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 \input{preamble} % OK, start here. % \begin{document} \title{Properties of Schemes} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we introduce some absolute properties of schemes. A foundational reference is \cite{EGA}. \section{Constructible sets} \label{section-constructible} \noindent Constructible and locally constructible sets are introduced in Topology, Section \ref{topology-section-constructible}. We may characterize locally constructible subsets of schemes as follows. \begin{lemma} \label{lemma-locally-constructible} Let $X$ be a scheme. A subset $E$ of $X$ is locally constructible in $X$ if and only if $E \cap U$ is constructible in $U$ for every affine open $U$ of $X$. \end{lemma} \begin{proof} Assume $E$ is locally constructible. Then there exists an open covering $X = \bigcup U_i$ such that $E \cap U_i$ is constructible in $U_i$ for each $i$. Let $V \subset X$ be any affine open. We can find a finite open affine covering $V = V_1 \cup \ldots \cup V_m$ such that for each $j$ we have $V_j \subset U_i$ for some $i = i(j)$. By Topology, Lemma \ref{topology-lemma-open-immersion-constructible-inverse-image} we see that each $E \cap V_j$ is constructible in $V_j$. Since the inclusions $V_j \to V$ are quasi-compact (see Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine}) we conclude that $E \cap V$ is constructible in $V$ by Topology, Lemma \ref{topology-lemma-collate-constructible}. The converse implication is immediate. \end{proof} \begin{lemma} \label{lemma-generic-point-in-constructible} Let $X$ be a scheme and let $E \subset X$ be a constructible subset. Let $\xi \in X$ be a generic point of an irreducible component of $X$. \begin{enumerate} \item If $\xi \in E$, then an open neighbourhood of $\xi$ is contained in $E$. \item If $\xi \not \in E$, then an open neighbourhood of $\xi$ is disjoint from $E$. \end{enumerate} \end{lemma} \begin{proof} As the complement of a locally constructible subset is locally constructible it suffices to show (2). We may assume $X$ is affine and hence $E$ constructible (Lemma \ref{lemma-locally-constructible}). In this case $X$ is a spectral space (Algebra, Lemma \ref{algebra-lemma-spec-spectral}). Then $\xi \not \in E$ implies $\xi \not \in \overline{E}$ by Topology, Lemma \ref{topology-lemma-constructible-stable-specialization-closed} and the fact that there are no points of $X$ different from $\xi$ which specialize to $\xi$. \end{proof} \begin{lemma} \label{lemma-quasi-separated-quasi-compact-open-retrocompact} Let $X$ be a quasi-separated scheme. The intersection of any two quasi-compact opens of $X$ is a quasi-compact open of $X$. Every quasi-compact open of $X$ is retrocompact in $X$. \end{lemma} \begin{proof} If $U$ and $V$ are quasi-compact open then $U \cap V = \Delta^{-1}(U \times V)$, where $\Delta : X \to X \times X$ is the diagonal. As $X$ is quasi-separated we see that $\Delta$ is quasi-compact. Hence we see that $U \cap V$ is quasi-compact as $U \times V$ is quasi-compact (details omitted; use Schemes, Lemma \ref{schemes-lemma-affine-covering-fibre-product} to see $U \times V$ is a finite union of affines). The other assertions follow from the first and Topology, Lemma \ref{topology-lemma-topology-quasi-separated-scheme}. \end{proof} \begin{lemma} \label{lemma-quasi-compact-quasi-separated-spectral} Let $X$ be a quasi-compact and quasi-separated scheme. Then the underlying topological space of $X$ is a spectral space. \end{lemma} \begin{proof} By Topology, Definition \ref{topology-definition-spectral-space} we have to check that $X$ is sober, quasi-compact, has a basis of quasi-compact opens, and the intersection of any two quasi-compact opens is quasi-compact. This follows from Schemes, Lemma \ref{schemes-lemma-scheme-sober} and \ref{schemes-lemma-basis-affine-opens} and Lemma \ref{lemma-quasi-separated-quasi-compact-open-retrocompact} above. \end{proof} \begin{lemma} \label{lemma-constructible-quasi-compact-quasi-separated} Let $X$ be a quasi-compact and quasi-separated scheme. Any locally constructible subset of $X$ is constructible. \end{lemma} \begin{proof} As $X$ is quasi-compact we can choose a finite affine open covering $X = V_1 \cup \ldots \cup V_m$. As $X$ is quasi-separated each $V_i$ is retrocompact in $X$ by Lemma \ref{lemma-quasi-separated-quasi-compact-open-retrocompact}. Hence by Topology, Lemma \ref{topology-lemma-collate-constructible} we see that $E \subset X$ is constructible in $X$ if and only if $E \cap V_j$ is constructible in $V_j$. Thus we win by Lemma \ref{lemma-locally-constructible}. \end{proof} \begin{lemma} \label{lemma-retrocompact} Let $X$ be a scheme. A subset $Z$ of $X$ is retrocompact in $X$ if and only if $E \cap U$ is quasi-compact for every affine open $U$ of $X$. \end{lemma} \begin{proof} Immediate from the fact that every quasi-compact open of $X$ is a finite union of affine opens. \end{proof} \section{Integral, irreducible, and reduced schemes} \label{section-integral} \begin{definition} \label{definition-integral} Let $X$ be a scheme. We say $X$ is {\it integral} if it is nonempty and for every nonempty affine open $\Spec(R) = U \subset X$ the ring $R$ is an integral domain. \end{definition} \begin{lemma} \label{lemma-characterize-reduced} Let $X$ be a scheme. The following are equivalent. \begin{enumerate} \item The scheme $X$ is reduced, see Schemes, Definition \ref{schemes-definition-reduced}. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\Gamma(U_i, \mathcal{O}_X)$ is reduced. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is reduced. \item For every open $U \subset X$ the ring $\mathcal{O}_X(U)$ is reduced. \end{enumerate} \end{lemma} \begin{proof} See Schemes, Lemmas \ref{schemes-lemma-reduced} and \ref{schemes-lemma-affine-reduced}. \end{proof} \begin{lemma} \label{lemma-characterize-irreducible} Let $X$ be a scheme. The following are equivalent. \begin{enumerate} \item The scheme $X$ is irreducible. \item There exists an affine open covering $X = \bigcup_{i \in I} U_i$ such that $I$ is not empty, $U_i$ is irreducible for all $i \in I$, and $U_i \cap U_j \not = \emptyset$ for all $i, j \in I$. \item The scheme $X$ is nonempty and every nonempty affine open $U \subset X$ is irreducible. \end{enumerate} \end{lemma} \begin{proof} Assume (1). By Schemes, Lemma \ref{schemes-lemma-scheme-sober} we see that $X$ has a unique generic point $\eta$. Then $X = \overline{\{\eta\}}$. Hence $\eta$ is an element of every nonempty affine open $U \subset X$. This implies that $U = \overline{\{\eta\}}$ and that any two nonempty affines meet. Thus (1) implies both (2) and (3). \medskip\noindent Assume (2). Suppose $X = Z_1 \cup Z_2$ is a union of two closed subsets. For every $i$ we see that either $U_i \subset Z_1$ or $U_i \subset Z_2$. Pick some $i \in I$ and assume $U_i \subset Z_1$ (possibly after renumbering $Z_1$, $Z_2$). For any $j \in I$ the open subset $U_i \cap U_j$ is dense in $U_j$ and contained in the closed subset $Z_1 \cap U_j$. We conclude that also $U_j \subset Z_1$. Thus $X = Z_1$ as desired. \medskip\noindent Assume (3). Choose an affine open covering $X = \bigcup_{i \in I} U_i$. We may assume that each $U_i$ is nonempty. Since $X$ is nonempty we see that $I$ is not empty. By assumption each $U_i$ is irreducible. Suppose $U_i \cap U_j = \emptyset$ for some pair $i, j \in I$. Then the open $U_i \amalg U_j = U_i \cup U_j$ is affine, see Schemes, Lemma \ref{schemes-lemma-disjoint-union-affines}. Hence it is irreducible by assumption which is absurd. We conclude that (3) implies (2). The lemma is proved. \end{proof} \begin{lemma} \label{lemma-characterize-integral} A scheme $X$ is integral if and only if it is reduced and irreducible. \end{lemma} \begin{proof} If $X$ is irreducible, then every affine open $\Spec(R) = U \subset X$ is irreducible. If $X$ is reduced, then $R$ is reduced, by Lemma \ref{lemma-characterize-reduced} above. Hence $R$ is reduced and $(0)$ is a prime ideal, i.e., $R$ is an integral domain. \medskip\noindent If $X$ is integral, then for every nonempty affine open $\Spec(R) = U \subset X$ the ring $R$ is reduced and hence $X$ is reduced by Lemma \ref{lemma-characterize-reduced}. Moreover, every nonempty affine open is irreducible. Hence $X$ is irreducible, see Lemma \ref{lemma-characterize-irreducible}. \end{proof} \noindent In Examples, Section \ref{examples-section-connected-locally-integral-not-integral} we construct a connected affine scheme all of whose local rings are domains, but which is not integral. \section{Types of schemes defined by properties of rings} \label{section-properties-rings} \noindent In this section we study what properties of rings allow one to define local properties of schemes. \begin{definition} \label{definition-property-local} Let $P$ be a property of rings. We say that $P$ is {\it local} if the following hold: \begin{enumerate} \item For any ring $R$, and any $f \in R$ we have $P(R) \Rightarrow P(R_f)$. \item For any ring $R$, and $f_i \in R$ such that $(f_1, \ldots, f_n) = R$ then $\forall i, P(R_{f_i}) \Rightarrow P(R)$. \end{enumerate} \end{definition} \begin{definition} \label{definition-locally-P} Let $P$ be a property of rings. Let $X$ be a scheme. We say $X$ is {\it locally $P$} if for any $x \in X$ there exists an affine open neighbourhood $U$ of $x$ in $X$ such that $\mathcal{O}_X(U)$ has property $P$. \end{definition} \noindent This is only a good notion if the property is local. Even if $P$ is a local property we will not automatically use this definition to say that a scheme is locally $P$'' unless we also explicitly state the definition elsewhere. \begin{lemma} \label{lemma-locally-P} Let $X$ be a scheme. Let $P$ be a local property of rings. The following are equivalent: \begin{enumerate} \item The scheme $X$ is locally $P$. \item For every affine open $U \subset X$ the property $P(\mathcal{O}_X(U))$ holds. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ satisfies $P$. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is locally $P$. \end{enumerate} Moreover, if $X$ is locally $P$ then every open subscheme is locally $P$. \end{lemma} \begin{proof} Of course (1) $\Leftrightarrow$ (3) and (2) $\Rightarrow$ (1). If (3) $\Rightarrow$ (2), then the final statement of the lemma holds and it follows easily that (4) is also equivalent to (1). Thus we show (3) $\Rightarrow$ (2). \medskip\noindent Let $X = \bigcup U_i$ be an affine open covering, say $U_i = \Spec(R_i)$. Assume $P(R_i)$. Let $\Spec(R) = U \subset X$ be an arbitrary affine open. By Schemes, Lemma \ref{schemes-lemma-good-subcover} there exists a standard covering of $U = \Spec(R)$ by standard opens $D(f_j)$ such that each ring $R_{f_j}$ is a principal localization of one of the rings $R_i$. By Definition \ref{definition-property-local} (1) we get $P(R_{f_j})$. Whereupon $P(R)$ by Definition \ref{definition-property-local} (2). \end{proof} \noindent Here is a sample application. \begin{lemma} \label{lemma-reduced-is-locally-reduced} Let $X$ be a scheme. Then $X$ is reduced if and only if $X$ is locally reduced'' in the sense of Definition \ref{definition-locally-P}. \end{lemma} \begin{proof} This is clear from Lemma \ref{lemma-characterize-reduced}. \end{proof} \begin{lemma} \label{lemma-properties-local} The following properties of a ring $R$ are local. \begin{enumerate} \item (Cohen-Macaulay.) The ring $R$ is Noetherian and CM, see Algebra, Definition \ref{algebra-definition-ring-CM}. \item (Regular.) The ring $R$ is Noetherian and regular, see Algebra, Definition \ref{algebra-definition-regular}. \item (Absolutely Noetherian.) The ring $R$ is of finite type over $Z$. \item Add more here as needed.\footnote{But we only list those properties here which we have not already dealt with separately somewhere else.} \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \section{Noetherian schemes} \label{section-noetherian} \noindent Recall that a ring $R$ is {\it Noetherian} if it satisfies the ascending chain condition of ideals. Equivalently every ideal of $R$ is finitely generated. \begin{definition} \label{definition-noetherian} Let $X$ be a scheme. \begin{enumerate} \item We say $X$ is {\it locally Noetherian} if every $x \in X$ has an affine open neighbourhood $\Spec(R) = U \subset X$ such that the ring $R$ is Noetherian. \item We say $X$ is {\it Noetherian} if $X$ is locally Noetherian and quasi-compact. \end{enumerate} \end{definition} \noindent Here is the standard result characterizing locally Noetherian schemes. \begin{lemma} \label{lemma-locally-Noetherian} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is locally Noetherian. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Noetherian. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Noetherian. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is locally Noetherian. \end{enumerate} Moreover, if $X$ is locally Noetherian then every open subscheme is locally Noetherian. \end{lemma} \begin{proof} To show this it suffices to show that being Noetherian is a local property of rings, see Lemma \ref{lemma-locally-P}. Any localization of a Noetherian ring is Noetherian, see Algebra, Lemma \ref{algebra-lemma-Noetherian-permanence}. By Algebra, Lemma \ref{algebra-lemma-cover} we see the second property to Definition \ref{definition-property-local}. \end{proof} \begin{lemma} \label{lemma-immersion-into-noetherian} Any immersion $Z \to X$ with $X$ locally Noetherian is quasi-compact. \end{lemma} \begin{proof} A closed immersion is clearly quasi-compact. A composition of quasi-compact morphisms is quasi-compact, see Topology, Lemma \ref{topology-lemma-composition-quasi-compact}. Hence it suffices to show that an open immersion into a locally Noetherian scheme is quasi-compact. Using Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine} we reduce to the case where $X$ is affine. Any open subset of the spectrum of a Noetherian ring is quasi-compact (for example combine Algebra, Lemma \ref{algebra-lemma-Noetherian-topology} and Topology, Lemmas \ref{topology-lemma-Noetherian} and \ref{topology-lemma-Noetherian-quasi-compact}). \end{proof} \begin{lemma} \label{lemma-locally-Noetherian-quasi-separated} A locally Noetherian scheme is quasi-separated. \end{lemma} \begin{proof} By Schemes, Lemma \ref{schemes-lemma-characterize-quasi-separated} we have to show that the intersection $U \cap V$ of two affine opens of $X$ is quasi-compact. This follows from Lemma \ref{lemma-immersion-into-noetherian} above on considering the open immersion $U \cap V \to U$ for example. (But really it is just because any open of the spectrum of a Noetherian ring is quasi-compact.) \end{proof} \begin{lemma} \label{lemma-Noetherian-topology} A (locally) Noetherian scheme has a (locally) Noetherian underlying topological space, see Topology, Definition \ref{topology-definition-noetherian}. \end{lemma} \begin{proof} This is because a Noetherian scheme is a finite union of spectra of Noetherian rings and Algebra, Lemma \ref{algebra-lemma-Noetherian-topology} and Topology, Lemma \ref{topology-lemma-finite-union-Noetherian}. \end{proof} \begin{lemma} \label{lemma-locally-closed-in-Noetherian} Any locally closed subscheme of a (locally) Noetherian scheme is (locally) Noetherian. \end{lemma} \begin{proof} Omitted. Hint: Any quotient, and any localization of a Noetherian ring is Noetherian. For the Noetherian case use again that any subset of a Noetherian space is a Noetherian space (with induced topology). \end{proof} \begin{lemma} \label{lemma-Noetherian-irreducible-components} A Noetherian scheme has a finite number of irreducible components. \end{lemma} \begin{proof} The underlying topological space of a Noetherian scheme is Noetherian (Lemma \ref{lemma-Noetherian-topology}) and we conclude because a Noetherian topological space has only finitely many irreducible components (Topology, Lemma \ref{topology-lemma-Noetherian}). \end{proof} \begin{lemma} \label{lemma-morphism-Noetherian-schemes-quasi-compact} Any morphism of schemes $f : X \to Y$ with $X$ Noetherian is quasi-compact. \end{lemma} \begin{proof} Use Lemma \ref{lemma-Noetherian-topology} and use that any subset of a Noetherian topological space is quasi-compact (see Topology, Lemmas Lemmas \ref{topology-lemma-Noetherian} and \ref{topology-lemma-Noetherian-quasi-compact}). \end{proof} \noindent Here is a fun lemma. It says that every locally Noetherian scheme has plenty of closed points (at least one in every closed subset). \begin{lemma} \label{lemma-locally-Noetherian-closed-point} Any nonempty locally Noetherian scheme has a closed point. Any nonempty closed subset of a locally Noetherian scheme has a closed point. Equivalently, any point of a locally Noetherian scheme specializes to a closed point. \end{lemma} \begin{proof} The second assertion follows from the first (using Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme} and Lemma \ref{lemma-locally-closed-in-Noetherian}). Consider any nonempty affine open $U \subset X$. Let $x \in U$ be a closed point. If $x$ is a closed point of $X$ then we are done. If not, let $X_0 \subset X$ be the reduced induced closed subscheme structure on $\overline{\{x\}}$. Then $U_0 = U \cap X_0$ is an affine open of $X_0$ by Schemes, Lemma \ref{schemes-lemma-closed-subspace-scheme} and $U_0 = \{x\}$. Let $y \in X_0$, $y \not = x$ be a specialization of $x$. Consider the local ring $R = \mathcal{O}_{X_0, y}$. This is a Noetherian local ring as $X_0$ is Noetherian by Lemma \ref{lemma-locally-closed-in-Noetherian}. Denote $V \subset \Spec(R)$ the inverse image of $U_0$ in $\Spec(R)$ by the canonical morphism $\Spec(R) \to X_0$ (see Schemes, Section \ref{schemes-section-points}.) By construction $V$ is a singleton with unique point corresponding to $x$ (use Schemes, Lemma \ref{schemes-lemma-specialize-points}). By Algebra, Lemma \ref{algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens} we see that $\dim(R) = 1$. In other words, we see that $y$ is an immediate specialization of $x$ (see Topology, Definition \ref{topology-definition-dimension-function}). In other words, any point $y \not = x$ such that $x \leadsto y$ is an immediate specialization of $x$. Clearly each of these points is a closed point as desired. \end{proof} \begin{lemma} \label{lemma-locally-Noetherian-specialization-dvr} Let $X$ be a locally Noetherian scheme. Let $x' \leadsto x$ be a specialization of points of $X$. Then \begin{enumerate} \item there exists a discrete valuation ring $R$ and a morphism $f : \Spec(R) \to X$ such that the generic point $\eta$ of $\Spec(R)$ maps to $x'$ and the special point maps to $x$, and \item given a finitely generated field extension $\kappa(x') \subset K$ we may arrange it so that the extension $\kappa(x') \subset \kappa(\eta)$ induced by $f$ is isomorphic to the given one. \end{enumerate} \end{lemma} \begin{proof} Let $x' \leadsto x$ be a specialization in $X$, and let $\kappa(x') \subset K$ be a finitely generated extension of fields. By Schemes, Lemma \ref{schemes-lemma-specialize-points} and the discussion following Schemes, Lemma \ref{schemes-lemma-characterize-points} this leads to ring maps $\mathcal{O}_{X, x} \to \kappa(x') \to K$. Let $R \subset K$ be any discrete valuation ring whose field of fractions is $K$ and which dominates the image of $\mathcal{O}_{X, x} \to K$, see Algebra, Lemma \ref{algebra-lemma-exists-dvr}. The ring map $\mathcal{O}_{X, x} \to R$ induces the morphism $f : \Spec(R) \to X$, see Schemes, Lemma \ref{schemes-lemma-morphism-from-spec-local-ring}. This morphism has all the desired properties by construction. \end{proof} \begin{lemma} \label{lemma-thin-infinite-sequence} Let $S$ be a Noetherian scheme. Let $T \subset S$ be an infinite subset. Then there exists an infinite subset $T' \subset T$ such that there are no nontrivial specializations among the points $T'$. \end{lemma} \begin{proof} Let $T_0 \subset T$ be the set of $t \in T$ which do not specialize to another point of $T$. If $T_0$ is infinite, then $T' = T_0$ works. Hence we may and do assume $T_0$ is finite. Inductively, for $i > 0$, consider the set $T_i \subset T$ of $t \in T$ such that \begin{enumerate} \item $t \not \in T_{i - 1} \cup T_{i - 2} \cup \ldots \cup T_0$, \item there exist a nontrivial specialization $t \leadsto t'$ with $t' \in T_{i - 1}$, and \item for any nontrivial specialization $t \leadsto t'$ with $t' \in T$ we have $t' \in T_{i - 1} \cup T_{i - 2} \cup \ldots \cup T_0$. \end{enumerate} Again, if $T_i$ is infinite, then $T' = T_i$ works. Let $d$ be the maximum of the dimensions of the local rings $\mathcal{O}_{S, t}$ for $t \in T_0$; then $d$ is an integer because $T_0$ is finite and the dimensions of the local rings are finite by Algebra, Proposition \ref{algebra-proposition-dimension}. Then $T_i = \emptyset$ for $i > d$. Namely, if $t \in T_i$ then we can find a sequence of nontrivial specializations $t = t_i \leadsto t_{i - 1} \leadsto \ldots \leadsto t_0$ with $t_0 \in T_0$. As the points $t = t_i, t_{i - 1}, \ldots, t_0$ are in $\Spec(\mathcal{O}_{S, t_0})$ (Schemes, Lemma \ref{schemes-lemma-specialize-points}), we see that $i \leq d$. Thus $\bigcup T_i = T_d \cup \ldots \cup T_0$ is a finite subset of $T$. \medskip\noindent Suppose $t \in T$ is not in $\bigcup T_i$. Then there must be a specialization $t \leadsto t'$ with $t' \in T$ and $t' \not \in \bigcup T_i$. (Namely, if every specialization of $t$ is in the finite set $T_d \cup \ldots \cup T_0$, then there is a maximum $i$ such that there is some specialization $t \leadsto t'$ with $t' \in T_i$ and then $t \in T_{i + 1}$ by construction.) Hence we get an infinite sequence $$t \leadsto t' \leadsto t'' \leadsto \ldots$$ of nontrivial specializations between points of $T \setminus \bigcup T_i$. This is impossible because the underlying topological space of $S$ is Noetherian by Lemma \ref{lemma-locally-Noetherian-quasi-separated}. \end{proof} \section{Jacobson schemes} \label{section-jacobson} \noindent Recall that a space is said to be {\it Jacobson} if the closed points are dense in every closed subset, see Topology, Section \ref{topology-section-space-jacobson}. \begin{definition} \label{definition-jacobson} A scheme $S$ is said to be {\it Jacobson} if its underlying topological space is Jacobson. \end{definition} \noindent Recall that a ring $R$ is Jacobson if every radical ideal of $R$ is the intersection of maximal ideals, see Algebra, Definition \ref{algebra-definition-ring-jacobson}. \begin{lemma} \label{lemma-affine-jacobson} An affine scheme $\Spec(R)$ is Jacobson if and only if the ring $R$ is Jacobson. \end{lemma} \begin{proof} This is Algebra, Lemma \ref{algebra-lemma-jacobson}. \end{proof} \noindent Here is the standard result characterizing Jacobson schemes. Intuitively it claims that Jacobson $\Leftrightarrow$ locally Jacobson. \begin{lemma} \label{lemma-locally-jacobson} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is Jacobson. \item The scheme $X$ is locally Jacobson'' in the sense of Definition \ref{definition-locally-P}. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Jacobson. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Jacobson. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is Jacobson. \end{enumerate} Moreover, if $X$ is Jacobson then every open subscheme is Jacobson. \end{lemma} \begin{proof} The final assertion of the lemma holds by Topology, Lemma \ref{topology-lemma-jacobson-inherited}. The equivalence of (5) and (1) is Topology, Lemma \ref{topology-lemma-jacobson-local}. Hence, using Lemma \ref{lemma-affine-jacobson}, we see that (1) $\Leftrightarrow$ (2). To finish proving the lemma it suffices to show that Jacobson'' is a local property of rings, see Lemma \ref{lemma-locally-P}. Any localization of a Jacobson ring at an element is Jacobson, see Algebra, Lemma \ref{algebra-lemma-Jacobson-invert-element}. Suppose $R$ is a ring, $f_1, \ldots, f_n \in R$ generate the unit ideal and each $R_{f_i}$ is Jacobson. Then we see that $\Spec(R) = \bigcup D(f_i)$ is a union of open subsets which are all Jacobson, and hence $\Spec(R)$ is Jacobson by Topology, Lemma \ref{topology-lemma-jacobson-local} again. This proves the second property of Definition \ref{definition-property-local}. \end{proof} \noindent Many schemes used commonly in algebraic geometry are Jacobson, see Morphisms, Lemma \ref{morphisms-lemma-ubiquity-Jacobson-schemes}. We mention here the following interesting case. \begin{lemma} \label{lemma-complement-closed-point-Jacobson} Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak m$. In this case the scheme $S = \Spec(R) \setminus \{\mathfrak m\}$ is Jacobson. \end{lemma} \begin{proof} Since $\Spec(R)$ is a Noetherian scheme, hence $S$ is a Noetherian scheme (Lemma \ref{lemma-locally-closed-in-Noetherian}). Hence $S$ is a sober, Noetherian topological space (use Schemes, Lemma \ref{schemes-lemma-scheme-sober}). Assume $S$ is not Jacobson to get a contradiction. By Topology, Lemma \ref{topology-lemma-non-jacobson-Noetherian-characterize} there exists some non-closed point $\xi \in S$ such that $\{\xi\}$ is locally closed. This corresponds to a prime $\mathfrak p \subset R$ such that (1) there exists a prime $\mathfrak q$, $\mathfrak p \subset \mathfrak q \subset \mathfrak m$ with both inclusions strict, and (2) $\{\mathfrak p\}$ is open in $\Spec(R/\mathfrak p)$. This is impossible by Algebra, Lemma \ref{algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens}. \end{proof} \section{Normal schemes} \label{section-normal} \noindent Recall that a ring $R$ is said to be normal if all its local rings are normal domains, see Algebra, Definition \ref{algebra-definition-ring-normal}. A normal domain is a domain which is integrally closed in its field of fractions, see Algebra, Definition \ref{algebra-definition-domain-normal}. Thus it makes sense to define a normal scheme as follows. \begin{definition} \label{definition-normal} A scheme $X$ is {\it normal} if and only if for all $x \in X$ the local ring $\mathcal{O}_{X, x}$ is a normal domain. \end{definition} \noindent This seems to be the definition used in EGA, see \cite[0, 4.1.4]{EGA}. Suppose $X = \Spec(A)$, and $A$ is reduced. Then saying that $X$ is normal is not equivalent to saying that $A$ is integrally closed in its total ring of fractions. However, if $A$ is Noetherian then this is the case (see Algebra, Lemma \ref{algebra-lemma-characterize-reduced-ring-normal}). \begin{lemma} \label{lemma-locally-normal} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is normal. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is normal. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is normal. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is normal. \end{enumerate} Moreover, if $X$ is normal then every open subscheme is normal. \end{lemma} \begin{proof} This is clear from the definitions. \end{proof} \begin{lemma} \label{lemma-normal-reduced} A normal scheme is reduced. \end{lemma} \begin{proof} Immediate from the definitions. \end{proof} \begin{lemma} \label{lemma-integral-normal} Let $X$ be an integral scheme. Then $X$ is normal if and only if for every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is a normal domain. \end{lemma} \begin{proof} This follows from Algebra, Lemma \ref{algebra-lemma-normality-is-local}. \end{proof} \begin{lemma} \label{lemma-normal-locally-finite-nr-irreducibles} Let $X$ be a scheme such that any quasi-compact open has a finite number of irreducible components. The following are equivalent: \begin{enumerate} \item $X$ is normal, and \item $X$ is a disjoint union of normal integral schemes. \end{enumerate} \end{lemma} \begin{proof} It is immediate from the definitions that (2) implies (1). Let $X$ be a normal scheme such that every quasi-compact open has a finite number of irreducible components. If $X$ is affine then $X$ satisfies (2) by Algebra, Lemma \ref{algebra-lemma-characterize-reduced-ring-normal}. For a general $X$, let $X = \bigcup X_i$ be an affine open covering. Note that also each $X_i$ has but a finite number of irreducible components, and the lemma holds for each $X_i$. Let $T \subset X$ be an irreducible component. By the affine case each intersection $T \cap X_i$ is open in $X_i$ and an integral normal scheme. Hence $T \subset X$ is open, and an integral normal scheme. This proves that $X$ is the disjoint union of its irreducible components, which are integral normal schemes. There are only finitely many by assumption. \end{proof} \begin{lemma} \label{lemma-normal-Noetherian} Let $X$ be a Noetherian scheme. The following are equivalent: \begin{enumerate} \item $X$ is normal, and \item $X$ is a finite disjoint union of normal integral schemes. \end{enumerate} \end{lemma} \begin{proof} This is a special case of Lemma \ref{lemma-normal-locally-finite-nr-irreducibles} because a Noetherian scheme has a Noetherian underlying topological space (Lemma \ref{lemma-Noetherian-topology} and Topology, Lemma \ref{topology-lemma-Noetherian}. \end{proof} \begin{lemma} \label{lemma-normal-locally-Noetherian} Let $X$ be a locally Noetherian scheme. The following are equivalent: \begin{enumerate} \item $X$ is normal, and \item $X$ is a disjoint union of integral normal schemes. \end{enumerate} \end{lemma} \begin{proof} Omitted. Hint: This is purely topological from Lemma \ref{lemma-normal-Noetherian}. \end{proof} \begin{remark} \label{remark-normal-connected-irreducible} Let $X$ be a normal scheme. If $X$ is locally Noetherian then we see that $X$ is integral if and only if $X$ is connected, see Lemma \ref{lemma-normal-locally-Noetherian}. But there exists a connected affine scheme $X$ such that $\mathcal{O}_{X, x}$ is a domain for all $x \in X$, but $X$ is not irreducible, see Examples, Section \ref{examples-section-connected-locally-integral-not-integral}. This example is even a normal scheme (proof omitted), so beware! \end{remark} \begin{lemma} \label{lemma-normal-integral-sections} Let $X$ be an integral normal scheme. Then $\Gamma(X, \mathcal{O}_X)$ is a normal domain. \end{lemma} \begin{proof} Set $R = \Gamma(X, \mathcal{O}_X)$. It is clear that $R$ is a domain. Suppose $f = a/b$ is an element of its fraction field which is integral over $R$. Say we have $f^d + \sum_{i = 1, \ldots, d} a_i f^i = 0$ with $a_i \in R$. Let $U \subset X$ be affine open. Since $b \in R$ is not zero and since $X$ is integral we see that also $b|_U \in \mathcal{O}_X(U)$ is not zero. Hence $a/b$ is an element of the fraction field of $\mathcal{O}_X(U)$ which is integral over $\mathcal{O}_X(U)$ (because we can use the same polynomial $f^d + \sum_{i = 1, \ldots, d} a_i|_U f^i = 0$ on $U$). Since $\mathcal{O}_X(U)$ is a normal domain (Lemma \ref{lemma-locally-normal}), we see that $f_U = (a|_U)/(b|_U) \in \mathcal{O}_X(U)$. It is easy to see that $f_U|_V = f_V$ whenever $V \subset U \subset X$ are affine open. Hence the local sections $f_U$ glue to a global section $f$ as desired. \end{proof} \section{Cohen-Macaulay schemes} \label{section-Cohen-Macaulay} \noindent Recall, see Algebra, Definition \ref{algebra-definition-local-ring-CM}, that a local Noetherian ring $(R, \mathfrak m)$ is said to be Cohen-Macaulay if $\text{depth}_{\mathfrak m}(R) = \dim(R)$. Recall that a Noetherian ring $R$ is said to be Cohen-Macaulay if every local ring $R_{\mathfrak p}$ of $R$ is Cohen-Macaulay, see Algebra, Definition \ref{algebra-definition-ring-CM}. \begin{definition} \label{definition-Cohen-Macaulay} Let $X$ be a scheme. We say $X$ is {\it Cohen-Macaulay} if for every $x \in X$ there exists an affine open neighbourhood $U \subset X$ of $x$ such that the ring $\mathcal{O}_X(U)$ is Noetherian and Cohen-Macaulay. \end{definition} \begin{lemma} \label{lemma-characterize-Cohen-Macaulay} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item $X$ is Cohen-Macaulay, \item $X$ is locally Noetherian and all of its local rings are Cohen-Macaulay, and \item $X$ is locally Noetherian and for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ is Cohen-Macaulay. \end{enumerate} \end{lemma} \begin{proof} Algebra, Lemma \ref{algebra-lemma-localize-CM} says that the localization of a Cohen-Macaulay local ring is Cohen-Macaulay. The lemma follows by combining this with Lemma \ref{lemma-locally-Noetherian}, with the existence of closed points on locally Noetherian schemes (Lemma \ref{lemma-locally-Noetherian-closed-point}), and the definitions. \end{proof} \begin{lemma} \label{lemma-locally-Cohen-Macaulay} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is Cohen-Macaulay. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Noetherian and Cohen-Macaulay. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Noetherian and Cohen-Macaulay. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is Cohen-Macaulay. \end{enumerate} Moreover, if $X$ is Cohen-Macaulay then every open subscheme is Cohen-Macaulay. \end{lemma} \begin{proof} Combine Lemmas \ref{lemma-locally-Noetherian} and \ref{lemma-characterize-Cohen-Macaulay}. \end{proof} \noindent More information on Cohen-Macaulay schemes and depth can be found in Cohomology of Schemes, Section \ref{coherent-section-depth}. \section{Regular schemes} \label{section-regular} \noindent Recall, see Algebra, Definition \ref{algebra-definition-regular-local}, that a local Noetherian ring $(R, \mathfrak m)$ is said to be {\it regular} if $\mathfrak m$ can be generated by $\dim(R)$ elements. Recall that a Noetherian ring $R$ is said to be {\it regular} if every local ring $R_{\mathfrak p}$ of $R$ is regular, see Algebra, Definition \ref{algebra-definition-regular}. \begin{definition} \label{definition-regular} Let $X$ be a scheme. We say $X$ is {\it regular}, or {\it nonsingular} if for every $x \in X$ there exists an affine open neighbourhood $U \subset X$ of $x$ such that the ring $\mathcal{O}_X(U)$ is Noetherian and regular. \end{definition} \begin{lemma} \label{lemma-characterize-regular} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item $X$ is regular, \item $X$ is locally Noetherian and all of its local rings are regular, and \item $X$ is locally Noetherian and for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ is regular. \end{enumerate} \end{lemma} \begin{proof} By the discussion in Algebra preceding Algebra, Definition \ref{algebra-definition-regular} we know that the localization of a regular local ring is regular. The lemma follows by combining this with Lemma \ref{lemma-locally-Noetherian}, with the existence of closed points on locally Noetherian schemes (Lemma \ref{lemma-locally-Noetherian-closed-point}), and the definitions. \end{proof} \begin{lemma} \label{lemma-locally-regular} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is regular. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Noetherian and regular. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Noetherian and regular. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is regular. \end{enumerate} Moreover, if $X$ is regular then every open subscheme is regular. \end{lemma} \begin{proof} Combine Lemmas \ref{lemma-locally-Noetherian} and \ref{lemma-characterize-regular}. \end{proof} \begin{lemma} \label{lemma-regular-normal} A regular scheme is normal. \end{lemma} \begin{proof} See Algebra, Lemma \ref{algebra-lemma-regular-normal}. \end{proof} \section{Dimension} \label{section-dimension} \noindent The dimension of a scheme is just the dimension of its underlying topological space. \begin{definition} \label{definition-dimension} Let $X$ be a scheme. \begin{enumerate} \item The {\it dimension} of $X$ is just the dimension of $X$ as a topological spaces, see Topology, Definition \ref{topology-definition-Krull}. \item For $x \in X$ we denote $\dim_x(X)$ the dimension of the underlying topological space of $X$ at $x$ as in Topology, Definition \ref{topology-definition-Krull}. We say $\dim_x(X)$ is the {\it dimension of $X$ at $x$}. \end{enumerate} \end{definition} \noindent As a scheme has a sober underlying topological space (Schemes, Lemma \ref{schemes-lemma-scheme-sober}) we may compute the dimension of $X$ as the supremum of the lengths $n$ of chains $$T_0 \subset T_1 \subset \ldots \subset T_n$$ of irreducible closed subsets of $X$, or as the supremum of the lengths $n$ of chains of specializations $$\xi_n \leadsto \xi_{n - 1} \leadsto \ldots \leadsto \xi_0$$ of points of $X$. \begin{lemma} \label{lemma-dimension} Let $X$ be a scheme. The following are equal \begin{enumerate} \item The dimension of $X$. \item The supremum of the dimensions of the local rings of $X$. \item The supremum of $\dim_x(X)$ for $x \in X$. \end{enumerate} \end{lemma} \begin{proof} Note that given a chain of specializations $$\xi_n \leadsto \xi_{n - 1} \leadsto \ldots \leadsto \xi_0$$ of points of $X$ all of the points $\xi_i$ correspond to prime ideals of the local ring of $X$ at $\xi_0$ by Schemes, Lemma \ref{schemes-lemma-specialize-points}. Hence we see that the dimension of $X$ is the supremum of the dimensions of its local rings. In particular $\dim_x(X) \geq \dim(\mathcal{O}_{X, x})$ as $\dim_x(X)$ is the minimum of the dimensions of open neighbourhoods of $x$. Thus $\sup_{x \in X} \dim_x(X) \geq \dim(X)$. On the other hand, it is clear that $\sup_{x \in X} \dim_x(X) \leq \dim(X)$ as $\dim(U) \leq \dim(X)$ for any open subset of $X$. \end{proof} \begin{lemma} \label{lemma-codimension-local-ring} Let $X$ be a scheme. Let $Y \subset X$ be an irreducible closed subset. Let $\xi \in Y$ be the generic point. Then $$\text{codim}(Y, X) = \dim(\mathcal{O}_{X, \xi})$$ where the codimension is as defined in Topology, Definition \ref{topology-definition-codimension}. \end{lemma} \begin{proof} By Topology, Lemma \ref{topology-lemma-codimension-at-generic-point} we may replace $X$ by an affine open neighbourhood of $\xi$. In this case the result follows easily from Algebra, Lemma \ref{algebra-lemma-irreducible-components-containing-x}. \end{proof} \begin{lemma} \label{lemma-generic-point} Let $X$ be a scheme. Let $x \in X$. Then $x$ is a generic point of an irreducible component of $X$ if and only if $\dim(\mathcal{O}_{X, x}) = 0$. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-codimension-local-ring} for example. \end{proof} \begin{lemma} \label{lemma-locally-Noetherian-dimension-0} A locally Noetherian scheme of dimension $0$ is a disjoint union of spectra of Artinian local rings. \end{lemma} \begin{proof} A Noetherian ring of dimension $0$ is a finite product of Artinian local rings, see Algebra, Proposition \ref{algebra-proposition-dimension-zero-ring}. Hence an affine open of a locally Noetherian scheme $X$ of dimension $0$ has discrete underlying topological space. This implies that the topology on $X$ is discrete. The lemma follows easily from these remarks. \end{proof} \begin{lemma} \label{lemma-dimension-zero} \begin{reference} Email from Ofer Gabber dated June 4, 2016 \end{reference} Let $X$ be a scheme of dimension zero. The following are equivalent \begin{enumerate} \item $X$ is quasi-separated, \item $X$ is separated, \item $X$ is Hausdorff, \item every affine open is closed. \end{enumerate} In this case the connected components of $X$ are points. \end{lemma} \begin{proof} As the dimension of $X$ is zero, we see that for any affine open $U \subset X$ the space $U$ is profinite and satisfies a bunch of other properties which we will use freely below, see Algebra, Lemma \ref{algebra-lemma-ring-with-only-minimal-primes}. We choose an affine open covering $X = \bigcup U_i$. \medskip\noindent If (4) holds, then $U_i \cap U_j$ is a closed subset of $U_i$, hence quasi-compact, hence $X$ is quasi-separated, by Schemes, Lemma \ref{schemes-lemma-characterize-quasi-separated}, hence (1) holds. \medskip\noindent If (1) holds, then $U_i \cap U_j$ is a quasi-compact open of $U_i$ hence closed in $U_i$. Then $U_i \cap U_j \to U_i$ is an open immersion whose image is closed, hence it is a closed immersion. In particular $U_i \cap U_j$ is affine and $\mathcal{O}(U_i) \to \mathcal{O}_X(U_i \cap U_j)$ is surjective. Thus $X$ is separated by Schemes, Lemma \ref{schemes-lemma-characterize-quasi-separated}, hence (2) holds. \medskip\noindent Assume (2) and let $x, y \in X$. Say $x \in U_i$. If $y \in U_i$ too, then we can find disjoint open neighbourhoods of $x$ and $y$ because $U_i$ is Hausdorff. Say $y \not \in U_i$ and $y \in U_j$. Then $y \not \in U_i \cap U_j$ which is an affine open of $U_j$ and hence closed in $U_j$. Thus we can find an open neighbourhood of $y$ not meeting $U_i$ and we conclude that $X$ is Hausdorff, hence (3) holds. \medskip\noindent Assume (3). Let $U \subset X$ be affine open. Then $U$ is closed in $X$ by Topology, Lemma \ref{topology-lemma-quasi-compact-in-Hausdorff}. This proves (4) holds. \medskip\noindent We omit the proof of the final statement. \end{proof} \section{Catenary schemes} \label{section-catenary} \noindent Recall that a topological space $X$ is called {\it catenary} if for every pair of irreducible closed subsets $T \subset T'$ there exist a maximal chain of irreducible closed subsets $$T = T_0 \subset T_1 \subset \ldots \subset T_e = T'$$ and every such chain has the same length. See Topology, Definition \ref{topology-definition-catenary}. \begin{definition} \label{definition-catenary} Let $S$ be a scheme. We say $S$ is {\it catenary} if the underlying topological space of $S$ is catenary. \end{definition} \noindent Recall that a ring $A$ is called {\it catenary} if for any pair of prime ideals $\mathfrak p \subset \mathfrak q$ there exists a maximal chain of primes $$\mathfrak p = \mathfrak p_0 \subset \ldots \subset \mathfrak p_e = \mathfrak q$$ and all of these have the same length. See Algebra, Definition \ref{algebra-definition-catenary}. \begin{lemma} \label{lemma-catenary-local} Let $S$ be a scheme. The following are equivalent \begin{enumerate} \item $S$ is catenary, \item there exists an open covering of $S$ all of whose members are catenary schemes, \item for every affine open $\Spec(R) = U \subset S$ the ring $R$ is catenary, and \item there exists an affine open covering $S = \bigcup U_i$ such that each $U_i$ is the spectrum of a catenary ring. \end{enumerate} Moreover, in this case any locally closed subscheme of $S$ is catenary as well. \end{lemma} \begin{proof} Combine Topology, Lemma \ref{topology-lemma-catenary}, and Algebra, Lemma \ref{algebra-lemma-catenary}. \end{proof} \begin{lemma} \label{lemma-catenary-dimension-function} Let $S$ be a locally Noetherian scheme. The following are equivalent: \begin{enumerate} \item $S$ is catenary, and \item locally in the Zariski topology there exists a dimension function on $S$ (see Topology, Definition \ref{topology-definition-dimension-function}). \end{enumerate} \end{lemma} \begin{proof} This follows from Topology, Lemmas \ref{topology-lemma-catenary}, \ref{topology-lemma-dimension-function-catenary}, and \ref{topology-lemma-locally-dimension-function}, Schemes, Lemma \ref{schemes-lemma-scheme-sober} and finally Lemma \ref{lemma-Noetherian-topology}. \end{proof} \noindent It turns out that a scheme is catenary if and only if its local rings are catenary. \begin{lemma} \label{lemma-catenary-local-rings-catenary} Let $X$ be a scheme. The following are equivalent \begin{enumerate} \item $X$ is catenary, and \item for any $x \in X$ the local ring $\mathcal{O}_{X, x}$ is catenary. \end{enumerate} \end{lemma} \begin{proof} Assume $X$ is catenary. Let $x \in X$. By Lemma \ref{lemma-catenary-local} we may replace $X$ by an affine open neighbourhood of $x$, and then $\Gamma(X, \mathcal{O}_X)$ is a catenary ring. By Algebra, Lemma \ref{algebra-lemma-localization-catenary} any localization of a catenary ring is catenary. Whence $\mathcal{O}_{X, x}$ is catenary. \medskip\noindent Conversely assume all local rings of $X$ are catenary. Let $Y \subset Y'$ be an inclusion of irreducible closed subsets of $X$. Let $\xi \in Y$ be the generic point. Let $\mathfrak p \subset \mathcal{O}_{X, \xi}$ be the prime corresponding to the generic point of $Y'$, see Schemes, Lemma \ref{schemes-lemma-specialize-points}. By that same lemma the irreducible closed subsets of $X$ in between $Y$ and $Y'$ correspond to primes $\mathfrak q \subset \mathcal{O}_{X, \xi}$ with $\mathfrak p \subset \mathfrak q \subset \mathfrak m_{\xi}$. Hence we see all maximal chains of these are finite and have the same length as $\mathcal{O}_{X, \xi}$ is a catenary ring. \end{proof} \section{Serre's conditions} \label{section-Rk} \noindent Here are two technical notions that are often useful. See also Cohomology of Schemes, Section \ref{coherent-section-depth}. \begin{definition} \label{definition-Rk} Let $X$ be a locally Noetherian scheme. Let $k \geq 0$. \begin{enumerate} \item We say $X$ is {\it regular in codimension $k$}, or we say $X$ has property {\it $(R_k)$} if for every $x \in X$ we have $$\dim(\mathcal{O}_{X, x}) \leq k \Rightarrow \mathcal{O}_{X, x}\text{ is regular}$$ \item We say $X$ has property {\it $(S_k)$} if for every $x \in X$ we have $\text{depth}(\mathcal{O}_{X, x}) \geq \min(k, \dim(\mathcal{O}_{X, x}))$. \end{enumerate} \end{definition} \noindent The phrase regular in codimension $k$'' makes sense since we have seen in Section \ref{section-catenary} that if $Y \subset X$ is irreducible closed with generic point $x$, then $\dim(\mathcal{O}_{X, x}) = \text{codim}(Y, X)$. For example condition $(R_0)$ means that for every generic point $\eta \in X$ of an irreducible component of $X$ the local ring $\mathcal{O}_{X, \eta}$ is a field. But for general Noetherian schemes it can happen that the regular locus of $X$ is badly behaved, so care has to be taken. \begin{lemma} \label{lemma-scheme-regular-iff-all-Rk} Let $X$ be a locally Noetherian scheme. Then $X$ is regular if and only if $X$ has $(R_k)$ for all $k \geq 0$. \end{lemma} \begin{proof} Follows from Lemma \ref{lemma-characterize-regular} and the definitions. \end{proof} \begin{lemma} \label{lemma-scheme-CM-iff-all-Sk} Let $X$ be a locally Noetherian scheme. Then $X$ is Cohen-Macaulay if and only if $X$ has $(S_k)$ for all $k \geq 0$. \end{lemma} \begin{proof} By Lemma \ref{lemma-characterize-Cohen-Macaulay} we reduce to looking at local rings. Hence the lemma is true because a Noetherian local ring is Cohen-Macaulay if and only if it has depth equal to its dimension. \end{proof} \begin{lemma} \label{lemma-criterion-reduced} Let $X$ be a locally Noetherian scheme. Then $X$ is reduced if and only if $X$ has properties $(S_1)$ and $(R_0)$. \end{lemma} \begin{proof} This is Algebra, Lemma \ref{algebra-lemma-criterion-reduced}. \end{proof} \begin{lemma} \label{lemma-criterion-normal} Let $X$ be a locally Noetherian scheme. Then $X$ is normal if and only if $X$ has properties $(S_2)$ and $(R_1)$. \end{lemma} \begin{proof} This is Algebra, Lemma \ref{algebra-lemma-criterion-normal}. \end{proof} \begin{lemma} \label{lemma-normal-dimension-1-regular} Let $X$ be a locally Noetherian scheme which is normal and has dimension $\leq 1$. Then $X$ is regular. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-criterion-normal} and the definitions. \end{proof} \begin{lemma} \label{lemma-normal-dimension-2-Cohen-Macaulay} Let $X$ be a locally Noetherian scheme which is normal and has dimension $\leq 2$. Then $X$ is Cohen-Macaulay. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-criterion-normal} and the definitions. \end{proof} \section{Japanese and Nagata schemes} \label{section-nagata} \noindent The notions considered in this section are not prominently defined in EGA. A universally Japanese scheme'' is mentioned and defined in \cite[IV Corollary 5.11.4]{EGA}. A Japanese scheme'' is mentioned in \cite[IV Remark 10.4.14 (ii)]{EGA} but no definition is given. A Nagata scheme (as given below) occurs in a few places in the literature (see for example \cite[Definition 8.2.30]{Liu} and \cite[Page 142]{Greco}). \medskip\noindent We briefly recall that a domain $R$ is called {\it Japanese} if the integral closure of $R$ in any finite extension of its fraction field is finite over $R$. A ring $R$ is called {\it universally Japanese} if for any finite type ring map $R \to S$ with $S$ a domain $S$ is Japanese. A ring $R$ is called {\it Nagata} if it is Noetherian and $R/\mathfrak p$ is Japanese for every prime $\mathfrak p$ of $R$. \begin{definition} \label{definition-nagata} Let $X$ be a scheme. \begin{enumerate} \item Assume $X$ integral. We say $X$ is {\it Japanese} if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_X(U)$ is Japanese (see Algebra, Definition \ref{algebra-definition-N}). \item We say $X$ is {\it universally Japanese} if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_X(U)$ is universally Japanese (see Algebra, Definition \ref{algebra-definition-nagata}). \item We say $X$ is {\it Nagata} if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_X(U)$ is Nagata (see Algebra, Definition \ref{algebra-definition-nagata}). \end{enumerate} \end{definition} \noindent Being Nagata is the same thing as being locally Noetherian and universally Japanese, see Lemma \ref{lemma-nagata-universally-Japanese}. \begin{remark} \label{remark-non-integral-Japanese} In \cite{Hoobler-finite} a (locally Noetherian) scheme $X$ is called Japanese if for every $x \in X$ and every associated prime $\mathfrak p$ of $\mathcal{O}_{X, x}$ the ring $\mathcal{O}_{X, x}/\mathfrak p$ is Japanese. We do not use this definition since there exists a one dimensional noetherian domain with excellent (in particular Japanese) local rings whose normalization is not finite. See \cite[Example 1]{Hochster-loci} or \cite{Heinzer-Levy} or \cite[Expos\'e XIX]{Traveaux}. On the other hand, we could circumvent this problem by calling a scheme $X$ Japanese if for every affine open $\Spec(A) \subset X$ the ring $A/\mathfrak p$ is Japanese for every associated prime $\mathfrak p$ of $A$. \end{remark} \begin{lemma} \label{lemma-nagata-locally-Noetherian} A Nagata scheme is locally Noetherian. \end{lemma} \begin{proof} This is true because a Nagata ring is Noetherian by definition. \end{proof} \begin{lemma} \label{lemma-locally-Japanese} Let $X$ be an integral scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is Japanese. \item For every affine open $U \subset X$ the domain $\mathcal{O}_X(U)$ is Japanese. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Japanese. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is Japanese. \end{enumerate} Moreover, if $X$ is Japanese then every open subscheme is Japanese. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-locally-P} and Algebra, Lemmas \ref{algebra-lemma-localize-N} and \ref{algebra-lemma-Japanese-local}. \end{proof} \begin{lemma} \label{lemma-locally-universally-Japanese} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is universally Japanese. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is universally Japanese. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is universally Japanese. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is universally Japanese. \end{enumerate} Moreover, if $X$ is universally Japanese then every open subscheme is universally Japanese. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-locally-P} and Algebra, Lemmas \ref{algebra-lemma-universally-japanese} and \ref{algebra-lemma-nagata-local}. \end{proof} \begin{lemma} \label{lemma-locally-nagata} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is Nagata. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Nagata. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Nagata. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is Nagata. \end{enumerate} Moreover, if $X$ is Nagata then every open subscheme is Nagata. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-locally-P} and Algebra, Lemmas \ref{algebra-lemma-nagata-localize} and \ref{algebra-lemma-nagata-local}. \end{proof} \begin{lemma} \label{lemma-characterize-nagata} Let $X$ be a locally Noetherian scheme. Then $X$ is Nagata if and only if every integral closed subscheme $Z \subset X$ is Japanese. \end{lemma} \begin{proof} Assume $X$ is Nagata. Let $Z \subset X$ be an integral closed subscheme. Let $z \in Z$. Let $\Spec(A) = U \subset X$ be an affine open containing $z$ such that $A$ is Nagata. Then $Z \cap U \cong \Spec(A/\mathfrak p)$ for some prime $\mathfrak p$, see Schemes, Lemma \ref{schemes-lemma-closed-subspace-scheme} (and Definition \ref{definition-integral}). By Algebra, Definition \ref{algebra-definition-nagata} we see that $A/\mathfrak p$ is Japanese. Hence $Z$ is Japanese by definition. \medskip\noindent Assume every integral closed subscheme of $X$ is Japanese. Let $\Spec(A) = U \subset X$ be any affine open. As $X$ is locally Noetherian we see that $A$ is Noetherian (Lemma \ref{lemma-locally-Noetherian}). Let $\mathfrak p \subset A$ be a prime ideal. We have to show that $A/\mathfrak p$ is Japanese. Let $T \subset U$ be the closed subset $V(\mathfrak p) \subset \Spec(A)$. Let $\overline{T} \subset X$ be the closure. Then $\overline{T}$ is irreducible as the closure of an irreducible subset. Hence the reduced closed subscheme defined by $\overline{T}$ is an integral closed subscheme (called $\overline{T}$ again), see Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme}. In other words, $\Spec(A/\mathfrak p)$ is an affine open of an integral closed subscheme of $X$. This subscheme is Japanese by assumption and by Lemma \ref{lemma-locally-Japanese} we see that $A/\mathfrak p$ is Japanese. \end{proof} \begin{lemma} \label{lemma-nagata-universally-Japanese} Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item $X$ is Nagata, and \item $X$ is locally Noetherian and universally Japanese. \end{enumerate} \end{lemma} \begin{proof} This is Algebra, Proposition \ref{algebra-proposition-nagata-universally-japanese}. \end{proof} \noindent This discussion will be continued in Morphisms, Section \ref{morphisms-section-nagata}. \section{The singular locus} \label{section-singular-locus} \noindent Here is the definition. \begin{definition} \label{definition-singular-locus} Let $X$ be a locally Noetherian scheme. The {\it regular locus} $\text{Reg}(X)$ of $X$ is the set of $x \in X$ such that $\mathcal{O}_{X, x}$ is a regular local ring. The {\it singular locus} $\text{Sing}(X)$ is the complement $X \setminus \text{Reg}(X)$, i.e., the set of points $x \in X$ such that $\mathcal{O}_{X, x}$ is not a regular local ring. \end{definition} \noindent The regular locus of a locally Noetherian scheme is stable under generalizations, see the discussion preceding Algebra, Definition \ref{algebra-definition-regular}. However, for general locally Noetherian schemes the regular locus need not be open. In More on Algebra, Section \ref{more-algebra-section-singular-locus} the reader can find some criteria for when this is the case. We will discuss this further in Morphisms, Section \ref{morphisms-section-singular-locus}. \section{Local irreducibility} \label{section-unibranch} \noindent Recall that in More on Algebra, Section \ref{more-algebra-section-unibranch} we introduced the notion of a (geometrically) unibranch local ring. \begin{definition} \label{definition-unibranch} \begin{reference} \cite[Chapter IV (6.15.1)]{EGA4} \end{reference} Let $X$ be a scheme. Let $x \in X$. We say $X$ is {\it unibranch at $x$} if the local ring $\mathcal{O}_{X, x}$ is unibranch. We say $X$ is {\it geometrically unibranch at $x$} if the local ring $\mathcal{O}_{X, x}$ is geometrically unibranch. We say $X$ is {\it unibranch} if $X$ is unibranch at all of its points. We say $X$ is {\it geometrically unibranch} if $X$ is geometrically unibranch at all of its points. \end{definition} \noindent To be sure, it can happen that a local ring $A$ is geometrically unibranch (in the sense of More on Algebra, Definition \ref{more-algebra-definition-unibranch}) but the scheme $\Spec(A)$ is not geometrically unibranch in the sense of Definition \ref{definition-unibranch}. For example this happens if $A$ is the local ring at the vertex of the cone over an irreducible plane curve which has ordinary double point singularity (a node). \begin{lemma} \label{lemma-normal-geometrically-unibranch} A normal scheme is geometrically unibranch. \end{lemma} \begin{proof} This follows from the definitions. Namely, a scheme is normal if the local rings are normal domains. It is immediate from the More on Algebra, Definition \ref{more-algebra-definition-unibranch} that a local normal domain is geometrically unibranch. \end{proof} \begin{lemma} \label{lemma-geometrically-unibranch} \begin{reference} Compare with \cite[Proposition 2.3]{Etale-coverings} \end{reference} Let $X$ be a Noetherian scheme. The following are equivalent \begin{enumerate} \item $X$ is geometrically unibranch (Definition \ref{definition-unibranch}), \item for every point $x \in X$ which is not the generic point of an irreducible component of $X$, the punctured spectrum of the strict henselization $\mathcal{O}_{X, x}^{sh}$ is connected. \end{enumerate} \end{lemma} \begin{proof} More on Algebra, Lemma \ref{more-algebra-lemma-geometrically-unibranch} shows that (1) implies that the punctured spectra in (2) are irreducible and in particular connected. \medskip\noindent Assume (2). Let $x \in X$. We have to show that $\mathcal{O}_{X, x}$ is geometrically unibranch. By induction on $\dim(\mathcal{O}_{X, x})$ we may assume that the result holds for every nontrivial generalization of $x$. We may replace $X$ by $\Spec(\mathcal{O}_{X, x})$. In other words, we may assume that $X = \Spec(A)$ with $A$ local and that $A_\mathfrak p$ is geometrically unibranch for each nonmaximal prime $\mathfrak p \subset A$. \medskip\noindent Let $A^{sh}$ be the strict henselization of $A$. If $\mathfrak q \subset A^{sh}$ is a prime lying over $\mathfrak p \subset A$, then $A_\mathfrak p \to A^{sh}_\mathfrak q$ is a filtered colimit of \'etale algebras. Hence the strict henselizations of $A_\mathfrak p$ and $A^{sh}_\mathfrak q$ are isomorphic. Thus by More on Algebra, Lemma \ref{more-algebra-lemma-geometrically-unibranch} we conclude that $A^{sh}_\mathfrak q$ has a unique minimal prime ideal for every nonmaximal prime $\mathfrak q$ of $A^{sh}$. \medskip\noindent Let $\mathfrak q_1, \ldots, \mathfrak q_r$ be the minimal primes of $A^{sh}$. We have to show that $r = 1$. By the above we see that $V(\mathfrak q_1) \cap V(\mathfrak q_j) = \{\mathfrak m^{sh}\}$ for $j = 2, \ldots, r$. Hence $V(\mathfrak q_1) \setminus \{\mathfrak m^{sh}\}$ is an open and closed subset of the punctured spectrum of $A^{sh}$ which is a contradiction with the assumption that this punctured spectrum is connected unless $r = 1$. \end{proof} \begin{definition} \label{definition-number-of-branches} Let $X$ be a scheme. Let $x \in X$. The {\it number of branches of $X$ at $x$} is the number of branches of the local ring $\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition \ref{more-algebra-definition-number-of-branches}. The {\it number of geometric branches of $X$ at $x$} is the number of geometric branches of the local ring $\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition \ref{more-algebra-definition-number-of-branches}. \end{definition} \noindent Often we want to compare this with the branches of the complete local ring, but the comparison is not straightforward in general; some information on this topic can be found in More on Algebra, Section \ref{more-algebra-section-branches-completion}. \begin{lemma} \label{lemma-number-of-branches-1} Let $X$ be a scheme. Let $x \in X$. \begin{enumerate} \item The number of branches of $X$ at $x$ is $1$ if and only if $X$ is unibranch at $x$. \item The number of geometric branches of $X$ at $x$ is $1$ if and only if $X$ is geometrically unibranch at $x$. \end{enumerate} \end{lemma} \begin{proof} This lemma follows immediately from the definitions and the corresponding result for rings, see More on Algebra, Lemma \ref{more-algebra-lemma-number-of-branches-1}. \end{proof} \section{Characterizing modules of finite type and finite presentation} \label{section-characterizing-finite-type-presentation} \noindent Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The following lemma implies that $\mathcal{F}$ is of finite type (see Modules, Definition \ref{modules-definition-finite-type}) if and only if $\mathcal{F}$ is on each open affine $\Spec(A) = U \subset X$ of the form $\widetilde M$ for some finite type $A$-module $M$. Similarly, $\mathcal{F}$ is of finite presentation (see Modules, Definition \ref{modules-definition-finite-presentation}) if and only if $\mathcal{F}$ is on each open affine $\Spec(A) = U \subset X$ of the form $\widetilde M$ for some finitely presented $A$-module $M$. \begin{lemma} \label{lemma-finite-type-module} Let $X = \Spec(R)$ be an affine scheme. The quasi-coherent sheaf of $\mathcal{O}_X$-modules $\widetilde M$ is a finite type $\mathcal{O}_X$-module if and only if $M$ is a finite $R$-module. \end{lemma} \begin{proof} Assume $\widetilde M$ is a finite type $\mathcal{O}_X$-module. This means there exists an open covering of $X$ such that $\widetilde M$ restricted to the members of this covering is globally generated by finitely many sections. Thus there also exists a standard open covering $X = \bigcup_{i = 1, \ldots, n} D(f_i)$ such that $\widetilde M|_{D(f_i)}$ is generated by finitely many sections. Thus $M_{f_i}$ is finitely generated for each $i$. Hence we conclude by Algebra, Lemma \ref{algebra-lemma-cover}. \end{proof} \begin{lemma} \label{lemma-finite-presentation-module} Let $X = \Spec(R)$ be an affine scheme. The quasi-coherent sheaf of $\mathcal{O}_X$-modules $\widetilde M$ is an $\mathcal{O}_X$-module of finite presentation if and only if $M$ is an $R$-module of finite presentation. \end{lemma} \begin{proof} Assume $\widetilde M$ is an $\mathcal{O}_X$-module of finite presentation. By Lemma \ref{lemma-finite-type-module} we see that $M$ is a finite $R$-module. Choose a surjection $R^n \to M$ with kernel $K$. By Schemes, Lemma \ref{schemes-lemma-spec-sheaves} there is a short exact sequence $$0 \to \widetilde{K} \to \bigoplus \mathcal{O}_X^{\oplus n} \to \widetilde{M} \to 0$$ By Modules, Lemma \ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation} we see that $\widetilde{K}$ is a finite type $\mathcal{O}_X$-module. Hence by Lemma \ref{lemma-finite-type-module} again we see that $K$ is a finite $R$-module. Hence $M$ is an $R$-module of finite presentation. \end{proof} \section{Sections over principal opens} \label{section-principal-opens} \noindent Here is a typical result of this kind. We will use a more naive but more direct method of proof in later lemmas. \begin{lemma} \label{lemma-invert-f-sections} \begin{slogan} Sections of quasi-coherent sheaves have only meromorphic singularities at infinity. \end{slogan} Let $X$ be a scheme. Let $f \in \Gamma(X, \mathcal{O}_X)$. Denote $X_f \subset X$ the open where $f$ is invertible, see Schemes, Lemma \ref{schemes-lemma-f-open}. If $X$ is quasi-compact and quasi-separated, the canonical map $$\Gamma(X, \mathcal{O}_X)_f \longrightarrow \Gamma(X_f, \mathcal{O}_X)$$ is an isomorphism. Moreover, if $\mathcal{F}$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules the map $$\Gamma(X, \mathcal{F})_f \longrightarrow \Gamma(X_f, \mathcal{F})$$ is an isomorphism. \end{lemma} \begin{proof} Write $R = \Gamma(X, \mathcal{O}_X)$. Consider the canonical morphism $$\varphi : X \longrightarrow \Spec(R)$$ of schemes, see Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}. Then the inverse image of the standard open $D(f)$ on the right hand side is $X_f$ on the left hand side. Moreover, since $X$ is assumed quasi-compact and quasi-separated the morphism $\varphi$ is quasi-compact and quasi-separated, see Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine} and \ref{schemes-lemma-compose-after-separated}. Hence by Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} we see that $\varphi_*\mathcal{F}$ is quasi-coherent. Hence we see that $\varphi_*\mathcal{F} = \widetilde M$ with $M = \Gamma(X, \mathcal{F})$ as an $R$-module. Thus we see that $$\Gamma(X_f, \mathcal{F}) = \Gamma(D(f), \varphi_*\mathcal{F}) = \Gamma(D(f), \widetilde M) = M_f$$ which is exactly the content of the lemma. The first displayed isomorphism of the lemma follows by taking $\mathcal{F} = \mathcal{O}_X$. \end{proof} \noindent Recall that given a scheme $X$, an invertible sheaf $\mathcal{L}$ on $X$, and a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we get a graded ring $\Gamma_*(X, \mathcal{L}) = \bigoplus\nolimits_{n \geq 0} \Gamma(X, \mathcal{L}^{\otimes n})$ and a graded $\Gamma_*(X, \mathcal{L})$-module $\Gamma_*(X, \mathcal{L}, \mathcal{F}) = \bigoplus\nolimits_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})$ see Modules, Definition \ref{modules-definition-gamma-star}. If we have moreover a section $s \in \Gamma(X, \mathcal{L})$, then we obtain a map \begin{equation} \label{equation-module-invert-s} \Gamma_*(X, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow \Gamma(X_s, \mathcal{F}|_{X_s}) \end{equation} which sends $t/s^n$ where $t \in \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})$ to $t|_{X_s} \otimes s|_{X_s}^{-n}$. This makes sense because $X_s \subset X$ is by definition the open over which $s$ has an inverse, see Modules, Lemma \ref{modules-lemma-s-open}. \begin{lemma} \label{lemma-invert-s-sections} Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf on $X$. Let $s \in \Gamma(X, \mathcal{L})$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item If $X$ is quasi-compact, then (\ref{equation-module-invert-s}) is injective, and \item if $X$ is quasi-compact and quasi-separated, then (\ref{equation-module-invert-s}) is an isomorphism. \end{enumerate} In particular, the canonical map $$\Gamma_*(X, \mathcal{L})_{(s)} \longrightarrow \Gamma(X_s, \mathcal{O}_X),\quad a/s^n \longmapsto a \otimes s^{-n}$$ is an isomorphism if $X$ is quasi-compact and quasi-separated. \end{lemma} \begin{proof} Assume $X$ is quasi-compact. Choose a finite affine open covering $X = U_1 \cup \ldots \cup U_m$ with $U_j$ affine and $\mathcal{L}|_{U_j} \cong \mathcal{O}_{U_j}$. Via this isomorphism, the image $s|_{U_j}$ corresponds to some $f_j \in \Gamma(U_j, \mathcal{O}_{U_j})$. Then $X_s \cap U_j = D(f_j)$. \medskip\noindent Proof of (1). Let $t/s^n$ be an element in the kernel of (\ref{equation-module-invert-s}). Then $t|_{X_s} = 0$. Hence $(t|_{U_j})|_{D(f_j)} = 0$. By Lemma \ref{lemma-invert-f-sections} we conclude that $f_j^{e_j} t|_{U_j} = 0$ for some $e_j \geq 0$. Let $e = \max(e_j)$. Then we see that $t \otimes s^e$ restricts to zero on $U_j$ for all $j$, hence is zero. Since $t/s^n$ is equal to $t \otimes s^e/s^{n + e}$ in $\Gamma_*(X, \mathcal{L}, \mathcal{F})_{(s)}$ we conclude that $t/s^n = 0$ as desired. \medskip\noindent Proof of (2). Assume $X$ is quasi-compact and quasi-separated. Then $U_j \cap U_{j'}$ is quasi-compact for all pairs $j, j'$, see Schemes, Lemma \ref{schemes-lemma-characterize-quasi-separated}. By part (1) we know (\ref{equation-module-invert-s}) is injective. Let $t' \in \Gamma(X_s, \mathcal{F}|_{X_s})$. For every $j$, there exist an integer $n_j \geq 0$ and $t'_j \in \Gamma(U_j, \mathcal{F}|_{U_j})$ such that $t'|_{D(f_j)}$ corresponds to $t'_j/f_j^{e_j}$ via the isomorphism of Lemma \ref{lemma-invert-f-sections}. Set $e = \max(e_j)$ and $$t_j = t'_j \otimes s|_{U_j}^e \in \Gamma(U_j, (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes e})|_{U_j})$$ Then we see that $t_j|_{U_j \cap U_{j'}}$ and $t_{j'}|_{U_j \cap U_{j'}}$ map to the same section of $\mathcal{F}$ over $U_j \cap U_{j'} \cap X_s$. By quasi-compactness of $U_j \cap U_{j'}$ and part (1) there exists an integer $e' \geq 0$ such that $$t_j|_{U_j \cap U_{j'}} \otimes s^{e'}|_{U_j \cap U_{j'}} = t_{j'}|_{U_j \cap U_{j'}} \otimes s^{e'}|_{U_j \cap U_{j'}}$$ as sections of $\mathcal{F} \otimes \mathcal{L}^{\otimes e + e'}$ over $U_j \cap U_{j'}$. We may choose the same $e'$ to work for all pairs $j, j'$. Then the sheaf conditions implies there is a section $t \in \Gamma(X, \mathcal{F} \otimes \mathcal{L}^{\otimes e + e'})$ whose restriction to $U_j$ is $t_j \otimes s^{e'}|_{U_j}$. A simple computation shows that $t/s^{e + e'}$ maps to $t'$ as desired. \end{proof} \noindent Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $\mathcal{F}$ and $\mathcal{G}$ be quasi-coherent $\mathcal{O}_X$-modules. Consider the graded $\Gamma_*(X, \mathcal{L})$-module $$M = \bigoplus\nolimits_{n \in \mathbf{Z}} \Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})$$ Next, let $s \in \Gamma(X, \mathcal{L})$ be a section. Then there is a canonical map \begin{equation} \label{equation-hom-invert-s} M_{(s)} \longrightarrow \Hom_{\mathcal{O}_{X_s}}(\mathcal{F}|_{X_s}, \mathcal{G}|_{X_s}) \end{equation} which sends $\alpha/s^n$ to the map $\alpha|_{X_s} \otimes s|_{X_s}^{-n}$. The following lemma, combined with Lemma \ref{lemma-extend-finite-presentation}, says roughly that, if $X$ is quasi-compact and quasi-separated, the category of finitely presented $\mathcal{O}_{X_s}$-modules is the category of finitely presented $\mathcal{O}_X$-modules with the multiplicative system of maps $s^n: \mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}$ inverted. \begin{lemma} \label{lemma-section-maps-backwards} Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$ be a section. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_X$-modules. \begin{enumerate} \item If $X$ is quasi-compact and $\mathcal{F}$ is of finite type, then (\ref{equation-hom-invert-s}) is injective, and \item if $X$ is quasi-compact and quasi-separated and $\mathcal{F}$ is of finite presentation, then (\ref{equation-hom-invert-s}) is bijective. \end{enumerate} \end{lemma} \begin{proof} We first prove the lemma in case $X = \Spec(A)$ is affine and $\mathcal{L} = \mathcal{O}_X$. In this case $s$ corresponds to an element $f \in A$. Say $\mathcal{F} = \widetilde{M}$ and $\mathcal{G} = \widetilde{N}$ for some $A$-modules $M$ and $N$. Then the lemma translates (via Lemmas \ref{lemma-finite-type-module} and \ref{lemma-finite-presentation-module}) into the following algebra statements \begin{enumerate} \item If $M$ is a finite $A$-module and $\varphi : M \to N$ is an $A$-module map such that the induced map $M_f \to N_f$ is zero, then $f^n\varphi = 0$ for some $n$. \item If $M$ is a finitely presented $A$-module, then $\Hom_A(M, N)_f = \Hom_{A_f}(M_f, N_f)$. \end{enumerate} The second statement is Algebra, Lemma \ref{algebra-lemma-hom-from-finitely-presented} and we omit the proof of the first statement. \medskip\noindent Next, we prove (1) for general $X$. Assume $X$ is quasi-compact and hoose a finite affine open covering $X = U_1 \cup \ldots \cup U_m$ with $U_j$ affine and $\mathcal{L}|_{U_j} \cong \mathcal{O}_{U_j}$. Via this isomorphism, the image $s|_{U_j}$ corresponds to some $f_j \in \Gamma(U_j, \mathcal{O}_{U_j})$. Then $X_s \cap U_j = D(f_j)$. Let $\alpha/s^n$ be an element in the kernel of (\ref{equation-hom-invert-s}). Then $\alpha|_{X_s} = 0$. Hence $(\alpha|_{U_j})|_{D(f_j)} = 0$. By the affine case treated above we conclude that $f_j^{e_j} \alpha|_{U_j} = 0$ for some $e_j \geq 0$. Let $e = \max(e_j)$. Then we see that $\alpha \otimes s^e$ restricts to zero on $U_j$ for all $j$, hence is zero. Since $\alpha/s^n$ is equal to $\alpha \otimes s^e/s^{n + e}$ in $M_{(s)}$ we conclude that $\alpha/s^n = 0$ as desired. \medskip\noindent Proof of (2). Since $\mathcal{F}$ is of finite presentation, the sheaf $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is quasi-coherent, see Schemes, Section \ref{schemes-section-quasi-coherent}. Moreover, it is clear that $$\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}$$ for all $n$. Hence in this case the statement follows from Lemma \ref{lemma-invert-s-sections} applied to $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$. \end{proof} \section{Quasi-affine schemes} \label{section-quasi-affine} \begin{definition} \label{definition-quasi-affine} A scheme $X$ is called {\it quasi-affine} if it is quasi-compact and isomorphic to an open subscheme of an affine scheme. \end{definition} \begin{lemma} \label{lemma-invert-f-affine} Let $X$ be a scheme. Let $f \in \Gamma(X, \mathcal{O}_X)$. Assume $X$ is quasi-compact and quasi-separated and assume that $X_f$ is affine. Then the canonical morphism $$j : X \longrightarrow \Spec(\Gamma(X, \mathcal{O}_X))$$ from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} induces an isomorphism of $X_f = j^{-1}(D(f))$ onto the standard affine open $D(f) \subset \Spec(\Gamma(X, \mathcal{O}_X))$. \end{lemma} \begin{proof} This is clear as $j$ induces an isomorphism of rings $\Gamma(X, \mathcal{O}_X)_f \to \mathcal{O}_X(X_f)$ by Lemma \ref{lemma-invert-f-sections} above. \end{proof} \begin{lemma} \label{lemma-quasi-affine} Let $X$ be a scheme. Then $X$ is quasi-affine if and only if the canonical morphism $$X \longrightarrow \Spec(\Gamma(X, \mathcal{O}_X))$$ from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} is a quasi-compact open immersion. \end{lemma} \begin{proof} If the displayed morphism is a quasi-compact open immersion then $X$ is isomorphic to a quasi-compact open subscheme of $\Spec(\Gamma(X, \mathcal{O}_X))$ and clearly $X$ is quasi-affine. \medskip\noindent Assume $X$ is quasi-affine, say $X \subset \Spec(R)$ is quasi-compact open. This in particular implies that $X$ is separated, see Schemes, Lemma \ref{schemes-lemma-subscheme-of-separated-scheme}. Let $A = \Gamma(X, \mathcal{O}_X)$. Consider the ring map $R \to A$ coming from $R = \Gamma(\Spec(R), \mathcal{O}_{\Spec(R)})$ and the restriction mapping of the sheaf $\mathcal{O}_{\Spec(R)}$. By Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} we obtain a factorization: $$X \longrightarrow \Spec(A) \longrightarrow \Spec(R)$$ of the inclusion morphism. Let $x \in X$. Choose $r \in R$ such that $x \in D(r)$ and $D(r) \subset X$. Denote $f \in A$ the image of $r$ in $A$. The open $X_f$ of Lemma \ref{lemma-invert-f-sections} above is equal to $D(r) \subset X$ and hence $A_f \cong R_r$ by the conclusion of that lemma. Hence $D(r) \to \Spec(A)$ is an isomorphism onto the standard affine open $D(f)$ of $\Spec(A)$. Since $X$ can be covered by such affine opens $D(f)$ we win. \end{proof} \begin{lemma} \label{lemma-cartesian-diagram-quasi-affine} Let $U \to V$ be an open immersion of quasi-affine schemes. Then $$\xymatrix{ U \ar[d] \ar[rr]_-j & & \Spec(\Gamma(U, \mathcal{O}_U)) \ar[d] \\ U \ar[r] & V \ar[r]^-{j'} & \Spec(\Gamma(V, \mathcal{O}_V)) }$$ is cartesian. \end{lemma} \begin{proof} The diagram is commutative by Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}. Write $A = \Gamma(U, \mathcal{O}_U)$ and $B = \Gamma(V, \mathcal{O}_V)$. Let $g \in B$ be such that $V_g$ is affine and contained in $U$. This means that if $f$ is the image of $g$ in $A$, then $U_f = V_g$. By Lemma \ref{lemma-invert-f-affine} we see that $j'$ induces an isomorphism of $V_g$ with the standard open $D(g)$ of $\Spec(B)$. Thus $V_g \times_{\Spec(B)} \Spec(A) \to \Spec(A)$ is an isomorphism onto $D(f) \subset \Spec(A)$. By Lemma \ref{lemma-invert-f-affine} again $j$ maps $U_f$ isomorphically to $D(f)$. Thus we see that $U_f = U_f \times_{\Spec(B)} \Spec(A)$. Since by Lemma \ref{lemma-quasi-affine} we can cover $U$ by $V_g = U_f$ as above, we see that $U \to U \times_{\Spec(B)} \Spec(A)$ is an isomorphism. \end{proof} \section{Flat modules} \label{section-flat} \noindent On any ringed space $(X, \mathcal{O}_X)$ we know what it means for an $\mathcal{O}_X$-module to be flat (at a point), see Modules, Definition \ref{modules-definition-flat} (Definition \ref{modules-definition-flat-at-point}). For quasi-coherent sheaves on an affine scheme this matches the notion defined in the algebra chapter. \begin{lemma} \label{lemma-flat-module} \begin{slogan} Flatness is the same for modules and sheaves. \end{slogan} Let $X = \Spec(R)$ be an affine scheme. Let $\mathcal{F} = \widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\mathcal{F}$ is a flat $\mathcal{O}_X$-module if and only if $M$ is a flat $R$-module. \end{lemma} \begin{proof} Flatness of $\mathcal{F}$ may be checked on the stalks, see Modules, Lemma \ref{modules-lemma-flat-stalks-flat}. The same is true in the case of modules over a ring, see Algebra, Lemma \ref{algebra-lemma-flat-localization}. And since $\mathcal{F}_x = M_{\mathfrak p}$ if $x$ corresponds to $\mathfrak p$ the lemma is true. \end{proof} \section{Locally free modules} \label{section-finite-locally-free} \noindent On any ringed space we know what it means for an $\mathcal{O}_X$-module to be (finite) locally free. On an affine scheme this matches the notion defined in the algebra chapter. \begin{lemma} \label{lemma-locally-free-module} Let $X = \Spec(R)$ be an affine scheme. Let $\mathcal{F} = \widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\mathcal{F}$ is a (finite) locally free $\mathcal{O}_X$-module of if and only if $M$ is a (finite) locally free $R$-module. \end{lemma} \begin{proof} Follows from the definitions, see Modules, Definition \ref{modules-definition-locally-free} and Algebra, Definition \ref{algebra-definition-locally-free}. \end{proof} \noindent We can characterize finite locally free modules in many different ways. \begin{lemma} \label{lemma-finite-locally-free} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The following are equivalent: \begin{enumerate} \item $\mathcal{F}$ is a flat $\mathcal{O}_X$-module of finite presentation, \item $\mathcal{F}$ is $\mathcal{O}_X$-module of finite presentation and for all $x \in X$ the stalk $\mathcal{F}_x$ is a free $\mathcal{O}_{X, x}$-module, \item $\mathcal{F}$ is a locally free, finite type $\mathcal{O}_X$-module, \item $\mathcal{F}$ is a finite locally free $\mathcal{O}_X$-module, and \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite type, for every $x \in X$ the stalk $\mathcal{F}_x$ is a free $\mathcal{O}_{X, x}$-module, and the function $$\rho_\mathcal{F} : X \to \mathbf{Z}, \quad x \longmapsto \dim_{\kappa(x)} \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \kappa(x)$$ is locally constant in the Zariski topology on $X$. \end{enumerate} \end{lemma} \begin{proof} This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma \ref{algebra-lemma-finite-projective}. The translation uses Lemmas \ref{lemma-finite-type-module}, \ref{lemma-finite-presentation-module}, \ref{lemma-flat-module}, and \ref{lemma-locally-free-module}. \end{proof} \section{Locally projective modules} \label{section-locally-projective} \noindent A consequence of the work done in the algebra chapter is that it makes sense to define a locally projective module as follows. \begin{definition} \label{definition-locally-projective} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. We say $\mathcal{F}$ is {\it locally projective} if for every affine open $U \subset X$ the $\mathcal{O}_X(U)$-module $\mathcal{F}(U)$ is projective. \end{definition} \begin{lemma} \label{lemma-locally-projective} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is locally projective, and \item there exists an affine open covering $X = \bigcup U_i$ such that the $\mathcal{O}_X(U_i)$-module $\mathcal{F}(U_i)$ is projective for every $i$. \end{enumerate} In particular, if $X = \Spec(A)$ and $\mathcal{F} = \widetilde{M}$ then $\mathcal{F}$ is locally projective if and only if $M$ is a projective $A$-module. \end{lemma} \begin{proof} First, note that if $M$ is a projective $A$-module and $A \to B$ is a ring map, then $M \otimes_A B$ is a projective $B$-module, see Algebra, Lemma \ref{algebra-lemma-ascend-properties-modules}. Hence if $U$ is an affine open such that $\mathcal{F}(U)$ is a projective $\mathcal{O}_X(U)$-module, then the standard open $D(f)$ is an affine open such that $\mathcal{F}(D(f))$ is a projective $\mathcal{O}_X(D(f))$-module for all $f \in \mathcal{O}_X(U)$. Assume (2) holds. Let $U \subset X$ be an arbitrary affine open. We can find an open covering $U = \bigcup_{j = 1, \ldots, m} D(f_j)$ by finitely many standard opens $D(f_j)$ such that for each $j$ the open $D(f_j)$ is a standard open of some $U_i$, see Schemes, Lemma \ref{schemes-lemma-standard-open-two-affines}. Hence, if we set $A = \mathcal{O}_X(U)$ and if $M$ is an $A$-module such that $\mathcal{F}|_U$ corresponds to $M$, then we see that $M_{f_j}$ is a projective $A_{f_j}$-module. It follows that $A \to B = \prod A_{f_j}$ is a faithfully flat ring map such that $M \times_A B$ is a projective $B$-module. Hence $M$ is projective by Algebra, Theorem \ref{algebra-theorem-ffdescent-projectivity}. \end{proof} \begin{lemma} \label{lemma-locally-projective-pullback} Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. If $\mathcal{G}$ is locally projective on $Y$, then $f^*\mathcal{G}$ is locally projective on $X$. \end{lemma} \begin{proof} See Algebra, Lemma \ref{algebra-lemma-ascend-properties-modules}. \end{proof} \section{Extending quasi-coherent sheaves} \label{section-extending-quasi-coherent-sheaves} \noindent It is sometimes useful to be able to show that a given quasi-coherent sheaf on an open subscheme extends to the whole scheme. \begin{lemma} \label{lemma-extend-trivial} Let $j : U \to X$ be a quasi-compact open immersion of schemes. \begin{enumerate} \item Any quasi-coherent sheaf on $U$ extends to a quasi-coherent sheaf on $X$. \item Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}|_U$ be a quasi-coherent subsheaf. There exists a quasi-coherent subsheaf $\mathcal{H}$ of $\mathcal{F}$ such that $\mathcal{H}|_U = \mathcal{G}$ as subsheaves of $\mathcal{F}|_U$. \item Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $U$. Let $\varphi : \mathcal{G} \to \mathcal{F}|_U$ be a morphism of $\mathcal{O}_U$-modules. There exists a quasi-coherent sheaf $\mathcal{H}$ of $\mathcal{O}_X$-modules and a map $\psi : \mathcal{H} \to \mathcal{F}$ such that $\mathcal{H}|_U = \mathcal{G}$ and that $\psi|_U = \varphi$. \end{enumerate} \end{lemma} \begin{proof} An immersion is separated (see Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms}) and $j$ is quasi-compact by assumption. Hence for any quasi-coherent sheaf $\mathcal{G}$ on $U$ the sheaf $j_*\mathcal{G}$ is an extension to $X$. See Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} and Sheaves, Section \ref{sheaves-section-open-immersions}. \medskip\noindent Assume $\mathcal{F}$, $\mathcal{G}$ are as in (2). Then $j_*\mathcal{G}$ is a quasi-coherent sheaf on $X$ (see above). It is a subsheaf of $j_*j^*\mathcal{F}$. Hence the kernel $$\mathcal{H} = \Ker(\mathcal{F} \oplus j_* \mathcal{G} \longrightarrow j_*j^*\mathcal{F})$$ is quasi-coherent as well, see Schemes, Section \ref{schemes-section-quasi-coherent}. It is formal to check that $\mathcal{H} \subset \mathcal{F}$ and that $\mathcal{H}|_U = \mathcal{G}$ (using the material in Sheaves, Section \ref{sheaves-section-open-immersions} again). \medskip\noindent The same proof as above works. Just take $\mathcal{H} = \Ker(\mathcal{F} \oplus j_* \mathcal{G} \to j_*j^*\mathcal{F})$ with its obvious map to $\mathcal{F}$ and its obvious identification with $\mathcal{G}$ over $U$. \end{proof} \begin{lemma} \label{lemma-extend} Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G} \subset \mathcal{F}|_U$ be a quasi-coherent $\mathcal{O}_U$-submodule which is of finite type. Then there exists a quasi-coherent submodule $\mathcal{G}' \subset \mathcal{F}$ which is of finite type such that $\mathcal{G}'|_U = \mathcal{G}$. \end{lemma} \begin{proof} Let $n$ be the minimal number of affine opens $U_i \subset X$, $i = 1, \ldots , n$ such that $X = U \cup \bigcup U_i$. (Here we use that $X$ is quasi-compact.) Suppose we can prove the lemma for the case $n = 1$. Then we can successively extend $\mathcal{G}$ to a $\mathcal{G}_1$ over $U \cup U_1$ to a $\mathcal{G}_2$ over $U \cup U_1 \cup U_2$ to a $\mathcal{G}_3$ over $U \cup U_1 \cup U_2 \cup U_3$, and so on. Thus we reduce to the case $n = 1$. \medskip\noindent Thus we may assume that $X = U \cup V$ with $V$ affine. Since $X$ is quasi-separated and $U$, $V$ are quasi-compact open, we see that $U \cap V$ is a quasi-compact open. It suffices to prove the lemma for the system $(V, U \cap V, \mathcal{F}|_V, \mathcal{G}|_{U \cap V})$ since we can glue the resulting sheaf $\mathcal{G}'$ over $V$ to the given sheaf $\mathcal{G}$ over $U$ along the common value over $U \cap V$. Thus we reduce to the case where $X$ is affine. \medskip\noindent Assume $X = \Spec(R)$. Write $\mathcal{F} = \widetilde M$ for some $R$-module $M$. By Lemma \ref{lemma-extend-trivial} above we may find a quasi-coherent subsheaf $\mathcal{H} \subset \mathcal{F}$ which restricts to $\mathcal{G}$ over $U$. Write $\mathcal{H} = \widetilde N$ for some $R$-module $N$. For every $u \in U$ there exists an $f \in R$ such that $u \in D(f) \subset U$ and such that $N_f$ is finitely generated, see Lemma \ref{lemma-finite-type-module}. Since $U$ is quasi-compact we can cover it by finitely many $D(f_i)$ such that $N_{f_i}$ is generated by finitely many elements, say $x_{i, 1}/f_i^N, \ldots, x_{i, r_i}/f_i^N$. Let $N' \subset N$ be the submodule generated by the elements $x_{i, j}$. Then the subsheaf $\mathcal{G} := \widetilde{N'} \subset \mathcal{H} \subset \mathcal{F}$ works. \end{proof} \begin{lemma} \label{lemma-quasi-coherent-colimit-finite-type} Let $X$ be a quasi-compact and quasi-separated scheme. Any quasi-coherent sheaf of $\mathcal{O}_X$-modules is the directed colimit of its quasi-coherent $\mathcal{O}_X$-submodules which are of finite type. \end{lemma} \begin{proof} The colimit is directed because if $\mathcal{G}_1$, $\mathcal{G}_2$ are quasi-coherent subsheaves of finite type, then $\mathcal{G}_1 + \mathcal{G}_2 \subset \mathcal{F}$ is a quasi-coherent subsheaf of finite type. Let $U \subset X$ be any affine open, and let $s \in \Gamma(U, \mathcal{F})$ be any section. Let $\mathcal{G} \subset \mathcal{F}|_U$ be the subsheaf generated by $s$. Then clearly $\mathcal{G}$ is quasi-coherent and has finite type as an $\mathcal{O}_U$-module. By Lemma \ref{lemma-extend} we see that $\mathcal{G}$ is the restriction of a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{F}$ which has finite type. Since $X$ has a basis for the topology consisting of affine opens we conclude that every local section of $\mathcal{F}$ is locally contained in a quasi-coherent submodule of finite type. Thus we win. \end{proof} \begin{lemma} \label{lemma-extend-finite-presentation} (Variant of Lemma \ref{lemma-extend} dealing with modules of finite presentation.) Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{G}$ be an $\mathcal{O}_U$-module which of finite presentation. Let $\varphi : \mathcal{G} \to \mathcal{F}|_U$ be a morphism of $\mathcal{O}_U$-modules. Then there exists an $\mathcal{O}_X$-module $\mathcal{G}'$ of finite presentation, and a morphism of $\mathcal{O}_X$-modules $\varphi' : \mathcal{G}' \to \mathcal{F}$ such that $\mathcal{G}'|_U = \mathcal{G}$ and such that $\varphi'|_U = \varphi$. \end{lemma} \begin{proof} The beginning of the proof is a repeat of the beginning of the proof of Lemma \ref{lemma-extend}. We write it out carefuly anyway. \medskip\noindent Let $n$ be the minimal number of affine opens $U_i \subset X$, $i = 1, \ldots , n$ such that $X = U \cup \bigcup U_i$. (Here we use that $X$ is quasi-compact.) Suppose we can prove the lemma for the case $n = 1$. Then we can successively extend the pair $(\mathcal{G}, \varphi)$ to a pair $(\mathcal{G}_1, \varphi_1)$ over $U \cup U_1$ to a pair $(\mathcal{G}_2, \varphi_2)$ over $U \cup U_1 \cup U_2$ to a pair $(\mathcal{G}_3, \varphi_3)$ over $U \cup U_1 \cup U_2 \cup U_3$, and so on. Thus we reduce to the case $n = 1$. \medskip\noindent Thus we may assume that $X = U \cup V$ with $V$ affine. Since $X$ is quasi-separated and $U$ quasi-compact, we see that $U \cap V \subset V$ is quasi-compact. Suppose we prove the lemma for the system $(V, U \cap V, \mathcal{F}|_V, \mathcal{G}|_{U \cap V}, \varphi|_{U \cap V})$ thereby producing $(\mathcal{G}', \varphi')$ over $V$. Then we can glue $\mathcal{G}'$ over $V$ to the given sheaf $\mathcal{G}$ over $U$ along the common value over $U \cap V$, and similarly we can glue the map $\varphi'$ to the map $\varphi$ along the common value over $U \cap V$. Thus we reduce to the case where $X$ is affine. \medskip\noindent Assume $X = \Spec(R)$. By Lemma \ref{lemma-extend-trivial} above we may find a quasi-coherent sheaf $\mathcal{H}$ with a map $\psi : \mathcal{H} \to \mathcal{F}$ over $X$ which restricts to $\mathcal{G}$ and $\varphi$ over $U$. By Lemma \ref{lemma-extend} we can find a finite type quasi-coherent $\mathcal{O}_X$-submodule $\mathcal{H}' \subset \mathcal{H}$ such that $\mathcal{H}'|_U = \mathcal{G}$. Thus after replacing $\mathcal{H}$ by $\mathcal{H}'$ and $\psi$ by the restriction of $\psi$ to $\mathcal{H}'$ we may assume that $\mathcal{H}$ is of finite type. By Lemma \ref{lemma-finite-presentation-module} we conclude that $\mathcal{H} = \widetilde{N}$ with $N$ a finitely generated $R$-module. Hence there exists a surjection as in the following short exact sequence of quasi-coherent $\mathcal{O}_X$-modules $$0 \to \mathcal{K} \to \mathcal{O}_X^{\oplus n} \to \mathcal{H} \to 0$$ where $\mathcal{K}$ is defined as the kernel. Since $\mathcal{G}$ is of finite presentation and $\mathcal{H}|_U = \mathcal{G}$ by Modules, Lemma \ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation} the restriction $\mathcal{K}|_U$ is an $\mathcal{O}_U$-module of finite type. Hence by Lemma \ref{lemma-extend} again we see that there exists a finite type quasi-coherent $\mathcal{O}_X$-submodule $\mathcal{K}' \subset \mathcal{K}$ such that $\mathcal{K}'|_U = \mathcal{K}|_U$. The solution to the problem posed in the lemma is to set $$\mathcal{G}' = \mathcal{O}_X^{\oplus n}/\mathcal{K}'$$ which is clearly of finite presentation and restricts to give $\mathcal{G}$ on $U$ with $\varphi'$ equal to the composition $$\mathcal{G}' = \mathcal{O}_X^{\oplus n}/\mathcal{K}' \to \mathcal{O}_X^{\oplus n}/\mathcal{K} = \mathcal{H} \xrightarrow{\psi} \mathcal{F}.$$ This finishes the proof of the lemma. \end{proof} \noindent The following lemma says that every quasi-coherent sheaf on a quasi-compact and quasi-separated scheme is a filtered colimit of $\mathcal{O}$-modules of finite presentation. Actually, we reformulate this in (perhaps more familiar) terms of directed colimits over directed sets in the next lemma. \begin{lemma} \label{lemma-directed-colimit-diagram-finite-presentation} \begin{slogan} Quasi-coherent modules on quasi-compact and quasi-separated schemes are filtered colimits of finitely presented modules. \end{slogan} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. There exist \begin{enumerate} \item a filtered index category $\mathcal{I}$ (see Categories, Definition \ref{categories-definition-directed}), \item a diagram $\mathcal{I} \to \textit{Mod}(\mathcal{O}_X)$ (see Categories, Section \ref{categories-section-limits}), $i \mapsto \mathcal{F}_i$, \item morphisms of $\mathcal{O}_X$-modules $\varphi_i : \mathcal{F}_i \to \mathcal{F}$ \end{enumerate} such that each $\mathcal{F}_i$ is of finite presentation and such that the morphisms $\varphi_i$ induce an isomorphism $$\colim_i \mathcal{F}_i = \mathcal{F}.$$ \end{lemma} \begin{proof} Choose a set $I$ and for each $i \in I$ an $\mathcal{O}_X$-module of finite presentation and a homomorphism of $\mathcal{O}_X$-modules $\varphi_i : \mathcal{F}_i \to \mathcal{F}$ with the following property: For any $\psi : \mathcal{G} \to \mathcal{F}$ with $\mathcal{G}$ of finite presentation there is an $i \in I$ such that there exists an isomorphism $\alpha : \mathcal{F}_i \to \mathcal{G}$ with $\varphi_i = \psi \circ \alpha$. It is clear from Modules, Lemma \ref{modules-lemma-set-isomorphism-classes-finite-type-modules} that such a set exists (see also its proof). We denote $\mathcal{I}$ the category with $\Ob(\mathcal{I}) = I$ and given $i, i' \in I$ we set $$\Mor_\mathcal{I}(i, i') = \{\alpha : \mathcal{F}_i \to \mathcal{F}_{i'} \mid \alpha \circ \varphi_{i'} = \varphi_i \}.$$ We claim that $\mathcal{I}$ is a filtered category and that $\mathcal{F} = \colim_i \mathcal{F}_i$. \medskip\noindent Let $i, i' \in I$. Then we can consider the morphism $$\mathcal{F}_i \oplus \mathcal{F}_{i'} \longrightarrow \mathcal{F}$$ which is the direct sum of $\varphi_i$ and $\varphi_{i'}$. Since a direct sum of finitely presented $\mathcal{O}_X$-modules is finitely presented we see that there exists some $i'' \in I$ such that $\varphi_{i''} : \mathcal{F}_{i''} \to \mathcal{F}$ is isomorphic to the displayed arrow towards $\mathcal{F}$ above. Since there are commutative diagrams $$\xymatrix{ \mathcal{F}_i \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \\ \mathcal{F}_i \oplus \mathcal{F}_{i'} \ar[r] & \mathcal{F} } \quad \text{and} \quad \xymatrix{ \mathcal{F}_{i'} \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \\ \mathcal{F}_i \oplus \mathcal{F}_{i'} \ar[r] & \mathcal{F} }$$ we see that there are morphisms $i \to i''$ and $i' \to i''$ in $\mathcal{I}$. Next, suppose that we have $i, i' \in I$ and morphisms $\alpha, \beta : i \to i'$ (corresponding to $\mathcal{O}_X$-module maps $\alpha, \beta : \mathcal{F}_i \to \mathcal{F}_{i'}$). In this case consider the coequalizer $$\mathcal{G} = \Coker( \mathcal{F}_i \xrightarrow{\alpha - \beta} \mathcal{F}_{i'} )$$ Note that $\mathcal{G}$ is an $\mathcal{O}_X$-module of finite presentation. Since by definition of morphisms in the category $\mathcal{I}$ we have $\varphi_{i'} \circ \alpha = \varphi_{i'} \circ \beta$ we see that we get an induced map $\psi : \mathcal{G} \to \mathcal{F}$. Hence again the pair $(\mathcal{G}, \psi)$ is isomorphic to the pair $(\mathcal{F}_{i''}, \varphi_{i''})$ for some $i''$. Hence we see that there exists a morphism $i' \to i''$ in $\mathcal{I}$ which equalizes $\alpha$ and $\beta$. Thus we have shown that the category $\mathcal{I}$ is filtered. \medskip\noindent We still have to show that the colimit of the diagram is $\mathcal{F}$. By definition of the colimit, and by our definition of the category $\mathcal{I}$ there is a canonical map $$\varphi : \colim_i \mathcal{F}_i \longrightarrow \mathcal{F}.$$ Pick $x \in X$. Let us show that $\varphi_x$ is an isomorphism. Recall that $$(\colim_i \mathcal{F}_i)_x = \colim_i \mathcal{F}_{i, x},$$ see Sheaves, Section \ref{sheaves-section-limits-sheaves}. First we show that the map $\varphi_x$ is injective. Suppose that $s \in \mathcal{F}_{i, x}$ is an element such that $s$ maps to zero in $\mathcal{F}_x$. Then there exists a quasi-compact open $U$ such that $s$ comes from $s \in \mathcal{F}_i(U)$ and such that $\varphi_i(s) = 0$ in $\mathcal{F}(U)$. By Lemma \ref{lemma-extend} we can find a finite type quasi-coherent subsheaf $\mathcal{K} \subset \Ker(\varphi_i)$ which restricts to the quasi-coherent $\mathcal{O}_U$-submodule of $\mathcal{F}_i$ generated by $s$: $\mathcal{K}|_U = \mathcal{O}_U\cdot s \subset \mathcal{F}_i|_U$. Clearly, $\mathcal{F}_i/\mathcal{K}$ is of finite presentation and the map $\varphi_i$ factors through the quotient map $\mathcal{F}_i \to \mathcal{F}_i/\mathcal{K}$. Hence we can find an $i' \in I$ and a morphism $\alpha : \mathcal{F}_i \to \mathcal{F}_{i'}$ in $\mathcal{I}$ which can be identified with the quotient map $\mathcal{F}_i \to \mathcal{F}_i/\mathcal{K}$. Then it follows that the section $s$ maps to zero in $\mathcal{F}_{i'}(U)$ and in particular in $(\colim_i \mathcal{F}_i)_x = \colim_i \mathcal{F}_{i, x}$. The injectivity follows. Finally, we show that the map $\varphi_x$ is surjective. Pick $s \in \mathcal{F}_x$. Choose a quasi-compact open neighbourhood $U \subset X$ of $x$ such that $s$ corresponds to a section $s \in \mathcal{F}(U)$. Consider the map $s : \mathcal{O}_U \to \mathcal{F}$ (multiplication by $s$). By Lemma \ref{lemma-extend-finite-presentation} there exists an $\mathcal{O}_X$-module $\mathcal{G}$ of finite presentation and an $\mathcal{O}_X$-module map $\mathcal{G} \to \mathcal{F}$ such that $\mathcal{G}|_U \to \mathcal{F}|_U$ is identified with $s : \mathcal{O}_U \to \mathcal{F}$. Again by definition of $\mathcal{I}$ there exists an $i \in I$ such that $\mathcal{G} \to \mathcal{F}$ is isomorphic to $\varphi_i : \mathcal{F}_i \to \mathcal{F}$. Clearly there exists a section $s' \in \mathcal{F}_i(U)$ mapping to $s \in \mathcal{F}(U)$. This proves surjectivity and the proof of the lemma is complete. \end{proof} \begin{lemma} \label{lemma-directed-colimit-finite-presentation} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. There exist \begin{enumerate} \item a directed set $I$ (see Categories, Definition \ref{categories-definition-directed-set}), \item a system $(\mathcal{F}_i, \varphi_{ii'})$ over $I$ in $\textit{Mod}(\mathcal{O}_X)$ (see Categories, Definition \ref{categories-definition-system-over-poset}) \item morphisms of $\mathcal{O}_X$-modules $\varphi_i : \mathcal{F}_i \to \mathcal{F}$ \end{enumerate} such that each $\mathcal{F}_i$ is of finite presentation and such that the morphisms $\varphi_i$ induce an isomorphism $$\colim_i \mathcal{F}_i = \mathcal{F}.$$ \end{lemma} \begin{proof} This is a direct consequence of Lemma \ref{lemma-directed-colimit-diagram-finite-presentation} and Categories, Lemma \ref{categories-lemma-directed-category-system} (combined with the fact that colimits exist in the category of sheaves of $\mathcal{O}_X$-modules, see Sheaves, Section \ref{sheaves-section-limits-sheaves}). \end{proof} \begin{lemma} \label{lemma-directed-colimit-finite-type} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is the directed colimit of its finite type quasi-coherent submodules. \end{lemma} \begin{proof} If $\mathcal{G}, \mathcal{H} \subset \mathcal{F}$ are finite type quasi-coherent $\mathcal{O}_X$-submodules then the image of $\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}$ is another finite type quasi-coherent $\mathcal{O}_X$-submodule which contains both of them. In this way we see that the system is directed. To show that $\mathcal{F}$ is the colimit of this system, write $\mathcal{F} = \colim_i \mathcal{F}_i$ as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma \ref{lemma-directed-colimit-finite-presentation}. Then the images $\mathcal{G}_i = \Im(\mathcal{F}_i \to \mathcal{F})$ are finite type quasi-coherent subsheaves of $\mathcal{F}$. Since $\mathcal{F}$ is the colimit of these the result follows. \end{proof} \begin{lemma} \label{lemma-finite-directed-colimit-surjective-maps} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Then we can write $\mathcal{F} = \lim \mathcal{F}_i$ with $\mathcal{F}_i$ of finite presentation and all transition maps $\mathcal{F}_i \to \mathcal{F}_{i'}$ surjective. \end{lemma} \begin{proof} Write $\mathcal{F} = \colim \mathcal{G}_i$ as a filtered colimit of finitely presented $\mathcal{O}_X$-modules (Lemma \ref{lemma-directed-colimit-finite-presentation}). We claim that $\mathcal{G}_i \to \mathcal{F}$ is surjective for some $i$. Namely, choose a finite affine open covering $X = U_1 \cup \ldots \cup U_m$. Choose sections $s_{jl} \in \mathcal{F}(U_j)$ generating $\mathcal{F}|_{U_j}$, see Lemma \ref{lemma-finite-type-module}. By Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections} we see that $s_{jl}$ is in the image of $\mathcal{G}_i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{G}_i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{G}_i$ be the kernel of the map $\mathcal{G}_i \to \mathcal{F}$. Write $\mathcal{K} = \colim \mathcal{K}_a$ as the filtered colimit of its finite type quasi-coherent submodules (Lemma \ref{lemma-directed-colimit-finite-type}). Then $\mathcal{F} = \colim \mathcal{G}_i/\mathcal{K}_a$ is a solution to the problem posed by the lemma. \end{proof} \begin{lemma} \label{lemma-application-directed-colimit} Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $U \subset X$ be a quasi-compact open such that $\mathcal{F}|_U$ is of finite presentation. Then there exists a map of $\mathcal{O}_X$-modules $\varphi : \mathcal{G} \to \mathcal{F}$ with (a) $\mathcal{G}$ of finite presentation, (b) $\varphi$ is surjective, and (c) $\varphi|_U$ is an isomorphism. \end{lemma} \begin{proof} Write $\mathcal{F} = \colim \mathcal{F}_i$ as a directed colimit with each $\mathcal{F}_i$ of finite presentation, see Lemma \ref{lemma-directed-colimit-finite-presentation}. Choose a finite affine open covering $X = \bigcup V_j$ and choose finitely many sections $s_{jl} \in \mathcal{F}(V_j)$ generating $\mathcal{F}|_{V_j}$, see Lemma \ref{lemma-finite-type-module}. By Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections} we see that $s_{jl}$ is in the image of $\mathcal{F}_i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{F}_i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{F}_i$ be the kernel of the map $\mathcal{F}_i \to \mathcal{F}$. Since $\mathcal{F}_U$ is of finite presentation, we see that $\mathcal{K}|_U$ is of finite type, see Modules, Lemma \ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation}. Hence we can find a finite type quasi-coherent submodule $\mathcal{K}' \subset \mathcal{K}$ with $\mathcal{K}'|_U = \mathcal{K}|_U$, see Lemma \ref{lemma-extend}. Then $\mathcal{G} = \mathcal{F}_i/\mathcal{K}'$ with the given map $\mathcal{G} \to \mathcal{F}$ is a solution. \end{proof} \noindent Let $X$ be a scheme. In the following lemma we use the notion of a {\it quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{A}$ of finite presentation}. This means that for every affine open $\Spec(R) \subset X$ we have $\mathcal{A} = \widetilde{A}$ where $A$ is a (commutative) $R$-algebra which is of finite presentation as an $R$-algebra. \begin{lemma} \label{lemma-algebra-directed-colimit-finite-presentation} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. There exist \begin{enumerate} \item a directed set $I$ (see Categories, Definition \ref{categories-definition-directed-set}), \item a system $(\mathcal{A}_i, \varphi_{ii'})$ over $I$ in the category of $\mathcal{O}_X$-algebras, \item morphisms of $\mathcal{O}_X$-algebras $\varphi_i : \mathcal{A}_i \to \mathcal{A}$ \end{enumerate} such that each $\mathcal{A}_i$ is a quasi-coherent $\mathcal{O}_X$-algebra of finite presentation and such that the morphisms $\varphi_i$ induce an isomorphism $$\colim_i \mathcal{A}_i = \mathcal{A}.$$ \end{lemma} \begin{proof} First we write $\mathcal{A} = \colim_i \mathcal{F}_i$ as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma \ref{lemma-directed-colimit-finite-presentation}. For each $i$ let $\mathcal{B}_i = \text{Sym}(\mathcal{F}_i)$ be the symmetric algebra on $\mathcal{F}_i$ over $\mathcal{O}_X$. Write $\mathcal{I}_i = \Ker(\mathcal{B}_i \to \mathcal{A})$. Write $\mathcal{I}_i = \colim_j \mathcal{F}_{i, j}$ where $\mathcal{F}_{i, j}$ is a finite type quasi-coherent submodule of $\mathcal{I}_i$, see Lemma \ref{lemma-directed-colimit-finite-type}. Set $\mathcal{I}_{i, j} \subset \mathcal{I}_i$ equal to the $\mathcal{B}_i$-ideal generated by $\mathcal{F}_{i, j}$. Set $\mathcal{A}_{i, j} = \mathcal{B}_i/\mathcal{I}_{i, j}$. Then $\mathcal{A}_{i, j}$ is a quasi-coherent finitely presented $\mathcal{O}_X$-algebra. Define $(i, j) \leq (i', j')$ if $i \leq i'$ and the map $\mathcal{B}_i \to \mathcal{B}_{i'}$ maps the ideal $\mathcal{I}_{i, j}$ into the ideal $\mathcal{I}_{i', j'}$. Then it is clear that $\mathcal{A} = \colim_{i, j} \mathcal{A}_{i, j}$. \end{proof} \noindent Let $X$ be a scheme. In the following lemma we use the notion of a {\it quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{A}$ of finite type}. This means that for every affine open $\Spec(R) \subset X$ we have $\mathcal{A} = \widetilde{A}$ where $A$ is a (commutative) $R$-algebra which is of finite type as an $R$-algebra. \begin{lemma} \label{lemma-algebra-directed-colimit-finite-type} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A}$ is the directed colimit of its finite type quasi-coherent $\mathcal{O}_X$-subalgebras. \end{lemma} \begin{proof} Omitted. Hint: Compare with the proof of Lemma \ref{lemma-directed-colimit-finite-type}. \end{proof} \noindent Let $X$ be a scheme. In the following lemma we use the notion of a {\it finite (resp.\ integral) quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{A}$}. This means that for every affine open $\Spec(R) \subset X$ we have $\mathcal{A} = \widetilde{A}$ where $A$ is a (commutative) $R$-algebra which is finite (resp.\ integral) as an $R$-algebra. \begin{lemma} \label{lemma-finite-algebra-directed-colimit-finite-finitely-presented} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be a finite quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A} = \colim \mathcal{A}_i$ is a directed colimit of finite and finitely presented quasi-coherent $\mathcal{O}_X$-algebras such that all transition maps $\mathcal{A}_{i'} \to \mathcal{A}_i$ are surjective. \end{lemma} \begin{proof} By Lemma \ref{lemma-finite-directed-colimit-surjective-maps} there exists a finitely presented $\mathcal{O}_X$-module $\mathcal{F}$ and a surjection $\mathcal{F} \to \mathcal{A}$. Using the algebra structure we obtain a surjection $$\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}) \longrightarrow \mathcal{A}$$ Denote $\mathcal{J}$ the kernel. Write $\mathcal{J} = \colim \mathcal{E}_i$ as a filtered colimit of finite type $\mathcal{O}_X$-submodules $\mathcal{E}_i$ (Lemma \ref{lemma-directed-colimit-finite-type}). Set $$\mathcal{A}_i = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})/(\mathcal{E}_i)$$ where $(\mathcal{E}_i)$ indicates the ideal sheaf generated by the image of $\mathcal{E}_i \to \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})$. Then each $\mathcal{A}_i$ is a finitely presented $\mathcal{O}_X$-algebra, the transition maps are surjections, and $\mathcal{A} = \colim \mathcal{A}_i$. To finish the proof we still have to show that $\mathcal{A}_i$ is a finite $\mathcal{O}_X$-algebra for $i$ sufficiently large. To do this we choose an affine open covering $X = U_1 \cup \ldots \cup U_m$. Take generators $f_{j, 1}, \ldots, f_{j, N_j} \in \Gamma(U_i, \mathcal{F})$. As $\mathcal{A}(U_j)$ is a finite $\mathcal{O}_X(U_j)$-algebra we see that for each $k$ there exists a monic polynomial $P_{j, k} \in \mathcal{O}(U_j)[T]$ such that $P_{j, k}(f_{j, k})$ is zero in $\mathcal{A}(U_j)$. Since $\mathcal{A} = \colim \mathcal{A}_i$ by construction, we have $P_{j, k}(f_{j, k}) = 0$ in $\mathcal{A}_i(U_j)$ for all sufficiently large $i$. For such $i$ the algebras $\mathcal{A}_i$ are finite. \end{proof} \begin{lemma} \label{lemma-integral-algebra-directed-colimit-finite} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be an integral quasi-coherent $\mathcal{O}_X$-algebra. Then \begin{enumerate} \item $\mathcal{A}$ is the directed colimit of its finite quasi-coherent $\mathcal{O}_X$-subalgebras, and \item $\mathcal{A}$ is a direct colimit of finite and finitely presented quasi-coherent $\mathcal{O}_X$-algebras. \end{enumerate} \end{lemma} \begin{proof} By Lemma \ref{lemma-algebra-directed-colimit-finite-type} we have $\mathcal{A} = \colim \mathcal{A}_i$ where $\mathcal{A}_i \subset \mathcal{A}$ runs through the quasi-coherent $\mathcal{O}_X$-algebras of finite type. Any finite type quasi-coherent $\mathcal{O}_X$-subalgebra of $\mathcal{A}$ is finite (apply Algebra, Lemma \ref{algebra-lemma-characterize-finite-in-terms-of-integral} to $\mathcal{A}_i(U) \subset \mathcal{A}(U)$ for affine opens $U$ in $X$). This proves (1). \medskip\noindent To prove (2), write $\mathcal{A} = \colim \mathcal{F}_i$ as a colimit of finitely presented $\mathcal{O}_X$-modules using Lemma \ref{lemma-directed-colimit-finite-presentation}. For each $i$, let $\mathcal{J}_i$ be the kernel of the map $$\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i) \longrightarrow \mathcal{A}$$ For $i' \geq i$ there is an induced map $\mathcal{J}_i \to \mathcal{J}_{i'}$ and we have $\mathcal{A} = \colim \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i$. Moreover, the quasi-coherent $\mathcal{O}_X$-algebras $\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i$ are finite (see above). Write $\mathcal{J}_i = \colim \mathcal{E}_{ik}$ as a colimit of finitely presented $\mathcal{O}_X$-modules. Given $i' \geq i$ and $k$ there exists a $k'$ such that we have a map $\mathcal{E}_{ik} \to \mathcal{E}_{i'k'}$ making $$\xymatrix{ \mathcal{J}_i \ar[r] & \mathcal{J}_{i'} \\ \mathcal{E}_{ik} \ar[u] \ar[r] & \mathcal{E}_{i'k'} \ar[u] }$$ commute. This follows from Modules, Lemma \ref{modules-lemma-finite-presentation-quasi-compact-colimit}. This induces a map $$\mathcal{A}_{ik} = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/(\mathcal{E}_{ik}) \longrightarrow \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_{i'})/(\mathcal{E}_{i'k'}) = \mathcal{A}_{i'k'}$$ where $(\mathcal{E}_{ik})$ denotes the ideal generated by $\mathcal{E}_{ik}$. The quasi-coherent $\mathcal{O}_X$-algebras $\mathcal{A}_{ki}$ are of finite presentation and finite for $k$ large enough (see proof of Lemma \ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented}). Finally, we have $$\colim \mathcal{A}_{ik} = \colim \mathcal{A}_i = \mathcal{A}$$ Namely, the first equality was shown in the proof of Lemma \ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented} and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_i$. \end{proof} \section{Gabber's result} \label{section-gabber} \noindent In this section we prove a result of Gabber which guarantees that on every scheme there exists a cardinal $\kappa$ such that every quasi-coherent module $\mathcal{F}$ is the union of its quasi-coherent $\kappa$-generated subsheaves. It follows that the category of quasi-coherent sheaves on a scheme is a Grothendieck abelian category having limits and enough injectives\footnote{Nicely explained in a \href{http://amathew.wordpress.com/2011/07/30/quasi-coherent-sheaves-presentable-categories-and-a-result-of-gabber/}{blog post} by Akhil Mathew.}. \begin{definition} \label{definition-kappa-generated} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\kappa$ be an infinite cardinal. We say a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is {\it $\kappa$-generated} if there exists an open covering $X = \bigcup U_i$ such that $\mathcal{F}|_{U_i}$ is generated by a subset $R_i \subset \mathcal{F}(U_i)$ whose cardinality is at most $\kappa$. \end{definition} \noindent Note that a direct sum of at most $\kappa$ $\kappa$-generated modules is again $\kappa$-generated because $\kappa \otimes \kappa = \kappa$, see Sets, Section \ref{sets-section-cardinals}. In particular this holds for the direct sum of two $\kappa$-generated modules. Moreover, a quotient of a $\kappa$-generated sheaf is $\kappa$-generated. (But the same needn't be true for submodules.) \begin{lemma} \label{lemma-set-of-iso-classes} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\kappa$ be a cardinal. There exists a set $T$ and a family $(\mathcal{F}_t)_{t \in T}$ of $\kappa$-generated $\mathcal{O}_X$-modules such that every $\kappa$-generated $\mathcal{O}_X$-module is isomorphic to one of the $\mathcal{F}_t$. \end{lemma} \begin{proof} There is a set of coverings of $X$ (provided we disallow repeats). Suppose $X = \bigcup U_i$ is a covering and suppose $\mathcal{F}_i$ is an $\mathcal{O}_{U_i}$-module. Then there is a set of isomorphism classes of $\mathcal{O}_X$-modules $\mathcal{F}$ with the property that $\mathcal{F}|_{U_i} \cong \mathcal{F}_i$ since there is a set of glueing maps. This reduces us to proving there is a set of (isomorphism classes of) quotients $\oplus_{k \in \kappa} \mathcal{O}_X \to \mathcal{F}$ for any ringed space $X$. This is clear. \end{proof} \noindent Here is the result the title of this section refers to. \begin{lemma} \label{lemma-colimit-kappa} Let $X$ be a scheme. There exists a cardinal $\kappa$ such that every quasi-coherent module $\mathcal{F}$ is the directed colimit of its quasi-coherent $\kappa$-generated quasi-coherent subsheaves. \end{lemma} \begin{proof} Choose an affine open covering $X = \bigcup_{i \in I} U_i$. For each pair $i, j$ choose an affine open covering $U_i \cap U_j = \bigcup_{k \in I_{ij}} U_{ijk}$. Write $U_i = \Spec(A_i)$ and $U_{ijk} = \Spec(A_{ijk})$. Let $\kappa$ be any infinite cardinal $\geq$ than the cardinality of any of the sets $I$, $I_{ij}$. \medskip\noindent Let $\mathcal{F}$ be a quasi-coherent sheaf. Set $M_i = \mathcal{F}(U_i)$ and $M_{ijk} = \mathcal{F}(U_{ijk})$. Note that $$M_i \otimes_{A_i} A_{ijk} = M_{ijk} = M_j \otimes_{A_j} A_{ijk}.$$ see Schemes, Lemma \ref{schemes-lemma-widetilde-pullback}. Using the axiom of choice we choose a map $$(i, j, k, m) \mapsto S(i, j, k, m)$$ which associates to every $i, j \in I$, $k \in I_{ij}$ and $m \in M_i$ a finite subset $S(i, j, k, m) \subset M_j$ such that we have $$m \otimes 1 = \sum\nolimits_{m' \in S(i, j, k, m)} m' \otimes a_{m'}$$ in $M_{ijk}$ for some $a_{m'} \in A_{ijk}$. Moreover, let's agree that $S(i, i, k, m) = \{m\}$ for all $i, j = i, k, m$ as above. Fix such a map. \medskip\noindent Given a family $\mathcal{S} = (S_i)_{i \in I}$ of subsets $S_i \subset M_i$ of cardinality at most $\kappa$ we set $\mathcal{S}' = (S'_i)$ where $$S'_j = \bigcup\nolimits_{(i, j, k, m)\text{ such that }m \in S_i} S(i, j, k, m)$$ Note that $S_i \subset S'_i$. Note that $S'_i$ has cardinality at most $\kappa$ because it is a union over a set of cardinality at most $\kappa$ of finite sets. Set $\mathcal{S}^{(0)} = \mathcal{S}$, $\mathcal{S}^{(1)} = \mathcal{S}'$ and by induction $\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'$. Then set $\mathcal{S}^{(\infty)} = \bigcup_{n \geq 0} \mathcal{S}^{(n)}$. Writing $\mathcal{S}^{(\infty)} = (S^{(\infty)}_i)$ we see that for any element $m \in S^{(\infty)}_i$ the image of $m$ in $M_{ijk}$ can be written as a finite sum $\sum m' \otimes a_{m'}$ with $m' \in S_j^{(\infty)}$. In this way we see that setting $$N_i = A_i\text{-submodule of }M_i\text{ generated by }S^{(\infty)}_i$$ we have $$N_i \otimes_{A_i} A_{ijk} = N_j \otimes_{A_j} A_{ijk}.$$ as submodules of $M_{ijk}$. Thus there exists a quasi-coherent subsheaf $\mathcal{G} \subset \mathcal{F}$ with $\mathcal{G}(U_i) = N_i$. Moreover, by construction the sheaf $\mathcal{G}$ is $\kappa$-generated. \medskip\noindent Let $\{\mathcal{G}_t\}_{t \in T}$ be the set of $\kappa$-generated quasi-coherent subsheaves. If $t, t' \in T$ then $\mathcal{G}_t + \mathcal{G}_{t'}$ is also a $\kappa$-generated quasi-coherent subsheaf as it is the image of the map $\mathcal{G}_t \oplus \mathcal{G}_{t'} \to \mathcal{F}$. Hence the system (ordered by inclusion) is directed. The arguments above show that every section of $\mathcal{F}$ over $U_i$ is in one of the $\mathcal{G}_t$ (because we can start with $\mathcal{S}$ such that the given section is an element of $S_i$). Hence $\colim_t \mathcal{G}_t \to \mathcal{F}$ is both injective and surjective as desired. \end{proof} \begin{proposition} \label{proposition-coherator} Let $X$ be a scheme. \begin{enumerate} \item The category $\QCoh(\mathcal{O}_X)$ is a Grothendieck abelian category. Consequently, $\QCoh(\mathcal{O}_X)$ has enough injectives and all limits. \item The inclusion functor $\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$ has a right adjoint\footnote{This functor is sometimes called the {\it coherator}.} $$Q : \textit{Mod}(\mathcal{O}_X) \longrightarrow \QCoh(\mathcal{O}_X)$$ such that for every quasi-coherent sheaf $\mathcal{F}$ the adjunction mapping $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism. \end{enumerate} \end{proposition} \begin{proof} Part (1) means $\QCoh(\mathcal{O}_X)$ (a) has all colimits, (b) filtered colimits are exact, and (c) has a generator, see Injectives, Section \ref{injectives-section-grothendieck-conditions}. By Schemes, Section \ref{schemes-section-quasi-coherent} colimits in $\QCoh(\mathcal{O}_X)$ exist and agree with colimits in $\textit{Mod}(\mathcal{O}_X)$. By Modules, Lemma \ref{modules-lemma-limits-colimits} filtered colimits are exact. Hence (a) and (b) hold. To construct a generator $U$, pick a cardinal $\kappa$ as in Lemma \ref{lemma-colimit-kappa}. Pick a collection $(\mathcal{F}_t)_{t \in T}$ of $\kappa$-generated quasi-coherent sheaves as in Lemma \ref{lemma-set-of-iso-classes}. Set $U = \bigoplus_{t \in T} \mathcal{F}_t$. Since every object of $\QCoh(\mathcal{O}_X)$ is a filtered colimit of $\kappa$-generated quasi-coherent modules, i.e., of objects isomorphic to $\mathcal{F}_t$, it is clear that $U$ is a generator. The assertions on limits and injectives hold in any Grothendieck abelian category, see Injectives, Theorem \ref{injectives-theorem-injective-embedding-grothendieck} and Lemma \ref{injectives-lemma-grothendieck-products}. \medskip\noindent Proof of (2). To construct $Q$ we use the following general procedure. Given an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_X)$ we consider the functor $$\QCoh(\mathcal{O}_X)^{opp} \longrightarrow \textit{Sets},\quad \mathcal{G} \longmapsto \Hom_X(\mathcal{G}, \mathcal{F})$$ This functor transforms colimits into limits, hence is representable, see Injectives, Lemma \ref{injectives-lemma-grothendieck-brown}. Thus there exists a quasi-coherent sheaf $Q(\mathcal{F})$ and a functorial isomorphism $\Hom_X(\mathcal{G}, \mathcal{F}) = \Hom_X(\mathcal{G}, Q(\mathcal{F}))$ for $\mathcal{G}$ in $\QCoh(\mathcal{O}_X)$. By the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}) the construction $\mathcal{F} \leadsto Q(\mathcal{F})$ is functorial in $\mathcal{F}$. By construction $Q$ is a right adjoint to the inclusion functor. The fact that $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism when $\mathcal{F}$ is quasi-coherent is a formal consequence of the fact that the inclusion functor $\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$ is fully faithful. \end{proof} \section{Sections with support in a closed subset} \label{section-sections-with-support-in-closed} \noindent Given any topological space $X$, a closed subset $Z \subset X$, and an abelian sheaf $\mathcal{F}$ you can take the subsheaf of sections whose support is contained in $Z$. If $X$ is a scheme, $Z$ a closed subscheme, and $\mathcal{F}$ a quasi-coherent module there is a variant where you take sections which are scheme theoretically supported on $Z$. However, in the scheme setting you have to be careful because the resulting $\mathcal{O}_X$-module may not be quasi-coherent. \begin{lemma} \label{lemma-quasi-coherent-finite-type-ideals} Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be an open subscheme. The following are equivalent: \begin{enumerate} \item $U$ is retrocompact in $X$, \item $U$ is quasi-compact, \item $U$ is a finite union of affine opens, and \item there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ such that $X \setminus U = V(\mathcal{I})$ (set theoretically). \end{enumerate} \end{lemma} \begin{proof} The equivalence of (1), (2), and (3) follows from Lemma \ref{lemma-quasi-separated-quasi-compact-open-retrocompact}. Assume (1), (2), (3). Let $T = X \setminus U$. By Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme} there exists a unique quasi-coherent sheaf of ideals $\mathcal{J}$ cutting out the reduced induced closed subscheme structure on $T$. Note that $\mathcal{J}|_U = \mathcal{O}_U$ which is an $\mathcal{O}_U$-modules of finite type. By Lemma \ref{lemma-extend} there exists a quasi-coherent subsheaf $\mathcal{I} \subset \mathcal{J}$ which is of finite type and has the property that $\mathcal{I}|_U = \mathcal{J}|_U$. Then $X \setminus U = V(\mathcal{I})$ and we obtain (4). Conversely, if $\mathcal{I}$ is as in (4) and $W = \Spec(R) \subset X$ is an affine open, then $\mathcal{I}|_W = \widetilde{I}$ for some finitely generated ideal $I \subset R$, see Lemma \ref{lemma-finite-type-module}. It follows that $U \cap W = \Spec(R) \setminus V(I)$ is quasi-compact, see Algebra, Lemma \ref{algebra-lemma-qc-open}. Hence $U \subset X$ is retrocompact by Lemma \ref{lemma-retrocompact}. \end{proof} \begin{lemma} \label{lemma-sections-annihilated-by-ideal} Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider the sheaf of $\mathcal{O}_X$-modules $\mathcal{F}'$ which associates to every open $U \subset X$ $$\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}s = 0\}$$ Assume $\mathcal{I}$ is of finite type. Then \begin{enumerate} \item $\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules, \item on any affine open $U \subset X$ we have $\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}(U)s = 0\}$, and \item $\mathcal{F}'_x = \{s \in \mathcal{F}_x \mid \mathcal{I}_x s = 0\}$. \end{enumerate} \end{lemma} \begin{proof} It is clear that the rule defining $\mathcal{F}'$ gives a subsheaf of $\mathcal{F}$ (the sheaf condition is easy to verify). Hence we may work locally on $X$ to verify the other statements. In other words we may assume that $X = \Spec(A)$, $\mathcal{F} = \widetilde{M}$ and $\mathcal{I} = \widetilde{I}$. It is clear that in this case $\mathcal{F}'(U) = \{x \in M \mid Ix = 0\} =: M'$ because $\widetilde{I}$ is generated by its global sections $I$ which proves (2). To show $\mathcal{F}'$ is quasi-coherent it suffices to show that for every $f \in A$ we have $\{x \in M_f \mid I_f x = 0\} = (M')_f$. Write $I = (g_1, \ldots, g_t)$, which is possible because $\mathcal{I}$ is of finite type, see Lemma \ref{lemma-finite-type-module}. If $x = y/f^n$ and $I_fx = 0$, then that means that for every $i$ there exists an $m \geq 0$ such that $f^mg_ix = 0$. We may choose one $m$ which works for all $i$ (and this is where we use that $I$ is finitely generated). Then we see that $f^mx \in M'$ and $x/f^n = f^mx/f^{n + m}$ in $(M')_f$ as desired. The proof of (3) is similar and omitted. \end{proof} \begin{definition} \label{definition-subsheaf-sections-annihilated-by-ideal} Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma \ref{lemma-sections-annihilated-by-ideal} above is called the {\it subsheaf of sections annihilated by $\mathcal{I}$}. \end{definition} \begin{lemma} \label{lemma-push-sections-annihilated-by-ideal} Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{I} \subset \mathcal{O}_Y$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections annihilated by $f^{-1}\mathcal{I}\mathcal{O}_X$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections annihilated by $\mathcal{I}$. \end{lemma} \begin{proof} Omitted. (Hint: The assumption that $f$ is quasi-compact and quasi-separated implies that $f_*\mathcal{F}$ is quasi-coherent so that Lemma \ref{lemma-sections-annihilated-by-ideal} applies to $\mathcal{I}$ and $f_*\mathcal{F}$.) \end{proof} \noindent For an abelian sheaf on a topological space we have discussed the subsheaf of sections with support in a closed subset in Modules, Lemma \ref{modules-lemma-sections-support-in-closed}. For quasi-coherent modules this submodule isn't always a quasi-coherent module, but if the closed subset has a retrocompact complement, then it is. \begin{lemma} \label{lemma-sections-supported-on-closed-subset} Let $X$ be a scheme. Let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider the sheaf of $\mathcal{O}_X$-modules $\mathcal{F}'$ which associates to every open $U \subset X$  \mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \text{the support