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\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