If all the morphisms $f_{ii'} : S_i \to S_{i'}$ are affine, then the limit $S = \lim_i S_i$ exists in the category of schemes. Moreover, \begin{enumerate} \item each of the morphisms $f_i : S \to S_i$ is affine, \item for an element $0 \in I$ and any open subscheme $U_0 \subset S_0$ we have $$f_0^{-1}(U_0) = \lim_{i \geq 0} f_{i0}^{-1}(U_0)$$ in the category of schemes. \end{enumerate} \end{lemma} \begin{proof} Choose an element $0 \in I$. Note that $I$ is nonempty as the limit is directed. For every $i \geq 0$ consider the quasi-coherent sheaf of $\mathcal{O}_{S_0}$-algebras $\mathcal{A}_i = f_{i0, *}\mathcal{O}_{S_i}$. Recall that $S_i = \underline{\Spec}_{S_0}(\mathcal{A}_i)$, see Morphisms, Lemma \ref{morphisms-lemma-characterize-affine}. Set $\mathcal{A} = \colim_{i \geq 0} \mathcal{A}_i$. This is a quasi-coherent sheaf of $\mathcal{O}_{S_0}$-algebras, see Schemes, Section \ref{schemes-section-quasi-coherent}. Set $S = \underline{\Spec}_{S_0}(\mathcal{A})$. By Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-algebras} we get for $i \geq 0$ morphisms $f_i : S \to S_i$ compatible with the transition morphisms. Note that the morphisms $f_i$ are affine by Morphisms, Lemma \ref{morphisms-lemma-affine-permanence} for example. By Lemma \ref{lemma-directed-inverse-system-affine-schemes-has-limit} above we see that for any affine open $U_0 \subset S_0$ the inverse image $U = f_0^{-1}(U_0) \subset S$ is the limit of the system of opens $U_i = f_{i0}^{-1}(U_0)$, $i \geq 0$ in the category of schemes. \medskip\noindent Let $T$ be a scheme. Let $g_i : T \to S_i$ be a compatible system of morphisms. To show that $S = \lim_i S_i$ we have to prove there is a unique morphism $g : T \to S$ with $g_i = f_i \circ g$ for all $i \in I$. For every $t \in T$ there exists an affine open $U_0 \subset S_0$ containing $g_0(t)$. Let $V \subset g_0^{-1}(U_0)$ be an affine open neighbourhood containing $t$. By the remarks above we obtain a unique morphism $g_V : V \to U = f_0^{-1}(U_0)$ such that $f_i \circ g_V = g_i|_{U_i}$ for all $i$. The open sets $V \subset T$ so constructed form a basis for the topology of $T$. The morphisms $g_V$ glue to a morphism $g : T \to S$ because of the uniqueness property. This gives the desired morphism $g : T \to S$. \medskip\noindent The final statement is clear from the construction of the limit above. \end{proof} \begin{lemma} \label{lemma-scheme-over-limit} Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. Assume all the morphisms $f_{ii'} : S_i \to S_{i'}$ are affine, Let $S = \lim_i S_i$. Let $0 \in I$. Suppose that $T$ is a scheme over $S_0$. Then $$T \times_{S_0} S = \lim_{i \geq 0} T \times_{S_0} S_i$$ \end{lemma} \begin{proof} The right hand side is a scheme by Lemma \ref{lemma-directed-inverse-system-has-limit}. The equality is formal, see Categories, Lemma \ref{categories-lemma-colimits-commute}. \end{proof} \section{Infinite products} \label{section-inifinite-products} \noindent Infinite products of schemes usually do not exist. For example in Examples, Section \ref{examples-section-not-algebraic} it is shown that an infinite product of copies of $\mathbf{P}^1$ is not even an algebraic space. \medskip\noindent On the other hand, infinite products of affine schemes do exist and are affine. Using Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} this corresponds to the fact that in the category of rings we have infinite coproducts: if $I$ is a set and $R_i$ is a ring for each $i$, then we can consider the ring $$R = \otimes R_i = \colim_{\{i_1, \ldots, i_n\} \subset I} R_{i_1} \otimes_\mathbf{Z} \ldots \otimes_\mathbf{Z} R_{i_n}$$ Given another ring $A$ a map $R \to A$ is the same thing as a collection of ring maps $R_i \to A$ for all $i \in I$ as follows from the corresponding property of finite tensor products. \begin{lemma} \label{lemma-infinite-product} Let $S$ be a scheme. Let $I$ be a set and for each $i \in I$ let $f_i : T_i \to S$ be an affine morphism. Then the product $T = \prod T_i$ exists in the category of schemes over $S$. In fact, we have $$T = \lim_{\{i_1, \ldots, i_n\} \subset I} T_{i_1} \times_S \ldots \times_S T_{i_n}$$ and the projection morphisms $T \to T_{i_1} \times_S \ldots \times_S T_{i_n}$ are affine. \end{lemma} \begin{proof} Omitted. Hint: Argue as in the discussion preceding the lemma and use Lemma \ref{lemma-directed-inverse-system-has-limit} for existence of the limit. \end{proof} \begin{lemma} \label{lemma-infinite-product-surjective} Let $S$ be a scheme. Let $I$ be a set and for each $i \in I$ let $f_i : T_i \to S$ be a surjective affine morphism. Then the product $T = \prod T_i$ in the category of schemes over $S$ (Lemma \ref{lemma-infinite-product}) maps surjectively to $S$. \end{lemma} \begin{proof} Let $s \in S$. Choose $t_i \in T_i$ mapping to $s$. Choose a huge field extension $K/\kappa(s)$ such that $\kappa(s_i)$ embeds into $K$ for each $i$. Then we get morphisms $\Spec(K) \to T_i$ with image $s_i$ agreeing as morphisms to $S$. Whence a morphism $\Spec(K) \to T$ which proves there is a point of $T$ mapping to $s$. \end{proof} \begin{lemma} \label{lemma-infinite-product-integral} Let $S$ be a scheme. Let $I$ be a set and for each $i \in I$ let $f_i : T_i \to S$ be an integral morphism. Then the product $T = \prod T_i$ in the category of schemes over $S$ (Lemma \ref{lemma-infinite-product}) is integral over $S$. \end{lemma} \begin{proof} Omitted. Hint: On affine pieces this reduces to the following algebra fact: if $A \to B_i$ is integral for all $i$, then $A \to \otimes_A B_i$ is integral. \end{proof} \section{Descending properties} \label{section-descent} \noindent First some basic lemmas describing the topology of a limit. \begin{lemma} \label{lemma-inverse-limit-sets} Let $S = \lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). Then $S_{set} = \lim_i S_{i, set}$ where $S_{set}$ indicates the underlying set of the scheme $S$. \end{lemma} \begin{proof} Pick $i \in I$. Take $U_i \subset S_i$ an affine open. Denote $U_{i'} = f_{i'i}^{-1}(U_i)$ and $U = f_i^{-1}(U_i)$. Here $f_{i'i} : S_{i'} \to S_i$ is the transtion morphism and $f_i : S \to S_i$ is the projection. By Lemma \ref{lemma-directed-inverse-system-has-limit} we have $U = \lim_{i' \geq i} U_i$. Suppose we can show that $U_{set} = \lim_{i' \geq i} U_{i', set}$. Then the lemma follows by a simple argument using an affine covering of $S_i$. Hence we may assume all $S_i$ and $S$ affine. This reduces us to the algebra question considered in the next paragraph. \medskip\noindent Suppose given a system of rings $(A_i, \varphi_{ii'})$ over $I$. Set $A = \colim_i A_i$ with canonical maps $\varphi_i : A_i \to A$. Then $$\Spec(A) = \lim_i \Spec(A_i)$$ Namely, suppose that we are given primes $\mathfrak p_i \subset A_i$ such that $\mathfrak p_i = \varphi_{ii'}^{-1}(\mathfrak p_{i'})$ for all $i' \geq i$. Then we simply set $$\mathfrak p = \{x \in A \mid \exists i, x_i \in \mathfrak p_i \text{ with }\varphi_i(x_i) = x\}$$ It is clear that this is an ideal and has the property that $\varphi_i^{-1}(\mathfrak p) = \mathfrak p_i$. Then it follows easily that it is a prime ideal as well. \end{proof} \begin{lemma} \label{lemma-inverse-limit-top} Let $S = \lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). Then $S_{top} = \lim_i S_{i, top}$ where $S_{top}$ indicates the underlying topological space of the scheme $S$. \end{lemma} \begin{proof} We will use the criterion of Topology, Lemma \ref{topology-lemma-characterize-limit}. We have seen that $S_{set} = \lim_i S_{i, set}$ in Lemma \ref{lemma-inverse-limit-sets}. The maps $f_i : S \to S_i$ are morphisms of schemes hence continuous. Thus $f_i^{-1}(U_i)$ is open for each open $U_i \subset S_i$. Finally, let $s \in S$ and let $s \in V \subset S$ be an open neighbourhood. Choose $0 \in I$ and choose an affine open neighbourhood $U_0 \subset S_0$ of the image of $s$. Then $f_0^{-1}(U_0) = \lim_{i \geq 0} f_{i0}^{-1}(U_0)$, see Lemma \ref{lemma-directed-inverse-system-has-limit}. Then $f_0^{-1}(U_0)$ and $f_{i0}^{-1}(U_0)$ are affine and $$\mathcal{O}_S(f_0^{-1}(U_0)) = \colim_{i \geq 0} \mathcal{O}_{S_i}(f_{i0}^{-1}(U_0))$$ either by the proof of Lemma \ref{lemma-directed-inverse-system-has-limit} or by Lemma \ref{lemma-directed-inverse-system-affine-schemes-has-limit}. Choose $a \in \mathcal{O}_S(f_0^{-1}(U_0))$ such that $s \in D(a) \subset V$. This is possible because the principal opens form a basis for the topology on the affine scheme $f_0^{-1}(U_0)$. Then we can pick an $i \geq 0$ and $a_i \in \mathcal{O}_{S_i}(f_{i0}^{-1}(U_0))$ mapping to $a$. It follows that $D(a_i) \subset f_{i0}^{-1}(U_0) \subset S_i$ is an open subset whose inverse image in $S$ is $D(a)$. This finishes the proof. \end{proof} \begin{lemma} \label{lemma-limit-nonempty} Let $S = \lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). If all the schemes $S_i$ are nonempty and quasi-compact, then the limit $S = \lim_i S_i$ is nonempty. \end{lemma} \begin{proof} Choose $0 \in I$. Note that $I$ is nonempty as the limit is directed. Choose an affine open covering $S_0 = \bigcup_{j = 1, \ldots, m} U_j$. Since $I$ is directed there exists a $j \in \{1, \ldots, m\}$ such that $f_{i0}^{-1}(U_j) \not = \emptyset$ for all $i \geq 0$. Hence $\lim_{i \geq 0} f_{i0}^{-1}(U_j)$ is not empty since a directed colimit of nonzero rings is nonzero (because $1 \not = 0$). As $\lim_{i \geq 0} f_{i0}^{-1}(U_j)$ is an open subscheme of the limit we win. \end{proof} \begin{lemma} \label{lemma-inverse-limit-irreducibles} Let $S = \lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). Let $s \in S$ with images $s_i \in S_i$. Then $\overline{\{s\}} = \lim_i \overline{\{s_i\}}$ as sets and as schemes if endowed with the reduced induced scheme structure. \end{lemma} \begin{proof} Choose $0 \in I$ and an affine open covering $U_0 = \bigcup_{j \in J} U_{0, j}$. For $i \geq 0$ let $U_{i, j} = f_{i, 0}^{-1}(U_{0, j})$ and set $U_j = f_0^{-1}(U_{0, j})$. Here $f_{i'i} : S_{i'} \to S_i$ is the transtion morphism and $f_i : S \to S_i$ is the projection. For $j \in J$ the following are equivalent: (a) $x \in U_j$, (b) $x_0 \in U_{0, j}$, (c) $x_i \in U_{i, j}$ for all $i \geq 0$. Let $J' \subset J$ be the set of indices for which (a), (b), (c) are true. Then $\overline{\{s\}} = \bigcup_{j \in J'} (\overline{\{s\}} \cap U_j)$ and simiarly for $\overline{\{s_i\}}$ for $i \geq 0$. Note that $\overline{\{s\}} \cap U_j$ is the closure of the set $\{s\}$ in the topological space $U_j$. Similarly for $\overline{\{s_i\}} \cap U_{i, j}$ for $i \geq 0$. Hence it suffices to prove the lemma in the case $S$ and $S_i$ affine for all $i$. This reduces us to the algebra question considered in the next paragraph. \medskip\noindent Suppose given a system of rings $(A_i, \varphi_{ii'})$ over $I$. Set $A = \colim_i A_i$ with canonical maps $\varphi_i : A_i \to A$. Let $\mathfrak p \subset A$ be a prime and set $\mathfrak p_i = \varphi_i^{-1}(\mathfrak p)$. Then $$V(\mathfrak p) = \lim_i V(\mathfrak p_i)$$ This follows from Lemma \ref{lemma-inverse-limit-sets} because $A/\mathfrak p = \colim A_i/\mathfrak p_i$. This equality of rings also shows the final statement about reduced induced scheme structures holds true. \end{proof} \noindent In the rest of this section we work in the following situation. \begin{situation} \label{situation-descent} Let $S = \lim_{i \in I} S_i$ be the limit of a directed system of schemes with affine transition morphisms $f_{i'i} : S_{i'} \to S_i$ (Lemma \ref{lemma-directed-inverse-system-has-limit}). We assume that $S_i$ is quasi-compact and quasi-separated for all $i \in I$. We denote $f_i : S \to S_i$ the projection. We also choose an element $0 \in I$. \end{situation} \noindent The type of result we are looking for is the following: If we have an object over $S$, then for some $i$ there is a similar object over $S_i$. \begin{lemma} \label{lemma-topology-limit} In Situation \ref{situation-descent}. \begin{enumerate} \item We have $S_{set} = \lim_i S_{i, set}$ where $S_{set}$ indicates the underlying set of the scheme $S$. \item We have $S_{top} = \lim_i S_{i, top}$ where $S_{top}$ indicates the underlying topological space of the scheme $S$. \item If $s, s' \in S$ and $s'$ is not a specialization of $s$ then for some $i \in I$ the image $s'_i \in S_i$ of $s'$ is not a specialization of the image $s_i \in S_i$ of $s$. \item Add more easy facts on topology of $S$ here. (Requirement: whatever is added should be easy in the affine case.) \end{enumerate} \end{lemma} \begin{proof} Part (1) is a special case of Lemma \ref{lemma-inverse-limit-sets}. \medskip\noindent Part (2) is a special case of Lemma \ref{lemma-inverse-limit-top}. \medskip\noindent Part (3) is a special case of Lemma \ref{lemma-inverse-limit-irreducibles}. \end{proof} \begin{lemma} \label{lemma-descend-section} In Situation \ref{situation-descent}. Suppose that $\mathcal{F}_0$ is a quasi-coherent sheaf on $S_0$. Set $\mathcal{F}_i = f_{i0}^*\mathcal{F}_0$ for $i \geq 0$ and set $\mathcal{F} = f_0^*\mathcal{F}_0$. Then $$\Gamma(S, \mathcal{F}) = \colim_{i \geq 0} \Gamma(S_i, \mathcal{F}_i)$$ \end{lemma} \begin{proof} Write $\mathcal{A}_j = f_{i0, *} \mathcal{O}_{S_i}$. This is a quasi-coherent sheaf of $\mathcal{O}_{S_0}$-algebras (see Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-algebras}) and $S_i$ is the relative spectrum of $\mathcal{A}_i$ over $S_0$. In the proof of Lemma \ref{lemma-directed-inverse-system-has-limit} we constructed $S$ as the relative spectrum of $\mathcal{A} = \colim_{i \geq 0} \mathcal{A}_i$ over $S_0$. Set $$\mathcal{M}_i = \mathcal{F}_0 \otimes_{\mathcal{O}_{S_0}} \mathcal{A}_i$$ and $$\mathcal{M} = \mathcal{F}_0 \otimes_{\mathcal{O}_{S_0}} \mathcal{A}.$$ Then we have $f_{i0, *} \mathcal{F}_i = \mathcal{M}_i$ and $f_{0, *}\mathcal{F} = \mathcal{M}$. Since $\mathcal{A}$ is the colimit of the sheaves $\mathcal{A}_i$ and since tensor product commutes with directed colimits, we conclude that $\mathcal{M} = \colim_{i \geq 0} \mathcal{M}_i$. Since $S_0$ is quasi-compact and quasi-separated we see that \begin{eqnarray*} \Gamma(S, \mathcal{F}) & = & \Gamma(S_0, \mathcal{M}) \\ & = & \Gamma(S_0, \colim_{i \geq 0} \mathcal{M}_i) \\ & = & \colim_{i \geq 0} \Gamma(S_0, \mathcal{M}_i) \\ & = & \colim_{i \geq 0} \Gamma(S_i, \mathcal{F}_i) \end{eqnarray*} see Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections} and Topology, Lemma \ref{topology-lemma-topology-quasi-separated-scheme} for the middle equality. \end{proof} \begin{lemma} \label{lemma-limit-closed-nonempty} In Situation \ref{situation-descent}. Suppose for each $i$ we are given a nonempty closed subset $Z_i \subset S_i$ with $f_{ii'}(Z_i) \subset Z_{i'}$. Then there exists a point $s \in S$ with $f_i(s) \in Z_i$ for all $i$. \end{lemma} \begin{proof} Let $Z_i \subset S_i$ also denote the reduced closed subscheme associated to $Z_i$, see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}. A closed immersion is affine, and a composition of affine morphisms is affine (see Morphisms, Lemmas \ref{morphisms-lemma-closed-immersion-affine} and \ref{morphisms-lemma-composition-affine}), and hence $Z_i \to S_{i'}$ is affine when $i \geq i'$. We conclude that the morphism $f_{ii'} : Z_i \to Z_{i'}$ is affine by Morphisms, Lemma \ref{morphisms-lemma-affine-permanence}. Each of the schemes $Z_i$ is quasi-compact as a closed subscheme of a quasi-compact scheme. Hence we may apply Lemma \ref{lemma-limit-nonempty} to see that $Z = \lim_i Z_i$ is nonempty. Since there is a canonical morphism $Z \to S$ we win. \end{proof} \begin{lemma} \label{lemma-limit-fibre-product-empty} In Situation \ref{situation-descent}. Suppose we are given an $i$ and a morphism $T \to S_i$ such that \begin{enumerate} \item $T \times_{S_i} S = \emptyset$, and \item $T$ is quasi-compact. \end{enumerate} Then $T \times_{S_i} S_{i'} = \emptyset$ for all sufficiently large $i'$. \end{lemma} \begin{proof} By Lemma \ref{lemma-scheme-over-limit} we see that $T \times_{S_i} S = \lim_{i' \geq i} T \times_{S_i} S_{i'}$. Hence the result follows from Lemma \ref{lemma-limit-nonempty}. \end{proof} \begin{lemma} \label{lemma-limit-contained-in-constructible} In Situation \ref{situation-descent}. Suppose we are given an $i$ and a locally constructible subset $E \subset S_i$ such that $f_i(S) \subset E$. Then $f_{ii'}(S_{i'}) \subset E$ for all sufficiently large $i'$. \end{lemma} \begin{proof} Writing $S_i$ as a finite union of open affine subschemes reduces the question to the case that $S_i$ is affine and $E$ is constructible, see Lemma \ref{lemma-directed-inverse-system-has-limit} and Properties, Lemma \ref{properties-lemma-locally-constructible}. In this case the complement $S_i \setminus E$ is constructible too. Hence there exists an affine scheme $T$ and a morphism $T \to S_i$ whose image is $S_i \setminus E$, see Algebra, Lemma \ref{algebra-lemma-constructible-is-image}. By Lemma \ref{lemma-limit-fibre-product-empty} we see that $T \times_{S_i} S_{i'}$ is empty for all sufficiently large $i'$, and hence $f_{ii'}(S_{i'}) \subset E$ for all sufficiently large $i'$. \end{proof} \begin{lemma} \label{lemma-descend-opens} In Situation \ref{situation-descent} we have the following: \begin{enumerate} \item Given any quasi-compact open $V \subset S = \lim_i S_i$ there exists an $i \in I$ and a quasi-compact open $V_i \subset S_i$ such that $f_i^{-1}(V_i) = V$. \item Given $V_i \subset S_i$ and $V_{i'} \subset S_{i'}$ quasi-compact opens such that $f_i^{-1}(V_i) = f_{i'}^{-1}(V_{i'})$ there exists an index $i'' \geq i, i'$ such that $f_{i''i}^{-1}(V_i) = f_{i''i'}^{-1}(V_{i'})$. \item If $V_{1, i}, \ldots, V_{n, i} \subset S_i$ are quasi-compact opens and $S = f_i^{-1}(V_{1, i}) \cup \ldots \cup f_i^{-1}(V_{n, i})$ then $S_{i'} = f_{i'i}^{-1}(V_{1, i}) \cup \ldots \cup f_{i'i}^{-1}(V_{n, i})$ for some $i' \geq i$. \end{enumerate} \end{lemma} \begin{proof} Choose $i_0 \in I$. Note that $I$ is nonempty as the limit is directed. For convenience we write $S_0 = S_{i_0}$ and $i_0 = 0$. Choose an affine open covering $S_0 = U_{1, 0} \cup \ldots \cup U_{m, 0}$. Denote $U_{j, i} \subset S_i$ the inverse image of $U_{j, 0}$ under the transition morphism for $i \geq 0$. Denote $U_j$ the inverse image of $U_{j, 0}$ in $S$. Note that $U_j = \lim_i U_{j, i}$ is a limit of affine schemes. \medskip\noindent We first prove the uniqueness statement: Let $V_i \subset S_i$ and $V_{i'} \subset S_{i'}$ quasi-compact opens such that $f_i^{-1}(V_i) = f_{i'}^{-1}(V_{i'})$. It suffices to show that $f_{i''i}^{-1}(V_i \cap U_{j, i''})$ and $f_{i''i'}^{-1}(V_{i'} \cap U_{j, i''})$ become equal for $i''$ large enough. Hence we reduce to the case of a limit of affine schemes. In this case write $S = \Spec(R)$ and $S_i = \Spec(R_i)$ for all $i \in I$. We may write $V_i = S_i \setminus V(h_1, \ldots, h_m)$ and $V_{i'} = S_{i'} \setminus V(g_1, \ldots, g_n)$. The assumption means that the ideals $\sum g_jR$ and $\sum h_jR$ have the same radical in $R$. This means that $g_j^N = \sum a_{jj'}h_{j'}$ and $h_j^N = \sum b_{jj'} g_{j'}$ for some $N \gg 0$ and $a_{jj'}$ and $b_{jj'}$ in $R$. Since $R = \colim_i R_i$ we can chose an index $i'' \geq i$ such that the equations $g_j^N = \sum a_{jj'}h_{j'}$ and $h_j^N = \sum b_{jj'} g_{j'}$ hold in $R_{i''}$ for some $a_{jj'}$ and $b_{jj'}$ in $R_{i''}$. This implies that the ideals $\sum g_jR_{i''}$ and $\sum h_jR_{i''}$ have the same radical in $R_{i''}$ as desired. \medskip\noindent We prove existence: If $S_0$ is affine, then $S_i = \Spec(R_i)$ for all $i \geq 0$ and $S = \Spec(R)$ with $R = \colim R_i$. Then $V = S \setminus V(g_1, \ldots, g_n)$ for some $g_1, \ldots, g_n \in R$. Choose any $i$ large enough so that each of the $g_j$ comes from an element $g_{j, i} \in R_i$ and take $V_i = S_i \setminus V(g_{1, i}, \ldots, g_{n, i})$. If $S_0$ is general, then the opens $V \cap U_j$ are quasi-compact because $S$ is quasi-separated. Hence by the affine case we see that for each $j = 1, \ldots, m$ there exists an $i_j \in I$ and a quasi-compact open $V_{i_j} \subset U_{j, i_j}$ whose inverse image in $U_j$ is $V \cap U_j$. Set $i = \max(i_1, \ldots, i_m)$ and let $V_i = \bigcup f_{ii_j}^{-1}(V_{i_j})$. \medskip\noindent The statement on coverings follows from the uniqueness statement for the opens $V_{1, i} \cup \ldots \cup V_{n, i}$ and $S_i$ of $S_i$. \end{proof} \begin{lemma} \label{lemma-limit-quasi-affine} In Situation \ref{situation-descent} if $S$ is quasi-affine, then for some $i_0 \in I$ the schemes $S_i$ for $i \geq i_0$ are quasi-affine. \end{lemma} \begin{proof} Choose $i_0 \in I$. Note that $I$ is nonempty as the limit is directed. For convenience we write $S_0 = S_{i_0}$ and $i_0 = 0$. Let $s \in S$. We may choose an affine open $U_0 \subset S_0$ containing $f_0(s)$. Since $S$ is quasi-affine we may choose an element $a \in \Gamma(S, \mathcal{O}_S)$ such that $s \in D(a) \subset f_0^{-1}(U_0)$, and such that $D(a)$ is affine. By Lemma \ref{lemma-descend-section} there exists an $i \geq 0$ such that $a$ comes from an element $a_i \in \Gamma(S_i, \mathcal{O}_{S_i})$. For any index $j \geq i$ we denote $a_j$ the image of $a_i$ in the global sections of the structure sheaf of $S_j$. Consider the opens $D(a_j) \subset S_j$ and $U_j = f_{j0}^{-1}(U_0)$. Note that $U_j$ is affine and $D(a_j)$ is a quasi-compact open of $S_j$, see Properties, Lemma \ref{properties-lemma-affine-cap-s-open} for example. Hence we may apply Lemma \ref{lemma-descend-opens} to the opens $U_j$ and $U_j \cup D(a_j)$ to conclude that $D(a_j) \subset U_j$ for some $j \geq i$. For such an index $j$ we see that $D(a_j) \subset S_j$ is an affine open (because $D(a_j)$ is a standard affine open of the affine open $U_j$) containing the image $f_j(s)$. \medskip\noindent We conclude that for every $s \in S$ there exist an index $i \in I$, and a global section $a \in \Gamma(S_i, \mathcal{O}_{S_i})$ such that $D(a) \subset S_i$ is an affine open containing $f_i(s)$. Because $S$ is quasi-compact we may choose a single index $i \in I$ and global sections $a_1, \ldots, a_m \in \Gamma(S_i, \mathcal{O}_{S_i})$ such that each $D(a_j) \subset S_i$ is affine open and such that $f_i : S \to S_i$ has image contained in the union $W_i = \bigcup_{j = 1, \ldots, m} D(a_j)$. For $i' \geq i$ set $W_{i'} = f_{i'i}^{-1}(W_i)$. Since $f_i^{-1}(W_i)$ is all of $S$ we see (by Lemma \ref{lemma-descend-opens} again) that for a suitable $i' \geq i$ we have $S_{i'} = W_{i'}$. Thus we may replace $i$ by $i'$ and assume that $S_i = \bigcup_{j = 1, \ldots, m} D(a_j)$. This implies that $\mathcal{O}_{S_i}$ is an ample invertible sheaf on $S_i$ (see Properties, Definition \ref{properties-definition-ample}) and hence that $S_i$ is quasi-affine, see Properties, Lemma \ref{properties-lemma-quasi-affine-O-ample}. Hence we win. \end{proof} \begin{lemma} \label{lemma-limit-affine} In Situation \ref{situation-descent} if $S$ is affine, then for some $i_0 \in I$ the schemes $S_i$ for $i \geq i_0$ are affine. \end{lemma} \begin{proof} By Lemma \ref{lemma-limit-quasi-affine} we may assume that $S_0$ is quasi-affine for some $0 \in I$. Set $R_0 = \Gamma(S_0, \mathcal{O}_{S_0})$. Then $S_0$ is a quasi-compact open of $T_0 = \Spec(R_0)$. Denote $j_0 : S_0 \to T_0$ the corresponding quasi-compact open immersion. For $i \geq 0$ set $\mathcal{A}_i = f_{0i, *}\mathcal{O}_{S_i}$. Since $f_{0i}$ is affine we see that $S_i = \underline{\Spec}_{S_0}(\mathcal{A}_i)$. Set $T_i = \underline{\Spec}_{T_0}(j_{0, *}\mathcal{A}_i)$. Then $T_i \to T_0$ is affine, hence $T_i$ is affine. Thus $T_i$ is the spectrum of $$R_i = \Gamma(T_0, j_{0, *}\mathcal{A}_i) = \Gamma(S_0, \mathcal{A}_i) = \Gamma(S_i, \mathcal{O}_{S_i}).$$ Write $S = \Spec(R)$. We have $R = \colim_i R_i$ by Lemma \ref{lemma-descend-section}. Hence also $S = \lim_i T_i$. As formation of the relative spectrum commutes with base change, the inverse image of the open $S_0 \subset T_0$ in $T_i$ is $S_i$. Let $Z_0 = T_0 \setminus S_0$ and let $Z_i \subset T_i$ be the inverse image of $Z_0$. As $S_i = T_i \setminus Z_i$, it suffices to show that $Z_i$ is empty for some $i$. Assume $Z_i$ is nonempty for all $i$ to get a contradiction. By Lemma \ref{lemma-limit-closed-nonempty} there exists a point $s$ of $S = \lim T_i$ which maps to a point of $Z_i$ for every $i$. But $S = \lim_i S_i$, and hence we arrive at a contradiction by Lemma \ref{lemma-topology-limit}. \end{proof} \begin{lemma} \label{lemma-limit-separated} In Situation \ref{situation-descent} if $S$ is separated, then for some $i_0 \in I$ the schemes $S_i$ for $i \geq i_0$ are separated. \end{lemma} \begin{proof} Choose a finite affine open covering $S_0 = U_{0, 1} \cup \ldots \cup U_{0, m}$. Set $U_{i, j} \subset S_i$ and $U_j \subset S$ equal to the inverse image of $U_{0, j}$. Note that $U_{i, j}$ and $U_j$ are affine. As $S$ is separated the intersections $U_{j_1} \cap U_{j_2}$ are affine. Since $U_{j_1} \cap U_{j_2} = \lim_{i \geq 0} U_{i, j_1} \cap U_{i, j_2}$ we see that $U_{i, j_1} \cap U_{i, j_2}$ is affine for large $i$ by Lemma \ref{lemma-limit-affine}. To show that $S_i$ is separated for large $i$ it now suffices to show that $$\mathcal{O}_{S_i}(V_{i, j_1}) \otimes_{\mathcal{O}_S(S)} \mathcal{O}_{S_i}(V_{i, j_2}) \longrightarrow \mathcal{O}_{S_i}(V_{i, j_1} \cap V_{i, j_2})$$ is surjective for large $i$ (Schemes, Lemma \ref{schemes-lemma-characterize-separated}). \medskip\noindent To get rid of the annoying indices, assume we have affine opens $U, V \subset S_0$ such that $U \cap V$ is affine too. Let $U_i, V_i \subset S_i$, resp.\ $U, V \subset S$ be the inverse images. We have to show that $\mathcal{O}(U_i) \otimes \mathcal{O}(V_i) \to \mathcal{O}(U_i \cap V_i)$ is surjective for $i$ large enough and we know that $\mathcal{O}(U_) \otimes \mathcal{O}(V) \to \mathcal{O}(U \cap V)$ is surjective. Note that $\mathcal{O}(U_0) \otimes \mathcal{O}(V_0) \to \mathcal{O}(U_0 \cap V_0)$ is of finite type, as the diagonal morphism $S_i \to S_i \times S_i$ is an immersion (Schemes, Lemma \ref{schemes-lemma-diagonal-immersion}) hence locally of finite type (Morphisms, Lemmas \ref{morphisms-lemma-locally-finite-type-characterize} and \ref{morphisms-lemma-immersion-locally-finite-type}). Thus we can choose elements $f_{0, 1}, \ldots, f_{0, n} \in \mathcal{O}(U_0 \cap V_0)$ which generate $\mathcal{O}(U_0 \cap V_0)$ over $\mathcal{O}(U_0) \otimes \mathcal{O}(V_0)$. Observe that for $i \geq 0$ the diagram of schemes $$\xymatrix{ U_i \cap V_i \ar[r] \ar[d] & U_i \ar[d] \\ U_0 \cap V_0 \ar[r] & U_0 }$$ is cartesian. Thus we see that the images $f_{i, 1}, \ldots, f_{i, n} \in \mathcal{O}(U_i \cap V_i)$ generate $\mathcal{O}(U_i \cap V_i)$ over $\mathcal{O}(U_i) \otimes \mathcal{O}(V_0)$ and a fortiori over $\mathcal{O}(U_i) \otimes \mathcal{O}(V_i)$. By assumption the images $f_1, \ldots, f_n \in \mathcal{O}(U \otimes V)$ are in the image of the map $\mathcal{O}(U) \otimes \mathcal{O}(V) \to \mathcal{O}(U \cap V)$. Since $\mathcal{O}(U) \otimes \mathcal{O}(V) = \colim \mathcal{O}(U_i) \otimes \mathcal{O}(V_i)$ we see that they are in the image of the map at some finite level and the lemma is proved. \end{proof} \begin{lemma} \label{lemma-limit-ample} In Situation \ref{situation-descent} let $\mathcal{L}_0$ be an invertible sheaf of modules on $S_0$. If the pullback $\mathcal{L}$ to $S$ is ample, then for some $i \in I$ the pullback $\mathcal{L}_i$ to $S_i$ is ample. \end{lemma} \begin{proof} The assumption means there are finitely many sections $s_1, \ldots, s_m \in \Gamma(S, \mathcal{L})$ such that $S_{s_j}$ is affine and such that $S = \bigcup S_{s_j}$, see Properties, Definition \ref{properties-definition-ample}. By Lemma \ref{lemma-descend-section} we can find an $i \in I$ and sections $s_{i, j} \in \Gamma(S_i, \mathcal{L}_i)$ mapping to $s_j$. By Lemma \ref{lemma-limit-affine} we may, after increasing $i$, assume that $(S_i)_{s_{i, j}}$ is affine for $j = 1, \ldots, m$. By Lemma \ref{lemma-descend-opens} we may, after increasing $i$ a last time, assume that $S_i = \bigcup (S_i)_{s_{i, j}}$. Then $\mathcal{L}_i$ is ample by definition. \end{proof} \begin{lemma} \label{lemma-finite-type-eventually-closed} Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Let $Y \to X$ be a morphism of schemes over $S$. \begin{enumerate} \item If $Y \to X$ is a closed immersion, $X_i$ quasi-compact, and $Y$ locally of finite type over $S$, 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, $Y \to S$ locally of finite type, and $Y$ quasi-compact, then $Y \to X_i$ is an immersion for $i$ large enough. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). Choose $0 \in I$ and a finite affine open covering $X_0 = U_{0, 1} \cup \ldots \cup U_{0, m}$ with the property that $U_{0, j}$ maps into an affine open $W_j \subset S$. Let $V_j \subset Y$, resp.\ $U_{i, j} \subset X_i$, $i \geq 0$, resp. $U_j \subset X$ be the inverse image of $U_{0, j}$. It suffices to prove that $V_j \to U_{i, j}$ is a closed immersion for $i$ sufficiently large and we know that $V_j \to U_j$ is a closed immersion. Thus we reduce to the following algebra fact: If $A = \colim A_i$ is a directed colimit of $R$-algebras, $A \to B$ is a surjection of $R$-algebras, and $B$ is a finitely generated $R$-algebra, then $A_i \to B$ is surjective for $i$ sufficiently large. \medskip\noindent Proof of (2). Choose $0 \in I$. Choose a quasi-compact open $X'_0 \subset X_0$ 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. \end{proof} \begin{lemma} \label{lemma-eventually-separated} Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \begin{enumerate} \item $S$ quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item $X \to S$ separated. \end{enumerate} Then $X_i \to S$ is separated for all $i$ large enough. \end{lemma} \begin{proof} Let $0 \in I$. Note that $I$ is nonempty as the limit is directed. As $X_0$ is quasi-compact we can find finitely many affine opens $U_1, \ldots, U_n \subset S$ such that $X_0 \to S$ maps into $U_1 \cup \ldots \cup U_n$. Denote $h_i : X_i \to S$ the structure morphism. It suffices to check that for some $i \geq 0$ the morphisms $h_i^{-1}(U_j) \to U_j$ are separated for $j = 1, \ldots, n$. Since $S$ is quasi-separated the morphisms $U_j \to S$ are quasi-compact. Hence $h_i^{-1}(U_j)$ is quasi-compact and quasi-separated. In this way we reduce to the case $S$ affine. In this case we have to show that $X_i$ is separated and we know that $X$ is separated. Thus the lemma follows from Lemma \ref{lemma-limit-separated}. \end{proof} \begin{lemma} \label{lemma-eventually-affine} Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \begin{enumerate} \item $S$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item $X \to S$ affine. \end{enumerate} Then $X_i \to S$ is affine for $i$ large enough. \end{lemma} \begin{proof} Choose a finite affine open covering $S = \bigcup_{j = 1, \ldots, n} V_j$. Denote $f : X \to S$ and $f_i : X_i \to S$ the structure morphisms. For each $j$ the scheme $f^{-1}(V_j) = \lim_i f_i^{-1}(V_j)$ is affine (as a finite morphism is affine by definition). Hence by Lemma \ref{lemma-limit-affine} there exists an $i \in I$ such that each $f_i^{-1}(V_j)$ is affine. In other words, $f_i : X_i \to S$ is affine for $i$ large enough, see Morphisms, Lemma \ref{morphisms-lemma-characterize-affine}. \end{proof} \begin{lemma} \label{lemma-eventually-finite} Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \begin{enumerate} \item $S$ 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 $X_i \to S$ locally of finite type \item $X \to S$ integral. \end{enumerate} Then $X_i \to S$ is finite for $i$ large enough. \end{lemma} \begin{proof} By Lemma \ref{lemma-eventually-affine} we may assume $X_i \to S$ is affine for all $i$. Choose a finite affine open covering $S = \bigcup_{j = 1, \ldots, n} V_j$. Denote $f : X \to S$ and $f_i : X_i \to S$ the structure morphisms. It suffices to show that there exists an $i$ such that $f_i^{-1}(V_j)$ is finite over $V_j$ for $j = 1, \ldots, m$ (Morphisms, Lemma \ref{morphisms-lemma-finite-local}). Namely, for $i' \geq i$ the composition $X_{i'} \to X_i \to S$ will be finite as a composition of finite morphisms (Morphisms, Lemma \ref{morphisms-lemma-composition-finite}). This reduces us to the affine case: Let $R$ be a ring and $A = \colim A_i$ with $R \to A$ integral and $A_i \to A_{i'}$ finite for all $i \leq i'$. Moreover $R \to A_i$ is of finite type for all $i$. Goal: Show that $A_i$ is finite over $R$ for some $i$. To prove this choose an $i \in I$ and pick generators $x_1, \ldots, x_m \in A_i$ of $A_i$ as an $R$-algebra. Since $A$ is integral over $R$ we can find monic polynomials $P_j \in R[T]$ such that $P_j(x_j) = 0$ in $A$. Thus there exists an $i' \geq i$ such that $P_j(x_j) = 0$ in $A_{i'}$ for $j = 1, \ldots, m$. Then the image $A'_i$ of $A_i$ in $A_{i'}$ is finite over $R$ by Algebra, Lemma \ref{algebra-lemma-characterize-finite-in-terms-of-integral}. Since $A'_i \subset A_{i'}$ is finite too we conclude that $A_{i'}$ is finite over $R$ by Algebra, Lemma \ref{algebra-lemma-finite-transitive}. \end{proof} \begin{lemma} \label{lemma-eventually-closed-immersion} Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \begin{enumerate} \item $S$ 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 $X_i \to S$ locally of finite type \item $X \to S$ a closed immersion. \end{enumerate} Then $X_i \to S$ is a closed immersion for $i$ large enough. \end{lemma} \begin{proof} By Lemma \ref{lemma-eventually-affine} we may assume $X_i \to S$ is affine for all $i$. Choose a finite affine open covering $S = \bigcup_{j = 1, \ldots, n} V_j$. Denote $f : X \to S$ and $f_i : X_i \to S$ the structure morphisms. It suffices to show that there exists an $i$ such that $f_i^{-1}(V_j)$ is a closed subscheme of $V_j$ for $j = 1, \ldots, m$ (Morphisms, Lemma \ref{morphisms-lemma-closed-immersion}). This reduces us to the affine case: Let $R$ be a ring and $A = \colim A_i$ with $R \to A$ surjective and $A_i \to A_{i'}$ surjective for all $i \leq i'$. Moreover $R \to A_i$ is of finite type for all $i$. Goal: Show that $R \to A_i$ is surjective for some $i$. To prove this choose an $i \in I$ and pick generators $x_1, \ldots, x_m \in A_i$ of $A_i$ as an $R$-algebra. Since $R \to A$ is surjective we can find $r_j \in R$ such that $r_j$ maps to $x_j$ in $A$. Thus there exists an $i' \geq i$ such that $r_j$ maps to the image of $x_j$ in $A_{i'}$ for $j = 1, \ldots, m$. Since $A_i \to A_{i'}$ is surjective this implies that $R \to A_{i'}$ is surjective. \end{proof} \section{Absolute Noetherian Approximation} \label{section-approximation} \noindent A nice reference for this section is Appendix C of the article by Thomason and Trobaugh \cite{TT}. See Categories, Section \ref{categories-section-posets-limits} for our conventions regarding directed systems. We will use the existence result and properties of the limit from Section \ref{section-limits} without further mention. \begin{lemma} \label{lemma-quasi-affine-finite-type-over-Z} Let $W$ be a quasi-affine scheme of finite type over $\mathbf{Z}$. Suppose $W \to \Spec(R)$ is an open immersion into an affine scheme. There exists a finite type $\mathbf{Z}$-algebra $A \subset R$ which induces an open immersion $W \to \Spec(A)$. Moreover, $R$ is the directed colimit of such subalgebras. \end{lemma} \begin{proof} Choose an affine open covering $W = \bigcup_{i = 1, \ldots, n} W_i$ such that each $W_i$ is a standard affine open in $\Spec(R)$. In other words, if we write $W_i = \Spec(R_i)$ then $R_i = R_{f_i}$ for some $f_i \in R$. Choose finitely many $x_{ij} \in R_i$ which generate $R_i$ over $\mathbf{Z}$. Pick an $N \gg 0$ such that each $f_i^Nx_{ij}$ comes from an element of $R$, say $y_{ij} \in R$. Set $A$ equal to the $\mathbf{Z}$-algebra generated by the $f_i$ and the $y_{ij}$ and (optionally) finitely many additional elements of $R$. Then $A$ works. Details omitted. \end{proof} \begin{lemma} \label{lemma-diagram} Suppose given a cartesian diagram of rings $$\xymatrix{ B \ar[r]_s & R \\ B'\ar[u] \ar[r] & R' \ar[u]_t }$$ Let $W' \subset \Spec(R')$ be an open of the form $W' = D(f_1) \cup \ldots \cup D(f_n)$ such that $t(f_i) = s(g_i)$ for some $g_i \in B$ and $B_{g_i} \cong R_{s(g_i)}$. Then $B' \to R'$ induces an open immersion of $W'$ into $\Spec(B')$. \end{lemma} \begin{proof} Set $h_i = (g_i, f_i) \in B'$. More on Algebra, Lemma \ref{more-algebra-lemma-diagram-localize} shows that $(B')_{h_i} \cong (R')_{f_i}$ as desired. \end{proof} \noindent The following lemma is a precise statement of Noetherian approximation. \begin{lemma} \label{lemma-approximate} Let $S$ be a quasi-compact and quasi-separated scheme. Let $V \subset S$ be a quasi-compact open. Let $I$ be a directed set and let $(V_i, f_{ii'})$ be an inverse system of schemes over $I$ with affine transition maps, with each $V_i$ of finite type over $\mathbf{Z}$, and with $V = \lim V_i$. Then there exist \begin{enumerate} \item a directed set $J$, \item an inverse system of schemes $(S_j, g_{jj'})$ over $J$, \item an order preserving map $\alpha : J \to I$, \item open subschemes $V'_j \subset S_j$, and \item isomorphisms $V'_j \to V_{\alpha(j)}$ \end{enumerate} such that \begin{enumerate} \item the transition morphisms $g_{jj'} : S_j \to S_{j'}$ are affine, \item each $S_j$ is of finite type over $\mathbf{Z}$, \item $g_{jj'}^{-1}(V_{j'}) = V_j$, \item $S = \lim S_j$ and $V = \lim V_j$, and \item the diagrams $$\vcenter{ \xymatrix{ V \ar[d] \ar[rd] \\ V'_j \ar[r] & V_{\alpha(j)} } } \quad\text{and}\quad \vcenter{ \xymatrix{ V_j \ar[r] \ar[d] & V_{\alpha(j)} \ar[d] \\ V_{j'} \ar[r] & V_{\alpha(j')} } }$$ are commutative. \end{enumerate} \end{lemma} \begin{proof} Set $Z = S \setminus V$. Choose affine opens $U_1, \ldots, U_m \subset S$ such that $Z \subset \bigcup_{l = 1, \ldots, m} U_l$. Consider the opens $$V \subset V \cup U_1 \subset V \cup U_1 \cup U_2 \subset \ldots \subset V \cup \bigcup\nolimits_{l = 1, \ldots, m} U_l = S$$ If we can prove the lemma successively for each of the cases $$V \cup U_1 \cup \ldots \cup U_l \subset V \cup U_1 \cup \ldots \cup U_{l + 1}$$ then the lemma will follow for $V \subset S$. In each case we are adding one affine open. Thus we may assume \begin{enumerate} \item $S = U \cup V$, \item $U$ affine open in $S$, \item $V$ quasi-compact open in $S$, and \item $V = \lim_i V_i$ with $(V_i, f_{ii'})$ an inverse system over a directed set $I$, each $f_{ii'}$ affine and each $V_i$ of finite type over $\mathbf{Z}$. \end{enumerate} Set $W = U \cap V$. As $S$ is quasi-separated, this is a quasi-compact open of $V$. By Lemma \ref{lemma-descend-opens} (and after shrinking $I$) we may assume that there exist opens $W_i \subset V_i$ such that $f_{ij}^{-1}(W_j) = W_i$ and such that $f_i^{-1}(W_i) = W$. Since $W$ is a quasi-compact open of $U$ it is quasi-affine. Hence we may assume (after shrinking $I$ again) that $W_i$ is quasi-affine for all $i$, see Lemma \ref{lemma-limit-quasi-affine}. \medskip\noindent Write $U = \Spec(B)$. Set $R = \Gamma(W, \mathcal{O}_W)$, and $R_i = \Gamma(W_i, \mathcal{O}_{W_i})$. By Lemma \ref{lemma-descend-section} we have $R = \colim_i R_i$. Now we have the maps of rings $$\xymatrix{ B \ar[r]_s & R \\ & R_i \ar[u]_{t_i} }$$ We set $B_i = \{(b, r) \in B \times R_i \mid s(b) = t_i(t)\}$ so that we have a cartesian diagram $$\xymatrix{ B \ar[r]_s & R \\ B_i \ar[u] \ar[r] & R_i \ar[u]_{t_i} }$$ for each $i$. The transition maps $R_i \to R_{i'}$ induce maps $B_i \to B_{i'}$. It is clear that $B = \colim_i B_i$. In the next paragraph we show that for all sufficiently large $i$ the composition $W_i \to \Spec(R_i) \to \Spec(B_i)$ is an open immersion. \medskip\noindent As $W$ is a quasi-compact open of $U = \Spec(B)$ we can find a finitely many elements $g_l \in B$, $l = 1, \ldots, m$ such that $D(g_l) \subset W$ and such that $W = \bigcup_{l = 1, \ldots, m} D(g_l)$. Note that this implies $D(g_l) = W_{s(g_l)}$ as open subsets of $U$, where $W_{s(g_l)}$ denotes the largest open subset of $W$ on which $s(g_l)$ is invertible. Hence $$B_{g_l} = \Gamma(D(g_l), \mathcal{O}_U) = \Gamma(W_{s(g_l)}, \mathcal{O}_W) = R_{s(g_l)},$$ where the last equality is Properties, Lemma \ref{properties-lemma-invert-f-sections}. Since $W_{s(g_l)}$ is affine this also implies that $D(s(g_l)) = W_{s(g_l)}$ as open subsets of $\Spec(R)$. Since $R = \colim_i R_i$ we can (after shrinking $I$) assume there exist $g_{l, i} \in R_i$ for all $i \in I$ such that $s(g_l) = t_i(g_{l, i})$. Of course we choose the $g_{l, i}$ such that $g_{l, i}$ maps to $g_{l, i'}$ under the transition maps $R_i \to R_{i'}$. Then, by Lemma \ref{lemma-descend-opens} we can (after shrinking $I$ again) assume the corresponding opens $D(g_{l, i}) \subset \Spec(R_i)$ are contained in $W_i$ for $l = 1, \ldots, m$ and cover $W_i$. We conclude that the morphism $W_i \to \Spec(R_i) \to \Spec(B_i)$ is an open immersion, see Lemma \ref{lemma-diagram} \medskip\noindent By Lemma \ref{lemma-quasi-affine-finite-type-over-Z} we can write $B_i$ as a directed colimit of subalgebras $A_{i, p} \subset B_i$, $p \in P_i$ each of finite type over $\mathbf{Z}$ and such that $W_i$ is identified with an open subscheme of $\Spec(A_{i, p})$. Let $S_{i, p}$ be the scheme obtained by glueing $V_i$ and $\Spec(A_{i, p})$ along the open $W_i$, see Schemes, Section \ref{schemes-section-glueing-schemes}. Here is the resulting commutative diagram of schemes: $$\xymatrix{ & & V \ar[lld] \ar[d] & W \ar[l] \ar[lld] \ar[d] \\ V_i \ar[d] & W_i \ar[l] \ar[d] & S \ar[lld] & U \ar[lld] \ar[l] \\ S_{i, p} & \Spec(A_{i, p}) \ar[l] }$$ The morphism $S \to S_{i, p}$ arises because the upper right square is a pushout in the category of schemes. Note that $S_{i, p}$ is of finite type over $\mathbf{Z}$ since it has a finite affine open covering whose members are spectra of finite type $\mathbf{Z}$-algebras. We define a preorder on $J = \coprod_{i \in I} P_i$ by the rule $(i', p') \geq (i, p)$ if and only if $i' \geq i$ and the map $B_i \to B_{i'}$ maps $A_{i, p}$ into $A_{i', p'}$. This is exactly the condition needed to define a morphism $S_{i', p'} \to S_{i, p}$: namely make a commutative diagram as above using the transition morphisms $V_{i'} \to V_i$ and $W_{i'} \to W_i$ and the morphism $\Spec(A_{i', p'}) \to \Spec(A_{i, p})$ induced by the ring map $A_{i, p} \to A_{i', p'}$. The relevant commutativities have been built into the constructions. We claim that $S$ is the directed limit of the schemes $S_{i, p}$. Since by construction the schemes $V_i$ have limit $V$ this boils down to the fact that $B$ is the limit of the rings $A_{i, p}$ which is true by construction. The map $\alpha : J \to I$ is given by the rule $j = (i, p) \mapsto i$. The open subscheme $V'_j$ is just the image of $V_i \to S_{i, p}$ above. The commutativity of the diagrams in (5) is clear from the construction. This finishes the proof of the lemma. \end{proof} \begin{proposition} \label{proposition-approximate} Let $S$ be a quasi-compact and quasi-separated scheme. There exist a directed set $I$ and an inverse system of schemes $(S_i, f_{ii'})$ over $I$ such that \begin{enumerate} \item the transition morphisms $f_{ii'}$ are affine \item each $S_i$ is of finite type over $\mathbf{Z}$, and \item $S = \lim_i S_i$. \end{enumerate} \end{proposition} \begin{proof} This is a special case of Lemma \ref{lemma-approximate} with $V = \emptyset$. \end{proof} \section{Limits and morphisms of finite presentation} \label{section-finite-presentation} \noindent The following is a generalization of Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation}. \begin{proposition} \label{proposition-characterize-locally-finite-presentation} Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is locally of finite presentation. \item For any directed set $I$, and any inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$ with each $T_i$ affine, we have $$\Mor_S(\lim_i T_i, X) = \colim_i \Mor_S(T_i, X)$$ \item For any directed set $I$, and any inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$ with each $f_{ii'}$ affine and every $T_i$ quasi-compact and quasi-separated as a scheme, we have $$\Mor_S(\lim_i T_i, X) = \colim_i \Mor_S(T_i, X)$$ \end{enumerate} \end{proposition} \begin{proof} It is clear that (3) implies (2). \medskip\noindent Let us prove that (2) implies (1). Assume (2). Choose any affine opens $U \subset X$ and $V \subset S$ such that $f(U) \subset V$. We have to show that $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation. Let $(A_i, \varphi_{ii'})$ be a directed system of $\mathcal{O}_S(V)$-algebras. Set $A = \colim_i A_i$. According to Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation} we have to show that $$\Hom_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A) = \colim_i \Hom_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A_i)$$ Consider the schemes $T_i = \Spec(A_i)$. They form an inverse system of $V$-schemes over $I$ with transition morphisms $f_{ii'} : T_i \to T_{i'}$ induced by the $\mathcal{O}_S(V)$-algebra maps $\varphi_{i'i}$. Set $T := \Spec(A) = \lim_i T_i$. The formula above becomes in terms of morphism sets of schemes $$\Mor_V(\lim_i T_i, U) = \colim_i \Mor_V(T_i, U).$$ We first observe that $\Mor_V(T_i, U) = \Mor_S(T_i, U)$ and $\Mor_V(T, U) = \Mor_S(T, U)$. Hence we have to show that $$\Mor_S(\lim_i T_i, U) = \colim_i \Mor_S(T_i, U)$$ and we are given that $$\Mor_S(\lim_i T_i, X) = \colim_i \Mor_S(T_i, X).$$ Hence it suffices to prove that given a morphism $g_i : T_i \to X$ over $S$ such that the composition $T \to T_i \to X$ ends up in $U$ there exists some $i' \geq i$ such that the composition $g_{i'} : T_{i'} \to T_i \to X$ ends up in $U$. Denote $Z_{i'} = g_{i'}^{-1}(X \setminus U)$. Assume each $Z_{i'}$ is nonempty to get a contradiction. By Lemma \ref{lemma-limit-closed-nonempty} there exists a point $t$ of $T$ which is mapped into $Z_{i'}$ for all $i' \geq i$. Such a point is not mapped into $U$. A contradiction. \medskip\noindent Finally, let us prove that (1) implies (3). Assume (1). Let an inverse directed system $(T_i, f_{ii'})$ of $S$-schemes be given. Assume the morphisms $f_{ii'}$ are affine and each $T_i$ is quasi-compact and quasi-separated as a scheme. Let $T = \lim_i T_i$. Denote $f_i : T \to T_i$ the projection morphisms. We have to show: \begin{enumerate} \item[(a)] Given morphisms $g_i, g'_i : T_i \to X$ over $S$ such that $g_i \circ f_i = g'_i \circ f_i$, then there exists an $i' \geq i$ such that $g_i \circ f_{i'i} = g'_i \circ f_{i'i}$. \item[(b)] Given any morphism $g : T \to X$ over $S$ there exists an $i \in I$ and a morphism $g_i : T_i \to X$ such that $g = f_i \circ g_i$. \end{enumerate} \noindent First let us prove the uniqueness part (a). Let $g_i, g'_i : T_i \to X$ be morphisms such that $g_i \circ f_i = g'_i \circ f_i$. For any $i' \geq i$ we set $g_{i'} = g_i \circ f_{i'i}$ and $g'_{i'} = g'_i \circ f_{i'i}$. We also set $g = g_i \circ f_i = g'_i \circ f_i$. Consider the morphism $(g_i, g'_i) : T_i \to X \times_S X$. Set $$W = \bigcup\nolimits_{U \subset X\text{ affine open}, V \subset S\text{ affine open}, f(U) \subset V} U \times_V U.$$ This is an open in $X \times_S X$, with the property that the morphism $\Delta_{X/S}$ factors through a closed immersion into $W$, see the proof of Schemes, Lemma \ref{schemes-lemma-diagonal-immersion}. Note that the composition $(g_i, g'_i) \circ f_i : T \to X \times_S X$ is a morphism into $W$ because it factors through the diagonal by assumption. Set $Z_{i'} = (g_{i'}, g'_{i'})^{-1}(X \times_S X \setminus W)$. If each $Z_{i'}$ is nonempty, then by Lemma \ref{lemma-limit-closed-nonempty} there exists a point $t \in T$ which maps to $Z_{i'}$ for all $i' \geq i$. This is a contradiction with the fact that $T$ maps into $W$. Hence we may increase $i$ and assume that $(g_i, g'_i) : T_i \to X \times_S X$ is a morphism into $W$. By construction of $W$, and since $T_i$ is quasi-compact we can find a finite affine open covering $T_i = T_{1, i} \cup \ldots \cup T_{n, i}$ such that $(g_i, g'_i)|_{T_{j, i}}$ is a morphism into $U \times_V U$ for some pair $(U, V)$ as in the definition of $W$ above. Since it suffices to prove that $g_{i'}$ and $g'_{i'}$ agree on each of the $f_{i'i}^{-1}(T_{j, i})$ this reduces us to the affine case. The affine case follows from Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation} and the fact that the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation (see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-presentation-characterize}). \medskip\noindent Finally, we prove the existence part (b). Let $g : T \to X$ be a morphism of schemes over $S$. We can find a finite affine open covering $T = W_1 \cup \ldots \cup W_n$ such that for each $j \in \{1, \ldots, n\}$ there exist affine opens $U_j \subset X$ and $V_j \subset S$ with $f(U_j) \subset V_j$ and $g(W_j) \subset U_j$. By Lemmas \ref{lemma-descend-opens} and \ref{lemma-limit-affine} (after possibly shrinking $I$) we may assume that there exist affine open coverings $T_i = W_{1, i} \cup \ldots \cup W_{n, i}$ compatible with transition maps such that $W_j = \lim_i W_{j, i}$. We apply Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation} to the rings corresponding to the affine schemes $U_j$, $V_j$, $W_{j, i}$ and $W_j$ using that $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_j)$ is of finite presentation (see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-presentation-characterize}). Thus we can find for each $j$ an index $i_j \in I$ and a morphism $g_{j, i_j} : W_{j, i_j} \to X$ such that $g_{j, i_j} \circ f_i|_{W_j} : W_j \to W_{j, i} \to X$ equals $g|_{W_j}$. By part (a) proved above, using the quasi-compactness of $W_{j_1, i} \cap W_{j_2, i}$ which follows as $T_i$ is quasi-separated, we can find an index $i' \in I$ larger than all $i_j$ such that $$g_{j_1, i_{j_1}} \circ f_{i'i_{j_1}}|_{W_{j_1, i'} \cap W_{j_2, i'}} = g_{j_2, i_{j_2}} \circ f_{i'i_{j_2}}|_{W_{j_1, i'} \cap W_{j_2, i'}}$$ for all $j_1, j_2 \in \{1, \ldots, n\}$. Hence the morphisms $g_{j, i_j} \circ f_{i'i_j}|_{W_{j, i'}}$ glue to given the desired morphism $T_{i'} \to X$. \end{proof} \begin{remark} \label{remark-limit-preserving} Let $S$ be a scheme. Let us say that a functor $F : (\Sch/S)^{opp} \to \textit{Sets}$ is {\it limit preserving} if for every directed inverse system $\{T_i\}_{i \in I}$ of affine schemes with limit $T$ we have $F(T) = \colim_i F(T_i)$. Let $X$ be a scheme over $S$, and let $h_X : (\Sch/S)^{opp} \to \textit{Sets}$ be its functor of points, see Schemes, Section \ref{schemes-section-representable}. In this terminology Proposition \ref{proposition-characterize-locally-finite-presentation} says that a scheme $X$ is locally of finite presentation over $S$ if and only if $h_X$ is limit preserving. \end{remark} \begin{lemma} \label{lemma-surjection-is-enough} Let $f : X \to S$ be a morphism of schemes. If for every directed limit $T = \lim_{i \in I} T_i$ of affine schemes over $S$ the map $$\colim \Mor_S(T_i, X) \longrightarrow \Mor_S(T, X)$$ is surjective, then $f$ is locally of finite presentation. In other words, in Proposition \ref{proposition-characterize-locally-finite-presentation} parts (2) and (3) it suffices to check surjectivity of the map. \end{lemma} \begin{proof} The proof is exactly the same as the proof of the implication (2) implies (1)'' in Proposition \ref{proposition-characterize-locally-finite-presentation}. Choose any affine opens $U \subset X$ and $V \subset S$ such that $f(U) \subset V$. We have to show that $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation. Let $(A_i, \varphi_{ii'})$ be a directed system of $\mathcal{O}_S(V)$-algebras. Set $A = \colim_i A_i$. According to Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation} it suffices to show that $$\colim_i \Hom_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A_i) \to \Hom_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A)$$ is surjective. Consider the schemes $T_i = \Spec(A_i)$. They form an inverse system of $V$-schemes over $I$ with transition morphisms $f_{ii'} : T_i \to T_{i'}$ induced by the $\mathcal{O}_S(V)$-algebra maps $\varphi_{i'i}$. Set $T := \Spec(A) = \lim_i T_i$. The formula above becomes in terms of morphism sets of schemes $$\colim_i \Mor_V(T_i, U) \to \Mor_V(\lim_i T_i, U)$$ We first observe that $\Mor_V(T_i, U) = \Mor_S(T_i, U)$ and $\Mor_V(T, U) = \Mor_S(T, U)$. Hence we have to show that $$\colim_i \Mor_S(T_i, U) \to \Mor_S(\lim_i T_i, U)$$ is surjective and we are given that $$\colim_i \Mor_S(T_i, X) \to \Mor_S(\lim_i T_i, X)$$ is surjective. Hence it suffices to prove that given a morphism $g_i : T_i \to X$ over $S$ such that the composition $T \to T_i \to X$ ends up in $U$ there exists some $i' \geq i$ such that the composition $g_{i'} : T_{i'} \to T_i \to X$ ends up in $U$. Denote $Z_{i'} = g_{i'}^{-1}(X \setminus U)$. Assume each $Z_{i'}$ is nonempty to get a contradiction. By Lemma \ref{lemma-limit-closed-nonempty} there exists a point $t$ of $T$ which is mapped into $Z_{i'}$ for all $i' \geq i$. Such a point is not mapped into $U$. A contradiction. \end{proof} \section{Relative approximation} \label{section-relative-approximation} \noindent The title of this section refers to results of the following type. \begin{lemma} \label{lemma-relative-approximation} Let $f : X \to S$ be a morphism of schemes. Assume that \begin{enumerate} \item $X$ is quasi-compact and quasi-separated, and \item $S$ is quasi-separated. \end{enumerate} Then $X = \lim X_i$ is a limit of a directed system of schemes $X_i$ of finite presentation over $S$ with affine transition morphisms over $S$. \end{lemma} \begin{proof} Since $f(X)$ is quasi-compact we may replace $S$ by a quasi-compact open containing $f(X)$. Hence we may assume $S$ is quasi-compact as well. Write $X = \lim X_a$ and $S = \lim S_b$ as in Proposition \ref{proposition-approximate}, i.e., with $X_a$ and $S_b$ 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 $f_{a, b} : X_a \to S_b$ making the diagram $$\xymatrix{ X \ar[d] \ar[r] & S \ar[d] \\ X_a \ar[r] & S_b }$$ 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 preorder $$(a, b, f_{a, b}) \geq (a', b', f_{a', b'}) \Leftrightarrow a \geq a',\ b' \geq b,\text{ and } f_{a', b'} \circ h_{a, a'} = g_{b', b} \circ f_{a, b}$$ where $h_{a, a'} : X_a \to X_{a'}$ and $g_{b', b} : S_{b'} \to S_b$ are the transition morphisms. The remarks above show that this system is directed. It follows formally from the equalities $X = \lim X_a$ and $S = \lim S_b$ that $$X = \lim_{(a, b, f_{a, b})} X_a \times_{f_{a, b}, S_b} S.$$ where the limit is over our directed system above. The transition morphisms $X_a \times_{S_b} S \to X_{a'} \times_{S_{b'}} S$ are affine as the composition $$X_a \times_{S_b} S \to X_a \times_{S_{b'}} S \to X_{a'} \times_{S_{b'}} S$$ where the first morphism is a closed immersion (by Schemes, Lemma \ref{schemes-lemma-fibre-product-after-map}) and the second is a base change of an affine morphism (Morphisms, Lemma \ref{morphisms-lemma-base-change-affine}) and the composition of affine morphisms is affine (Morphisms, Lemma \ref{morphisms-lemma-composition-affine}). The morphisms $f_{a, b}$ are of finite presentation (Morphisms, Lemmas \ref{morphisms-lemma-noetherian-finite-type-finite-presentation} and \ref{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, Lemma \ref{morphisms-lemma-base-change-finite-presentation}). \end{proof} \begin{lemma} \label{lemma-integral-limit-finite-and-finite-presentation} Let $X \to S$ be an integral morphism with $S$ quasi-compact and quasi-separated. Then $X = \lim X_i$ with $X_i \to S$ finite and of finite presentation. \end{lemma} \begin{proof} Consider the sheaf $\mathcal{A} = f_*\mathcal{O}_X$. This is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras, see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. Combining Properties, Lemma \ref{properties-lemma-integral-algebra-directed-colimit-finite} we can write $\mathcal{A} = \colim_i \mathcal{A}_i$ as a filtered colimit of finite and finitely presented $\mathcal{O}_S$-algebras. Then $$X_i = \underline{\Spec}_S(\mathcal{A}_i) \longrightarrow S$$ is a finite and finitely presented morphism of schemes. By construction $X = \lim_i X_i$ which proves the lemma. \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 over $S$. We will work in the following situation. \begin{situation} \label{situation-descent-property} Let $S = \lim S_i$ be a limit of a directed system of schemes 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 schemes over $S_0$. Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated. Let $f_i : X_i \to Y_i$ be the base change of $f_0$ to $S_i$ and let $f : X \to Y$ be the base change of $f_0$ to $S$. \end{situation} \begin{lemma} \label{lemma-descend-affine-finite-presentation} Notation and assumptions as in Situation \ref{situation-descent-property}. If $f$ is affine, then there exists an index $i \geq 0$ such that $f_i$ is affine. \end{lemma} \begin{proof} Let $Y_0 = \bigcup_{j = 1, \ldots, m} V_{j, 0}$ be a finite affine open covering. Set $U_{j, 0} = f_0^{-1}(V_{j, 0})$. For $i \geq 0$ we denote $V_{j, i}$ the inverse image of $V_{j, 0}$ in $Y_i$ and $U_{j, i} = f_i^{-1}(V_{j, i})$. Similarly we have $U_j = f^{-1}(V_j)$. Then $U_j = \lim_{i \geq 0} U_{j, i}$ (see Lemma \ref{lemma-directed-inverse-system-has-limit}). Since $U_j$ is affine by assumption we see that each $U_{j, i}$ is affine for $i$ large enough, see Lemma \ref{lemma-limit-affine}. As there are finitely many $j$ we can pick an $i$ which works for all $j$. Thus $f_i$ is affine for $i$ large enough, see Morphisms, Lemma \ref{morphisms-lemma-characterize-affine}. \end{proof} \begin{lemma} \label{lemma-descend-finite-finite-presentation} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a finite morphism, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is finite. \end{lemma} \begin{proof} A finite morphism is affine, see Morphisms, Definition \ref{morphisms-definition-integral}. Hence by Lemma \ref{lemma-descend-affine-finite-presentation} above after increasing $0$ we may assume that $f_0$ is affine. By writing $Y_0$ as a finite union of affines we reduce to proving the result when $X_0$ and $Y_0$ are affine and map into a common affine $W \subset S_0$. The corresponding algebra statement follows from Algebra, Lemma \ref{algebra-lemma-colimit-finite}. \end{proof} \begin{lemma} \label{lemma-descend-unramified} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is unramified, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is unramified. \end{lemma} \begin{proof} Choose a finite affine open covering $Y_0 = \bigcup_{j = 1, \ldots, m} Y_{j, 0}$ such that each $Y_{j, 0}$ maps into an affine open $S_{j, 0} \subset S_0$. For each $j$ let $f_0^{-1}Y_{j, 0} = \bigcup_{k = 1, \ldots, n_j} X_{k, 0}$ be a finite affine open covering. Since the property of being unramified is local we see that it suffices to prove the lemma for the morphisms of affines $X_{k, i} \to Y_{j, i} \to S_{j, i}$ which are the base changes of $X_{k, 0} \to Y_{j, 0} \to S_{j, 0}$ to $S_i$. Thus we reduce to the case that $X_0, Y_0, S_0$ are affine \medskip\noindent In the affine case we reduce to the following algebra result. Suppose that $R = \colim_{i \in I} R_i$. For some $0 \in I$ suppose given an $R_0$-algebra map $A_i \to B_i$ of finite type. If $R \otimes_{R_0} A_0 \to R \otimes_{R_0} B_0$ is unramified, then for some $i \geq 0$ the map $R_i \otimes_{R_0} A_0 \to R_i \otimes_{R_0} B_0$ is unramified. This follows from Algebra, Lemma \ref{algebra-lemma-colimit-unramified}. \end{proof} \begin{lemma} \label{lemma-descend-closed-immersion-finite-presentation} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a closed immersion, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is a closed immersion. \end{lemma} \begin{proof} A closed immersion is affine, see Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-affine}. Hence by Lemma \ref{lemma-descend-affine-finite-presentation} above after increasing $0$ we may assume that $f_0$ is affine. By writing $Y_0$ as a finite union of affines we reduce to proving the result when $X_0$ and $Y_0$ are affine and map into a common affine $W \subset S_0$. The corresponding algebra statement is a consequence of Algebra, Lemma \ref{algebra-lemma-colimit-surjective}. \end{proof} \begin{lemma} \label{lemma-descend-separated-finite-presentation} 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-finite-presentation} to the diagonal morphism $\Delta_{X_0/S_0} : X_0 \to X_0 \times_{S_0} X_0$. (This is permissible as diagonal morphisms are locally of finite type and the fibre product $X_0 \times_{S_0} X_0$ is quasi-compact and quasi-separated, see Schemes, Lemma \ref{schemes-lemma-diagonal-immersion}, Morphisms, Lemma \ref{morphisms-lemma-immersion-locally-finite-type}, and Schemes, Remark \ref{schemes-remark-quasi-compact-and-quasi-separated}. \end{proof} \begin{lemma} \label{lemma-descend-flat-finite-presentation} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is flat, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is flat for some $i \geq 0$. \end{lemma} \begin{proof} Choose a finite affine open covering $Y_0 = \bigcup_{j = 1, \ldots, m} Y_{j, 0}$ such that each $Y_{j, 0}$ maps into an affine open $S_{j, 0} \subset S_0$. For each $j$ let $f_0^{-1}Y_{j, 0} = \bigcup_{k = 1, \ldots, n_j} X_{k, 0}$ be a finite affine open covering. Since the property of being flat is local we see that it suffices to prove the lemma for the morphisms of affines $X_{k, i} \to Y_{j, i} \to S_{j, i}$ which are the base changes of $X_{k, 0} \to Y_{j, 0} \to S_{j, 0}$ to $S_i$. Thus we reduce to the case that $X_0, Y_0, S_0$ are affine \medskip\noindent In the affine case we reduce to the following algebra result. Suppose that $R = \colim_{i \in I} R_i$. For some $0 \in I$ suppose given an $R_0$-algebra map $A_i \to B_i$ of finite presentation. If $R \otimes_{R_0} A_0 \to R \otimes_{R_0} B_0$ is flat, then for some $i \geq 0$ the map $R_i \otimes_{R_0} A_0 \to R_i \otimes_{R_0} B_0$ is flat. This follows from Algebra, Lemma \ref{algebra-lemma-flat-finite-presentation-limit-flat} part (3). \end{proof} \begin{lemma} \label{lemma-descend-finite-locally-free} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is finite locally free (of degree $d$), \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is finite locally free (of degree $d$) for some $i \geq 0$. \end{lemma} \begin{proof} By Lemmas \ref{lemma-descend-flat-finite-presentation} and \ref{lemma-descend-finite-finite-presentation} we find an $i$ such that $f_i$ is flat and finite. On the other hand, $f_i$ is locally of finite presentation. Hence $f_i$ is finite locally free by Morphisms, Lemma \ref{morphisms-lemma-finite-flat}. If moreover $f$ is finite locally free of degree $d$, then the image of $Y \to Y_i$ is contained in the open and closed locus $W_d \subset Y_i$ over which $f_i$ has degree $d$. By Lemma \ref{lemma-limit-contained-in-constructible} we see that for some $i' \geq i$ the image of $Y_{i'} \to Y_i$ is contained in $W_d$. Then $f_{i'}$ will be finite locally free of degree $d$. \end{proof} \begin{lemma} \label{lemma-descend-smooth} 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} Being smooth is local on the source and the target (Morphisms, Lemma \ref{morphisms-lemma-smooth-characterize}) hence we may assume $S_0, X_0, Y_0$ affine (details omitted). The corresponding algebra fact is Algebra, Lemma \ref{algebra-lemma-colimit-smooth}. \end{proof} \begin{lemma} \label{lemma-descend-etale} 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} Being \'etale is local on the source and the target (Morphisms, Lemma \ref{morphisms-lemma-etale-characterize}) hence we may assume $S_0, X_0, Y_0$ affine (details omitted). The corresponding algebra fact is Algebra, Lemma \ref{algebra-lemma-colimit-etale}. \end{proof} \begin{lemma} \label{lemma-descend-isomorphism} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is an isomorphism, and \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is an isomorphism for some $i \geq 0$. \end{lemma} \begin{proof} By Lemmas \ref{lemma-descend-etale} and \ref{lemma-descend-closed-immersion-finite-presentation} we can find an $i$ such that $f_i$ is flat and a closed immersion. Then $f_i$ identifies $X_i$ with an open and closed subscheme of $Y_i$, see Morphisms, Lemma \ref{morphisms-lemma-flat-closed-immersions-finite-presentation}. By assumption the image of $Y \to Y_i$ maps into $f_i(X_i)$. Thus by Lemma \ref{lemma-limit-contained-in-constructible} we find that $Y_{i'}$ maps into $f_i(X_i)$ for some $i' \geq i$. It follows that $X_{i'} \to Y_{i'}$ is surjective and we win. \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, and \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 of schemes $V \to W$ is a monomorphism if and only if the diagonal $V \to V \times_W V$ is an isomorphism (Schemes, Lemma \ref{schemes-lemma-monomorphism}). The morphism $X_0 \to X_0 \times_{Y_0} X_0$ is locally of finite presentation by Morphisms, Lemma \ref{morphisms-lemma-diagonal-morphism-finite-type}. Since $X_0 \times_{Y_0} X_0$ is quasi-compact and quasi-separated (Schemes, Remark \ref{schemes-remark-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-surjective} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is surjective, and \item $f_0$ is locally of finite presentation, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is surjective. \end{lemma} \begin{proof} The morphism $f_0$ is of finite presentation. Hence $E = f_0(X_0)$ is a constructible subset of $Y_0$, see Morphisms, Lemma \ref{morphisms-lemma-chevalley}. Since $f_i$ is the base change of $f_0$ by $Y_i \to Y_0$ we see that the image of $f_i$ is the inverse image of $E$ in $Y_i$. Moreover, we know that $Y \to Y_0$ maps into $E$. Hence we win by Lemma \ref{lemma-limit-contained-in-constructible}. \end{proof} \begin{lemma} \label{lemma-descend-syntomic} Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is syntomic, and \item $f_0$ is locally of finite presentation, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is syntomic. \end{lemma} \begin{proof} Choose a finite affine open covering $Y_0 = \bigcup_{j = 1, \ldots, m} Y_{j, 0}$ such that each $Y_{j, 0}$ maps into an affine open $S_{j, 0} \subset S_0$. For each $j$ let $f_0^{-1}Y_{j, 0} = \bigcup_{k = 1, \ldots, n_j} X_{k, 0}$ be a finite affine open covering. Since the property of being syntomic is local we see that it suffices to prove the lemma for the morphisms of affines $X_{k, i} \to Y_{j, i} \to S_{j, i}$ which are the base changes of $X_{k, 0} \to Y_{j, 0} \to S_{j, 0}$ to $S_i$. Thus we reduce to the case that $X_0, Y_0, S_0$ are affine \medskip\noindent In the affine case we reduce to the following algebra result. Suppose that $R = \colim_{i \in I} R_i$. For some $0 \in I$ suppose given an $R_0$-algebra map $A_i \to B_i$ of finite presentation. If $R \otimes_{R_0} A_0 \to R \otimes_{R_0} B_0$ is syntomic, then for some $i \geq 0$ the map $R_i \otimes_{R_0} A_0 \to R_i \otimes_{R_0} B_0$ is syntomic. This follows from Algebra, Lemma \ref{algebra-lemma-colimit-syntomic}. \end{proof} \section{Finite type closed in finite presentation} \label{section-finite-type-closed-in-finite-presentation} \noindent A result of this type is \cite[Satz 2.10]{Kiehl}. Another reference is \cite{Conrad-Nagata}. \begin{lemma} \label{lemma-locally-finite-type-in-finite-presentation} Let $f : X \to S$ be a morphism of schemes. Assume: \begin{enumerate} \item The morphism $f$ is locally of finite type. \item The scheme $X$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a morphism of finite presentation $f' : X' \to S$ and an immersion $X \to X'$ of schemes over $S$. \end{lemma} \begin{proof} By Proposition \ref{proposition-approximate} we can write $X = \lim_i X_i$ with each $X_i$ of finite type over $\mathbf{Z}$ and with transition morphisms $f_{ii'} : X_i \to X_{i'}$ affine. Consider the commutative diagram $$\xymatrix{ X \ar[r] \ar[rd] & X_{i, S} \ar[r] \ar[d] & X_i \ar[d] \\ & S \ar[r] & \Spec(\mathbf{Z}) }$$ Note that $X_i$ is of finite presentation over $\Spec(\mathbf{Z})$, see Morphisms, Lemma \ref{morphisms-lemma-noetherian-finite-type-finite-presentation}. Hence the base change $X_{i, S} \to S$ is of finite presentation by Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-presentation}. Thus it suffices to show that the arrow $X \to X_{i, S}$ is an immersion for $i$ sufficiently large. \medskip\noindent To do this we choose a finite affine open covering $X = V_1 \cup \ldots \cup V_n$ such that $f$ maps each $V_j$ into an affine open $U_j \subset S$. Let $h_{j, a} \in \mathcal{O}_X(V_j)$ be a finite set of elements which generate $\mathcal{O}_X(V_j)$ as an $\mathcal{O}_S(U_j)$-algebra, see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-type-characterize}. By Lemmas \ref{lemma-descend-opens} and \ref{lemma-limit-affine} (after possibly shrinking $I$) we may assume that there exist affine open coverings $X_i = V_{1, i} \cup \ldots \cup V_{n, i}$ compatible with transition maps such that $V_j = \lim_i V_{j, i}$. By Lemma \ref{lemma-descend-section} we can choose $i$ so large that each $h_{j, a}$ comes from an element $h_{j, a, i} \in \mathcal{O}_{X_i}(V_{j, i})$. Thus the arrow in $$V_j \longrightarrow U_j \times_{\Spec(\mathbf{Z})} V_{j, i} = (V_{j, i})_{U_j} \subset (V_{j, i})_S \subset X_{i, S}$$ is a closed immersion. Since $\bigcup (V_{j, i})_{U_j}$ forms an open of $X_{i, S}$ and since the inverse image of $(V_{j, i})_{U_j}$ in $X$ is $V_j$ it follows that $X \to X_{i, S}$ is an immersion. \end{proof} \begin{remark} \label{remark-cannot-do-better} We cannot do better than this if we do not assume more on $S$ and the morphism $f : X \to S$. For example, in general it will not be possible to find a {\it closed} immersion $X \to X'$ as in the lemma. The reason is that this would imply that $f$ is quasi-compact which may not be the case. An example is to take $S$ to be infinite dimensional affine space with $0$ doubled and $X$ to be one of the two infinite dimensional affine spaces. \end{remark} \begin{lemma} \label{lemma-finite-type-closed-in-finite-presentation} Let $f : X \to S$ be a morphism of schemes. Assume: \begin{enumerate} \item The morphism $f$ is of locally of finite type. \item The scheme $X$ is quasi-compact and quasi-separated, and \item The scheme $S$ is quasi-separated. \end{enumerate} Then there exists a morphism of finite presentation $f' : X' \to S$ and a closed immersion $X \to X'$ of schemes over $S$. \end{lemma} \begin{proof} By Lemma \ref{lemma-locally-finite-type-in-finite-presentation} above there exists a morphism $Y \to S$ of finite presentation and an immersion $i : X \to Y$ of schemes over $S$. For every point $x \in X$, there exists an affine open $V_x \subset Y$ such that $i^{-1}(V_x) \to V_x$ is a closed immersion. Since $X$ is quasi-compact we can find finitely may affine opens $V_1, \ldots, V_n \subset Y$ such that $i(X) \subset V_1 \cup \ldots \cup V_n$ and $i^{-1}(V_j) \to V_j$ is a closed immersion. In other words such that $i : X \to X' = V_1 \cup \ldots \cup V_n$ is a closed immersion of schemes over $S$. Since $S$ is quasi-separated and $Y$ is quasi-separated over $S$ we deduce that $Y$ is quasi-separated, see Schemes, Lemma \ref{schemes-lemma-separated-permanence}. Hence the open immersion $X' = V_1 \cup \ldots \cup V_n \to Y$ is quasi-compact. This implies that $X' \to Y$ is of finite presentation, see Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-open-immersion-finite-presentation}. We conclude since then $X' \to Y \to S$ is a composition of morphisms of finite presentation, and hence of finite presentation (see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-presentation}). \end{proof} \begin{lemma} \label{lemma-closed-is-limit-closed-and-finite-presentation} Let $X \to Y$ be a closed immersion of schemes. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \lim X_i$ of schemes over $Y$ where $X_i \to Y$ is a closed immersion 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 subscheme of $Y$. By Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} we can write $\mathcal{I}$ as a directed colimit $\mathcal{I} = \colim_{i \in I} \mathcal{I}_i$ of its quasi-coherent sheaves of ideals of finite type. Let $X_i \subset Y$ be the closed subscheme defined by $\mathcal{I}_i$. These form an inverse system of schemes indexed by $I$. The transition morphisms $X_i \to X_{i'}$ are affine because they are closed immersions. Each $X_i$ is quasi-compact and quasi-separated since it is a closed subscheme of $Y$ and $Y$ is quasi-compact and quasi-separated by our assumptions. We have $X = \lim_i X_i$ as follows directly from the fact that $\mathcal{I} = \colim_{i \in I} \mathcal{I}_a$. Each of the morphisms $X_i \to Y$ is of finite presentation, see Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-finite-presentation}. \end{proof} \begin{lemma} \label{lemma-finite-type-is-limit-finite-presentation} Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item The morphism $f$ is of locally of finite type. \item The scheme $X$ is quasi-compact and quasi-separated, and \item The scheme $S$ is quasi-separated. \end{enumerate} Then $X = \lim X_i$ where the $X_i \to S$ are of finite presentation, the $X_i$ are quasi-compact and quasi-separated, and the transition morphisms $X_{i'} \to X_i$ are closed immersions (which implies that $X \to X_i$ are closed immersions for all $i$). \end{lemma} \begin{proof} By Lemma \ref{lemma-finite-type-closed-in-finite-presentation} there is a closed immersion $X \to Y$ with $Y \to S$ of finite presentation. Then $Y$ is quasi-separated by Schemes, Lemma \ref{schemes-lemma-separated-permanence}. Since $X$ is quasi-compact, we may assume $Y$ is quasi-compact by replacing $Y$ with a quasi-compact open containing $X$. We see that $X = \lim X_i$ with $X_i \to Y$ a closed immersion of finite presentation by Lemma \ref{lemma-closed-is-limit-closed-and-finite-presentation}. The morphisms $X_i \to S$ are of finite presentation by Morphisms, Lemma \ref{morphisms-lemma-composition-finite-presentation}. \end{proof} \begin{proposition} \label{proposition-separated-closed-in-finite-presentation} Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is of finite type and separated, and \item $S$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a separated morphism of finite presentation $f' : X' \to S$ and a closed immersion $X \to X'$ of schemes over $S$. \end{proposition} \begin{proof} Apply Lemma \ref{lemma-finite-type-is-limit-finite-presentation} and note that $X_i \to S$ is separated for large $i$ by Lemma \ref{lemma-eventually-separated} as we have assumed that $X \to S$ is separated. \end{proof} \begin{lemma} \label{lemma-finite-closed-in-finite-finite-presentation} Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is finite, and \item $S$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a morphism which is finite and of finite presentation $f' : X' \to S$ and a closed immersion $X \to X'$ of schemes over $S$. \end{lemma} \begin{proof} We may write $X = \lim X_i$ as in Lemma \ref{lemma-finite-type-is-limit-finite-presentation}. Applying Lemma \ref{lemma-eventually-finite} we see that $X_i \to S$ is finite for large enough $i$. \end{proof} \begin{lemma} \label{lemma-finite-in-finite-and-finite-presentation} Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is finite, and \item $S$ quasi-compact and quasi-separated. \end{enumerate} Then $X$ is 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 $S$. \end{lemma} \begin{proof} We may write $X = \lim X_i$ as in Lemma \ref{lemma-finite-type-is-limit-finite-presentation}. Applying Lemma \ref{lemma-eventually-finite} we see that $X_i \to S$ is finite for large enough $i$. \end{proof} \section{Descending relative objects} \label{section-descending-relative} \noindent The following lemma is typical of the type of results in this section. We write out the standard'' proof completely. It may be faster to convince yourself that the result is true than to read this proof. \begin{lemma} \label{lemma-descend-finite-presentation} Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. Assume \begin{enumerate} \item the morphisms $f_{ii'} : S_i \to S_{i'}$ are affine, \item the schemes $S_i$ are quasi-compact and quasi-separated. \end{enumerate} Let $S = \lim_i S_i$. Then we have the following: \begin{enumerate} \item For any morphism of finite presentation $X \to S$ there exists an index $i \in I$ and a morphism of finite presentation $X_i \to S_i$ such that $X \cong X_{i, S}$ as schemes over $S$. \item Given an index $i \in I$, schemes $X_i$, $Y_i$ of finite presentation over $S_i$, and a morphism $\varphi : X_{i, S} \to Y_{i, S}$ over $S$, there exists an index $i' \geq i$ and a morphism $\varphi_{i'} : X_{i, S_{i'}} \to Y_{i, S_{i'}}$ whose base change to $S$ is $\varphi$. \item Given an index $i \in I$, schemes $X_i$, $Y_i$ of finite presentation over $S_i$ and a pair of morphisms $\varphi_i, \psi_i : X_i \to Y_i$ whose base changes $\varphi_{i, S} = \psi_{i, S}$ are equal, there exists an index $i' \geq i$ such that $\varphi_{i, S_{i'}} = \psi_{i, S_{i'}}$. \end{enumerate} In other words, the category of schemes of finite presentation over $S$ is the colimit over $I$ of the categories of schemes of finite presentation over $S_i$. \end{lemma} \begin{proof} In case each of the schemes $S_i$ is affine, and we consider only affine schemes of finite presentation over $S_i$, resp.\ $S$ this lemma is equivalent to Algebra, Lemma \ref{algebra-lemma-colimit-category-fp-algebras}. We claim that the affine case implies the lemma in general. \medskip\noindent Let us prove (3). Suppose given an index $i \in I$, schemes $X_i$, $Y_i$ of finite presentation over $S_i$ and a pair of morphisms $\varphi_i, \psi_i : X_i \to Y_i$. Assume that the base changes are equal: $\varphi_{i, S} = \psi_{i, S}$. We will use the notation $X_{i'} = X_{i, S_{i'}}$ and $Y_{i'} = Y_{i, S_{i'}}$ for $i' \geq i$. We also set $X = X_{i, S}$ and $Y = Y_{i, S}$. Note that according to Lemma \ref{lemma-scheme-over-limit} we have $X = \lim_{i' \geq i} X_{i'}$ and similarly for $Y$. Additionally we denote $\varphi_{i'}$ and $\psi_{i'}$ (resp.\ $\varphi$ and $\psi$) the base change of $\varphi_i$ and $\psi_i$ to $S_{i'}$ (resp.\ $S$). So our assumption means that $\varphi = \psi$. Since $Y_i$ and $X_i$ are of finite presentation over $S_i$, and since $S_i$ is quasi-compact and quasi-separated, also $X_i$ and $Y_i$ are quasi-compact and quasi-separated (see Morphisms, Lemma \ref{morphisms-lemma-finite-presentation-quasi-compact-quasi-separated}). Hence we may choose a finite affine open covering $Y_i = \bigcup V_{j, i}$ such that each $V_{j, i}$ maps into an affine open of $S$. As above, denote $V_{j, i'}$ the inverse image of $V_{j, i}$ in $Y_{i'}$ and $V_j$ the inverse image in $Y$. The immersions $V_{j, i'} \to Y_{i'}$ are quasi-compact, and the inverse images $U_{j, i'} = \varphi_i^{-1}(V_{j, i'})$ and $U_{j, i'}' = \psi_i^{-1}(V_{j, i'})$ are quasi-compact opens of $X_{i'}$. By assumption the inverse images of $V_j$ under $\varphi$ and $\psi$ in $X$ are equal. Hence by Lemma \ref{lemma-descend-opens} there exists an index $i' \geq i$ such that of $U_{j, i'} = U_{j, i'}'$ in $X_{i'}$. Choose an finite affine open covering $U_{j, i'} = U_{j, i'}' = \bigcup W_{j, k, i'}$ which induce coverings $U_{j, i''} = U_{j, i''}' = \bigcup W_{j, k, i''}$ for all $i'' \geq i'$. By the affine case there exists an index $i''$ such that $\varphi_{i''}|_{W_{j, k, i''}} = \psi_{i''}|_{W_{j, k, i''}}$ for all $j, k$. Then $i''$ is an index such that $\varphi_{i''} = \psi_{i''}$ and (3) is proved. \medskip\noindent Let us prove (2). Suppose given an index $i \in I$, schemes $X_i$, $Y_i$ of finite presentation over $S_i$ and a morphism $\varphi : X_{i, S} \to Y_{i, S}$. We will use the notation $X_{i'} = X_{i, S_{i'}}$ and $Y_{i'} = Y_{i, S_{i'}}$ for $i' \geq i$. We also set $X = X_{i, S}$ and $Y = Y_{i, S}$. Note that according to Lemma \ref{lemma-scheme-over-limit} we have $X = \lim_{i' \geq i} X_{i'}$ and similarly for $Y$. Since $Y_i$ and $X_i$ are of finite presentation over $S_i$, and since $S_i$ is quasi-compact and quasi-separated, also $X_i$ and $Y_i$ are quasi-compact and quasi-separated (see Morphisms, Lemma \ref{morphisms-lemma-finite-presentation-quasi-compact-quasi-separated}). Hence we may choose a finite affine open covering $Y_i = \bigcup V_{j, i}$ such that each $V_{j, i}$ maps into an affine open of $S$. As above, denote $V_{j, i'}$ the inverse image of $V_{j, i}$ in $Y_{i'}$ and $V_j$ the inverse image in $Y$. The immersions $V_j \to Y$ are quasi-compact, and the inverse images $U_j = \varphi^{-1}(V_j)$ are quasi-compact opens of $X$. Hence by Lemma \ref{lemma-descend-opens} there exists an index $i' \geq i$ and quasi-compact opens $U_{j, i'}$ of $X_{i'}$ whose inverse image in $X$ is $U_j$. Choose an finite affine open covering $U_{j, i'} = \bigcup W_{j, k, i'}$ which induce affine open coverings $U_{j, i''} = \bigcup W_{j, k, i''}$ for all $i'' \geq i'$ and an affine open covering $U_j = \bigcup W_{j, k}$. By the affine case there exists an index $i''$ and morphisms $\varphi_{j, k, i''} : W_{j, k, i''} \to V_{j, i''}$ such that $\varphi|_{W_{j, k}} = \varphi_{j, k, i'', S}$ for all $j, k$. By part (3) proved above, there is a further index $i''' \geq i''$ such that $$\varphi_{j_1, k_1, i'', S_{i'''}}|_{W_{j_1, k_1, i'''} \cap W_{j_2, k_2, i'''}} = \varphi_{j_2, k_2, i'', S_{i'''}}|_{W_{j_1, k_1, i'''} \cap W_{j_2, k_2, i'''}}$$ for all $j_1, j_2, k_1, k_2$. Then $i'''$ is an index such that there exists a morphism $\varphi_{i'''} : X_{i'''} \to Y_{i'''}$ whose base change to $S$ gives $\varphi$. Hence (2) holds. \medskip\noindent Let us prove (1). Suppose given a scheme $X$ of finite presentation over $S$. Since $X$ is of finite presentation over $S$, and since $S$ is quasi-compact and quasi-separated, also $X$ is quasi-compact and quasi-separated (see Morphisms, Lemma \ref{morphisms-lemma-finite-presentation-quasi-compact-quasi-separated}). Choose a finite affine open covering $X = \bigcup U_j$ such that each $U_j$ maps into an affine open $V_j \subset S$. Denote $U_{j_1j_2} = U_{j_1} \cap U_{j_2}$ and $U_{j_1j_2j_3} = U_{j_1} \cap U_{j_2} \cap U_{j_3}$. By Lemmas \ref{lemma-descend-opens} and \ref{lemma-limit-affine} we can find an index $i_1$ and affine opens $V_{j, i_1} \subset S_{i_1}$ such that each $V_j$ is the inverse of this in $S$. Let $V_{j, i}$ be the inverse image of $V_{j, i_1}$ in $S_i$ for $i \geq i_1$. By the affine case we may find an index $i_2 \geq i_1$ and affine schemes $U_{j, i_2} \to V_{j, i_2}$ such that $U_j = S \times_{S_{i_2}} U_{j, i_2}$ is the base change. Denote $U_{j, i} = S_i \times_{S_{i_2}} U_{j, i_2}$ for $i \geq i_2$. By Lemma \ref{lemma-descend-opens} there exists an index $i_3 \geq i_2$ and open subschemes $W_{j_1, j_2, i_3} \subset U_{j_1, i_3}$ whose base change to $S$ is equal to $U_{j_1j_2}$. Denote $W_{j_1, j_2, i} = S_i \times_{S_{i_3}} W_{j_1, j_2, i_3}$ for $i \geq i_3$. By part (2) shown above there exists an index $i_4 \geq i_3$ and morphisms $\varphi_{j_1, j_2, i_4} : W_{j_1, j_2, i_4} \to W_{j_2, j_1, i_4}$ whose base change to $S$ gives the identity morphism $U_{j_1j_2} = U_{j_2j_1}$ for all $j_1, j_2$. For all $i \geq i_4$ denote $\varphi_{j_1, j_2, i} = \text{id}_S \times \varphi_{j_1, j_2, i_4}$ the base change. We claim that for some $i_5 \geq i_4$ the system $((U_{j, i_5})_j, (W_{j_1, j_2, i_5})_{j_1, j_2}, (\varphi_{j_1, j_2, i_5})_{j_1, j_2})$ forms a glueing datum as in Schemes, Section \ref{schemes-section-glueing-schemes}. In order to see this we have to verify that for $i$ large enough we have $$\varphi_{j_1, j_2, i}^{-1}(W_{j_1, j_2, i} \cap W_{j_1, j_3, i}) = W_{j_1, j_2, i} \cap W_{j_1, j_3, i}$$ and that for large enough $i$ the cocycle condition holds. The first condition follows from Lemma \ref{lemma-descend-opens} and the fact that $U_{j_2j_1j_3} = U_{j_1j_2j_3}$. The second from part (1) of the lemma proved above and the fact that the cocycle condition holds for the maps $\text{id} : U_{j_1j_2} \to U_{j_2j_1}$. Ok, so now we can use Schemes, Lemma \ref{schemes-lemma-glue-schemes} to glue the system $((U_{j, i_5})_j, (W_{j_1, j_2, i_5})_{j_1, j_2}, (\varphi_{j_1, j_2, i_5})_{j_1, j_2})$ to get a scheme $X_{i_5} \to S_{i_5}$. By construction the base change of $X_{i_5}$ to $S$ is formed by glueing the open affines $U_j$ along the opens $U_{j_1} \leftarrow U_{j_1j_2} \rightarrow U_{j_2}$. Hence $S \times_{S_{i_5}} X_{i_5} \cong X$ as desired. \end{proof} \begin{lemma} \label{lemma-descend-modules-finite-presentation} Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. Assume \begin{enumerate} \item all the morphisms $f_{ii'} : S_i \to S_{i'}$ are affine, \item all the schemes $S_i$ are quasi-compact and quasi-separated. \end{enumerate} Let $S = \lim_i S_i$. Then we have the following: \begin{enumerate} \item For any sheaf of $\mathcal{O}_S$-modules $\mathcal{F}$ of finite presentation there exists an index $i \in I$ and a sheaf of $\mathcal{O}_{S_i}$-modules of finite presentation $\mathcal{F}_i$ such that $\mathcal{F} \cong f_i^*\mathcal{F}_i$. \item Suppose given an index $i \in I$, sheaves of $\mathcal{O}_{S_i}$-modules $\mathcal{F}_i$, $\mathcal{G}_i$ of finite presentation and a morphism $\varphi : f_i^*\mathcal{F}_i \to f_i^*\mathcal{G}_i$ over $S$. Then there exists an index $i' \geq i$ and a morphism $\varphi_{i'} : f_{i'i}^*\mathcal{F}_i \to f_{i'i}^*\mathcal{G}_i$ whose base change to $S$ is $\varphi$. \item Suppose given an index $i \in I$, sheaves of $\mathcal{O}_{S_i}$-modules $\mathcal{F}_i$, $\mathcal{G}_i$ of finite presentation and a pair of morphisms $\varphi_i, \psi_i : \mathcal{F}_i \to \mathcal{G}_i$. Assume that the base changes are equal: $f_i^*\varphi_i = f_i^*\psi_i$. Then there exists an index $i' \geq i$ such that $f_{i'i}^*\varphi_i = f_{i'i}^*\psi_i$. \end{enumerate} In other words, the category of modules of finite presentation over $S$ is the colimit over $I$ of the categories modules of finite presentation over $S_i$. \end{lemma} \begin{proof} Omitted. Since we have written out completely the proof of Lemma \ref{lemma-descend-finite-presentation} above it seems wise to use this here and not completely write this proof out also. For example we can use: \begin{enumerate} \item there is an equivalence of categories between quasi-coherent $\mathcal{O}_S$-modules and vector bundles over $S$, see Constructions, Section \ref{constructions-section-vector-bundle}. \item a vector bundle $\mathbf{V}(\mathcal{F}) \to S$ is of finite presentation over $S$ if and only if $\mathcal{F}$ is an $\mathcal{O}_S$-module of finite presentation. \end{enumerate} Then you can descend morphisms in terms of morphisms of the associated vectorbundles. Similarly for objects. \end{proof} \begin{lemma} \label{lemma-descend-invertible-modules} Let $S = \lim S_i$ be the limit of a directed system of quasi-compact and quasi-separated schemes $S_i$ with affine transition morphisms. Then any invertible $\mathcal{O}_S$-module is the pullback of an invertible $\mathcal{O}_{S_i}$-module for some $i$. \end{lemma} \begin{proof} Let $\mathcal{L}$ be an invertible $\mathcal{O}_S$-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 $S_i$ 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 $f_i^*(\mathcal{L}_i \otimes_{\mathcal{O}_{S_i}} \mathcal{N}_i)$ is isomorphic to $\mathcal{O}_S$. Since the tensor product of finitely presented modules is finitely presented, the same lemma implies that $f_{i'i}^*\mathcal{L}_i \otimes_{\mathcal{O}_{S_{i'}}} f_{i'i}^*\mathcal{N}_i$ is isomorphic to $\mathcal{O}_{S_{i'}}$ for some $i' \geq i$. It follows that $f_{i'i}^*\mathcal{L}_i$ is invertible (Modules, Lemma \ref{modules-lemma-invertible}) and the proof is complete. \end{proof} \begin{lemma} \label{lemma-descend-module-flat-finite-presentation} With notation and assumptions as in Lemma \ref{lemma-descend-finite-presentation}. Let $i \in I$. Suppose that $\varphi_i : X_i \to Y_i$ is a morphism of schemes of finite presentation over $S_i$ and that $\mathcal{F}_i$ is a quasi-coherent $\mathcal{O}_{X_i}$-module of finite presentation. If the pullback of $\mathcal{F}_i$ to $X_i \times_{S_i} S$ is flat over $Y_i \times_{S_i} S$, then there exists an index $i' \geq i$ such that the pullback of $\mathcal{F}_i$ to $X_i \times_{S_i} S_{i'}$ is flat over $Y_i \times_{S_i} S_{i'}$. \end{lemma} \begin{proof} (This lemma is the analogue of Lemma \ref{lemma-descend-flat-finite-presentation} for modules.) For $i' \geq i$ denote $X_{i'} = S_{i'} \times_{S_i} X_i$, $\mathcal{F}_{i'} = (X_{i'} \to X_i)^*\mathcal{F}_i$ and similarly for $Y_{i'}$. Denote $\varphi_{i'}$ the base change of $\varphi_i$ to $S_{i'}$. Also set $X = S \times_{S_i} X_i$, $Y =S \times_{S_i} X_i$, $\mathcal{F} = (X \to X_i)^*\mathcal{F}_i$ and $\varphi$ the base change of $\varphi_i$ to $S$. Let $Y_i = \bigcup_{j = 1, \ldots, m} V_{j, i}$ be a finite affine open covering such that each $V_{j, i}$ maps into some affine open of $S_i$. For each $j = 1, \ldots m$ let $\varphi_i^{-1}(V_{j, i}) = \bigcup_{k = 1, \ldots, m(j)} U_{k, j, i}$ be a finite affine open covering. For $i' \geq i$ we denote $V_{j, i'}$ the inverse image of $V_{j, i}$ in $Y_{i'}$ and $U_{k, j, i'}$ the inverse image of $U_{k, j, i}$ in $X_{i'}$. Similarly we have $U_{k, j} \subset X$ and $V_j \subset Y$. Then $U_{k, j} = \lim_{i' \geq i} U_{k, j, i'}$ and $V_j = \lim_{i' \geq i} V_j$ (see Lemma \ref{lemma-directed-inverse-system-has-limit}). Since $X_{i'} = \bigcup_{k, j} U_{k, j, i'}$ is a finite open covering it suffices to prove the lemma for each of the morphisms $U_{k, j, i} \to V_{j, i}$ and the sheaf $\mathcal{F}_i|_{U_{k, j, i}}$. Hence we see that the lemma reduces to the case that $X_i$ and $Y_i$ are affine and map into an affine open of $S_i$, i.e., we may also assume that $S$ is affine. \medskip\noindent In the affine case we reduce to the following algebra result. Suppose that $R = \colim_{i \in I} R_i$. For some $i \in I$ suppose given a map $A_i \to B_i$ of finitely presented $R_i$-algebras. Let $N_i$ be a finitely presented $B_i$-module. Then, if $R \otimes_{R_i} N_i$ is flat over $R \otimes_{R_i} A_i$, then for some $i' \geq i$ the module $R_{i'} \otimes_{R_i} N_i$ is flat over $R_{i'} \otimes_{R_i} A$. This is exactly the result proved in Algebra, Lemma \ref{algebra-lemma-flat-finite-presentation-limit-flat} part (3). \end{proof} \section{Characterizing affine schemes} \label{section-affine} \noindent If $f : X \to S$ is a surjective integral morphism of schemes such that $X$ is an affine scheme then $S$ is affine too. See \cite[A.2]{Conrad-Nagata}. Our proof relies on the Noetherian case which we stated and proved in Cohomology of Schemes, Lemma \ref{coherent-lemma-image-affine-finite-morphism-affine-Noetherian}. See also \cite[II 6.7.1]{EGA}. \begin{lemma} \label{lemma-affine} Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is surjective and finite, and assume that $X$ is affine. Then $S$ is affine. \end{lemma} \begin{proof} Since $f$ is surjective and $X$ is quasi-compact we see that $S$ is quasi-compact. Since $X$ is separated and $f$ is surjective and universally closed (Morphisms, Lemma \ref{morphisms-lemma-integral-universally-closed}), we see that $S$ is separated (Morphisms, Lemma \ref{morphisms-lemma-image-universally-closed-separated}). \medskip\noindent By Lemma \ref{lemma-finite-in-finite-and-finite-presentation} we can write $X = \lim_a X_a$ with $X_a \to S$ finite and of finite presentation. By Lemma \ref{lemma-limit-affine} we see that $X_a$ is affine for some $a \in A$. Replacing $X$ by $X_a$ we may assume that $X \to S$ is surjective, finite, of finite presentation and that $X$ is affine. \medskip\noindent By Proposition \ref{proposition-approximate} we may write $S = \lim_{i \in I} S_i$ as a directed limits as schemes of finite type over $\mathbf{Z}$. By Lemma \ref{lemma-descend-finite-presentation} we can after shrinking $I$ assume there exist schemes $X_i \to S_i$ of finite presentation such that $X_{i'} = X_i \times_S S_{i'}$ for $i' \geq i$ and such that $X = \lim_i X_i$. By Lemma \ref{lemma-descend-finite-finite-presentation} we may assume that $X_i \to S_i$ is finite for all $i \in I$ as well. By Lemma \ref{lemma-limit-affine} once again we may assume that $X_i$ is affine for all $i \in I$. Hence the result follows from the Noetherian case, see Cohomology of Schemes, Lemma \ref{coherent-lemma-image-affine-finite-morphism-affine-Noetherian}. \end{proof} \begin{proposition} \label{proposition-affine} Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is surjective and integral, and assume that $X$ is affine. Then $S$ is affine. \end{proposition} \begin{proof} Since $f$ is surjective and $X$ is quasi-compact we see that $S$ is quasi-compact. Since $X$ is separated and $f$ is surjective and universally closed (Morphisms, Lemma \ref{morphisms-lemma-integral-universally-closed}), we see that $S$ is separated (Morphisms, Lemma \ref{morphisms-lemma-image-universally-closed-separated}). \medskip\noindent By Lemma \ref{lemma-integral-limit-finite-and-finite-presentation} we can write $X = \lim_i X_i$ with $X_i \to S$ finite. By Lemma \ref{lemma-limit-affine} we see that for $i$ sufficiently large the scheme $X_i$ is affine. Moreover, since $X \to S$ factors through each $X_i$ we see that $X_i \to S$ is surjective. Hence we conclude that $S$ is affine by Lemma \ref{lemma-affine}. \end{proof} \begin{lemma} \label{lemma-affines-glued-in-closed-affine} Let $X$ be a scheme which is set theoretically the union of finitely many affine closed subschemes. Then $X$ is affine. \end{lemma} \begin{proof} Let $Z_i \subset X$, $i = 1, \ldots, n$ be affine closed subschemes such that $X = \bigcup Z_i$ set theoretically. Then $\coprod Z_i \to X$ is surjective and integral with affine source. Hence $X$ is affine by Proposition \ref{proposition-affine}. \end{proof} \begin{lemma} \label{lemma-ample-on-reduction} Let $i : Z \to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Let $\mathcal{L}$ be an invertible sheaf on $X$. Then $i^*\mathcal{L}$ is ample on $Z$, if and only if $\mathcal{L}$ is ample on $X$. \end{lemma} \begin{proof} If $\mathcal{L}$ is ample, then $i^*\mathcal{L}$ is ample for example by Morphisms, Lemma \ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}. Assume $i^*\mathcal{L}$ is ample. Then $Z$ is quasi-compact (Properties, Definition \ref{properties-definition-ample}) and separated (Properties, Lemma \ref{properties-lemma-ample-separated}). Since $i$ is surjective, we see that $X$ is quasi-compact. Since $i$ is universally closed and surjective, we see that $X$ is separated (Morphisms, Lemma \ref{morphisms-lemma-image-universally-closed-separated}). \medskip\noindent By Proposition \ref{proposition-approximate} we can write $X = \lim X_i$ as a directed limit of finite type schemes over $\mathbf{Z}$ with affine transition morphisms. We can find an $i$ and an invertible sheaf $\mathcal{L}_i$ on $X_i$ whose pullback to $X$ is isomorphic to $\mathcal{L}$, see Lemma \ref{lemma-descend-modules-finite-presentation}. \medskip\noindent For each $i$ let $Z_i \subset X_i$ be the scheme theoretic image of the morphism $Z \to X$. If $\Spec(A_i) \subset X_i$ is an affine open subscheme with inverse image of $\Spec(A)$ in $X$ and if $Z \cap \Spec(A)$ is defined by the ideal $I \subset A$, then $Z_i \cap \Spec(A_i)$ is defined by the ideal $I_i \subset A_i$ which is the inverse image of $I$ in $A_i$ under the ring map $A_i \to A$, see Morphisms, Example \ref{morphisms-example-scheme-theoretic-image}. Since $\colim A_i/I_i = A/I$ it follows that $\lim Z_i = Z$. By Lemma \ref{lemma-limit-ample} we see that $\mathcal{L}_i|_{Z_i}$ is ample for some $i$. Since $Z$ and hence $X$ maps into $Z_i$ set theoretically, we see that $X_{i'} \to X_i$ maps into $Z_i$ set theoretically for some $i' \geq i$, see Lemma \ref{lemma-limit-contained-in-constructible}. (Observe that since $X_i$ is Noetherian, every closed subset of $X_i$ is constructible.) Let $T \subset X_{i'}$ be the scheme theoretic inverse image of $Z_i$ in $X_{i'}$. Observe that $\mathcal{L}_{i'}|_T$ is the pullback of $\mathcal{L}_i|_{Z_i}$ and hence ample by Morphisms, Lemma \ref{morphisms-lemma-pullback-ample-tensor-relatively-ample} and the fact that $T \to Z_i$ is an affine morphism. Thus we see that $\mathcal{L}_{i'}$ is ample on $X_{i'}$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-ample-on-reduction}. Pulling back to $X$ (using the same lemma as above) we find that $\mathcal{L}$ is ample. \end{proof} \begin{lemma} \label{lemma-thickening-quasi-affine} Let $i : Z \to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Then $X$ is quasi-affine if and only if $Z$ is quasi-affine. \end{lemma} \begin{proof} Recall that a scheme is quasi-affine if and only if the structure sheaf is ample, see Properties, Lemma \ref{properties-lemma-quasi-affine-O-ample}. Hence if $Z$ is quasi-affine, then $\mathcal{O}_Z$ is ample, hence $\mathcal{O}_X$ is ample by Lemma \ref{lemma-ample-on-reduction}, hence $X$ is quasi-affine. A proof of the converse, which can also be seen in an elementary way, is gotten by reading the argument just given backwards. \end{proof} \section{Variants of Chow's Lemma} \label{section-chows-lemma} \noindent In this section we prove a number of variants of Chow's lemma. The most interesting version is probably just the Noetherian case, which we stated and proved in Cohomology of Schemes, Section \ref{coherent-section-chows-lemma}. \begin{lemma} \label{lemma-chow-finite-type} Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a separated morphism of finite type. Then there exists an $n \geq 0$ and a diagram $$\xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & }$$ where $X' \to \mathbf{P}^n_S$ is an immersion, and $\pi : X' \to X$ is proper and surjective. \end{lemma} \begin{proof} By Proposition \ref{proposition-separated-closed-in-finite-presentation} we can find a closed immersion $X \to Y$ where $Y$ is separated and of finite presentation over $S$. Clearly, if we prove the assertion for $Y$, then the result follows for $X$. Hence we may assume that $X$ is of finite presentation over $S$. \medskip\noindent Write $S = \lim_i S_i$ as a directed limit of Noetherian schemes, see Proposition \ref{proposition-approximate}. By Lemma \ref{lemma-descend-finite-presentation} we can find an index $i \in I$ and a scheme $X_i \to S_i$ of finite presentation so that $X = S \times_{S_i} X_i$. By Lemma \ref{lemma-descend-separated-finite-presentation} we may assume that $X_i \to S_i$ is separated. Clearly, if we prove the assertion for $X_i$ over $S_i$, then the assertion holds for $X$. The case $X_i \to S_i$ is treated by Cohomology of Schemes, Lemma \ref{coherent-lemma-chow-Noetherian}. \end{proof} \noindent Here is a variant of Chow's lemma where we assume the scheme on top has finitely many irreducible components. \begin{lemma} \label{lemma-chow-EGA} Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a separated morphism of finite type. Assume that $X$ has finitely many irreducible components. Then there exists an $n \geq 0$ and a diagram $$\xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & }$$ where $X' \to \mathbf{P}^n_S$ is an immersion, and $\pi : X' \to X$ is proper and surjective. Moreover, there exists an open dense subscheme $U \subset X$ such that $\pi^{-1}(U) \to U$ is an isomorphism of schemes. \end{lemma} \begin{proof} Let $X = Z_1 \cup \ldots \cup Z_n$ be the decomposition of $X$ into irreducible components. Let $\eta_j \in Z_j$ be the generic point. \medskip\noindent There are (at least) two ways to proceed with the proof. The first is to redo the proof of Cohomology of Schemes, Lemma \ref{coherent-lemma-chow-Noetherian} using the general Properties, Lemma \ref{properties-lemma-point-and-maximal-points-affine} to find suitable affine opens in $X$. (This is the standard'' proof.) The second is to use absolute Noetherian approximation as in the proof of Lemma \ref{lemma-chow-finite-type} above. This is what we will do here. \medskip\noindent By Proposition \ref{proposition-separated-closed-in-finite-presentation} we can find a closed immersion $X \to Y$ where $Y$ is separated and of finite presentation over $S$. Write $S = \lim_i S_i$ as a directed limit of Noetherian schemes, see Proposition \ref{proposition-approximate}. By Lemma \ref{lemma-descend-finite-presentation} we can find an index $i \in I$ and a scheme $Y_i \to S_i$ of finite presentation so that $Y = S \times_{S_i} Y_i$. By Lemma \ref{lemma-descend-separated-finite-presentation} we may assume that $Y_i \to S_i$ is separated. We have the following diagram $$\xymatrix{ \eta_j \in Z_j \ar[r] & X \ar[r] \ar[rd] & Y \ar[r] \ar[d] & Y_i \ar[d] \\ & & S \ar[r] & S_i }$$ Denote $h : X \to Y_i$ the composition. \medskip\noindent For $i' \geq i$ write $Y_{i'} = S_{i'} \times_{S_i} Y_i$. Then $Y = \lim_{i' \geq i} Y_{i'}$, see Lemma \ref{lemma-scheme-over-limit}. Choose $j, j' \in \{1, \ldots, n\}$, $j \not = j'$. Note that $\eta_j$ is not a specialization of $\eta_{j'}$. By Lemma \ref{lemma-topology-limit} we can replace $i$ by a bigger index and assume that $h(\eta_j)$ is not a specialization of $h(\eta_{j'})$ for all pairs $(j, j')$ as above. For such an index, let $Y' \subset Y_i$ be the scheme theoretic image of $h : X \to Y_i$, see Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-image}. The morphism $h$ is quasi-compact as the composition of the quasi-compact morphisms $X \to Y$ and $Y \to Y_i$ (which is affine). Hence by Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image} the morphism $X \to Y'$ is dominant. Thus the generic points of $Y'$ are all contained in the set $\{h(\eta_1), \ldots, h(\eta_n)\}$, see Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-dominant}. Since none of the $h(\eta_j)$ is the specialization of another we see that the points $h(\eta_1), \ldots, h(\eta_n)$ are pairwise distinct and are each a generic point of $Y'$. \medskip\noindent We apply Cohomology of Schemes, Lemma \ref{coherent-lemma-chow-Noetherian} above to the morphism $Y' \to S_i$. This gives a diagram $$\xymatrix{ Y' \ar[rd] & Y^* \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_{S_i} \ar[dl] \\ & S_i & }$$ such that $\pi$ is proper and surjective and an isomorphism over a dense open subscheme $V \subset Y'$. By our choice of $i$ above we know that $h(\eta_1), \ldots, h(\eta_n) \in V$. Consider the commutative diagram $$\xymatrix{ X' \ar@{=}[r] & X \times_{Y'} Y^* \ar[r] \ar[d] & Y^* \ar[r] \ar[d] & \mathbf{P}^n_{S_i} \ar[ddl] \\ & X \ar[r] \ar[d] & Y' \ar[d] & \\ & S \ar[r] & S_i & }$$ Note that $X' \to X$ is an isomorphism over the open subscheme $U = h^{-1}(V)$ which contains each of the $\eta_j$ and hence is dense in $X$. We conclude $X \leftarrow X' \rightarrow \mathbf{P}^n_S$ is a solution to the problem posed in the lemma. \end{proof} \section{Applications of Chow's lemma} \label{section-apply-chow} \noindent Here is a first application of Chow's lemma. \begin{lemma} \label{lemma-eventually-proper} \begin{slogan} If the base change of a scheme to a limit is proper, then already the base change is proper at a finite level. \end{slogan} 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} By Lemma \ref{lemma-descend-separated-finite-presentation} 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, see Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence}. By Lemma \ref{lemma-chow-finite-type} 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, Lemmas \ref{morphisms-lemma-composition-proper} and \ref{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, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). By Morphisms, Lemma \ref{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-finite-presentation} 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-proper-limit-of-proper-finite-presentation} Let $f : X \to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Then $X = \lim X_i$ is a directed limit of schemes $X_i$ proper and of finite presentation over $S$ such that all transition morphisms and the morphisms $X \to X_i$ are closed immersions. \end{lemma} \begin{proof} By Proposition \ref{proposition-separated-closed-in-finite-presentation} we can find a closed immersion $X \to Y$ with $Y$ separated and of finite presentation over $S$. By Lemma \ref{lemma-chow-finite-type} we can find a diagram $$\xymatrix{ Y \ar[rd] & Y' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & }$$ where $Y' \to \mathbf{P}^n_S$ is an immersion, and $\pi : Y' \to Y$ is proper and surjective. By Lemma \ref{lemma-closed-is-limit-closed-and-finite-presentation} we can write $X = \lim X_i$ with $X_i \to Y$ a closed immersion of finite presentation. Denote $X'_i \subset Y'$, resp.\ $X' \subset Y'$ the scheme theoretic inverse image of $X_i \subset Y$, resp.\ $X \subset Y$. Then $\lim X'_i = X'$. Since $X' \to S$ is proper (Morphisms, Lemmas \ref{morphisms-lemma-composition-proper}), we see that $X' \to \mathbf{P}^n_S$ is a closed immersion (Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). Hence for $i$ large enough we find that $X'_i \to \mathbf{P}^n_S$ is a closed immersion by Lemma \ref{lemma-eventually-closed-immersion}. Thus $X'_i$ is proper over $S$. For such $i$ the morphism $X_i \to S$ is proper by Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper}. \end{proof} \begin{lemma} \label{lemma-proper-limit-of-proper-finite-presentation-noetherian} Let $f : X \to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_i : X_i \to S_i)$ of morphisms of schemes over $I$, such that the transition morphisms $X_i \to X_{i'}$ and $S_i \to S_{i'}$ are affine, such that $f_i$ is proper, such that $S_i$ is of finite type over $\mathbf{Z}$, and such that $(X \to S) = \lim (X_i \to S_i)$. \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 $X_k \to S$ proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition \ref{proposition-approximate}) we can write $S = \lim_{j \in J} S_j$ with $S_j$ of finite type over $\mathbf{Z}$. For each $k$ there exists a $j$ and a morphism $X_{k, j} \to S_j$ of finite presentation with $X_k \cong S \times_{S_j} X_{k, j}$ as schemes over $S$, see Lemma \ref{lemma-descend-finite-presentation}. After increasing $j$ we may assume $X_{k, j} \to S_j$ is proper, see Lemma \ref{lemma-eventually-proper}. The set $I$ will be consist of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \to S_j$. For every $k' \geq k$ we can find a $j' \geq j$ and a morphism $X_{j', k'} \to X_{j, k}$ over $S_{j'} \to S_j$ whose base change to $S$ 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, Definition \ref{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, Lemma \ref{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, Lemmas \ref{morphisms-lemma-closed-immersion-proper} and \ref{morphisms-lemma-composition-proper}. \end{proof} \section{Universally closed morphisms} \label{section-universally-closed} \noindent In this section we discuss when a quasi-compact (but not necessarily separated) morphism is universally closed. We first prove a lemma which will allow us to check universal closedness after a base change which is locally of finite presentation. \begin{lemma} \label{lemma-separate} Let $f : X \to S$ be a quasi-compact morphism of schemes. Let $g : T \to S$ be a morphism of schemes. Let $t \in T$ be a point and $Z \subset X_T$ be a closed subscheme such that $Z \cap X_t = \emptyset$. Then there exists an open neighbourhood $V \subset T$ of $t$, a commutative diagram $$\xymatrix{ V \ar[d] \ar[r]_a & T' \ar[d]^b \\ T \ar[r]^g & S, }$$ and a closed subscheme $Z' \subset X_{T'}$ such that \begin{enumerate} \item the morphism $b : T' \to S$ is locally of finite presentation, \item with $t' = a(t)$ we have $Z' \cap X_{t'} = \emptyset$, and \item $Z \cap X_V$ maps into $Z'$ via the morphism $X_V \to X_{T'}$. \end{enumerate} Moreover, we may assume $V$ and $T'$ are affine. \end{lemma} \begin{proof} Let $s = g(t)$. During the proof we may always replace $T$ by an open neighbourhood of $t$. Hence we may also replace $S$ by an open neighbourhood of $s$. Thus we may and do assume that $T$ and $S$ are affine. Say $S = \Spec(A)$, $T = \Spec(B)$, $g$ is given by the ring map $A \to B$, and $t$ correspond to the prime ideal $\mathfrak q \subset B$. \medskip\noindent As $X \to S$ is quasi-compact and $S$ is affine we may write $X = \bigcup_{i = 1, \ldots, n} U_i$ as a finite union of affine opens. Write $U_i = \Spec(C_i)$. In particular we have $X_T = \bigcup_{i = 1, \ldots, n} U_{i, T} = \bigcup_{i = 1, \ldots n} \Spec(C_i \otimes_A B)$. Let $I_i \subset C_i \otimes_A B$ be the ideal corresponding to the closed subscheme $Z \cap U_{i, T}$. The condition that $Z \cap X_t = \emptyset$ signifies that $I_i$ generates the unit ideal in the ring $$C_i \otimes_A \kappa(\mathfrak q) = (B \setminus \mathfrak q)^{-1}\left( C_i \otimes_A B/\mathfrak q C_i \otimes_A B \right)$$ Since $I_i (B \setminus \mathfrak q)^{-1}(C_i \otimes_A B) = (B \setminus \mathfrak q)^{-1} I_i$ this means that $1 = x_i/g_i$ for some $x_i \in I_i$ and $g_i \in B$, $g_i \not \in \mathfrak q$. Thus, clearing denominators we can find a relation of the form $$x_i + \sum\nolimits_j f_{i, j}c_{i, j} = g_i$$ with $x_i \in I_i$, $f_{i, j} \in \mathfrak q$, $c_{i, j} \in C_i \otimes_A B$, and $g_i \in B$, $g_i \not \in \mathfrak q$. After replacing $B$ by $B_{g_1 \ldots g_n}$, i.e., after replacing $T$ by a smaller affine neighbourhood of $t$, we may assume the equations read $$x_i + \sum\nolimits_j f_{i, j}c_{i, j} = 1$$ with $x_i \in I_i$, $f_{i, j} \in \mathfrak q$, $c_{i, j} \in C_i \otimes_A B$. \medskip\noindent To finish the argument write $B$ as a colimit of finitely presented $A$-algebras $B_\lambda$ over a directed set $\Lambda$. For each $\lambda$ set $\mathfrak q_\lambda = (B_\lambda \to B)^{-1}(\mathfrak q)$. For sufficiently large $\lambda \in \Lambda$ we can find \begin{enumerate} \item an element $x_{i, \lambda} \in C_i \otimes_A B_\lambda$ which maps to $x_i$, \item elements $f_{i, j, \lambda} \in \mathfrak q_{i, \lambda}$ mapping to $f_{i, j}$, and \item elements $c_{i, j, \lambda} \in C_i \otimes_A B_\lambda$ mapping to $c_{i, j}$. \end{enumerate} After increasing $\lambda$ a bit more the equation $$x_{i, \lambda} + \sum\nolimits_j f_{i, j, \lambda}c_{i, j, \lambda} = 1$$ will hold. Fix such a $\lambda$ and set $T' = \Spec(B_\lambda)$. Then $t' \in T'$ is the point corresponding to the prime $\mathfrak q_\lambda$. Finally, let $Z' \subset X_{T'}$ be the scheme theoretic image of $Z \to X_T \to X_{T'}$. As $X_T \to X_{T'}$ is affine, we can compute $Z'$ on the affine open pieces $U_{i, T'}$ as the closed subscheme associated to $\Ker(C_i \otimes_A B_\lambda \to C_i \otimes_A B/I_i)$, see Morphisms, Example \ref{morphisms-example-scheme-theoretic-image}. Hence $x_{i, \lambda}$ is in the ideal defining $Z'$. Thus the last displayed equation shows that $Z' \cap X_{t'}$ is empty. \end{proof} \begin{lemma} \label{lemma-test-universally-closed} Let $f : X \to S$ be a quasi-compact morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is universally closed, \item for every morphism $S' \to S$ which is locally of finite presentation the base change $X_{S'} \to S'$ is closed, and \item for every $n$ the morphism $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is closed. \end{enumerate} \end{lemma} \begin{proof} It is clear that (1) implies (2). Let us prove that (2) implies (1). Suppose that the base change $X_T \to T$ is not closed for some scheme $T$ over $S$. By Schemes, Lemma \ref{schemes-lemma-quasi-compact-closed} this means that there exists some specialization $t_1 \leadsto t$ in $T$ and a point $\xi \in X_T$ mapping to $t_1$ such that $\xi$ does not specialize to a point in the fibre over $t$. Set $Z = \overline{\{\xi\}} \subset X_T$. Then $Z \cap X_t = \emptyset$. Apply Lemma \ref{lemma-separate}. We find an open neighbourhood $V \subset T$ of $t$, a commutative diagram $$\xymatrix{ V \ar[d] \ar[r]_a & T' \ar[d]^b \\ T \ar[r]^g & S, }$$ and a closed subscheme $Z' \subset X_{T'}$ such that \begin{enumerate} \item the morphism $b : T' \to S$ is locally of finite presentation, \item with $t' = a(t)$ we have $Z' \cap X_{t'} = \emptyset$, and \item $Z \cap X_V$ maps into $Z'$ via the morphism $X_V \to X_{T'}$. \end{enumerate} Clearly this means that $X_{T'} \to T'$ maps the closed subset $Z'$ to a subset of $T'$ which contains $a(t_1)$ but not $t' = a(t)$. Since $a(t_1) \leadsto a(t) = t'$ we conclude that $X_{T'} \to T'$ is not closed. Hence we have shown that $X \to S$ not universally closed implies that $X_{T'} \to T'$ is not closed for some $T' \to S$ which is locally of finite presentation. In order words (2) implies (1). \medskip\noindent Assume that $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is closed for every integer $n$. We want to prove that $X_T \to T$ is closed for every scheme $T$ which is locally of finite presentation over $S$. We may of course assume that $T$ is affine and maps into an affine open $V$ of $S$ (since $X_T \to T$ being a closed is local on $T$). In this case there exists a closed immersion $T \to \mathbf{A}^n \times V$ because $\mathcal{O}_T(T)$ is a finitely presented $\mathcal{O}_S(V)$-algebra, see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-presentation-characterize}. Then $T \to \mathbf{A}^n \times S$ is a locally closed immersion. Hence we get a cartesian diagram $$\xymatrix{ X_T \ar[d]_{f_T} \ar[r] & \mathbf{A}^n \times X \ar[d]^{f_n} \\ T \ar[r] & \mathbf{A}^n \times S }$$ of schemes where the horizontal arrows are locally closed immersions. Hence any closed subset $Z \subset X_T$ can be written as $X_T \cap Z'$ for some closed subset $Z' \subset \mathbf{A}^n \times X$. Then $f_T(Z) = T \cap f_n(Z')$ and we see that if $f_n$ is closed, then also $f_T$ is closed. \end{proof} \begin{lemma} \label{lemma-limited-base-change} Let $S$ be a scheme. Let $f : X \to S$ be a separated morphism of finite type. The following are equivalent: \begin{enumerate} \item The morphism $f$ is proper. \item For any morphism $S' \to S$ which is locally of finite type the base change $X_{S'} \to S'$ is closed. \item For every $n \geq 0$ the morphism $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is closed. \end{enumerate} \end{lemma} \begin{proof}[First proof] In view of the fact that a proper morphism is the same thing as a separated, finite type, and universally closed morphism, this lemma is a special case of Lemma \ref{lemma-test-universally-closed}. \end{proof} \begin{proof}[Second proof] Clearly (1) implies (2), and (2) implies (3), so we just need to show (3) implies (1). First we reduce to the case when $S$ is affine. Assume that (3) implies (1) when the base is affine. Now let $f: X \to S$ be a separated morphism of finite type. Being proper is local on the base (see Morphisms, Lemma \ref{morphisms-lemma-proper-local-on-the-base}), so if $S = \bigcup_\alpha S_\alpha$ is an open affine cover, and if we denote $X_\alpha := f^{-1}(S_\alpha)$, then it is enough to show that $f|_{X_\alpha}: X_\alpha \to S_\alpha$ is proper for all $\alpha$. Since $S_\alpha$ is affine, if the map $f|_{X_\alpha}$ satisfies (3), then it will satisfy (1) by assumption, and will be proper. To finish the reduction to the case $S$ is affine, we must show that if $f: X \to S$ is separated of finite type satisfying (3), then $f|_{X_\alpha} : X_\alpha \to S_\alpha$ is separated of finite type satisfying (3). Separatedness and finite type are clear. To see (3), notice that $\mathbf{A}^n \times X_\alpha$ is the open preimage of $\mathbf{A}^n \times S_\alpha$ under the map $1 \times f$. Fix a closed set $Z \subset \mathbf A^n \times X_\alpha$. Let $\bar Z$ denote the closure of $Z$ in $\mathbf{A}^n \times X$. Then for topological reasons, $$1 \times f(\bar Z) \cap \mathbf{A}^n \times S_\alpha = 1 \times f(Z).$$ Hence $1 \times f(Z)$ is closed, and we have reduced the proof of (3) $\Rightarrow$ (1) to the affine case. \medskip\noindent Assume $S$ affine, and $f : X \to S$ separated of finite type. We can apply Chow's Lemma \ref{lemma-chow-finite-type} to get $\pi : X' \to X$ proper surjective and $X' \to \mathbf{P}^n_S$ an immersion. If $X$ is proper over $S$, then $X' \to S$ is proper (Morphisms, Lemma \ref{morphisms-lemma-composition-proper}). Since $\mathbf{P}^n_S \to S$ is separated, we conclude that $X' \to \mathbf{P}^n_S$ is proper (Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}) and hence a closed immersion (Schemes, Lemma \ref{schemes-lemma-immersion-when-closed}). Conversely, assume $X' \to \mathbf{P}^n_S$ is a closed immersion. Consider the diagram: \begin{equation} \label{equation-check-proper} \xymatrix{ X' \ar[r] \ar@{->>}[d]_{\pi} & \mathbf{P}^n_S \ar[d] \\ X \ar[r]^f & S } \end{equation} All maps are a priori proper except for $X \to S$. Hence we conclude that $X \to S$ is proper by Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper}. Therefore, we have shown that $X \to S$ is proper if and only if $X' \to \mathbf{P}^n_S$ is a closed immersion. \medskip\noindent Assume $S$ is affine and (3) holds, and let $n, X', \pi$ be as above. Since being a closed morphism is local on the base, the map $X \times \mathbf{P}^n \to S \times \mathbf{P}^n$ is closed since by (3) $X \times \mathbf{A}^n \to S \times \mathbf{A}^n$ is closed and since projective space is covered by copies of affine $n$-space, see Constructions, Lemma \ref{constructions-lemma-standard-covering-projective-space}. By Morphisms, Lemma \ref{morphisms-lemma-base-change-proper} the morphism $$X' \times_S \mathbf{P}^n_S \to X \times_S \mathbf{P}^n_S = X \times \mathbf{P}^n$$ is proper. Since $\mathbf{P}^n$ is separated, the projection $$X' \times_S \mathbf{P}^n_S = \mathbf{P}^n_{X'} \to X'$$ will be separated as it is just a base change of a separated morphism. Therefore, the map $X' \to X' \times_S \mathbf{P}^n_S$ is proper, since it is a section to a separated map (see Schemes, Lemma \ref{schemes-lemma-section-immersion}). Composing these morphisms $$X' \to X' \times_S \mathbf{P}^n_S \to X \times_S \mathbf{P}^n_S = X \times \mathbf{P}^n \to S \times \mathbf{P}^n = \mathbf{P}^n_S$$ we find that the immersion $X' \to \mathbf{P}^n_S$ is closed, and hence a closed immersion. \end{proof} \section{Noetherian valuative criterion} \label{section-Noetherian-valuative-criterion} \noindent If the base is Noetherian we can show that the valuative criterion holds using only discrete valuation rings. \medskip\noindent Many of the results in this section can (and perhaps should) be proved by appealing to the following lemma, although we have not always done so. \begin{lemma} \label{lemma-reach-point-closure-Noetherian} Let $f : X \to Y$ be a morphism of schemes. Assume $f$ finite type and $Y$ locally Noetherian. Let $y \in Y$ be a point in the closure of the image of $f$. Then there exists a commutative diagram  \xymatrix{