Permalink
Find file
Fetching contributors…
Cannot retrieve contributors at this time
8215 lines (7393 sloc) 315 KB
\input{preamble}
% OK, start here.
%
\begin{document}
\title{Divisors}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
In this chapter we study some very basic questions related
to defining divisors, etc. A basic reference is \cite{EGA}.
\section{Associated points}
\label{section-associated}
\noindent
Let $R$ be a ring and let $M$ be an $R$-module.
Recall that a prime $\mathfrak p \subset R$ is {\it associated} to $M$
if there exists an element of $M$ whose annihilator is $\mathfrak p$.
See Algebra, Definition \ref{algebra-definition-associated}.
Here is the definition of associated points
for quasi-coherent sheaves on schemes
as given in \cite[IV Definition 3.1.1]{EGA}.
\begin{definition}
\label{definition-associated}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
\begin{enumerate}
\item We say $x \in X$ is {\it associated} to $\mathcal{F}$
if the maximal ideal
$\mathfrak m_x$ is associated to the $\mathcal{O}_{X, x}$-module
$\mathcal{F}_x$.
\item We denote $\text{Ass}(\mathcal{F})$ or $\text{Ass}_X(\mathcal{F})$
the set of associated points of $\mathcal{F}$.
\item The {\it associated points of $X$} are the associated
points of $\mathcal{O}_X$.
\end{enumerate}
\end{definition}
\noindent
These definitions are most useful when $X$ is locally Noetherian
and $\mathcal{F}$ of finite type.
For example it may happen that a generic point of an irreducible
component of $X$ is not associated to $X$, see
Example \ref{example-no-associated-prime}.
In the non-Noetherian case it may be more convenient to use weakly
associated points, see
Section \ref{section-weakly-associated}.
Let us link the scheme theoretic notion with the algebraic notion
on affine opens; note that this correspondence works perfectly only
for locally Noetherian schemes.
\begin{lemma}
\label{lemma-associated-affine-open}
Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
\begin{enumerate}
\item If $\mathfrak p$ is associated to $M$, then $x$ is associated
to $\mathcal{F}$.
\item If $\mathfrak p$ is finitely generated, then the converse holds
as well.
\end{enumerate}
In particular, if $X$ is locally Noetherian, then the equivalence
$$
\mathfrak p \in \text{Ass}(M) \Leftrightarrow x \in \text{Ass}(\mathcal{F})
$$
holds for all pairs $(\mathfrak p, x)$ as above.
\end{lemma}
\begin{proof}
This follows from
Algebra, Lemma \ref{algebra-lemma-associated-primes-localize}.
But we can also argue directly as follows.
Suppose $\mathfrak p$ is associated to $M$.
Then there exists an $m \in M$ whose annihilator is $\mathfrak p$.
Since localization is exact we see that
$\mathfrak pA_{\mathfrak p}$ is the annihilator of
$m/1 \in M_{\mathfrak p}$. Since $M_{\mathfrak p} = \mathcal{F}_x$
(Schemes, Lemma \ref{schemes-lemma-spec-sheaves})
we conclude that $x$ is associated to $\mathcal{F}$.
\medskip\noindent
Conversely, assume that $x$ is associated to $\mathcal{F}$,
and $\mathfrak p$ is finitely generated.
As $x$ is associated to $\mathcal{F}$
there exists an element $m' \in M_{\mathfrak p}$ whose
annihilator is $\mathfrak pA_{\mathfrak p}$. Write
$m' = m/f$ for some $f \in A$, $f \not \in \mathfrak p$.
The annihilator $I$ of $m$ is an ideal of $A$ such that
$IA_{\mathfrak p} = \mathfrak pA_{\mathfrak p}$. Hence
$I \subset \mathfrak p$, and $(\mathfrak p/I)_{\mathfrak p} = 0$.
Since $\mathfrak p$ is finitely generated,
there exists a $g \in A$, $g \not \in \mathfrak p$ such that
$g(\mathfrak p/I) = 0$. Hence the annihilator of $gm$ is
$\mathfrak p$ and we win.
\medskip\noindent
If $X$ is locally Noetherian, then $A$ is Noetherian
(Properties, Lemma \ref{properties-lemma-locally-Noetherian})
and $\mathfrak p$ is always finitely generated.
\end{proof}
\begin{lemma}
\label{lemma-ass-support}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Then $\text{Ass}(\mathcal{F}) \subset \text{Supp}(\mathcal{F})$.
\end{lemma}
\begin{proof}
This is immediate from the definitions.
\end{proof}
\begin{lemma}
\label{lemma-ses-ass}
Let $X$ be a scheme.
Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$
be a short exact sequence of quasi-coherent sheaves on $X$.
Then
$\text{Ass}(\mathcal{F}_2) \subset
\text{Ass}(\mathcal{F}_1) \cup \text{Ass}(\mathcal{F}_3)$
and
$\text{Ass}(\mathcal{F}_1) \subset \text{Ass}(\mathcal{F}_2)$.
\end{lemma}
\begin{proof}
For every point $x \in X$ the sequence of stalks
$0 \to \mathcal{F}_{1, x} \to \mathcal{F}_{2, x} \to \mathcal{F}_{3, x} \to 0$
is a short exact sequence of $\mathcal{O}_{X, x}$-modules.
Hence the lemma follows from
Algebra, Lemma \ref{algebra-lemma-ass}.
\end{proof}
\begin{lemma}
\label{lemma-finite-ass}
Let $X$ be a locally Noetherian scheme.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
Then $\text{Ass}(\mathcal{F}) \cap U$ is finite for
every quasi-compact open $U \subset X$.
\end{lemma}
\begin{proof}
This is true because the set of associated primes of a finite module over
a Noetherian ring is finite, see
Algebra, Lemma \ref{algebra-lemma-finite-ass}.
To translate from schemes to algebra use that $U$ is a finite union of
affine opens, each of these opens is the spectrum of a Noetherian ring
(Properties, Lemma \ref{properties-lemma-locally-Noetherian}),
$\mathcal{F}$ corresponds to a finite module over this ring
(Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}),
and finally use
Lemma \ref{lemma-associated-affine-open}.
\end{proof}
\begin{lemma}
\label{lemma-ass-zero}
Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a
quasi-coherent $\mathcal{O}_X$-module. Then
$$
\mathcal{F} = 0 \Leftrightarrow \text{Ass}(\mathcal{F}) = \emptyset.
$$
\end{lemma}
\begin{proof}
If $\mathcal{F} = 0$, then $\text{Ass}(\mathcal{F}) = \emptyset$
by definition. Conversely, if $\text{Ass}(\mathcal{F}) = \emptyset$,
then $\mathcal{F} = 0$ by
Algebra, Lemma \ref{algebra-lemma-ass-zero}.
To translate from schemes to algebra, restrict to any affine and use
Lemma \ref{lemma-associated-affine-open}.
\end{proof}
\begin{example}
\label{example-no-associated-prime}
Let $k$ be a field. The ring $R = k[x_1, x_2, x_3, \ldots]/(x_i^2)$
is local with locally nilpotent maximal ideal $\mathfrak m$.
There exists no element of $R$ which has annihilator $\mathfrak m$.
Hence $\text{Ass}(R) = \emptyset$, and $X = \Spec(R)$
is an example of a scheme which has no associated points.
\end{example}
\begin{lemma}
\label{lemma-restriction-injective-open-contains-ass}
Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. If $\text{Ass}(\mathcal{F}) \subset U \subset X$
is open, then $\Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F})$
is injective.
\end{lemma}
\begin{proof}
Let $s \in \Gamma(X, \mathcal{F})$ be a section which restricts to zero on $U$.
Let $\mathcal{F}' \subset \mathcal{F}$ be the image of the map
$\mathcal{O}_X \to \mathcal{F}$ defined by $s$. Then
$\text{Supp}(\mathcal{F}') \cap U = \emptyset$. On the other hand,
$\text{Ass}(\mathcal{F}') \subset \text{Ass}(\mathcal{F})$
by Lemma \ref{lemma-ses-ass}. Since also
$\text{Ass}(\mathcal{F}') \subset \text{Supp}(\mathcal{F}')$
(Lemma \ref{lemma-ass-support}) we conclude
$\text{Ass}(\mathcal{F}') = \emptyset$.
Hence $\mathcal{F}' = 0$ by Lemma \ref{lemma-ass-zero}.
\end{proof}
\begin{lemma}
\label{lemma-minimal-support-in-ass}
Let $X$ be a locally Noetherian scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $x \in \text{Supp}(\mathcal{F})$ be a point in the support
of $\mathcal{F}$ which is not a specialization of another point of
$\text{Supp}(\mathcal{F})$. Then $x \in \text{Ass}(\mathcal{F})$.
In particular, any generic point of an irreducible component of $X$
is an associated point of $X$.
\end{lemma}
\begin{proof}
Since $x \in \text{Supp}(\mathcal{F})$ the module $\mathcal{F}_x$
is not zero. Hence
$\text{Ass}(\mathcal{F}_x) \subset \Spec(\mathcal{O}_{X, x})$
is nonempty by
Algebra, Lemma \ref{algebra-lemma-ass-zero}.
On the other hand, by assumption
$\text{Supp}(\mathcal{F}_x) = \{\mathfrak m_x\}$.
Since
$\text{Ass}(\mathcal{F}_x) \subset \text{Supp}(\mathcal{F}_x)$
(Algebra, Lemma \ref{algebra-lemma-ass-support})
we see that $\mathfrak m_x$ is associated to $\mathcal{F}_x$
and we win.
\end{proof}
\noindent
The following lemma is the analogue of
More on Algebra, Lemma \ref{more-algebra-lemma-check-injective-on-ass}.
\begin{lemma}
\label{lemma-check-injective-on-ass}
Let $X$ be a locally Noetherian scheme. Let
$\varphi : \mathcal{F} \to \mathcal{G}$ be a map of
quasi-coherent $\mathcal{O}_X$-modules.
Assume that for every $x \in X$
at least one of the following happens
\begin{enumerate}
\item $\mathcal{F}_x \to \mathcal{G}_x$ is injective, or
\item $x \not \in \text{Ass}(\mathcal{F})$.
\end{enumerate}
Then $\varphi$ is injective.
\end{lemma}
\begin{proof}
The assumptions imply that $\text{WeakAss}(\Ker(\varphi)) = \emptyset$
and hence $\Ker(\varphi) = 0$ by Lemma \ref{lemma-ass-zero}.
\end{proof}
\begin{lemma}
\label{lemma-check-isomorphism-via-depth-and-ass}
Let $X$ be a locally Noetherian scheme. Let
$\varphi : \mathcal{F} \to \mathcal{G}$ be a map of
quasi-coherent $\mathcal{O}_X$-modules. Assume $\mathcal{F}$ is coherent
and that for every $x \in X$ one of the following happens
\begin{enumerate}
\item $\mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, or
\item $\text{depth}(\mathcal{F}_x) \geq 2$ and
$x \not \in \text{Ass}(\mathcal{G})$.
\end{enumerate}
Then $\varphi$ is an isomorphism.
\end{lemma}
\begin{proof}
This is a translation of More on Algebra, Lemma
\ref{more-algebra-lemma-check-isomorphism-via-depth-and-ass}
into the language of schemes.
\end{proof}
\section{Morphisms and associated points}
\label{section-morphisms-associated}
\begin{lemma}
\label{lemma-bourbaki}
Let $f : X \to S$ be a morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$ which is flat over $S$.
Let $\mathcal{G}$ be a quasi-coherent sheaf on $S$.
Then we have
$$
\text{Ass}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})
\supset
\bigcup\nolimits_{s \in \text{Ass}_S(\mathcal{G})}
\text{Ass}_{X_s}(\mathcal{F}_s)
$$
and equality holds if $S$ is locally Noetherian.
\end{lemma}
\begin{proof}
Let $x \in X$ and let $s = f(x) \in S$.
Set $B = \mathcal{O}_{X, x}$, $A = \mathcal{O}_{S, s}$,
$N = \mathcal{F}_x$, and $M = \mathcal{G}_s$.
Note that the stalk of $\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}$
at $x$ is equal to the $B$-module $M \otimes_A N$. Hence
$x \in \text{Ass}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})$
if and only if $\mathfrak m_B$ is in $\text{Ass}_B(M \otimes_A N)$.
Similarly $s \in \text{Ass}_S(\mathcal{G})$ and
$x \in \text{Ass}_{X_s}(\mathcal{F}_s)$ if and only if
$\mathfrak m_A \in \text{Ass}_A(M)$ and
$\mathfrak m_B/\mathfrak m_A B \in
\text{Ass}_{B \otimes \kappa(\mathfrak m_A)}(N \otimes \kappa(\mathfrak m_A))$.
Thus the lemma follows from
Algebra, Lemma \ref{algebra-lemma-bourbaki-fibres}.
\end{proof}
\section{Embedded points}
\label{section-embedded}
\noindent
Let $R$ be a ring and let $M$ be an $R$-module.
Recall that a prime $\mathfrak p \subset R$ is an
{\it embedded associated prime} of $M$ if it is an associated prime of
$M$ which is not minimal among the associated primes of $M$. See
Algebra, Definition \ref{algebra-definition-embedded-primes}.
Here is the definition of embedded associated points
for quasi-coherent sheaves on schemes
as given in \cite[IV Definition 3.1.1]{EGA}.
\begin{definition}
\label{definition-embedded}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
\begin{enumerate}
\item An {\it embedded associated point} of $\mathcal{F}$
is an associated point which is not maximal among the
associated points of $\mathcal{F}$, i.e., it is the specialization
of another associated point of $\mathcal{F}$.
\item A point $x$ of $X$ is called an {\it embedded point}
if $x$ is an embedded associated point of $\mathcal{O}_X$.
\item An {\it embedded component} of $X$ is an irreducible
closed subset $Z = \overline{\{x\}}$ where $x$ is an embedded
point of $X$.
\end{enumerate}
\end{definition}
\noindent
In the Noetherian case when $\mathcal{F}$ is coherent we have
the following.
\begin{lemma}
\label{lemma-embedded}
Let $X$ be a locally Noetherian scheme.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
Then
\begin{enumerate}
\item the generic points of irreducible components of
$\text{Supp}(\mathcal{F})$ are associated points of $\mathcal{F}$, and
\item an associated point of $\mathcal{F}$ is embedded if and only
if it is not a generic point of an irreducible component
of $\text{Supp}(\mathcal{F})$.
\end{enumerate}
In particular an embedded point of $X$ is an associated point of $X$
which is not a generic point of an irreducible component of $X$.
\end{lemma}
\begin{proof}
Recall that in this case $Z = \text{Supp}(\mathcal{F})$ is closed, see
Morphisms, Lemma \ref{morphisms-lemma-support-finite-type}
and that the generic points of irreducible components of $Z$ are
associated points of $\mathcal{F}$, see
Lemma \ref{lemma-minimal-support-in-ass}.
Finally, we have $\text{Ass}(\mathcal{F}) \subset Z$, by
Lemma \ref{lemma-ass-support}.
These results, combined with the fact that $Z$ is a sober topological
space and hence every point of $Z$ is a specialization of a generic
point of $Z$, imply (1) and (2).
\end{proof}
\begin{lemma}
\label{lemma-S1-no-embedded}
Let $X$ be a locally Noetherian scheme.
Let $\mathcal{F}$ be a coherent sheaf on $X$.
Then the following are equivalent:
\begin{enumerate}
\item $\mathcal{F}$ has no embedded associated points, and
\item $\mathcal{F}$ has property $(S_1)$.
\end{enumerate}
\end{lemma}
\begin{proof}
This is Algebra, Lemma \ref{algebra-lemma-criterion-no-embedded-primes},
combined with Lemma \ref{lemma-associated-affine-open} above.
\end{proof}
\begin{lemma}
\label{lemma-noetherian-dim-1-CM-no-embedded-points}
Let $X$ be a locally Noetherian scheme of dimension $\leq 1$.
The following are equivalent
\begin{enumerate}
\item $X$ is Cohen-Macaulay, and
\item $X$ has no embedded points.
\end{enumerate}
\end{lemma}
\begin{proof}
Follows from Lemma \ref{lemma-S1-no-embedded} and the definitions.
\end{proof}
\begin{lemma}
\label{lemma-scheme-theoretically-dense-contain-embedded-points}
Let $X$ be a locally Noetherian scheme. Let $U \subset X$ be an
open subscheme. The following are equivalent
\begin{enumerate}
\item $U$ is scheme theoretically dense in $X$
(Morphisms, Definition \ref{morphisms-definition-scheme-theoretically-dense}),
\item $U$ is dense in $X$ and $U$ contains all embedded points of $X$.
\end{enumerate}
\end{lemma}
\begin{proof}
The question is local on $X$, hence we may assume that $X = \Spec(A)$
where $A$ is a Noetherian ring. Then $U$ is quasi-compact
(Properties, Lemma \ref{properties-lemma-immersion-into-noetherian})
hence $U = D(f_1) \cup \ldots \cup D(f_n)$
(Algebra, Lemma \ref{algebra-lemma-qc-open}).
In this situation $U$ is scheme theoretically dense in $X$ if and only if
$A \to A_{f_1} \times \ldots \times A_{f_n}$ is injective, see
Morphisms, Example \ref{morphisms-example-scheme-theoretic-closure}.
Condition (2) translated into algebra means that for every associated
prime $\mathfrak p$ of $A$ there exists an $i$ with $f_i \not \in \mathfrak p$.
\medskip\noindent
Assume (1), i.e., $A \to A_{f_1} \times \ldots \times A_{f_n}$ is injective.
If $x \in A$ has annihilator a prime $\mathfrak p$, then $x$ maps
to a nonzero element of $A_{f_i}$ for some $i$ and hence
$f_i \not \in \mathfrak p$. Thus (2) holds.
Assume (2), i.e., every associated prime $\mathfrak p$ of $A$
corresponds to a prime of $A_{f_i}$ for some $i$. Then
$A \to A_{f_1} \times \ldots \times A_{f_n}$ is injective because
$A \to \prod_{\mathfrak p \in \text{Ass}(A)} A_\mathfrak p$ is injective
by Algebra, Lemma \ref{algebra-lemma-zero-at-ass-zero}.
\end{proof}
\begin{lemma}
\label{lemma-remove-embedded-points}
Let $X$ be a locally Noetherian scheme.
Let $\mathcal{F}$ be a coherent sheaf on $X$.
The set of coherent subsheaves
$$
\{
\mathcal{K} \subset \mathcal{F}
\mid
\text{Supp}(\mathcal{K})\text{ is nowhere dense in }\text{Supp}(\mathcal{F})
\}
$$
has a maximal element $\mathcal{K}$.
Setting $\mathcal{F}' = \mathcal{F}/\mathcal{K}$ we have the
following
\begin{enumerate}
\item $\text{Supp}(\mathcal{F}') = \text{Supp}(\mathcal{F})$,
\item $\mathcal{F}'$ has no embedded associated points, and
\item there exists a dense open $U \subset X$ such that
$U \cap \text{Supp}(\mathcal{F})$ is dense in $\text{Supp}(\mathcal{F})$
and $\mathcal{F}'|_U \cong \mathcal{F}|_U$.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows from
Algebra, Lemmas \ref{algebra-lemma-remove-embedded-primes} and
\ref{algebra-lemma-remove-embedded-primes-localize}.
Note that $U$ can be taken as the complement of the closure
of the set of embedded associated points of $\mathcal{F}$.
\end{proof}
\begin{lemma}
\label{lemma-no-embedded-points-endos}
Let $X$ be a locally Noetherian scheme.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module
without embedded associated points. Set
$$
\mathcal{I}
=
\Ker(\mathcal{O}_X
\longrightarrow
\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{F})).
$$
This is a coherent sheaf of ideals which defines a closed
subscheme $Z \subset X$ without embedded points. Moreover
there exists a coherent sheaf $\mathcal{G}$ on $Z$
such that (a) $\mathcal{F} = (Z \to X)_*\mathcal{G}$,
(b) $\mathcal{G}$ has no associated embedded points, and
(c) $\text{Supp}(\mathcal{G}) = Z$ (as sets).
\end{lemma}
\begin{proof}
Some of the statements we have seen in the proof of
Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-support-closed}.
The others follow from
Algebra, Lemma \ref{algebra-lemma-no-embedded-primes-endos}.
\end{proof}
\section{Weakly associated points}
\label{section-weakly-associated}
\noindent
Let $R$ be a ring and let $M$ be an $R$-module.
Recall that a prime $\mathfrak p \subset R$ is {\it weakly associated}
to $M$ if there exists an element $m$ of $M$ such that $\mathfrak p$ is
minimal among the primes containing the annihilator of $m$. See
Algebra, Definition \ref{algebra-definition-weakly-associated}.
If $R$ is a local ring with maximal ideal $\mathfrak m$, then
$\mathfrak m$ is associated to $M$ if and only if there exists an
element $m \in M$ whose annihilator has radical $\mathfrak m$, see
Algebra, Lemma \ref{algebra-lemma-weakly-ass-local}.
\begin{definition}
\label{definition-weakly-associated}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
\begin{enumerate}
\item We say $x \in X$ is {\it weakly associated} to $\mathcal{F}$
if the maximal ideal $\mathfrak m_x$ is weakly associated to the
$\mathcal{O}_{X, x}$-module $\mathcal{F}_x$.
\item We denote $\text{WeakAss}(\mathcal{F})$ the set of weakly associated
points of $\mathcal{F}$.
\item The {\it weakly associated points of $X$} are the weakly associated
points of $\mathcal{O}_X$.
\end{enumerate}
\end{definition}
\noindent
In this case, on any affine open, this corresponds exactly to the
weakly associated primes as defined above. Here is the precise statement.
\begin{lemma}
\label{lemma-weakly-associated-affine-open}
Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $\Spec(A) = U \subset X$ be an affine open, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
The following are equivalent
\begin{enumerate}
\item $\mathfrak p$ is weakly associated to $M$, and
\item $x$ is weakly associated to $\mathcal{F}$.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows from
Algebra, Lemma \ref{algebra-lemma-weakly-ass-local}.
\end{proof}
\begin{lemma}
\label{lemma-weakly-ass-support}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Then
$$
\text{Ass}(\mathcal{F}) \subset \text{WeakAss}(\mathcal{F}) \subset
\text{Supp}(\mathcal{F}).
$$
\end{lemma}
\begin{proof}
This is immediate from the definitions.
\end{proof}
\begin{lemma}
\label{lemma-ses-weakly-ass}
Let $X$ be a scheme.
Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$
be a short exact sequence of quasi-coherent sheaves on $X$.
Then
$\text{WeakAss}(\mathcal{F}_2) \subset
\text{WeakAss}(\mathcal{F}_1) \cup \text{WeakAss}(\mathcal{F}_3)$
and
$\text{WeakAss}(\mathcal{F}_1) \subset \text{WeakAss}(\mathcal{F}_2)$.
\end{lemma}
\begin{proof}
For every point $x \in X$ the sequence of stalks
$0 \to \mathcal{F}_{1, x} \to \mathcal{F}_{2, x} \to \mathcal{F}_{3, x} \to 0$
is a short exact sequence of $\mathcal{O}_{X, x}$-modules.
Hence the lemma follows from
Algebra, Lemma \ref{algebra-lemma-weakly-ass}.
\end{proof}
\begin{lemma}
\label{lemma-weakly-ass-zero}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Then
$$
\mathcal{F} = (0) \Leftrightarrow \text{WeakAss}(\mathcal{F}) = \emptyset
$$
\end{lemma}
\begin{proof}
Follows from
Lemma \ref{lemma-weakly-associated-affine-open}
and
Algebra, Lemma \ref{algebra-lemma-weakly-ass-zero}
\end{proof}
\begin{lemma}
\label{lemma-restriction-injective-open-contains-weakly-ass}
Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. If $\text{WeakAss}(\mathcal{F}) \subset U \subset X$
is open, then $\Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F})$
is injective.
\end{lemma}
\begin{proof}
Let $s \in \Gamma(X, \mathcal{F})$ be a section which restricts to zero on $U$.
Let $\mathcal{F}' \subset \mathcal{F}$ be the image of the map
$\mathcal{O}_X \to \mathcal{F}$ defined by $s$. Then
$\text{Supp}(\mathcal{F}') \cap U = \emptyset$. On the other hand,
$\text{WeakAss}(\mathcal{F}') \subset \text{WeakAss}(\mathcal{F})$
by Lemma \ref{lemma-ses-weakly-ass}. Since also
$\text{Ass}(\mathcal{F}') \subset \text{Supp}(\mathcal{F}')$
(Lemma \ref{lemma-weakly-ass-support}) we conclude
$\text{WeakAss}(\mathcal{F}') = \emptyset$.
Hence $\mathcal{F}' = 0$ by Lemma \ref{lemma-weakly-ass-zero}.
\end{proof}
\begin{lemma}
\label{lemma-minimal-support-in-weakly-ass}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $x \in \text{Supp}(\mathcal{F})$ be a point in the support
of $\mathcal{F}$ which is not a specialization of another point of
$\text{Supp}(\mathcal{F})$. Then
$x \in \text{WeakAss}(\mathcal{F})$.
In particular, any generic point of an irreducible component of $X$
is weakly associated to $\mathcal{O}_X$.
\end{lemma}
\begin{proof}
Since $x \in \text{Supp}(\mathcal{F})$ the module $\mathcal{F}_x$
is not zero. Hence
$\text{WeakAss}(\mathcal{F}_x) \subset \Spec(\mathcal{O}_{X, x})$
is nonempty by
Algebra, Lemma \ref{algebra-lemma-weakly-ass-zero}.
On the other hand, by assumption
$\text{Supp}(\mathcal{F}_x) = \{\mathfrak m_x\}$.
Since
$\text{WeakAss}(\mathcal{F}_x) \subset \text{Supp}(\mathcal{F}_x)$
(Algebra, Lemma \ref{algebra-lemma-weakly-ass-support})
we see that $\mathfrak m_x$ is weakly associated to $\mathcal{F}_x$
and we win.
\end{proof}
\begin{lemma}
\label{lemma-ass-weakly-ass}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
If $\mathfrak m_x$ is a finitely generated ideal of $\mathcal{O}_{X, x}$,
then
$$
x \in \text{Ass}(\mathcal{F}) \Leftrightarrow
x \in \text{WeakAss}(\mathcal{F}).
$$
In particular, if $X$ is locally Noetherian, then
$\text{Ass}(\mathcal{F}) = \text{WeakAss}(\mathcal{F})$.
\end{lemma}
\begin{proof}
See
Algebra, Lemma \ref{algebra-lemma-ass-weakly-ass}.
\end{proof}
\begin{lemma}
\label{lemma-weakass-pushforward}
Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $s \in S$ be a point which is not in the image of $f$. Then
$s$ is not weakly associated to $f_*\mathcal{F}$.
\end{lemma}
\begin{proof}
The question is local so we may assume $X = \Spec(A)$.
By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}
the sheaf $f_*\mathcal{F}$ is quasi-coherent, say corresponding
to the $A$-module $M$.
Say $s$ corresponds to $\mathfrak p \subset A$.
As $s$ is not in the image of $f$ we see that
$X = \bigcup_{a \in \mathfrak p} f^{-1}D(a)$ is an open covering.
Since $X$ is quasi-compact we can find $a_1, \ldots, a_n \in \mathfrak p$
such that $X = f^{-1}D(a_1) \cup \ldots \cup f^{-1}D(a_n)$. It follows
that
$$
M \to M_{a_1} \oplus \ldots \oplus M_{a_r}
$$
is injective. Hence for any nonzero element $m$ of the stalk $M_\mathfrak p$
there exists an $i$ such that $a_i^n m$ is nonzero for all $n \geq 0$.
Thus $\mathfrak pA_\mathfrak p$ is not weakly associated to $M_\mathfrak p$.
\end{proof}
\begin{lemma}
\label{lemma-check-injective-on-weakass}
Let $X$ be a scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of
quasi-coherent $\mathcal{O}_X$-modules. Assume that for every $x \in X$
at least one of the following happens
\begin{enumerate}
\item $\mathcal{F}_x \to \mathcal{G}_x$ is injective, or
\item $x \not \in \text{WeakAss}(\mathcal{F})$.
\end{enumerate}
Then $\varphi$ is injective.
\end{lemma}
\begin{proof}
The assumptions imply that $\text{WeakAss}(\Ker(\varphi)) = \emptyset$
and hence $\Ker(\varphi) = 0$ by Lemma \ref{lemma-weakly-ass-zero}.
\end{proof}
\section{Morphisms and weakly associated points}
\label{section-morphisms-weakly-associated}
\begin{lemma}
\label{lemma-weakly-ass-reverse-functorial}
Let $f : X \to S$ be an affine morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Then we have
$$
\text{WeakAss}_S(f_*\mathcal{F}) \subset f(\text{WeakAss}_X(\mathcal{F}))
$$
\end{lemma}
\begin{proof}
We may assume $X$ and $S$ affine, so $X \to S$ comes from a ring map
$A \to B$. Then $\mathcal{F} = \widetilde M$ for some $B$-module $M$. By
Lemma \ref{lemma-weakly-associated-affine-open}
the weakly associated points of $\mathcal{F}$ correspond exactly to the
weakly associated primes of $M$. Similarly, the weakly associated points
of $f_*\mathcal{F}$ correspond exactly to the weakly associated primes
of $M$ as an $A$-module. Hence the lemma follows from
Algebra, Lemma \ref{algebra-lemma-weakly-ass-reverse-functorial}.
\end{proof}
\begin{lemma}
\label{lemma-ass-functorial-equal}
Let $f : X \to S$ be an affine morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
If $X$ is locally Noetherian, then we have
$$
f(\text{Ass}_X(\mathcal{F})) =
\text{Ass}_S(f_*\mathcal{F}) =
\text{WeakAss}_S(f_*\mathcal{F}) =
f(\text{WeakAss}_X(\mathcal{F}))
$$
\end{lemma}
\begin{proof}
We may assume $X$ and $S$ affine, so $X \to S$ comes from a ring map
$A \to B$. As $X$ is locally Noetherian the ring $B$ is Noetherian, see
Properties, Lemma \ref{properties-lemma-locally-Noetherian}.
Write $\mathcal{F} = \widetilde M$ for some $B$-module $M$. By
Lemma \ref{lemma-associated-affine-open}
the associated points of $\mathcal{F}$ correspond exactly to the associated
primes of $M$, and any associated prime of $M$ as an $A$-module is an
associated points of $f_*\mathcal{F}$.
Hence the inclusion
$$
f(\text{Ass}_X(\mathcal{F})) \subset \text{Ass}_S(f_*\mathcal{F})
$$
follows from
Algebra, Lemma \ref{algebra-lemma-ass-functorial-Noetherian}.
We have the inclusion
$$
\text{Ass}_S(f_*\mathcal{F}) \subset \text{WeakAss}_S(f_*\mathcal{F})
$$
by
Lemma \ref{lemma-weakly-ass-support}.
We have the inclusion
$$
\text{WeakAss}_S(f_*\mathcal{F}) \subset f(\text{WeakAss}_X(\mathcal{F}))
$$
by
Lemma \ref{lemma-weakly-ass-reverse-functorial}.
The outer sets are equal by
Lemma \ref{lemma-ass-weakly-ass}
hence we have equality everywhere.
\end{proof}
\begin{lemma}
\label{lemma-weakly-associated-finite}
Let $f : X \to S$ be a finite morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Then $\text{WeakAss}(f_*\mathcal{F}) = f(\text{WeakAss}(\mathcal{F}))$.
\end{lemma}
\begin{proof}
We may assume $X$ and $S$ affine, so $X \to S$ comes from a finite ring map
$A \to B$. Write $\mathcal{F} = \widetilde M$ for some $B$-module $M$. By
Lemma \ref{lemma-weakly-associated-affine-open}
the weakly associated points of $\mathcal{F}$ correspond exactly to the
weakly associated primes of $M$. Similarly, the weakly associated points
of $f_*\mathcal{F}$ correspond exactly to the weakly associated primes
of $M$ as an $A$-module. Hence the lemma follows from
Algebra, Lemma \ref{algebra-lemma-weakly-ass-finite-ring-map}.
\end{proof}
\begin{lemma}
\label{lemma-weakly-ass-pullback}
Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{G}$ be a
quasi-coherent $\mathcal{O}_S$-module. Let $x \in X$ with $s = f(x)$.
If $f$ is flat at $x$, the point $x$ is a generic point of the fibre $X_s$, and
$s \in \text{WeakAss}_S(\mathcal{G})$, then
$x \in \text{WeakAss}(f^*\mathcal{G})$.
\end{lemma}
\begin{proof}
Let $A = \mathcal{O}_{S, s}$, $B = \mathcal{O}_{X, x}$, and
$M = \mathcal{G}_s$. Let $m \in M$ be an element whose annihilator
$I = \{a \in A \mid am = 0\}$ has radical $\mathfrak m_A$. Then
$m \otimes 1$ has annihilator $I B$ as $A \to B$ is
faithfully flat. Thus it suffices to see that $\sqrt{I B} = \mathfrak m_B$.
This follows from the fact that the maximal ideal of $B/\mathfrak m_AB$
is locally nilpotent (see
Algebra, Lemma \ref{algebra-lemma-minimal-prime-reduced-ring})
and the assumption that $\sqrt{I} = \mathfrak m_A$.
Some details omitted.
\end{proof}
\begin{lemma}
\label{lemma-weakly-ass-change-fields}
Let $K/k$ be a field extension. Let $X$ be a scheme over $k$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $y \in X_K$ with image $x \in X$. If $y$ is a weakly
associated point of the pullback $\mathcal{F}_K$, then $x$
is a weakly associated point of $\mathcal{F}$.
\end{lemma}
\begin{proof}
This is the translation of
Algebra, Lemma \ref{algebra-lemma-weakly-ass-change-fields}
into the language of schemes.
\end{proof}
\section{Relative assassin}
\label{section-relative-assassin}
\noindent
Let $A \to B$ be a ring map. Let $N$ be a $B$-module. Recall that
a prime $\mathfrak q \subset B$ is said to be in the relative assassin
of $N$ over $B/A$ if $\mathfrak q$ is an associated prime of
$N \otimes_A \kappa(\mathfrak p)$. Here $\mathfrak p = A \cap \mathfrak q$.
Here is the definition of the relative assassin for quasi-coherent
sheaves over a morphism of schemes.
\begin{definition}
\label{definition-relative-assassin}
Let $f : X \to S$ be a morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
The {\it relative assassin of $\mathcal{F}$ in $X$ over $S$}
is the set
$$
\text{Ass}_{X/S}(\mathcal{F}) =
\bigcup\nolimits_{s \in S} \text{Ass}_{X_s}(\mathcal{F}_s)
$$
where $\mathcal{F}_s = (X_s \to X)^*\mathcal{F}$ is the restriction
of $\mathcal{F}$ to the fibre of $f$ at $s$.
\end{definition}
\noindent
Again there is a caveat that this is best used when the fibres of $f$
are locally Noetherian and $\mathcal{F}$ is of finite type. In the general
case we should probably use the relative weak assassin (defined in the next
section). Let us link the scheme theoretic notion with the algebraic notion
on affine opens; note that this correspondence works perfectly only
for morphisms of schemes whose fibres are locally Noetherian.
\begin{lemma}
\label{lemma-relative-assassin-affine-open}
Let $f : X \to S$ be a morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
Let $U \subset X$ and $V \subset S$ be affine opens
with $f(U) \subset V$. Write $U = \Spec(A)$, $V = \Spec(R)$, and set
$M = \Gamma(U, \mathcal{F})$.
Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
Then
$$
\mathfrak p \in \text{Ass}_{A/R}(M) \Rightarrow
x \in \text{Ass}_{X/S}(\mathcal{F})
$$
If all fibres $X_s$ of $f$ are locally Noetherian, then
$\mathfrak p \in \text{Ass}_{A/R}(M) \Leftrightarrow
x \in \text{Ass}_{X/S}(\mathcal{F})$
for all pairs $(\mathfrak p, x)$ as above.
\end{lemma}
\begin{proof}
The set $\text{Ass}_{A/R}(M)$ is defined in
Algebra, Definition \ref{algebra-definition-relative-assassin}.
Choose a pair $(\mathfrak p, x)$. Let $s = f(x)$.
Let $\mathfrak r \subset R$ be the prime lying under $\mathfrak p$,
i.e., the prime corresponding to $s$.
Let $\mathfrak p' \subset A \otimes_R \kappa(\mathfrak r)$
be the prime whose inverse image is $\mathfrak p$, i.e.,
the prime corresponding to $x$ viewed as a point of its fibre $X_s$.
Then $\mathfrak p \in \text{Ass}_{A/R}(M)$ if and only if
$\mathfrak p'$ is an associated prime of
$M \otimes_R \kappa(\mathfrak r)$, see
Algebra, Lemma \ref{algebra-lemma-compare-relative-assassins}.
Note that the ring $A \otimes_R \kappa(\mathfrak r)$ corresponds to $U_s$
and the module $M \otimes_R \kappa(\mathfrak r)$ corresponds to the
quasi-coherent sheaf $\mathcal{F}_s|_{U_s}$.
Hence $x$ is an associated point of $\mathcal{F}_s$
by Lemma \ref{lemma-associated-affine-open}.
The reverse implication holds if $\mathfrak p'$ is finitely generated
which is how the last sentence is seen to be true.
\end{proof}
\begin{lemma}
\label{lemma-base-change-relative-assassin}
Let $f : X \to S$ be a morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $g : S' \to S$ be a morphism of schemes.
Consider the base change diagram
$$
\xymatrix{
X' \ar[d] \ar[r]_{g'} & X \ar[d] \\
S' \ar[r]^g & S
}
$$
and set $\mathcal{F}' = (g')^*\mathcal{F}$. Let $x' \in X'$ be a point
with images $x \in X$, $s' \in S'$ and $s \in S$.
Assume $f$ locally of finite type.
Then $x' \in \text{Ass}_{X'/S'}(\mathcal{F}')$ if and only if
$x \in \text{Ass}_{X/S}(\mathcal{F})$ and $x'$ corresponds to
a generic point of an irreducible component of
$\Spec(\kappa(s') \otimes_{\kappa(s)} \kappa(x))$.
\end{lemma}
\begin{proof}
Consider the morphism $X'_{s'} \to X_s$ of fibres. As
$X_{s'} = X_s \times_{\Spec(\kappa(s))} \Spec(\kappa(s'))$
this is a flat morphism. Moreover $\mathcal{F}'_{s'}$ is the pullback
of $\mathcal{F}_s$ via this morphism. As $X_s$ is locally of finite
type over the Noetherian scheme $\Spec(\kappa(s))$ we have that
$X_s$ is locally Noetherian, see
Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}.
Thus we may apply
Lemma \ref{lemma-bourbaki}
and we see that
$$
\text{Ass}_{X'_{s'}}(\mathcal{F}'_{s'}) =
\bigcup\nolimits_{x \in \text{Ass}(\mathcal{F}_s)} \text{Ass}((X'_{s'})_x).
$$
Thus to prove the lemma it suffices to show that the associated points
of the fibre $(X'_{s'})_x$ of the morphism $X'_{s'} \to X_s$ over $x$
are its generic points. Note that
$(X'_{s'})_x = \Spec(\kappa(s') \otimes_{\kappa(s)} \kappa(x))$
as schemes. By
Algebra, Lemma \ref{algebra-lemma-tensor-fields-CM}
the ring $\kappa(s') \otimes_{\kappa(s)} \kappa(x)$ is a Noetherian
Cohen-Macaulay ring. Hence its associated primes are its minimal primes, see
Algebra, Proposition \ref{algebra-proposition-minimal-primes-associated-primes}
(minimal primes are associated) and
Algebra, Lemma \ref{algebra-lemma-criterion-no-embedded-primes}
(no embedded primes).
\end{proof}
\begin{remark}
\label{remark-base-change-relative-assassin}
With notation and assumptions as in
Lemma \ref{lemma-base-change-relative-assassin}
we see that it is always the case that
$(g')^{-1}(\text{Ass}_{X/S}(\mathcal{F})) \supset
\text{Ass}_{X'/S'}(\mathcal{F}')$.
If the morphism $S' \to S$ is locally quasi-finite, then we actually have
$$
(g')^{-1}(\text{Ass}_{X/S}(\mathcal{F}))
=
\text{Ass}_{X'/S'}(\mathcal{F}')
$$
because in this case the field extensions $\kappa(s) \subset \kappa(s')$
are always finite. In fact, this holds more generally for any morphism
$g : S' \to S$ such that all the field extensions
$\kappa(s) \subset \kappa(s')$ are algebraic, because in this case all
prime ideals of $\kappa(s') \otimes_{\kappa(s)} \kappa(x)$ are
maximal (and minimal) primes, see
Algebra, Lemma \ref{algebra-lemma-integral-over-field}.
\end{remark}
\section{Relative weak assassin}
\label{section-relative-weak-assassin}
\begin{definition}
\label{definition-relative-weak-assassin}
Let $f : X \to S$ be a morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
The {\it relative weak assassin of $\mathcal{F}$ in $X$ over $S$}
is the set
$$
\text{WeakAss}_{X/S}(\mathcal{F}) =
\bigcup\nolimits_{s \in S} \text{WeakAss}(\mathcal{F}_s)
$$
where $\mathcal{F}_s = (X_s \to X)^*\mathcal{F}$ is the restriction
of $\mathcal{F}$ to the fibre of $f$ at $s$.
\end{definition}
\begin{lemma}
\label{lemma-relative-weak-assassin-assassin-finite-type}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Then $\text{WeakAss}_{X/S}(\mathcal{F}) = \text{Ass}_{X/S}(\mathcal{F})$.
\end{lemma}
\begin{proof}
This is true because the fibres of $f$ are locally Noetherian schemes,
and associated and weakly associated points agree on locally Noetherian
schemes, see
Lemma \ref{lemma-ass-weakly-ass}.
\end{proof}
\begin{lemma}
\label{lemma-relative-weak-assassin-finite}
Let $f : X \to S$ be a morphism of schemes.
Let $i : Z \to X$ be a finite morphism.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_Z$-module.
Then $\text{WeakAss}_{X/S}(i_*\mathcal{F}) =
i(\text{WeakAss}_{Z/S}(\mathcal{F}))$.
\end{lemma}
\begin{proof}
Let $i_s : Z_s \to X_s$ be the induced morphism between fibres.
Then $(i_*\mathcal{F})_s = i_{s, *}(\mathcal{F}_s)$ by
Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-base-change}
and the fact that $i$ is affine. Hence
we may apply Lemma \ref{lemma-weakly-associated-finite} to conclude.
\end{proof}
\section{Fitting ideals}
\label{section-fitting-ideals}
\noindent
This section is the continuation of the discussion in
More on Algebra, Section \ref{more-algebra-section-fitting-ideals}.
Let $S$ be a scheme. Let $\mathcal{F}$ be a
finite type quasi-coherent $\mathcal{O}_S$-module.
In this situation we can construct the Fitting ideals
$$
0 = \text{Fit}_{-1}(\mathcal{F}) \subset \text{Fit}_0(\mathcal{F}) \subset
\text{Fit}_1(\mathcal{F}) \subset \ldots \subset \mathcal{O}_S
$$
as the sequence of quasi-coherent ideals characterized by the following
property: for every affine open $U = \Spec(A)$ of $S$ if $\mathcal{F}|_U$
corresponds to the $A$-module $M$, then $\text{Fit}_i(\mathcal{F})|_U$
corresponds to the ideal $\text{Fit}_i(M) \subset A$.
This is well defined and a quasi-coherent sheaf of ideals because
if $f \in A$, then the $i$th Fitting ideal of $M_f$ over $A_f$
is equal to $\text{Fit}_i(M) A_f$ by
More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideal-basics}.
\medskip\noindent
Alternatively, we can construct the Fitting ideals in terms of local
presentations of $\mathcal{F}$. Namely, if $U \subset X$ is open, and
$$
\bigoplus\nolimits_{i \in I} \mathcal{O}_U \to
\mathcal{O}_U^{\oplus n} \to \mathcal{F}|_U \to 0
$$
is a presentation of $\mathcal{F}$ over $U$, then
$\text{Fit}_r(\mathcal{F})|_U$ is generated by the
$(n - r) \times (n - r)$-minors
of the matrix defining the first arrow of the presentation.
This is compatible with the construction above because this
is how the Fitting ideal of a module over a ring is actually defined.
Some details omitted.
\begin{lemma}
\label{lemma-base-change-fitting-ideal}
Let $f : T \to S$ be a morphism of schemes.
Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_S$-module.
Then
$f^{-1}\text{Fit}_i(\mathcal{F}) \cdot \mathcal{O}_T =
\text{Fit}_i(f^*\mathcal{F})$.
\end{lemma}
\begin{proof}
Follows immediately from More on Algebra, Lemma
\ref{more-algebra-lemma-fitting-ideal-basics} part (3).
\end{proof}
\begin{lemma}
\label{lemma-fitting-ideal-of-finitely-presented}
Let $S$ be a scheme.
Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_S$-module.
Then $\text{Fit}_r(\mathcal{F})$ is a quasi-coherent ideal of finite type.
\end{lemma}
\begin{proof}
Follows immediately from More on Algebra, Lemma
\ref{more-algebra-lemma-fitting-ideal-basics} part (4).
\end{proof}
\begin{lemma}
\label{lemma-on-subscheme-cut-out-by-Fit-0}
Let $S$ be a scheme.
Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_S$-module.
Let $Z_0 \subset S$ be the closed subscheme cut out by
$\text{Fit}_0(\mathcal{F})$.
Let $Z \subset S$ be the scheme theoretic support of $\mathcal{F}$.
Then
\begin{enumerate}
\item $Z \subset Z_0 \subset S$ as closed subschemes,
\item $Z = Z_0 = \text{Supp}(\mathcal{F})$ as closed subsets,
\item there exists a finite type, quasi-coherent $\mathcal{O}_{Z_0}$-module
$\mathcal{G}_0$ with
$$
(Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}.
$$
\end{enumerate}
\end{lemma}
\begin{proof}
Recall that $Z$ is locally cut out by the annihilator of $\mathcal{F}$, see
Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-support}
(which uses Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-support}
to define $Z$). Hence we see that $Z \subset Z_0$ scheme theoretically
by More on Algebra, Lemma
\ref{more-algebra-lemma-fitting-ideal-basics} part (6).
On the other hand we have $Z = \text{Supp}(\mathcal{F})$
set theoretically by
Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-support}
and we have $Z_0 = Z$ set theoretically by
More on Algebra, Lemma
\ref{more-algebra-lemma-fitting-ideal-basics} part (7).
Finally, to get $\mathcal{G}_0$ as in part (3) we can either use
that we have $\mathcal{G}$ on $Z$ as in
Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-support}
and set $\mathcal{G}_0 = (Z \to Z_0)_*\mathcal{G}$
or we can use Morphisms, Lemma \ref{morphisms-lemma-i-star-equivalence}
and the fact that $\text{Fit}_0(\mathcal{F})$ annihilates
$\mathcal{F}$ by More on Algebra, Lemma
\ref{more-algebra-lemma-fitting-ideal-basics} part (6).
\end{proof}
\begin{lemma}
\label{lemma-fitting-ideal-generate-locally}
Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent
$\mathcal{O}_S$-module. Let $s \in S$. Then $\mathcal{F}$ can be
generated by $r$ elements in a neighbourhood of $s$ if and only
if $\text{Fit}_r(\mathcal{F})_s = \mathcal{O}_{S, s}$.
\end{lemma}
\begin{proof}
Follows immediately from
More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideal-generate-locally}.
\end{proof}
\begin{lemma}
\label{lemma-fitting-ideal-finite-locally-free}
Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent
$\mathcal{O}_S$-module. Let $r \geq 0$. The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is finite locally free of rank $r$
\item $\text{Fit}_{r - 1}(\mathcal{F}) = 0$ and
$\text{Fit}_r(\mathcal{F}) = \mathcal{O}_S$, and
\item $\text{Fit}_k(\mathcal{F}) = 0$ for $k < r$ and
$\text{Fit}_k(\mathcal{F}) = \mathcal{O}_S$ for $k \geq r$.
\end{enumerate}
\end{lemma}
\begin{proof}
Follows immediately from
More on Algebra, Lemma
\ref{more-algebra-lemma-fitting-ideal-finite-locally-free}.
\end{proof}
\begin{lemma}
\label{lemma-locally-free-rank-r-pullback}
Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent
$\mathcal{O}_S$-module. The closed subschemes
$$
S = Z_{-1} \supset Z_0 \supset Z_1 \supset Z_2 \ldots
$$
defined by the Fitting ideals of $\mathcal{F}$ have the following
properties
\begin{enumerate}
\item The intersection $\bigcap Z_r$ is empty.
\item The functor $(\Sch/S)^{opp} \to \textit{Sets}$ defined by the rule
$$
T \longmapsto
\left\{
\begin{matrix}
\{*\} & \text{if }\mathcal{F}_T\text{ is locally generated by }
\leq r\text{ sections} \\
\emptyset & \text{otherwise}
\end{matrix}
\right.
$$
is representable by the open subscheme $S \setminus Z_r$.
\item The functor $F_r : (\Sch/S)^{opp} \to \textit{Sets}$ defined by the rule
$$
T \longmapsto
\left\{
\begin{matrix}
\{*\} & \text{if }\mathcal{F}_T\text{ locally free rank }r\\
\emptyset & \text{otherwise}
\end{matrix}
\right.
$$
is representable by the locally closed subscheme $Z_{r - 1} \setminus Z_r$
of $S$.
\end{enumerate}
If $\mathcal{F}$ is of finite presentation, then
$Z_r \to S$, $S \setminus Z_r \to S$, and $Z_{r - 1} \setminus Z_r \to S$
are of finite presentation.
\end{lemma}
\begin{proof}
Part (1) is true because over every affine open $U$ there is an integer $n$
such that $\text{Fit}_n(\mathcal{F})|_U = \mathcal{O}_U$. Namely, we can
take $n$ to be the number of generators of $\mathcal{F}$ over $U$, see
More on Algebra, Section \ref{more-algebra-section-fitting-ideals}.
\medskip\noindent
For any morphism $g : T \to S$ we see from
Lemmas \ref{lemma-base-change-fitting-ideal} and
\ref{lemma-fitting-ideal-generate-locally}
that $\mathcal{F}_T$ is locally generated by $\leq r$ sections if and only if
$\text{Fit}_r(\mathcal{F}) \cdot \mathcal{O}_T = \mathcal{O}_T$.
This proves (2).
\medskip\noindent
For any morphism $g : T \to S$ we see from
Lemmas \ref{lemma-base-change-fitting-ideal} and
\ref{lemma-fitting-ideal-finite-locally-free}
that $\mathcal{F}_T$ is free of rank $r$ if and only if
$\text{Fit}_r(\mathcal{F}) \cdot \mathcal{O}_T = \mathcal{O}_T$ and
$\text{Fit}_{r - 1}(\mathcal{F}) \cdot \mathcal{O}_T = 0$.
This proves (3).
\medskip\noindent
Part (4) follows from the fact that if
$\mathcal{F}$ is of finite presentation, then each of the morphisms
$Z_r \to S$ is of finite presentation as $\text{Fit}_r(\mathcal{F})$
is of finite type (Lemma \ref{lemma-fitting-ideal-of-finitely-presented} and
Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-finite-presentation}).
This implies that $Z_{r - 1} \setminus Z_r$ is a retrocompact open in $Z_r$
(Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals})
and hence the morphism $Z_{r - 1} \setminus Z_r \to Z_r$
is of finite presentation as well.
\end{proof}
\noindent
Lemma \ref{lemma-locally-free-rank-r-pullback} notwithstanding
the following lemma does not hold if $\mathcal{F}$ is a finite type
quasi-coherent module. Namely, the stratification still exists but
it isn't true that it represents the functor $F_{flat}$ in general.
\begin{lemma}
\label{lemma-finite-presentation-module}
Let $S$ be a scheme. Let $\mathcal{F}$ be an $\mathcal{O}_S$-module
of finite presentation. Let $S = Z_{-1} \subset Z_0 \subset Z_1 \subset \ldots$
be as in Lemma \ref{lemma-locally-free-rank-r-pullback}.
Set $S_r = Z_{r - 1} \setminus Z_r$.
Then $S' = \coprod_{r \geq 0} S_r$ represents the functor
$$
F_{flat} : \Sch/S \longrightarrow \textit{Sets},\quad\quad
T \longmapsto
\left\{
\begin{matrix}
\{*\} & \text{if }\mathcal{F}_T\text{ flat over }T\\
\emptyset & \text{otherwise}
\end{matrix}
\right.
$$
Moreover, $\mathcal{F}|_{S_r}$ is locally free of rank $r$ and the
morphisms $S_r \to S$ and $S' \to S$ are of finite presentation.
\end{lemma}
\begin{proof}
Suppose that $g : T \to S$ is a morphism of schemes such that the pullback
$\mathcal{F}_T = g^*\mathcal{F}$ is flat. Then $\mathcal{F}_T$ is a flat
$\mathcal{O}_T$-module of finite presentation. Hence
$\mathcal{F}_T$ is finite locally free, see
Properties, Lemma \ref{properties-lemma-finite-locally-free}.
Thus $T = \coprod_{r \geq 0} T_r$, where $\mathcal{F}_T|_{T_r}$ is locally
free of rank $r$. This implies that
$$
F_{flat} = \coprod\nolimits_{r \geq 0} F_r
$$
in the category of Zariski sheaves on $\Sch/S$ where $F_r$ is as in
Lemma \ref{lemma-locally-free-rank-r-pullback}. It follows
that $F_{flat}$ is represented by
$\coprod_{r \geq 0} (Z_{r - 1} \setminus Z_r)$ where
$Z_r$ is as in
Lemma \ref{lemma-locally-free-rank-r-pullback}.
The other statements also follow from the lemma.
\end{proof}
\section{The singular locus of a morphism}
\label{section-singular-locus-morphism}
\noindent
Let $f : X \to S$ be a finite type morphism of schemes. The set $U$ of points
where $f$ is smooth is an open of $X$
(by Morphisms, Definition \ref{morphisms-definition-smooth}).
In many situations it is useful to a have canonical closed
subscheme $\text{Sing}(f) \subset X$ whose complement is $U$
and whose formation commutes with arbitrary change of base.
\medskip\noindent
If $f$ is of finite presentation, then one choice would be to consider the
closed subscheme $Z$ cut out by functions which are affine locally
``strictly standard'' in the sense of
Smoothing Ring Maps, Definition \ref{smoothing-definition-strictly-standard}.
It follows from
Smoothing Ring Maps, Lemma \ref{smoothing-lemma-strictly-standard-base-change}
that if $f' : X' \to S'$ is the base change of $f$ by a morphism
$S' \to S$, then $Z' \subset S' \times_S Z$ where $Z'$ is the
closed subscheme of $X'$ cut out by functions which are affine
locally strictly standard. However, equality isn't clear.
The notion of a strictly standard element was useful in the chapter on
Popescu's theorem. The closed subscheme defined by these elements is
(as far as we know) not used in the literature\footnote{If $f$ is a
local complete intersection morphism
(More on Morphisms, Definition \ref{more-morphisms-definition-lci})
then the closed subscheme cut out by the locally strictly standard
elements is the correct thing to look at.}.
\medskip\noindent
If $f$ is flat, of finite presentation, and the fibres of $f$
all are equidimensional of dimension $d$, then the $d$th fitting ideal
of $\Omega_{X/S}$ is used to get a good closed subscheme. For any
morphism of finite type the closed subschemes of $X$ defined by the
fitting ideals of $\Omega_{X/S}$ define a stratification of $X$
in terms of the rank of $\Omega_{X/S}$ whose formation commutes with
base change. This can be helpful; it is related to embedding dimensions of
fibres, see Varieties, Section \ref{varieties-section-embedding-dimension}.
\begin{lemma}
\label{lemma-base-change-and-fitting-ideal-omega}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
Let $X = Z_{-1} \supset Z_0 \supset Z_1 \supset \ldots$
be the closed subschemes defined by the fitting ideals
of $\Omega_{X/S}$. Then the formation of $Z_i$ commutes
with arbitrary base change.
\end{lemma}
\begin{proof}
Observe that $\Omega_{X/S}$ is a finite type quasi-coherent
$\mathcal{O}_X$-module
(Morphisms, Lemma \ref{morphisms-lemma-finite-type-differentials})
hence the fitting ideals are defined. If $f' : X' \to S'$
is the base change of $f$ by $g : S' \to S$, then
$\Omega_{X'/S'} = (g')^*\Omega_{X/S}$ where $g' : X' \to X$
is the projection
(Morphisms, Lemma \ref{morphisms-lemma-base-change-differentials}).
Hence $(g')^{-1}\text{Fit}_i(\Omega_{X/S}) \cdot \mathcal{O}_{X'} =
\text{Fit}_i(\Omega_{X'/S'})$. This means that
$$
Z'_i = (g')^{-1}(Z_i) = Z_i \times_X X'
$$
scheme theoretically and this is the meaning of the statement of
the lemma.
\end{proof}
\noindent
The $0$th fitting ideal of $\Omega$
cuts out the ``ramified locus'' of the morphism.
\begin{lemma}
\label{lemma-zero-fitting-ideal-omega-unramified}
Let $f : X \to S$ be a morphism of schemes which is locally of finite type.
The closed subscheme $Z \subset X$ cut out by the $0$th fitting ideal of
$\Omega_{X/S}$ is exactly the set of points where $f$ is not unramified.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-on-subscheme-cut-out-by-Fit-0} the complement of $Z$
is exactly the locus where $\Omega_{X/S}$ is zero. This is exactly
the set of points where $f$ is unramified by
Morphisms, Lemma \ref{morphisms-lemma-unramified-omega-zero}.
\end{proof}
\begin{lemma}
\label{lemma-d-fitting-ideal-omega-smooth}
Let $f : X \to S$ be a morphism of schemes. Let $d \geq 0$ be an integer.
Assume
\begin{enumerate}
\item $f$ is flat,
\item $f$ is locally of finite presentation, and
\item every nonempty fibre of $f$ is equidimensional of dimension $d$.
\end{enumerate}
Let $Z \subset X$ be the closed subscheme cut out by the $d$th fitting
ideal of $\Omega_{X/S}$. Then $Z$ is exactly the set of points
where $f$ is not smooth.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-locally-free-rank-r-pullback} the complement of $Z$
is exactly the locus where $\Omega_{X/S}$ can be generated by at most
$d$ elements. Hence the lemma follows from
Morphisms, Lemma \ref{morphisms-lemma-smooth-at-point}.
\end{proof}
\section{Torsion free modules}
\label{section-torsion-free}
\noindent
This section is the analogue of
More on Algebra, Section \ref{more-algebra-section-torsion-flat}
for quasi-coherent modules.
\begin{lemma}
\label{lemma-torsion-sections}
Let $X$ be an integral scheme with generic point $\eta$. Let $\mathcal{F}$
be a quasi-coherent $\mathcal{O}_X$-module. Let $U \subset X$ be nonempty
open and $s \in \mathcal{F}(U)$. The following are equivalent
\begin{enumerate}
\item for some $x \in U$ the image of $s$ in $\mathcal{F}_x$ is torsion,
\item for all $x \in U$ the image of $s$ in $\mathcal{F}_x$ is torsion,
\item the image of $s$ in $\mathcal{F}_\eta$ is zero,
\item the image of $s$ in $j_*\mathcal{F}_\eta$ is zero, where $j : \eta \to X$
is the inclusion morphism.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{definition}
\label{definition-torsion}
Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module.
\begin{enumerate}
\item We say a local section of $\mathcal{F}$ is {\it torsion}
if it satisfies the equivalent conditions of Lemma \ref{lemma-torsion-sections}.
\item We say $\mathcal{F}$ is {\it torsion free} if every torsion section
of $\mathcal{F}$ is $0$.
\end{enumerate}
\end{definition}
\noindent
Here is the obligatory lemma comparing this to the usual algebraic notion.
\begin{lemma}
\label{lemma-check-torsion-on-affines}
Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is torsion free,
\item for $U \subset X$ affine open $\mathcal{F}(U)$
is a torsion free $\mathcal{O}(U)$-module.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-torsion}
Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. The torsion sections of $\mathcal{F}$ form
a quasi-coherent $\mathcal{O}_X$-submodule
$\mathcal{F}_{tors} \subset \mathcal{F}$.
The quotient module $\mathcal{F}/\mathcal{F}_{tors}$ is torsion free.
\end{lemma}
\begin{proof}
Omitted. See More on Algebra, Lemma \ref{more-algebra-lemma-torsion}
for the algebraic analogue.
\end{proof}
\begin{lemma}
\label{lemma-flat-torsion-free}
Let $X$ be an integral scheme. Any flat quasi-coherent $\mathcal{O}_X$-module
is torsion free.
\end{lemma}
\begin{proof}
Omitted. See More on Algebra, Lemma \ref{more-algebra-lemma-flat-torsion-free}.
\end{proof}
\begin{lemma}
\label{lemma-flat-pullback-torsion}
Let $f : X \to Y$ be a flat morphism of integral schemes.
Let $\mathcal{G}$ be a torsion free quasi-coherent $\mathcal{O}_Y$-module.
Then $f^*\mathcal{G}$ is a torsion free $\mathcal{O}_X$-module.
\end{lemma}
\begin{proof}
Omitted. See
More on Algebra, Lemma \ref{more-algebra-lemma-flat-pullback-torsion}
for the algebraic analogue.
\end{proof}
\begin{lemma}
\label{lemma-flat-over-integral-integral-fibre}
Let $f : X \to Y$ be a flat morphism of schemes. If $Y$ is integral
and the generic fibre of $f$ is integral, then $X$ is integral.
\end{lemma}
\begin{proof}
The algebraic analogue is this: let $A$ be a domain with fraction
field $K$ and let $B$ be a flat $A$-algebra such that $B \otimes_A K$
is a domain. Then $B$ is a domain. This is true because $B$ is
torsion free by More on Algebra, Lemma
\ref{more-algebra-lemma-flat-torsion-free}
and hence $B \subset B \otimes_A K$.
\end{proof}
\begin{lemma}
\label{lemma-check-torsion}
Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. Then $\mathcal{F}$ is torsion free if and only if
$\mathcal{F}_x$ is a torsion free $\mathcal{O}_{X, x}$-module for all $x \in X$.
\end{lemma}
\begin{proof}
Omitted. See More on Algebra, Lemma
\ref{more-algebra-lemma-check-torsion}.
\end{proof}
\begin{lemma}
\label{lemma-extension-torsion-free}
Let $X$ be an integral scheme. Let
$0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$
be a short exact sequence of quasi-coherent $\mathcal{O}_X$-modules.
If $\mathcal{F}$ and $\mathcal{F}''$ are torsion free, then $\mathcal{F}'$
is torsion free.
\end{lemma}
\begin{proof}
Omitted. See
More on Algebra, Lemma \ref{more-algebra-lemma-extension-torsion-free}
for the algebraic analogue.
\end{proof}
\begin{lemma}
\label{lemma-torsion-free-finite-noetherian-domain}
Let $X$ be a locally Noetherian integral scheme with generic point $\eta$.
Let $\mathcal{F}$ be a nonzero coherent $\mathcal{O}_X$-module.
The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is torsion free,
\item $\eta$ is the only associated prime of $\mathcal{F}$,
\item $\eta$ is in the support of $\mathcal{F}$ and $\mathcal{F}$
has property $(S_1)$, and
\item $\eta$ is in the support of $\mathcal{F}$ and $\mathcal{F}$
has no embedded associated prime.
\end{enumerate}
\end{lemma}
\begin{proof}
This is a translation of More on Algebra, Lemma
\ref{more-algebra-lemma-torsion-free-finite-noetherian-domain}
into the language of schemes. We omit the translation.
\end{proof}
\begin{lemma}
\label{lemma-torsion-free-over-regular-dim-1}
Let $X$ be an integral regular scheme of dimension $\leq 1$.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is torsion free,
\item $\mathcal{F}$ is finite locally free.
\end{enumerate}
\end{lemma}
\begin{proof}
It is clear that a finite locally free module is torsion free.
For the converse, we will show that if $\mathcal{F}$ is
torsion free, then $\mathcal{F}_x$ is a free $\mathcal{O}_{X, x}$-module
for all $x \in X$. This is enough by
Algebra, Lemma \ref{algebra-lemma-finite-projective}
and the fact that $\mathcal{F}$ is coherent.
If $\dim(\mathcal{O}_{X, x}) = 0$, then
$\mathcal{O}_{X, x}$ is a field and the statement is clear.
If $\dim(\mathcal{O}_{X, x}) = 1$, then $\mathcal{O}_{X, x}$
is a discrete valuation ring
(Algebra, Lemma \ref{algebra-lemma-characterize-dvr})
and $\mathcal{F}_x$ is torsion free.
Hence $\mathcal{F}_x$ is free by More on Algebra, Lemma
\ref{more-algebra-lemma-dedekind-torsion-free-flat}.
\end{proof}
\begin{lemma}
\label{lemma-hom-into-torsion-free}
Let $X$ be an integral scheme. Let $\mathcal{F}$, $\mathcal{G}$ be
quasi-coherent $\mathcal{O}_X$-modules.
If $\mathcal{G}$ is torsion free and $\mathcal{F}$ is of finite presentation,
then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is torsion free.
\end{lemma}
\begin{proof}
The statement makes sense because
$\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$
is quasi-coherent by Schemes, Section \ref{schemes-section-quasi-coherent}.
To see the statement is true, see
More on Algebra, Lemma \ref{more-algebra-lemma-hom-into-torsion-free}.
Some details omitted.
\end{proof}
\begin{lemma}
\label{lemma-isom-depth-2-torsion-free}
Let $X$ be an integral locally Noetherian scheme. Let
$\varphi : \mathcal{F} \to \mathcal{G}$ be a map of
quasi-coherent $\mathcal{O}_X$-modules. Assume $\mathcal{F}$ is coherent,
$\mathcal{G}$ is torsion free, and that for every $x \in X$ one of the
following happens
\begin{enumerate}
\item $\mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, or
\item $\text{depth}(\mathcal{F}_x) \geq 2$.
\end{enumerate}
Then $\varphi$ is an isomorphism.
\end{lemma}
\begin{proof}
This is a translation of More on Algebra, Lemma
\ref{more-algebra-lemma-isom-depth-2-torsion-free}
into the language of schemes.
\end{proof}
\section{Reflexive modules}
\label{section-reflexive}
\noindent
This section is the analogue of
More on Algebra, Section \ref{more-algebra-section-reflexive}
for coherent modules on locally Noetherian schemes. The reason for
working with coherent modules is that
$\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is coherent
for every pair of coherent $\mathcal{O}_X$-modules $\mathcal{F}, \mathcal{G}$,
see Modules, Lemma \ref{modules-lemma-internal-hom-locally-kernel-direct-sum}.
\begin{definition}
\label{definition-reflexive}
Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$
be a coherent $\mathcal{O}_X$-module. The {\it reflexive hull}
of $\mathcal{F}$ is the $\mathcal{O}_X$-module
$$
\mathcal{F}^{**} = \SheafHom_{\mathcal{O}_X}(
\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{O}_X), \mathcal{O}_X)
$$
We say $\mathcal{F}$ is {\it reflexive} if the natural map
$j : \mathcal{F} \longrightarrow \mathcal{F}^{**}$
is an isomorphism.
\end{definition}
\noindent
It follows from Lemma \ref{lemma-dual-reflexive} that the reflexive hull
is a reflexive $\mathcal{O}_X$-module.
You can use the same definition to define reflexive modules in more
general situations, but this does not seem to be very useful.
Here is the obligatory lemma comparing this to the usual algebraic notion.
\begin{lemma}
\label{lemma-check-reflexive-on-affines}
Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a
coherent $\mathcal{O}_X$-module. The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is reflexive,
\item for $U \subset X$ affine open $\mathcal{F}(U)$
is a reflexive $\mathcal{O}(U)$-module.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{remark}
\label{remark-different-reflexive}
If $X$ is a scheme of finite type over a field, then sometimes a different
notion of reflexive modules is used (see for example
\cite[bottom of page 5 and Definition 1.1.9]{HL}).
This other notion uses $R\SheafHom$ into a dualizing complex
$\omega_X^\bullet$ instead of into $\mathcal{O}_X$ and
should probably have a different name because it can be different
when $X$ is not Gorenstein. For example, if
$X = \Spec(k[t^3, t^4, t^5])$, then a computation shows the dualizing
sheaf $\omega_X$ is not reflexive in our sense, but it is reflexive in the
other sense as
$\omega_X \to \SheafHom(\SheafHom(\omega_X, \omega_X), \omega_X)$
is an isomorphism.
\end{remark}
\begin{lemma}
\label{lemma-reflexive-torsion-free}
Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$
be a coherent $\mathcal{O}_X$-module.
\begin{enumerate}
\item If $\mathcal{F}$ is reflexive, then $\mathcal{F}$ is torsion free.
\item The map $j : \mathcal{F} \longrightarrow \mathcal{F}^{**}$
is injective if and only if $\mathcal{F}$ is torsion free
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted. See More on Algebra, Lemma
\ref{more-algebra-lemma-reflexive-torsion-free}.
\end{proof}
\begin{lemma}
\label{lemma-check-reflexive}
Let $X$ be an integral locally Noetherian scheme.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is reflexive,
\item $\mathcal{F}_x$ is a reflexive $\mathcal{O}_{X, x}$-module
for all $x \in X$,
\item $\mathcal{F}_x$ is a reflexive $\mathcal{O}_{X, x}$-module
for all closed points $x \in X$.
\end{enumerate}
\end{lemma}
\begin{proof}
By Modules, Lemma \ref{modules-lemma-stalk-internal-hom} we see that
(1) and (2) are equivalent. Since every point of $X$ specializes to
a closed point
(Properties, Lemma \ref{properties-lemma-locally-Noetherian-closed-point})
we see that (2) and (3) are equivalent.
\end{proof}
\begin{lemma}
\label{lemma-dual-reflexive}
Let $X$ be an integral locally Noetherian scheme.
Let $\mathcal{F}$, $\mathcal{G}$ be
coherent $\mathcal{O}_X$-modules.
If $\mathcal{G}$ is reflexive,
then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is reflexive.
\end{lemma}
\begin{proof}
The statement makes sense because
$\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$
is coherent by Cohomology of Schemes, Lemma
\ref{coherent-lemma-tensor-hom-coherent}.
To see the statement is true, see
More on Algebra, Lemma \ref{more-algebra-lemma-dual-reflexive}.
Some details omitted.
\end{proof}
\begin{lemma}
\label{lemma-reflexive-depth-2}
Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$
be a coherent $\mathcal{O}_X$-module. The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is reflexive,
\item for each $x \in X$ one of the following happens
\begin{enumerate}
\item $\mathcal{F}_x$ is a reflexive $\mathcal{O}_{X, x}$-module, or
\item $\text{depth}(\mathcal{O}_{X, x}) \geq 2$ and
$\text{depth}(\mathcal{F}_x) \geq 2$.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted. See More on Algebra, Lemma \ref{more-algebra-lemma-reflexive-depth-2}.
\end{proof}
\noindent
If the scheme is normal, then reflexive is the same thing as
torsion free and $(S_2)$.
\begin{lemma}
\label{lemma-reflexive-over-normal}
Let $X$ be an integral locally Noetherian normal scheme.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is reflexive, and
\item $\mathcal{F}$ is torsion free and has property $(S_2)$.
\end{enumerate}
\end{lemma}
\begin{proof}
This is the scheme theoretic analogue of
More on Algebra, Lemma \ref{more-algebra-lemma-reflexive-over-normal}.
To translate into algebra use Lemma \ref{lemma-check-reflexive-on-affines}.
\end{proof}
\begin{lemma}
\label{lemma-describe-reflexive-hull}
Let $X$ be an integral locally Noetherian normal scheme with
generic point $\eta$. Let $\mathcal{F}$, $\mathcal{G}$ be coherent
$\mathcal{O}_X$-modules. Let $T : \mathcal{G}_\eta \to \mathcal{F}_\eta$
be a linear map. Then $T$ extends to a map
$\mathcal{G} \to \mathcal{F}^{**}$ of $\mathcal{O}_X$-modules
if and only if
\begin{itemize}
\item[$(*)$] for every $x \in X$ with $\dim(\mathcal{O}_{X, x}) = 1$
we have
$$
T\left(\Im(\mathcal{G}_x \to \mathcal{G}_\eta)\right) \subset
\Im(\mathcal{F}_x \to \mathcal{F}_\eta).
$$
\end{itemize}
\end{lemma}
\begin{proof}
Because $\mathcal{F}^{**}$ is torsion free and
$\mathcal{F}_\eta = \mathcal{F}^{**}_\eta$ an extension, if it exists,
is unique. Thus it suffices to prove the lemma over the members of an
open covering of $X$, i.e., we may assume $X$ is affine. In this case
we are asking the following algebra question: Let $R$ be a Noetherian
normal domain with fraction field $K$, let $M$, $N$ be finite $R$-modules,
let $T : M \otimes_R K \to N \otimes_R K$ be a $K$-linear map. When
does $T$ extend to a map $N \to M^{**}$? By More on Algebra, Lemma
\ref{more-algebra-lemma-describe-reflexive-hull}
this happens if and only if $N_\mathfrak p$ maps into
$(M/M_{tors})_\mathfrak p$ for every height $1$ prime $\mathfrak p$ of $R$.
This is exactly condition $(*)$ of the lemma.
\end{proof}
\begin{lemma}
\label{lemma-reflexive-over-regular-dim-2}
Let $X$ be a regular scheme of dimension $\leq 2$.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is reflexive,
\item $\mathcal{F}$ is finite locally free.
\end{enumerate}
\end{lemma}
\begin{proof}
It is clear that a finite locally free module is reflexive.
For the converse, we will show that if $\mathcal{F}$ is
reflexive, then $\mathcal{F}_x$ is a free $\mathcal{O}_{X, x}$-module
for all $x \in X$. This is enough by
Algebra, Lemma \ref{algebra-lemma-finite-projective}
and the fact that $\mathcal{F}$ is coherent.
If $\dim(\mathcal{O}_{X, x}) = 0$, then
$\mathcal{O}_{X, x}$ is a field and the statement is clear.
If $\dim(\mathcal{O}_{X, x}) = 1$, then $\mathcal{O}_{X, x}$
is a discrete valuation ring
(Algebra, Lemma \ref{algebra-lemma-characterize-dvr})
and $\mathcal{F}_x$ is torsion free.
Hence $\mathcal{F}_x$ is free by More on Algebra, Lemma
\ref{more-algebra-lemma-dedekind-torsion-free-flat}.
If $\dim(\mathcal{O}_{X, x}) = 2$, then $\mathcal{O}_{X, x}$
is a regular local ring of dimension $2$. By
More on Algebra, Lemma \ref{more-algebra-lemma-reflexive-over-normal}
we see that $\mathcal{F}_x$ has depth $\geq 2$.
Hence $\mathcal{F}$ is free by
Algebra, Lemma \ref{algebra-lemma-regular-mcm-free}.
\end{proof}
\section{Effective Cartier divisors}
\label{section-effective-Cartier-divisors}
\noindent
We define the notion of an effective Cartier divisor before any other type
of divisor.
\begin{definition}
\label{definition-effective-Cartier-divisor}
Let $S$ be a scheme.
\begin{enumerate}
\item A {\it locally principal closed subscheme} of $S$ is a closed subscheme
whose sheaf of ideals is locally generated by a single element.
\item An {\it effective Cartier divisor} on $S$ is a closed subscheme
$D \subset S$ whose ideal sheaf $\mathcal{I}_D \subset \mathcal{O}_S$
is an invertible $\mathcal{O}_S$-module.
\end{enumerate}
\end{definition}
\noindent
Thus an effective Cartier divisor is a locally principal closed subscheme,
but the converse is not always true. Effective Cartier divisors are closed
subschemes of pure codimension $1$ in the strongest possible sense. Namely
they are locally cut out by a single element which is a nonzerodivisor.
In particular they are nowhere dense.
\begin{lemma}
\label{lemma-characterize-effective-Cartier-divisor}
Let $S$ be a scheme.
Let $D \subset S$ be a closed subscheme.
The following are equivalent:
\begin{enumerate}
\item The subscheme $D$ is an effective Cartier divisor on $S$.
\item For every $x \in D$ there exists an affine open neighbourhood
$\Spec(A) = U \subset S$ of $x$ such that
$U \cap D = \Spec(A/(f))$ with $f \in A$ a nonzerodivisor.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume (1). For every $x \in D$ there exists an affine open neighbourhood
$\Spec(A) = U \subset S$ of $x$ such that
$\mathcal{I}_D|_U \cong \mathcal{O}_U$. In other words, there exists
a section $f \in \Gamma(U, \mathcal{I}_D)$ which freely generates the
restriction $\mathcal{I}_D|_U$. Hence $f \in A$, and the multiplication
map $f : A \to A$ is injective. Also, since $\mathcal{I}_D$ is
quasi-coherent we see that $D \cap U = \Spec(A/(f))$.
\medskip\noindent
Assume (2). Let $x \in D$. By assumption there exists an affine open
neighbourhood $\Spec(A) = U \subset S$ of $x$ such that
$U \cap D = \Spec(A/(f))$ with $f \in A$ a nonzerodivisor.
Then $\mathcal{I}_D|_U \cong \mathcal{O}_U$ since it is equal to
$\widetilde{(f)} \cong \widetilde{A} \cong \mathcal{O}_U$.
Of course $\mathcal{I}_D$ restricted to the open subscheme
$S \setminus D$ is isomorphic to $\mathcal{O}_{S \setminus D}$.
Hence $\mathcal{I}_D$ is an invertible $\mathcal{O}_S$-module.
\end{proof}
\begin{lemma}
\label{lemma-complement-locally-principal-closed-subscheme}
Let $S$ be a scheme. Let $Z \subset S$ be a locally principal closed
subscheme. Let $U = S \setminus Z$. Then $U \to S$ is an affine morphism.
\end{lemma}
\begin{proof}
The question is local on $S$, see
Morphisms, Lemmas \ref{morphisms-lemma-characterize-affine}.
Thus we may assume $S = \Spec(A)$ and $Z = V(f)$ for some $f \in A$.
In this case $U = D(f) = \Spec(A_f)$ is affine hence $U \to S$ is affine.
\end{proof}
\begin{lemma}
\label{lemma-complement-effective-Cartier-divisor}
Let $S$ be a scheme. Let $D \subset S$ be an effective Cartier divisor.
Let $U = S \setminus D$. Then $U \to S$ is an affine morphism and $U$
is scheme theoretically dense in $S$.
\end{lemma}
\begin{proof}
Affineness is Lemma \ref{lemma-complement-locally-principal-closed-subscheme}.
The density question is local on $S$, see
Morphisms, Lemma \ref{morphisms-lemma-characterize-scheme-theoretically-dense}.
Thus we may assume $S = \Spec(A)$ and $D$ corresponding to the
nonzerodivisor $f \in A$, see
Lemma \ref{lemma-characterize-effective-Cartier-divisor}.
Thus $A \subset A_f$ which implies that $U \subset S$ is
scheme theoretically dense, see
Morphisms, Example \ref{morphisms-example-scheme-theoretic-closure}.
\end{proof}
\begin{lemma}
\label{lemma-effective-Cartier-makes-dimension-drop}
Let $S$ be a scheme.
Let $D \subset S$ be an effective Cartier divisor.
Let $s \in D$.
If $\dim_s(S) < \infty$, then $\dim_s(D) < \dim_s(S)$.
\end{lemma}
\begin{proof}
Assume $\dim_s(S) < \infty$.
Let $U = \Spec(A) \subset S$ be an affine open neighbourhood
of $s$ such that $\dim(U) = \dim_s(S)$ and such that $D = V(f)$
for some nonzerodivisor $f \in A$ (see
Lemma \ref{lemma-characterize-effective-Cartier-divisor}).
Recall that $\dim(U)$ is the Krull dimension of the ring $A$
and that $\dim(U \cap D)$ is the Krull dimension of the ring $A/(f)$.
Then $f$ is not contained in any minimal prime of $A$.
Hence any maximal chain of primes in $A/(f)$, viewed as a chain
of primes in $A$, can be extended by adding a minimal prime.
\end{proof}
\begin{definition}
\label{definition-sum-effective-Cartier-divisors}
Let $S$ be a scheme. Given effective Cartier divisors
$D_1$, $D_2$ on $S$ we set $D = D_1 + D_2$ equal to the
closed subscheme of $S$ corresponding to the quasi-coherent
sheaf of ideals
$\mathcal{I}_{D_1}\mathcal{I}_{D_2} \subset \mathcal{O}_S$.
We call this the {\it sum of the effective Cartier divisors
$D_1$ and $D_2$}.
\end{definition}
\noindent
It is clear that we may define the sum $\sum n_iD_i$ given
finitely many effective Cartier divisors $D_i$ on $X$
and nonnegative integers $n_i$.
\begin{lemma}
\label{lemma-sum-effective-Cartier-divisors}
The sum of two effective Cartier divisors is an effective
Cartier divisor.
\end{lemma}
\begin{proof}
Omitted. Locally $f_1, f_2 \in A$ are nonzerodivisors, then also
$f_1f_2 \in A$ is a nonzerodivisor.
\end{proof}
\begin{lemma}
\label{lemma-difference-effective-Cartier-divisors}
Let $X$ be a scheme.
Let $D, D'$ be two effective Cartier divisors on $X$.
If $D \subset D'$ (as closed subschemes of $X$), then
there exists an effective Cartier divisor $D''$ such
that $D' = D + D''$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-sum-closed-subschemes-effective-Cartier}
Let $X$ be a scheme. Let $Z, Y$ be two closed subschemes of $X$
with ideal sheaves $\mathcal{I}$ and $\mathcal{J}$. If $\mathcal{I}\mathcal{J}$
defines an effective Cartier divisor $D \subset X$, then $Z$ and $Y$
are effective Cartier divisors and $D = Z + Y$.
\end{lemma}
\begin{proof}
Applying Lemma \ref{lemma-characterize-effective-Cartier-divisor} we obtain
the following algebra situation: $A$ is a ring, $I, J \subset A$
ideals and $f \in A$ a nonzerodivisor such that $IJ = (f)$.
Thus the result follows from
Algebra, Lemma \ref{algebra-lemma-product-ideals-principal}.
\end{proof}
\begin{lemma}
\label{lemma-sum-effective-Cartier-divisors-union}
Let $X$ be a scheme. Let $D, D' \subset X$ be effective Cartier divisors
such that the scheme theoretic intersection $D \cap D'$ is an effective
Cartier divisor on $D'$. Then $D + D'$ is the scheme theoretic
union of $D$ and $D'$.
\end{lemma}
\begin{proof}
See Morphisms, Definition
\ref{morphisms-definition-scheme-theoretic-intersection-union}
for the definition of scheme theoretic intersection and union.
To prove the lemma working locally
(using Lemma \ref{lemma-characterize-effective-Cartier-divisor})
we obtain the following algebra problem: Given a ring $A$
and nonzerodivisors $f_1, f_2 \in A$ such that $f_1$ maps
to a nonzerodivisor in $A/f_2A$, show that $f_1A \cap f_2A = f_1f_2A$.
We omit the straightforward argument.
\end{proof}
\noindent
Recall that we have defined the inverse image of a closed subscheme
under any morphism of schemes in
Schemes, Definition \ref{schemes-definition-inverse-image-closed-subscheme}.
\begin{lemma}
\label{lemma-pullback-locally-principal}
Let $f : S' \to S$ be a morphism of schemes. Let $Z \subset S$
be a locally principal closed subscheme. Then the inverse image
$f^{-1}(Z)$ is a locally principal closed subscheme of $S'$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{definition}
\label{definition-pullback-effective-Cartier-divisor}
Let $f : S' \to S$ be a morphism of schemes. Let $D \subset S$
be an effective Cartier divisor. We say the {\it pullback of
$D$ by $f$ is defined} if the closed subscheme $f^{-1}(D) \subset S'$
is an effective Cartier divisor. In this case we denote it either
$f^*D$ or $f^{-1}(D)$ and we call it the
{\it pullback of the effective Cartier divisor}.
\end{definition}
\noindent
The condition that $f^{-1}(D)$ is an effective Cartier divisor
is often satisfied in practice. Here is an example lemma.
\begin{lemma}
\label{lemma-pullback-effective-Cartier-defined}
Let $f : X \to Y$ be a morphism of schemes.
Let $D \subset Y$ be an effective Cartier divisor.
The pullback of $D$ by $f$ is defined in each of the following cases:
\begin{enumerate}
\item $X$, $Y$ integral and $f$ dominant,
\item $X$ reduced, and for any generic point $\xi$ of any
irreducible component of $X$ we have $f(\xi) \not \in D$,
\item $X$ is locally Noetherian and for any associated point
$x$ of $X$ we have $f(x) \not \in D$,
\item $X$ is locally Noetherian, has no embedded points, and
for any generic point $\xi$ of any irreducible component of
$X$ we have $f(\xi) \not \in D$,
\item $f$ is flat, and
\item add more here as needed.
\end{enumerate}
\end{lemma}
\begin{proof}
The question is local on $X$, and hence we reduce to the case
where $X = \Spec(A)$, $Y = \Spec(R)$, $f$ is
given by $\varphi : R \to A$ and
$D = \Spec(R/(t))$ where $t \in R$ is a nonzerodivisor.
The goal in each case is to show that $\varphi(t) \in A$
is a nonzerodivisor.
\medskip\noindent
In case (2) this follows as the intersection of all minimal
primes of a ring is the nilradical of the ring, see
Algebra, Lemma \ref{algebra-lemma-Zariski-topology}.
\medskip\noindent
Let us prove (3). By
Lemma \ref{lemma-associated-affine-open}
the associated points of $X$ correspond to the primes
$\mathfrak p \in \text{Ass}(A)$.
By Algebra, Lemma \ref{algebra-lemma-ass-zero-divisors} we have
$\bigcup_{\mathfrak p \in \text{Ass}(A)} \mathfrak p$ is
the set of zerodivisors of $A$. The hypothesis of
(3) is that $\varphi(t) \not \in \mathfrak p$ for
all $\mathfrak p \in \text{Ass}(A)$. Hence $\varphi(t)$
is a nonzerodivisor as desired.
\medskip\noindent
Part (4) follows from (3) and the definitions.
\end{proof}
\begin{lemma}
\label{lemma-pullback-effective-Cartier-divisors-additive}
Let $f : S' \to S$ be a morphism of schemes.
Let $D_1$, $D_2$ be effective Cartier divisors on $S$.
If the pullbacks of $D_1$ and $D_2$ are defined then the
pullback of $D = D_1 + D_2$ is defined and
$f^*D = f^*D_1 + f^*D_2$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\section{Effective Cartier divisors and invertible sheaves}
\label{section-effective-Cartier-invertible}
\noindent
Since an effective Cartier divisor has an invertible ideal sheaf
(Definition \ref{definition-effective-Cartier-divisor}) the
following definition makes sense.
\begin{definition}
\label{definition-invertible-sheaf-effective-Cartier-divisor}
Let $S$ be a scheme. Let $D \subset S$ be an effective Cartier divisor
with ideal sheaf $\mathcal{I}_D$.
\begin{enumerate}
\item The {\it invertible sheaf $\mathcal{O}_S(D)$ associated to $D$}
is defined by
$$
\mathcal{O}_S(D) =
\SheafHom_{\mathcal{O}_S}(\mathcal{I}_D, \mathcal{O}_S) =
\mathcal{I}_D^{\otimes -1}.
$$
\item The {\it canonical section}, usually denoted $1$ or $1_D$, is the
global section of $\mathcal{O}_S(D)$ corresponding to
the inclusion mapping $\mathcal{I}_D \to \mathcal{O}_S$.
\item We write
$\mathcal{O}_S(-D) = \mathcal{O}_S(D)^{\otimes -1} = \mathcal{I}_D$.
\item Given a second effective Cartier divisor $D' \subset S$ we define
$\mathcal{O}_S(D - D') =
\mathcal{O}_S(D) \otimes_{\mathcal{O}_S} \mathcal{O}_S(-D')$.
\end{enumerate}
\end{definition}
\noindent
Some comments. We will see below that the assignment
$D \mapsto \mathcal{O}_S(D)$ turns addition of effective Cartier
divisors (Definition \ref{definition-sum-effective-Cartier-divisors})
into addition in the Picard group of $S$
(Lemma \ref{lemma-invertible-sheaf-sum-effective-Cartier-divisors}).
However, the expression $D - D'$ in the definition above does not
have any geometric meaning. More precisely, we can think of the
set of effective Cartier divisors on $S$ as a commutative monoid
$\text{EffCart}(S)$ whose zero element is the empty effective Cartier divisor.
Then the assignment $(D, D') \mapsto \mathcal{O}_S(D - D')$ defines
a group homomorphism
$$
\text{EffCart}(S)^{gp} \longrightarrow \text{Pic}(S)
$$
where the left hand side is the group completion of
$\text{EffCart}(S)$. In other words, when we write $\mathcal{O}_S(D - D')$
we may think of $D - D'$ as an element of $\text{EffCart}(S)^{gp}$.
\begin{lemma}
\label{lemma-conormal-effective-Cartier-divisor}
Let $S$ be a scheme and let $D \subset S$ be an effective Cartier divisor.
Then the conormal sheaf is $\mathcal{C}_{D/S} = \mathcal{I}_D|D =
\mathcal{O}_S(-D)|_D$ and the normal sheaf is
$\mathcal{N}_{D/S} = \mathcal{O}_S(D)|_D$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-ses-add-divisor}
Let $X$ be a scheme. Let $D, C \subset X$ be
effective Cartier divisors with $C \subset D$ and let $D' = D + C$.
Then there is a short exact sequence
$$
0 \to \mathcal{O}_X(-D)|_C \to \mathcal{O}_{D'} \to \mathcal{O}_D \to 0
$$
of $\mathcal{O}_X$-modules.
\end{lemma}
\begin{proof}
In the statement of the lemma and in the proof we use the equivalence of
Morphisms, Lemma \ref{morphisms-lemma-i-star-equivalence} to think of
quasi-coherent modules on closed subschemes of $X$
as quasi-coherent modules on $X$. Let $\mathcal{I}$ be the ideal
sheaf of $D$ in $D'$. Then there is a short exact sequence
$$
0 \to \mathcal{I} \to \mathcal{O}_{D'} \to \mathcal{O}_D \to 0
$$
because $D \to D'$ is a closed immersion. There is a
canonical surjection
$\mathcal{I} \to \mathcal{I}/\mathcal{I}^2 = \mathcal{C}_{D/D'}$.
We have $\mathcal{C}_{D/X} = \mathcal{O}_X(-D)|_D$ by
Lemma \ref{lemma-conormal-effective-Cartier-divisor}
and there is a canonical surjective map
$$
\mathcal{C}_{D/X} \longrightarrow \mathcal{C}_{D/D'}
$$
see Morphisms, Lemmas \ref{morphisms-lemma-conormal-functorial} and
\ref{morphisms-lemma-conormal-functorial-flat}.
Thus it suffices to show: (a) $\mathcal{I}^2 = 0$ and (b)
$\mathcal{I}$ is an invertible $\mathcal{O}_C$-module.
Both (a) and (b) can be checked locally, hence we may assume
$X = \Spec(A)$, $D = \Spec(A/fA)$ and $C = \Spec(A/gA)$ where
$f, g \in A$ are nonzerodivisors
(Lemma \ref{lemma-characterize-effective-Cartier-divisor}).
Since $C \subset D$ we see
that $f \in gA$. Then $I = fA/fgA$ has square zero and is invertible
as an $A/gA$-module as desired.
\end{proof}
\begin{lemma}
\label{lemma-invertible-sheaf-sum-effective-Cartier-divisors}
Let $S$ be a scheme.
Let $D_1$, $D_2$ be effective Cartier divisors on $S$.
Let $D = D_1 + D_2$.
Then there is a unique isomorphism
$$
\mathcal{O}_S(D_1) \otimes_{\mathcal{O}_S} \mathcal{O}_S(D_2)
\longrightarrow
\mathcal{O}_S(D)
$$
which maps $1_{D_1} \otimes 1_{D_2}$ to $1_D$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-pullback-effective-Cartier-divisors}
Let $f : S' \to S$ be a morphism of schemes.
Let $D$ be a effective Cartier divisors on $S$.
If the pullback of $D$ is defined then
$f^*\mathcal{O}_S(D) = \mathcal{O}_{S'}(f^*D)$
and the canonical section $1_D$ pulls back to
the canonical section $1_{f^*D}$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{definition}
\label{definition-regular-section}
Let $(X, \mathcal{O}_X)$ be a locally ringed space.
Let $\mathcal{L}$ be an invertible sheaf on $X$.
A global section $s \in \Gamma(X, \mathcal{L})$ is called a
{\it regular section} if the map $\mathcal{O}_X \to \mathcal{L}$,
$f \mapsto fs$ is injective.
\end{definition}
\begin{lemma}
\label{lemma-regular-section-structure-sheaf}
Let $X$ be a locally ringed space. Let $f \in \Gamma(X, \mathcal{O}_X)$.
The following are equivalent:
\begin{enumerate}
\item $f$ is a regular section, and
\item for any $x \in X$ the image $f \in \mathcal{O}_{X, x}$
is a nonzerodivisor.
\end{enumerate}
If $X$ is a scheme these are also equivalent to
\begin{enumerate}
\item[(3)] for any affine open $\Spec(A) = U \subset X$
the image $f \in A$ is a nonzerodivisor,
\item[(4)] there exists an affine open covering
$X = \bigcup \Spec(A_i)$ such that
the image of $f$ in $A_i$ is a nonzerodivisor for all $i$.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\noindent
Note that a global section $s$ of an invertible $\mathcal{O}_X$-module
$\mathcal{L}$ may be seen as an $\mathcal{O}_X$-module map
$s : \mathcal{O}_X \to \mathcal{L}$. Its dual is therefore a
map $s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_X$.
(See Modules, Definition \ref{modules-definition-powers}
for the definition of the dual invertible sheaf.)
\begin{definition}
\label{definition-zero-scheme-s}
Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf.
Let $s \in \Gamma(X, \mathcal{L})$ be a global section.
The {\it zero scheme} of $s$ is the closed subscheme $Z(s) \subset X$
defined by the quasi-coherent sheaf of ideals
$\mathcal{I} \subset \mathcal{O}_X$ which is the image of the
map $s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_X$.
\end{definition}
\begin{lemma}
\label{lemma-zero-scheme}
Let $X$ be a scheme.
Let $\mathcal{L}$ be an invertible sheaf.
Let $s \in \Gamma(X, \mathcal{L})$.
\begin{enumerate}
\item Consider closed immersions $i : Z \to X$ such that
$i^*s \in \Gamma(Z, i^*\mathcal{L})$ is zero
ordered by inclusion. The zero scheme $Z(s)$ is the
maximal element of this ordered set.
\item For any morphism of schemes $f : Y \to X$ we have
$f^*s = 0$ in $\Gamma(Y, f^*\mathcal{L})$ if and only if
$f$ factors through $Z(s)$.
\item The zero scheme $Z(s)$ is a locally principal closed subscheme.
\item The zero scheme $Z(s)$ is an effective Cartier divisor
if and only if $s$ is a regular section of $\mathcal{L}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-characterize-OD}
\begin{slogan}
Effective Cartier divisors on a scheme are the same as invertible sheaves
with fixed regular global section.
\end{slogan}
Let $X$ be a scheme.
\begin{enumerate}
\item If $D \subset X$ is an effective Cartier divisor, then
the canonical section $1_D$ of $\mathcal{O}_X(D)$ is regular.
\item Conversely, if $s$ is a regular section of the invertible
sheaf $\mathcal{L}$, then there exists a unique effective
Cartier divisor $D = Z(s) \subset X$ and a unique isomorphism
$\mathcal{O}_X(D) \to \mathcal{L}$ which maps $1_D$ to $s$.
\end{enumerate}
The constructions
$D \mapsto (\mathcal{O}_X(D), 1_D)$ and $(\mathcal{L}, s) \mapsto Z(s)$
give mutually inverse maps
$$
\left\{
\begin{matrix}
\text{effective Cartier divisors on }X
\end{matrix}
\right\}
\leftrightarrow
\left\{
\begin{matrix}
\text{pairs }(\mathcal{L}, s)\text{ consisting of an invertible}\\
\mathcal{O}_X\text{-module and a regular global section}
\end{matrix}
\right\}
$$
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{remark}
\label{remark-ses-regular-section}
Let $X$ be a scheme, $\mathcal{L}$ an invertible $\mathcal{O}_X$-module,
and $s$ a regular section of $\mathcal{L}$. Then the zero scheme
$D = Z(s)$ is an effective Cartier divisor on $X$ and there are
short exact sequences
$$
0 \to \mathcal{O}_X \to \mathcal{L} \to i_*(\mathcal{L}|_D) \to 0
\quad\text{and}\quad
0 \to \mathcal{L}^{\otimes -1} \to \mathcal{O}_X \to i_*\mathcal{O}_D \to 0.
$$
Given an effective Cartier divisor $D \subset X$ using
Lemmas \ref{lemma-characterize-OD} and
\ref{lemma-conormal-effective-Cartier-divisor}
we get
$$
0 \to \mathcal{O}_X \to \mathcal{O}_X(D) \to i_*(\mathcal{N}_{D/X}) \to 0
\quad\text{and}\quad
0 \to \mathcal{O}_X(-D) \to \mathcal{O}_X \to i_*(\mathcal{O}_D) \to 0
$$
\end{remark}
\section{Effective Cartier divisors on Noetherian schemes}
\label{section-Noetherian-effective-Cartier}
\noindent
In the locally Noetherian setting most of the discussion of
effective Cartier divisors and regular sections simplifies somewhat.
\begin{lemma}
\label{lemma-regular-section-associated-points}
Let $X$ be a locally Noetherian scheme. Let $\mathcal{L}$ be an invertible
$\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. Then $s$
is a regular section if and only if $s$ does not vanish in the associated
points of $X$.
\end{lemma}
\begin{proof}
Omitted. Hint: reduce to the affine case and $\mathcal{L}$ trivial
and then use Lemma \ref{lemma-regular-section-structure-sheaf} and
Algebra, Lemma \ref{algebra-lemma-ass-zero-divisors}.
\end{proof}
\begin{lemma}
\label{lemma-effective-Cartier-in-points}
Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a closed subscheme
corresponding to the quasi-coherent ideal sheaf
$\mathcal{I} \subset \mathcal{O}_X$.
\begin{enumerate}
\item If for every $x \in D$ the ideal
$\mathcal{I}_x \subset \mathcal{O}_{X, x}$
can be generated by one element, then $D$ is locally principal.
\item If for every $x \in D$ the ideal
$\mathcal{I}_x \subset \mathcal{O}_{X, x}$
can be generated by a single nonzerodivisor, then $D$ is an
effective Cartier divisor.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\Spec(A)$ be an affine neighbourhood of a point $x \in D$.
Let $\mathfrak p \subset A$ be the prime corresponding to $x$.
Let $I \subset A$ be the ideal defining the trace of $D$ on
$\Spec(A)$. Since $A$ is Noetherian (as $X$ is Noetherian)
the ideal $I$ is generated by finitely many elements, say
$I = (f_1, \ldots, f_r)$. Under the assumption of (1) we have
$I_\mathfrak p = (f)$ for some $f \in A_\mathfrak p$.
Then $f_i = g_i f$ for some $g_i \in A_\mathfrak p$.
Write $g_i = a_i/h_i$ and $f = f'/h$ for some
$h_i, h \in A$, $h_i, h \not \in \mathfrak p$.
Then $I_{h_1 \ldots h_r h} \subset A_{h_1 \ldots h_r h}$ is
principal, because it is generated by $f'$. This proves (1).
For (2) we may assume $I = (f)$. The assumption implies
that the image of $f$ in $A_\mathfrak p$ is a nonzerodivisor.
Then $f$ is a nonzerodivisor on a neighbourhood of $x$ by
Algebra, Lemma \ref{algebra-lemma-regular-sequence-in-neighbourhood}.
This proves (2).
\end{proof}
\begin{lemma}
\label{lemma-effective-Cartier-codimension-1}
Let $X$ be a locally Noetherian scheme.
\begin{enumerate}
\item Let $D \subset X$ be a locally principal closed subscheme.
Let $\xi \in D$ be a generic point of an irreducible component of $D$.
Then $\dim(\mathcal{O}_{X, \xi}) \leq 1$.
\item Let $D \subset X$ be an effective Cartier divisor.
Let $\xi \in D$ be a generic point of an irreducible component of $D$.
Then $\dim(\mathcal{O}_{X, \xi}) = 1$.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof of (1). By assumption we may assume $X = \Spec(A)$ and
$D = \Spec(A/(f))$ where $A$ is a Noetherian ring and $f \in A$.
Let $\xi$ correspond to the prime ideal $\mathfrak p \subset A$.
The assumption that $\xi$ is a generic point of an irreducible
component of $D$ signifies $\mathfrak p$ is minimal over $(f)$.
Thus $\dim(A_\mathfrak p) \leq 1$ by
Algebra, Lemma \ref{algebra-lemma-minimal-over-1}.
\medskip\noindent
Proof of (2). By part (1) we see that $\dim(\mathcal{O}_{X, \xi}) \leq 1$.
On the other hand, the local equation $f$ is a nonzerodivisor in
$A_\mathfrak p$ by Lemma \ref{lemma-characterize-effective-Cartier-divisor}
which implies the dimension is at least $1$ (because there must be a
prime in $A_\mathfrak p$ not containing $f$ by the elementary
Algebra, Lemma \ref{algebra-lemma-Zariski-topology}).
\end{proof}
\begin{lemma}
\label{lemma-integral-effective-Cartier-divisor-dvr}
Let $X$ be a Noetherian scheme. Let $D \subset X$ be an
integral closed subscheme which is also an
effective Cartier divisor. Then the local ring of $X$
at the generic point of $D$ is a discrete valuation ring.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-characterize-effective-Cartier-divisor}
we may assume $X = \Spec(A)$ and $D = \Spec(A/(f))$
where $A$ is a Noetherian ring and $f \in A$ is a nonzerodivisor.
The assumption that $D$ is integral signifies that $(f)$ is prime.
Hence the local ring of $X$ at the generic point is $A_{(f)}$
which is a Noetherian local ring whose maximal ideal is generated by
a nonzerodivisor. Thus it is a discrete valuation ring by
Algebra, Lemma \ref{algebra-lemma-characterize-dvr}.
\end{proof}
\begin{lemma}
\label{lemma-effective-Cartier-divisor-Sk}
Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be an
effective Cartier divisor. If $X$ is $(S_k)$, then $D$ is $(S_{k - 1})$.
\end{lemma}
\begin{proof}
Let $x \in D$. Then $\mathcal{O}_{D, x} = \mathcal{O}_{X, x}/(f)$ where
$f \in \mathcal{O}_{X, x}$ is a nonzerodivisor. By assumption we have
$\text{depth}(\mathcal{O}_{X, x}) \geq \min(\dim(\mathcal{O}_{X, x}), k)$.
By Algebra, Lemma \ref{algebra-lemma-depth-drops-by-one} we have
$\text{depth}(\mathcal{O}_{D, x}) = \text{depth}(\mathcal{O}_{X, x}) - 1$
and by Algebra, Lemma \ref{algebra-lemma-one-equation}
$\dim(\mathcal{O}_{D, x}) = \dim(\mathcal{O}_{X, x}) - 1$.
It follows that
$\text{depth}(\mathcal{O}_{D, x}) \geq \min(\dim(\mathcal{O}_{D, x}), k - 1)$
as desired.
\end{proof}
\begin{lemma}
\label{lemma-normal-effective-Cartier-divisor-S1}
Let $X$ be a locally Noetherian normal scheme. Let $D \subset X$ be an
effective Cartier divisor. Then $D$ is $(S_1)$.
\end{lemma}
\begin{proof}
By Properties, Lemma \ref{properties-lemma-criterion-normal}
we see that $X$ is $(S_2)$. Thus we conclude by
Lemma \ref{lemma-effective-Cartier-divisor-Sk}.
\end{proof}
\begin{lemma}
\label{lemma-weil-divisor-is-cartier-UFD}
Let $X$ be a Noetherian scheme. Let $D \subset X$ be a integral
closed subscheme. Assume that
\begin{enumerate}
\item $D$ has codimension $1$ in $X$, and
\item $\mathcal{O}_{X, x}$ is a UFD for all $x \in D$.
\end{enumerate}
Then $D$ is an effective Cartier divisor.
\end{lemma}
\begin{proof}
Let $x \in D$ and set $A = \mathcal{O}_{X, x}$. Let $\mathfrak p \subset A$
correspond to the generic point of $D$. Then $A_\mathfrak p$ has dimension
$1$ by assumption (1). Thus $\mathfrak p$ is a prime ideal of height $1$.
Since $A$ is a UFD this implies that $\mathfrak p = (f)$ for some $f \in A$.
Of course $f$ is a nonzerodivisor and we conclude by
Lemma \ref{lemma-effective-Cartier-in-points}.
\end{proof}
\begin{lemma}
\label{lemma-codim-1-part}
Let $X$ be a Noetherian scheme. Let $Z \subset X$ be a closed subscheme.
Assume there exist integral effective Cartier divisors $D_i \subset X$
and a closed subset $Z' \subset X$ of codimension $\geq 2$ such that
$Z \subset Z' \cup \bigcup D_i$ set-theoretically.
Then there exists an effective Cartier divisor of the form
$$
D = \sum a_i D_i \subset Z
$$
such that $D \to Z$ is an isomorphism away from codimension $2$ in $X$.
The existence of the $D_i$ is guaranteed if $\mathcal{O}_{X, x}$
is a UFD for all $x \in Z$ or if $X$ is regular.
\end{lemma}
\begin{proof}
Let $\xi_i \in D_i$ be the generic point and let
$\mathcal{O}_i = \mathcal{O}_{X, \xi_i}$ be the local ring
which is a discrete valuation ring by
Lemma \ref{lemma-integral-effective-Cartier-divisor-dvr}.
Let $a_i \geq 0$ be the minimal valuation of an element of
$\mathcal{I}_{Z, \xi_i} \subset \mathcal{O}_i$.
We claim that the effective Cartier divisor $D = \sum a_i D_i$ works.
\medskip\noindent
Namely, suppose that $x \in X$. Let $A = \mathcal{O}_{X, x}$.
Let $f_i \in A$ be a local equation for $D_i$;
we only consider those $i$ such that $x \in D_i$. Then $f_i$ is
a prime element of $A$ and $\mathcal{O}_i = A_{(f_i)}$. Let
$I = \mathcal{I}_{Z, x} \subset A$. By our choice of $a_i$ we have
$I A_{(f_i)} = f_i^{a_i}A_{(f_i)}$. It follows that
$I \subset (\prod f_i^{a_i})$ because the $f_i$ are prime elements of $A$.
This proves that $\mathcal{I}_Z \subset \mathcal{I}_D$, i.e., that
$D \subset Z$. Moreover, we also see that $D$ and $Z$ agree at the $\xi_i$,
which proves the final assertion.
\medskip\noindent
To see the final statements we argue as follows. A regular local
ring is a UFD (More on Algebra, Lemma
\ref{more-algebra-lemma-regular-local-UFD}) hence it suffices
to argue in the UFD case. In that case, let
$D_i$ be the irreducible components of $Z$
which have codimension $1$ in $X$.
By Lemma \ref{lemma-weil-divisor-is-cartier-UFD} each $D_i$
is an effective Cartier divisor.
\end{proof}
\begin{lemma}
\label{lemma-codimension-1-is-effective-Cartier}
Let $Z \subset X$ be a closed subscheme of a Noetherian scheme. Assume
\begin{enumerate}
\item $Z$ has no embedded points,
\item every irreducible component of $Z$ has codimension $1$ in $X$,
\item every local ring $\mathcal{O}_{X, x}$, $x \in Z$ is
a UFD or $X$ is regular.
\end{enumerate}
Then $Z$ is an effective Cartier divisor.
\end{lemma}
\begin{proof}
Let $D = \sum a_i D_i$ be as in Lemma \ref{lemma-codim-1-part}
where $D_i \subset Z$ are the irreducible components of $Z$.
If $D \to Z$ is not an isomorphism, then $\mathcal{O}_Z \to \mathcal{O}_D$
has a nonzero kernel sitting in codimension $\geq 2$. This
would mean that $Z$ has embedded points, which is forbidden
by assumption (1). Hence $D \cong Z$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-UFD-one-equation-CM}
Let $R$ be a Noetherian UFD. Let $I \subset R$ be an ideal
such that $R/I$ has no embedded primes and such that
every minimal prime over $I$ has height $1$.
Then $I = (f)$ for some $f \in R$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-codimension-1-is-effective-Cartier}
the ideal sheaf $\tilde I$ is invertible on $\Spec(R)$.
By More on Algebra, Lemma \ref{more-algebra-lemma-UFD-Pic-trivial}
it is generated by a single element.
\end{proof}
\begin{lemma}
\label{lemma-effective-Cartier-divisor-is-a-sum}
Let $X$ be a Noetherian scheme. Let $D \subset X$ be an effective
Cartier divisor. Assume that there exist integral effective Cartier
divisors $D_i \subset X$ such that $D \subset \bigcup D_i$
set theoretically. Then $D = \sum a_i D_i$ for some $a_i \geq 0$.
The existence of the $D_i$ is guaranteed if $\mathcal{O}_{X, x}$
is a UFD for all $x \in D$ or if $X$ is regular.
\end{lemma}
\begin{proof}
Choose $a_i$ as in Lemma \ref{lemma-codim-1-part} and set $D' = \sum a_i D_i$.
Then $D' \to D$ is an inclusion of effective Cartier divisors which
is an isomorphism away from codimension $2$ on $X$. Pick $x \in X$.
Set $A = \mathcal{O}_{X, x}$ and let $f, f' \in A$ be the nonzerodivisor
generating the ideal of $D, D'$ in $A$. Then $f = gf'$ for some $g \in A$.
Moreover, for every prime $\mathfrak p$ of height $\leq 1$ of $A$ we see
that $g$ maps to a unit of $A_\mathfrak p$. This implies that $g$ is
a unit because the minimal primes over $(g)$ have height $1$
(Algebra, Lemma \ref{algebra-lemma-minimal-over-1}).
\end{proof}
\begin{lemma}
\label{lemma-quasi-projective-Noetherian-pic-effective-Cartier}
Let $X$ be a Noetherian scheme which has an ample invertible sheaf.
Then every invertible $\mathcal{O}_X$-module is isomorphic to
$$
\mathcal{O}_X(D - D') =
\mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D')^{\otimes -1}
$$
for some effective Cartier divisors $D, D'$ in $X$.
\end{lemma}
\begin{proof}
Let $x_1, \ldots, x_n$ be the associated points of $X$
(Lemma \ref{lemma-finite-ass}). Let $\mathcal{L}$ be an ample invertible sheaf.
There exists an $n > 0$ and a section
$s \in \Gamma(X, \mathcal{L}^{\otimes n})$ such that $X_s = \Spec(A)$
is affine and such that $x_i \in X_s$ for $i = 1, \ldots, n$
(Properties, Lemma \ref{properties-lemma-ample-finite-set-in-principal-affine}).
Let $\mathfrak p_1, \ldots, \mathfrak p_n \subset A$ be the
prime ideals corresponding to $x_1, \ldots, x_n$.
\medskip\noindent
Then $\mathcal{N}|_{X_s}$ corresponds to an invertible $A$-module $N$.
Choose an element $t \in N$, $t \not \in \mathfrak p_iN$ for all $i$.
Such an element exists. This is clear if $n = 1$. If $n > 1$ first
rearrange the primes such that $\mathfrak p_i \not \subset \mathfrak p_n$
for all $i < n$. Then using induction choose an element
$t \in N$ with $t \not \in \mathfrak p_i N$ for $i < n$.
Then we are done if $t \not \in \mathfrak p_nN$. Otherwise, pick an
$t' \in N$, $t' \not \in \mathfrak p_nN$ and $f_i \in \mathfrak p_i$,
$f_i \not \in \mathfrak p_n$. The element $t + f_1 f_2 \ldots f_{n - 1}t'$
will be as desired.
\medskip\noindent
By Properties, Lemma \ref{properties-lemma-invert-s-sections}
we see that for some $e \geq 0$ the section $s^e|_U t$ extends to
a global section $\tau$ of $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$.
Thus both $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$ and
$\mathcal{L}^{\otimes e}$ are invertible sheaves which have global sections
which generate the stalks at the associated points of $X$.
Thus these are regular sections by
Lemma \ref{lemma-regular-section-associated-points}.
Hence $\mathcal{L}^{\otimes e} \otimes \mathcal{N} \cong \mathcal{O}_X(D)$
and $\mathcal{L}^{\otimes e} \cong \mathcal{O}_X(D')$ for some
effective Cartier divisors, see Lemma \ref{lemma-characterize-OD}.
\end{proof}
\begin{lemma}
\label{lemma-wedge-product-ses}
Let $X$ be an integral regular scheme of dimension $2$.
Let $i : D \to X$ be the immersion of an effective Cartier divisor.
Let $\mathcal{F} \to \mathcal{F}' \to i_*\mathcal{G} \to 0$
be an exact sequence of coherent $\mathcal{O}_X$-modules.
Assume
\begin{enumerate}
\item $\mathcal{F}, \mathcal{F}'$ are locally free of rank $r$ on a nonempty
open of $X$,
\item $D$ is an integral scheme,
\item $\mathcal{G}$ is a finite locally free $\mathcal{O}_D$-module
of rank $s$.
\end{enumerate}
Then $\mathcal{L} = (\wedge^r\mathcal{F})^{**}$ and
$\mathcal{L}' = (\wedge^r \mathcal{F}')^{**}$
are invertible $\mathcal{O}_X$-modules and
$\mathcal{L}' \cong \mathcal{L}(k D)$ for some
$k \in \{0, \ldots, \min(s, r)\}$.
\end{lemma}
\begin{proof}
The first statement follows from Lemma \ref{lemma-reflexive-over-regular-dim-2}
as assumption (1) implies that $\mathcal{L}$ and $\mathcal{L}'$
have rank $1$. Taking $\wedge^r$ and double duals are functors, hence
we obtain a canonical map $\sigma : \mathcal{L} \to \mathcal{L}'$
which is an isomorphism over the nonempty open of (1), hence
nonzero. To finish the proof, it suffices to see that
$\sigma$ viewed as a global section of
$\mathcal{L}' \otimes \mathcal{L}^{\otimes -1}$ does not
vanish at any codimension point of $X$, except at the generic
point of $D$ and there with vanishing order at most $\min(s, r)$.
\medskip\noindent
Translated into algebra, we arrive at the following problem:
Let $(A, \mathfrak m, \kappa)$ be a discrete valuation ring
with fraction field $K$. Let $M \to M' \to N \to 0$ be an exact sequence
of finite $A$-modules with $\dim_K(M \otimes K) = \dim_K(M' \otimes K) = r$
and with $N \cong \kappa^{\oplus s}$. Show that the induced map
$L = \wedge^r(M)^{**} \to L' = \wedge^r(M')^{**}$ vanishes to
order at most $\min(s, r)$. We will use the structure theorem for
modules over $A$, see
More on Algebra, Lemma
\ref{more-algebra-lemma-generalized-valuation-ring-modules} or
\ref{more-algebra-lemma-modules-PID}.
Dividing out a finite $A$-module by a torsion submodule does not
change the double dual.
Thus we may replace $M$ by $M/M_{tors}$ and $M'$ by
$M'/\Im(M_{tors} \to M')$ and assume that $M$ is torsion free.
Then $M \to M'$ is injective and $M'_{tors} \to N$ is injective.
Hence we may replace $M'$ by $M'/M'_{tors}$ and $N$ by $N/M'_{tors}$.
Thus we reduce to the case where $M$ and $M'$ are free of rank $r$
and $N \cong \kappa^{\oplus s}$. In this case $\sigma$
is the determinant of $M \to M'$ and vanishes to order $s$
for example by Algebra, Lemma \ref{algebra-lemma-order-vanishing-determinant}.
\end{proof}
\section{Complements of affine opens}
\label{section-complement-affine-open}
\noindent
In this section we discuss the result that the complement of an
affine open in a variety has pure codimension $1$.
\begin{lemma}
\label{lemma-affine-punctured-spec}
Let $(A, \mathfrak m)$ be a Noetherian local ring.
The punctured spectrum $U = \Spec(A) \setminus \{\mathfrak m\}$
of $A$ is affine if and only if $\dim(A) \leq 1$.
\end{lemma}
\begin{proof}
If $\dim(A) = 0$, then $U$ is empty hence affine (equal to the spectrum of
the $0$ ring). If $\dim(A) = 1$, then we can choose an element
$f \in \mathfrak m$ not contained in any of the finite number of minimal
primes of $A$
(Algebra, Lemmas \ref{algebra-lemma-Noetherian-irreducible-components} and
\ref{algebra-lemma-silly}). Then $U = \Spec(A_f)$
is affine.
\medskip\noindent
The converse is more interesting. We will give a somewhat nonstandard proof
and discuss the standard argument in a remark below.
Assume $U = \Spec(B)$ is affine. Since affineness and dimension are not
affecting by going to the reduction we may replace $A$ by the quotient by
its ideal of nilpotent elements and assume $A$ is reduced.
Set $Q = B/A$ viewed as an $A$-module.
The support of $Q$ is $\{\mathfrak m\}$ as $A_\mathfrak p = B_\mathfrak p$
for all nonmaximal primes $\mathfrak p$ of $A$.
We may assume $\dim(A) \geq 1$, hence as above we can pick
$f \in \mathfrak m$ not contained in any of the minimal ideals of $A$.
Since $A$ is reduced this implies that $f$ is a nonzerodivisor.
In particular $\dim(A/fA) = \dim(A) - 1$, see
Algebra, Lemma \ref{algebra-lemma-one-equation}.
Applying the snake lemma to multiplication by $f$ on the short
exact sequence $0 \to A \to B \to Q \to 0$ we obtain
$$
0 \to Q[f] \to A/fA \to B/fB \to Q/fQ \to 0
$$
where $Q[f] = \Ker(f : Q \to Q)$.
This implies that $Q[f]$ is a finite $A$-module. Since the support of
$Q[f]$ is $\{\mathfrak m\}$ we see $l = \text{length}_A(Q[f]) < \infty$
(Algebra, Lemma \ref{algebra-lemma-support-point}).
Set $l_n = \text{length}_A(Q[f^n])$. The exact sequence
$$
0 \to Q[f^n] \to Q[f^{n + 1}] \xrightarrow{f^n} Q[f]
$$
shows inductively that $l_n < \infty$ and that $l_n \leq l_{n + 1}$.
Considering the exact sequence
$$
0 \to Q[f] \to Q[f^{n + 1}] \xrightarrow{f} Q[f^n] \to Q/fQ
$$
and we see that the image of $Q[f^n]$ in $Q/fQ$ has length
$l_n - l_{n + 1} + l \leq l$. Since $Q = \bigcup Q[f^n]$ we
find that the length of $Q/fQ$ is at most $l$, i.e., bounded.
Thus $Q/fQ$ is a finite $A$-module. Hence $A/fA \to B/fB$ is a
finite ring map, in particular induces a closed map on spectra
(Algebra, Lemmas \ref{algebra-lemma-integral-going-up} and
\ref{algebra-lemma-going-up-closed}).
On the other hand $\Spec(B/fB)$ is the punctured spectrum of $\Spec(A/fA)$.
This is a contradiction unless $\Spec(B/fB) = \emptyset$ which
means that $\dim(A/fA) = 0$ as desired.
\end{proof}
\begin{remark}
\label{remark-affine-punctured-spectrum-standard-proof}
If $(A, \mathfrak m)$ is a Noetherian local normal domain of
dimension $\geq 2$ and $U$
is the punctured spectrum of $A$, then $\Gamma(U, \mathcal{O}_U) = A$.
This algebraic version of Hartog's theorem follows from the fact that
$A = \bigcap_{\text{height}(\mathfrak p) = 1} A_\mathfrak p$
we've seen in Algebra, Lemma
\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.
Thus in this case $U$ cannot be affine (since it would force $\mathfrak m$
to be a point of $U$). This is often used as the starting point of
the proof of Lemma \ref{lemma-affine-punctured-spec}.
To reduce the case of a general Noetherian local ring to this case,
we first complete (to get a Nagata local ring),
then replace $A$ by $A/\mathfrak q$ for a suitable minimal prime,
and then normalize. Each of these steps does not change the
dimension and we obtain a contradiction.
You can skip the completion step, but then the normalization in
general is not a Noetherian domain. However, it is still a
Krull domain of the same dimension (this is proved using
Krull-Akizuki) and one can apply the same argument.
\end{remark}
\begin{remark}
\label{remark-affine-puctured-spectrum-general}
It is not clear how to characterize the non-Noetherian local
rings $(A, \mathfrak m)$ whose punctured spectrum is affine.
Such a ring has a finitely generated ideal $I$ with
$\mathfrak m = \sqrt{I}$. Of course if we can take $I$
generated by $1$ element, then $A$ has an affine puncture
spectrum; this gives lots of non-Noetherian examples.
Conversely, it follows from the argument in the proof of
Lemma \ref{lemma-affine-punctured-spec}
that such a ring cannot possess a nonzerodivisor $f \in \mathfrak m$
with $H^0_I(A/fA) = 0$ (so $A$ cannot have a regular sequence
of length $2$). Moreover, the same holds for any ring $A'$ which is
the target of a local homomorphism of local rings $A \to A'$ such that
$\mathfrak m_{A'} = \sqrt{\mathfrak mA'}$.
\end{remark}
\begin{lemma}
\label{lemma-complement-affine-open-immersion}
\begin{reference}
\cite[EGA IV, Corollaire 21.12.7]{EGA4}
\end{reference}
Let $X$ be a locally Noetherian scheme. Let $U \subset X$ be an open subscheme
such that the inclusion morphism $U \to X$ is affine.
For every generic point $\xi$ of an irreducible component of
$X \setminus U$ the local ring $\mathcal{O}_{X, \xi}$
has dimension $\leq 1$. If $U$ is dense or if $\xi$ is in the closure
of $U$, then $\dim(\mathcal{O}_{X, \xi}) = 1$.
\end{lemma}
\begin{proof}
Since $\xi$ is a generic point of $X \setminus U$, we see that
$$
U_\xi = U \times_X \Spec(\mathcal{O}_{X, \xi}) \subset
\Spec(\mathcal{O}_{X, \xi})
$$
is the punctured spectrum of $\mathcal{O}_{X, \xi}$ (hint: use
Schemes, Lemma \ref{schemes-lemma-specialize-points}).
As $U \to X$ is affine, we see that $U_\xi \to \Spec(\mathcal{O}_{X, \xi})$
is affine (Morphisms, Lemma \ref{morphisms-lemma-base-change-affine})
and we conclude that $U_\xi$ is affine.
Hence $\dim(\mathcal{O}_{X, \xi}) \leq 1$ by
Lemma \ref{lemma-affine-punctured-spec}.
If $\xi \in \overline{U}$, then there is a specialization
$\eta \to \xi$ where $\eta \in U$ (just take $\eta$ a generic
point of an irreducible component of $\overline{U}$ which
contains $\xi$; since $\overline{U}$ is locally Noetherian,
hence locally has finitely many irreducible components, we see that
$\eta \in U$). Then $\eta \in \Spec(\mathcal{O}_{X, \xi})$ and
we see that the dimension cannot be $0$.
\end{proof}
\begin{lemma}
\label{lemma-complement-affine-open}
Let $X$ be a separated locally Noetherian scheme. Let $U \subset X$ be an
affine open. For every generic point $\xi$ of an irreducible component of
$X \setminus U$ the local ring $\mathcal{O}_{X, \xi}$
has dimension $\leq 1$. If $U$ is dense or if $\xi$ is in the closure
of $U$, then $\dim(\mathcal{O}_{X, \xi}) = 1$.
\end{lemma}
\begin{proof}
This follows from Lemma \ref{lemma-complement-affine-open-immersion}
because the morphism $U \to X$ is affine by
Morphisms, Lemma \ref{morphisms-lemma-affine-permanence}.
\end{proof}
\noindent
The following lemma can sometimes be used to produce effective
Cartier divisors.
\begin{lemma}
\label{lemma-complement-open-affine-effective-cartier-divisor}
Let $X$ be a Noetherian separated scheme. Let $U \subset X$ be
a dense affine open. If $\mathcal{O}_{X, x}$ is a UFD for all
$x \in X \setminus U$, then there exists an effective Cartier
divisor $D \subset X$ with $U = X \setminus D$.
\end{lemma}
\begin{proof}
Since $X$ is Noetherian, the complement $X \setminus U$ has finitely
many irreducible components $D_1, \ldots, D_r$
(Properties, Lemma \ref{properties-lemma-Noetherian-irreducible-components}
applied to the reduced induced subscheme structure on $X \setminus U$).
Each $D_i \subset X$ has codimension $1$ by
Lemma \ref{lemma-complement-affine-open}
(and Properties, Lemma \ref{properties-lemma-codimension-local-ring}).
Thus $D_i$ is an effective Cartier divisor by
Lemma \ref{lemma-weil-divisor-is-cartier-UFD}.
Hence we can take $D = D_1 + \ldots + D_r$.
\end{proof}
\section{Norms}
\label{section-norms}
\noindent
Let $\pi : X \to Y$ be a finite morphism of schemes and let $d \geq 1$
be an integer. Let us say there exists a
{\it norm of degree $d$ for $\pi$}\footnote{This is nonstandard
notation.} if there exists a multiplicative map
$$
\text{Norm}_\pi : \pi_*\mathcal{O}_X \to \mathcal{O}_Y
$$
of sheaves such that
\begin{enumerate}
\item the composition
$\mathcal{O}_Y \xrightarrow{\pi^\sharp} \pi_*\mathcal{O}_X
\xrightarrow{\text{Norm}_\pi} \mathcal{O}_Y$ equals $g \mapsto g^d$, and
\item if $f \in \mathcal{O}_X(\pi^{-1}V)$
is zero at $x \in \pi^{-1}(V)$, then $\text{Norm}_\pi(f)$
is zero at $\pi(x)$.
\end{enumerate}
We observe that condition (1) forces $\pi$ to be surjective.
Since $\text{Norm}_\pi$ is multiplicative it sends units to units
hence, given $y \in Y$, if $f$ is a regular function on $X$
defined at but nonvanishing at any $x \in X$
with $\pi(x) = y$, then $\text{Norm}_\pi(f)$ is defined
and does not vanish at $y$. This holds without requiring (2);
in fact, the constructions in this section will only require condition (1)
and only certain vanishing properties (which are used in particular
in the proof of Lemma \ref{lemma-norm-ample}) will require property (2).
\begin{lemma}
\label{lemma-finite-trivialize-invertible-upstairs}
Let $\pi : X \to Y$ be a finite morphism of schemes.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
Let $y \in Y$. There exists an open neighbourhood
$V \subset Y$ of $y$ such that $\mathcal{L}|_{\pi^{-1}(V)}$ is trivial.
\end{lemma}
\begin{proof}
Clearly we may assume $Y$ and hence $X$ affine. Since $\pi$ is finite the
fibre $\pi^{-1}(\{y\})$ over $y$ is finite.
Since $X$ is affine, we can pick $s \in \Gamma(X, \mathcal{L})$
not vanishing in any point of $\pi^{-1}(\{y\})$. Namely, we can
pick a finite set $E \subset X$ of closed points such that
every $x \in \pi^{-1}(\{y\})$ specializes to some point of $E$.
For $x \in E$ denote $i_x : x \to X$ the closed immersion.
Then
$\mathcal{L} \to \bigoplus_{x \in E} i_{x, *}i_x^*\mathcal{L}$
is a surjective map of quasi-coherent $\mathcal{O}_X$-modules,
and hence the map
$$
\Gamma(X, \mathcal{L}) \to
\bigoplus\nolimits_{x \in E} \mathcal{L}_x/\mathfrak m_x\mathcal{L}_x
$$
is surjective (as taking global sections is an exact functor on the
category of quasi-coherent $\mathcal{O}_X$-modules, see
Schemes, Lemma \ref{schemes-lemma-equivalence-quasi-coherent}).
Thus we can find an $s \in \Gamma(X, \mathcal{L})$
not vanishing at any point specializing to a point of $E$.
Then $X_s \subset X$ is an open neighbourhood of $\pi^{-1}(\{y\})$.
Since $\pi$ is finite, hence closed, we conclude that there is an
open neighbourhood $V \subset Y$ of $y$ whose inverse image
is contained in $X_s$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-norm-invertible}
Let $\pi : X \to Y$ be a finite morphism of schemes. If there exists
a norm of degree $d$ for $\pi$, then there exists a homomorphism of
abelian groups
$$
\text{Norm}_\pi : \text{Pic}(X) \to \text{Pic}(Y)
$$
such that $\text{Norm}_\pi(\pi^*\mathcal{N}) \cong \mathcal{N}^{\otimes d}$
for all invertible $\mathcal{O}_Y$-modules $\mathcal{N}$.
\end{lemma}
\begin{proof}
We will use the correspondence between isomorphism classes of
invertible $\mathcal{O}_X$-modules and elements of
$H^1(X, \mathcal{O}_X^*)$ given in
Cohomology, Lemma \ref{cohomology-lemma-h1-invertible}
without further mention. We explain how to take the norm of an invertible
$\mathcal{O}_X$-module $\mathcal{L}$. Namely, by
Lemma \ref{lemma-finite-trivialize-invertible-upstairs}
there exists an open covering $Y = \bigcup V_j$ such that
$\mathcal{L}|_{\pi^{-1}V_j}$ is trivial. Choose a generating section
$s_j \in \mathcal{L}(\pi^{-1}V_j)$ for each $j$.
On the overlaps $\pi^{-1}V_j \cap \pi^{-1}V_{j'}$ we can write
$$
s_j = u_{jj'} s_{j'}
$$
for a unique $u_{jj'} \in \mathcal{O}^*_X(\pi^{-1}V_j \cap \pi^{-1}V_{j'})$.
Thus we can consider the elements
$$
v_{jj'} = \text{Norm}_\pi(u_{jj'}) \in \mathcal{O}_Y^*(V_j \cap V_{j'})
$$
These elements satisfy the cocycle condition (because the
$u_{jj'}$ do and $\text{Norm}_\pi$ is multiplicative) and
therefore define an invertible $\mathcal{O}_Y$-module.
We omit the verification that: this is well defined,
additive on Picard groups, and satisfies the property
$\text{Norm}_\pi(\pi^*\mathcal{N}) \cong \mathcal{N}^{\otimes d}$
for all invertible $\mathcal{O}_Y$-modules $\mathcal{N}$.
\end{proof}
\begin{lemma}
\label{lemma-norm-map-invertible}
Let $\pi : X \to Y$ be a finite morphism of schemes. Assume there exists
a norm of degree $d$ for $\pi$. For any $\mathcal{O}_X$-linear map
$\varphi : \mathcal{L} \to \mathcal{L}'$
of invertible $\mathcal{O}_X$-modules there is an $\mathcal{O}_Y$-linear
map
$$
\text{Norm}_\pi(\varphi) :
\text{Norm}_\pi(\mathcal{L})
\longrightarrow
\text{Norm}_\pi(\mathcal{L}')
$$
with $\text{Norm}_\pi(\mathcal{L})$, $\text{Norm}_\pi(\mathcal{L}')$
as in Lemma \ref{lemma-norm-invertible}. Moreover, for
$y \in Y$ the following are equivalent
\begin{enumerate}
\item $\varphi$ is zero at a point of $x \in X$ with $\pi(x) = y$, and
\item $\text{Norm}_\pi(\varphi)$ is zero at $y$.
\end{enumerate}
\end{lemma}
\begin{proof}
We choose an open covering $Y = \bigcup V_j$ such that
$\mathcal{L}$ and $\mathcal{L}'$ are trivial over the opens $\pi^{-1}V_j$.
This is possible by
Lemma \ref{lemma-finite-trivialize-invertible-upstairs}.
Choose generating sections
$s_j$ and $s'_j$ of $\mathcal{L}$ and $\mathcal{L}'$
over the opens $\pi^{-1}V_j$. Then $\varphi(s_j) = f_js'_j$
for some $f_j \in \mathcal{O}_X(\pi^{-1}V_j)$.
Define $\text{Norm}_\pi(\varphi)$ to be multiplication
by $\text{Norm}_\pi(f_j)$ on $V_j$. An simple
calculation involving the cocycles used to construct
$\text{Norm}_\pi(\mathcal{L})$, $\text{Norm}_\pi(\mathcal{L}')$
in the proof of Lemma \ref{lemma-norm-invertible}
shows that this defines
a map as stated in the lemma. The final statement follows
from condition (2) in the definition of a norm map of degree $d$.
Some details omitted.
\end{proof}
\begin{lemma}
\label{lemma-norm-ample}
Let $\pi : X \to Y$ be a finite morphism of schemes. Assume $X$ has
an ample invertible sheaf and there exists a norm of degree $d$
for $\pi$. Then $Y$ has an ample invertible sheaf.
\end{lemma}
\begin{proof}
Let $\mathcal{L}$ be the ample invertible sheaf on $X$ given to us
by assumption. We will prove that $\mathcal{N} = \text{Norm}_\pi(\mathcal{L})$
is ample on $Y$.
\medskip\noindent
Since $X$ is quasi-compact (Properties, Definition
\ref{properties-definition-ample}) and $X \to Y$ surjective
(by the existence of $\text{Norm}_\pi$)
we see that $Y$ is quasi-compact.
Let $y \in Y$ be a point. To finish the proof
we will show that there exists a section $t$ of some positive tensor
power of $\mathcal{N}$ which does not vanish at $y$ such that $Y_t$
is affine. To do this, choose an affine open neighbourhood $V \subset Y$
of $y$. Choose $n \gg 0$ and a section
$s \in \Gamma(X, \mathcal{L}^{\otimes n})$
such that
$$
\pi^{-1}(\{y\}) \subset X_s \subset \pi^{-1}V
$$
by
Properties, Lemma \ref{properties-lemma-ample-finite-set-in-principal-affine}.
Then $t = \text{Norm}_\pi(s)$ is a section of $\mathcal{N}^{\otimes n}$
which does not vanish at $x$ and with $Y_t \subset V$, see
Lemma \ref{lemma-norm-map-invertible}. Then $Y_t$
is affine by Properties, Lemma \ref{properties-lemma-affine-cap-s-open}.
\end{proof}
\begin{lemma}
\label{lemma-norm-quasi-affine}
Let $\pi : X \to Y$ be a finite morphism of schemes. Assume $X$ is quasi-affine
and there exists a norm of degree $d$ for $\pi$. Then $Y$ is quasi-affine.
\end{lemma}
\begin{proof}
By Properties, Lemma \ref{properties-lemma-quasi-affine-O-ample}
we see that $\mathcal{O}_X$ is an ample invertible sheaf on $X$.
The proof of Lemma \ref{lemma-norm-ample} shows that
$\text{Norm}_\pi(\mathcal{O}_X) = \mathcal{O}_Y$
is an ample invertible $\mathcal{O}_Y$-module. Hence
Properties, Lemma \ref{properties-lemma-quasi-affine-O-ample}
shows that $Y$ is quasi-affine.
\end{proof}
\begin{lemma}
\label{lemma-finite-locally-free-has-norm}
Let $\pi : X \to Y$ be a finite locally free morphism of degree $d \geq 1$.
Then there exists a canonical norm of degree $d$ whose formation commutes
with arbitrary base change.
\end{lemma}
\begin{proof}
Let $V \subset Y$ be an affine open such that $(\pi_*\mathcal{O}_X)|_V$
is finite free of rank $d$. Choosing a basis we obtain an isomorphism
$$
\mathcal{O}_V^{\oplus d} \cong (\pi_*\mathcal{O}_X)|_V
$$
For every $f \in \pi_*\mathcal{O}_X(V) = \mathcal{O}_X(\pi^{-1}(V))$
multiplication by $f$ defines a $\mathcal{O}_V$-linear endomorphism
$m_f$ of the displayed free vector bundle. Thus we get a $d \times d$
matrix $M_f \in \text{Mat}(d \times d, \mathcal{O}_Y(V))$ and we can set
$$
\text{Norm}_\pi(f) = \det(M_f)
$$
Since the determinant of a matrix is independent of the choice of
the basis chosen we see that this is well defined which also means
that this construction will glue to a global map as desired.
Compatibility with base change is straightforward from the construction.
\medskip\noindent
Property (1) follows from the fact that the determinant of a
$d \times d$ diagonal matrix with entries $g, g, \ldots, g$ is $g^d$.
To see property (2) we may base change and assume that $Y$ is the
spectrum of a field $k$. Then $X = \Spec(A)$ with $A$ a $k$-algebra
with $\dim_k(A) = d$. If there exists an $x \in X$ such that
$f \in A$ vanishes at $x$, then there exists a map $A \to \kappa$
into a field such that $f$ maps to zero in $\kappa$. Then
$f : A \to A$ cannot be surjective, hence $\det(f : A \to A) = 0$
as desired.
\end{proof}
\begin{lemma}
\label{lemma-norm-in-normal-case}
Let $\pi : X \to Y$ be a finite surjective morphism with $X$ and $Y$
integral and $Y$ normal. Then there exists a norm of degree
$[R(X) : R(Y)]$ for $\pi$.
\end{lemma}
\begin{proof}
Let $\Spec(B) \subset Y$ be an affine open subset and let
$\Spec(A) \subset X$ be its inverse image. Then $A$ and $B$
are domains. Let $K$ be the fraction
field of $A$ and $L$ the fraction field of $B$. Picture:
$$
\xymatrix{
L \ar[r] & K \\
B \ar[u] \ar[r] & A \ar[u]
}
$$
Since $K/L$ is a finite extension, there is a norm map
$\text{Norm}_{K/L} : K^* \to L^*$ of degree $d = [K : L]$; this is given by
mapping $f \in K$ to $\det_L(f : K \to K)$ as in the proof
of Lemma \ref{lemma-finite-locally-free-has-norm}.
Observe that the characteristic polynomial of $f : K \to K$
is a power of the minimal polynomial of $f$ over $L$;
in particular $\text{Norm}_{K/L}(f)$ is a power of the constant
coefficient of the minimal polynomial of $f$ over $L$. Hence by
Algebra, Lemma \ref{algebra-lemma-minimal-polynomial-normal-domain}
$\text{Norm}_{K/L}$ maps $A$ into $B$.
This determines a compatible system of maps
on sections over affines and hence a global norm map
$\text{Norm}_\pi$ of degree $d$.
\medskip\noindent
Property (1) is immediate from the construction.
To see property (2) let $f \in A$ be contained in the
prime ideal $\mathfrak p \subset A$. Let
$f^m + b_1 f^{m - 1} + \ldots + b_m$ be the minimal
polynomial of $f$ over $L$. By
Algebra, Lemma \ref{algebra-lemma-minimal-polynomial-normal-domain}
we have $b_i \in B$. Hence $b_0 \in B \cap \mathfrak p$.
Since $\text{Norm}_{K/L}(f) = b_0^{d/m}$ (see above)
we conclude that the norm vanishes in the image point of $\mathfrak p$.
\end{proof}
\begin{lemma}
\label{lemma-Frobenius-gives-norm-for-reduction}
Let $X$ be a Noetherian scheme. Let $p$ be a prime number such that
$p\mathcal{O}_X = 0$. Then for some $e > 0$ there exists a norm
of degree $p^e$ for $X_{red} \to X$ where $X_{red}$ is the reduction
of $X$.
\end{lemma}
\begin{proof}
Let $A$ be a Noetherian ring with $pA = 0$. Let $I \subset A$ be the
ideal of nilpotent elements. Then $I^n = 0$ for some $n$ (Algebra,
Lemma \ref{algebra-lemma-Noetherian-power}).
Pick $e$ such that $p^e \geq n$. Then
$$
A/I \longrightarrow A,\quad
f \bmod I \longmapsto f^{p^e}
$$
is well defined. This produces a norm of degree $p^e$ for
$\Spec(A/I) \to \Spec(A)$. Now if $X$ is obtained by glueing some
affine schemes $\Spec(A_i)$ then for some $e \gg 0$ these maps
glue to a norm map for $X_{red} \to X$. Details omitted.
\end{proof}
\begin{proposition}
\label{proposition-push-down-ample}
Let $f : X \to Y$ be a finite surjective morphism of schemes.
Assume that $X$ has an ample invertible $\mathcal{O}_X$-module. If
\begin{enumerate}
\item $\pi$ is finite locally free, or
\item $Y$ is an integral normal scheme, or
\item $Y$ is Noetherian, $p\mathcal{O}_Y = 0$, and $X = Y_{red}$,
\end{enumerate}
then $Y$ has an ample invertible $\mathcal{O}_Y$-module.
\end{proposition}
\begin{proof}
Case (1) follows from a combination of
Lemmas \ref{lemma-finite-locally-free-has-norm} and \ref{lemma-norm-ample}.
Case (3) follows from a combination of
Lemmas \ref{lemma-Frobenius-gives-norm-for-reduction} and
\ref{lemma-norm-ample}.
In case (2) we first replace $X$ by an irreducible component of $X$
which dominates $Y$ (viewed as a reduced closed subscheme of $X$).
Then we can apply Lemma \ref{lemma-norm-in-normal-case}.
\end{proof}
\begin{lemma}
\label{lemma-push-down-quasi-affine}
Let $f : X \to Y$ be a finite surjective morphism of schemes.
Assume that $X$ is quasi-affine. If either
\begin{enumerate}
\item $\pi$ is finite locally free, or
\item $Y$ is an integral normal scheme
\end{enumerate}
then $Y$ is quasi-affine.