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modules.tex
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\input{preamble}
% OK, start here.
%
\begin{document}
\title{Sheaves of Modules}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
In this chapter we work out basic notions of sheaves of modules.
This in particular includes the case of abelian sheaves, since
these may be viewed as sheaves of $\underline{\mathbf{Z}}$-modules.
Basic references are \cite{FAC}, \cite{EGA} and \cite{SGA4}.
\medskip\noindent
We work out what happens for sheaves of modules on ringed topoi
in another chapter (see
Modules on Sites, Section \ref{sites-modules-section-introduction}),
although there we will mostly just duplicate the discussion
from this chapter.
\section{Pathology}
\label{section-pathology}
\noindent
A ringed space is a pair consisting of a topological space $X$
and a sheaf of rings $\mathcal{O}$. We allow $\mathcal{O} = 0$
in the definition. In this case the category of modules has a
single object (namely $0$). It is still an abelian category etc,
but it is a little degenerate. Similarly the sheaf $\mathcal{O}$
may be zero over open subsets of $X$, etc.
\medskip\noindent
This doesn't happen when considering locally ringed spaces (as we
will do later).
\section{The abelian category of sheaves of modules}
\label{section-kernels}
\noindent
Let $(X, \mathcal{O}_X)$ be a ringed space, see
Sheaves, Definition \ref{sheaves-definition-ringed-space}.
Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}_X$-modules, see
Sheaves, Definition \ref{sheaves-definition-sheaf-modules}.
Let $\varphi, \psi : \mathcal{F} \to \mathcal{G}$
be morphisms of sheaves of $\mathcal{O}_X$-modules.
We define $\varphi + \psi : \mathcal{F} \to \mathcal{G}$
to be the map which on each open $U \subset X$ is the
sum of the maps induced by $\varphi$, $\psi$. This is
clearly again a map of sheaves of $\mathcal{O}_X$-modules.
It is also clear that composition of maps of
$\mathcal{O}_X$-modules is bilinear with respect to this
addition. Thus $\textit{Mod}(\mathcal{O}_X)$ is a pre-additive
category, see Homology, Definition \ref{homology-definition-preadditive}.
\medskip\noindent
We will denote $0$ the sheaf of $\mathcal{O}_X$-modules
which has constant value $\{0\}$ for all open $U \subset X$.
Clearly this is both a final and an initial object of
$\textit{Mod}(\mathcal{O}_X)$. Given a morphism
of $\mathcal{O}_X$-modules $\varphi : \mathcal{F} \to \mathcal{G}$
the following are equivalent:
(a) $\varphi$ is zero, (b) $\varphi$ factors through $0$,
(c) $\varphi$ is zero on sections over each open $U$, and
(d) $\varphi_x = 0$ for all $x \in X$. See
Sheaves, Lemma \ref{sheaves-lemma-points-exactness}.
\medskip\noindent
Moreover, given a pair
$\mathcal{F}$, $\mathcal{G}$ of sheaves of $\mathcal{O}_X$-modules
we may define the direct sum as
$$
\mathcal{F} \oplus \mathcal{G} = \mathcal{F} \times \mathcal{G}
$$
with obvious maps $(i, j, p, q)$ as in Homology, Definition
\ref{homology-definition-direct-sum}. Thus $\textit{Mod}(\mathcal{O}_X)$
is an additive category, see
Homology, Definition \ref{homology-definition-additive-category}.
\medskip\noindent
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism
of $\mathcal{O}_X$-modules. We may define $\Ker(\varphi)$
to be the subsheaf of $\mathcal{F}$ with sections
$$
\Ker(\varphi)(U) =
\{ s \in \mathcal{F}(U) \mid \varphi(s) = 0 \text{ in } \mathcal{G}(U)\}
$$
for all open $U \subset X$. It is easy to see that this is indeed
a kernel in the category of $\mathcal{O}_X$-modules. In other words,
a morphism $\alpha : \mathcal{H} \to \mathcal{F}$ factors
through $\Ker(\varphi)$ if and only if $\varphi \circ \alpha = 0$.
Moreover, on the level of stalks we have
$\Ker(\varphi)_x = \Ker(\varphi_x)$.
\medskip\noindent
On the other hand, we define
$\Coker(\varphi)$ as the sheaf of $\mathcal{O}_X$-modules
associated to the presheaf of $\mathcal{O}_X$-modules defined
by the rule
$$
U
\longmapsto
\Coker(\mathcal{G}(U)\to \mathcal{F}(U)) =
\mathcal{F}(U)/\varphi(\mathcal{G}(U)).
$$
Since taking stalks commutes with taking sheafification, see
Sheaves, Lemma \ref{sheaves-lemma-stalk-sheafification} we
see that $\Coker(\varphi)_x = \Coker(\varphi_x)$.
Thus the map $\mathcal{G} \to \Coker(\varphi)$ is surjective
(as a map of sheaves of sets),
see Sheaves, Section \ref{sheaves-section-exactness-points}.
To show that this is a cokernel, note that if
$\beta : \mathcal{G} \to \mathcal{H}$ is a morphism of $\mathcal{O}_X$-modules
such that $\beta \circ \varphi$ is zero, then you get for every
open $U \subset X$ a map induced by $\beta$ from
$\mathcal{G}(U)/\varphi(\mathcal{F}(U))$ into $\mathcal{H}(U)$.
By the universal property of sheafification (see
Sheaves, Lemma \ref{sheaves-lemma-sheafification-presheaf-modules})
we obtain a canonical map $\Coker(\varphi) \to \mathcal{H}$
such that the original $\beta$ is equal to
the composition
$\mathcal{G} \to \Coker(\varphi) \to \mathcal{H}$.
The morphism $\Coker(\varphi) \to \mathcal{H}$ is unique
because of the surjectivity mentioned above.
\begin{lemma}
\label{lemma-abelian}
Let $(X, \mathcal{O}_X)$ be a ringed space. The category
$\textit{Mod}(\mathcal{O}_X)$ is an abelian category. Moreover
a complex
$$
\mathcal{F} \to \mathcal{G} \to \mathcal{H}
$$
is exact at $\mathcal{G}$ if and only if for all $x \in X$ the
complex
$$
\mathcal{F}_x \to \mathcal{G}_x \to \mathcal{H}_x
$$
is exact at $\mathcal{G}_x$.
\end{lemma}
\begin{proof}
By Homology, Definition \ref{homology-definition-abelian-category}
we have to show that image and coimage agree. By Sheaves,
Lemma \ref{sheaves-lemma-points-exactness} it is enough to show
that image and coimage have the same stalk at every $x \in X$.
By the constructions of kernels and cokernels above these stalks
are the coimage and image in the categories of $\mathcal{O}_{X, x}$-modules.
Thus we get the result from the fact that the category of modules
over a ring is abelian.
\end{proof}
\noindent
Actually the category $\textit{Mod}(\mathcal{O}_X)$ has many more properties.
Here are two constructions we can do.
\begin{enumerate}
\item Given any set $I$ and for each $i \in I$ a $\mathcal{O}_X$-module
we can form the product
$$
\prod\nolimits_{i \in I} \mathcal{F}_i
$$
which is the sheaf that associates to each open $U$ the
product of the modules $\mathcal{F}_i(U)$. This is also the
categorical product, as in
Categories, Definition \ref{categories-definition-product}.
\item Given any set $I$ and for each $i \in I$ a $\mathcal{O}_X$-module
we can form the direct sum
$$
\bigoplus\nolimits_{i \in I} \mathcal{F}_i
$$
which is the {\it sheafification} of the presheaf
that associates to each open $U$ the
direct sum of the modules $\mathcal{F}_i(U)$.
This is also the categorical coproduct, as in
Categories, Definition \ref{categories-definition-coproduct}.
To see this you use the universal property of sheafification.
\end{enumerate}
Using these we conclude that all limits and colimits exist in
$\textit{Mod}(\mathcal{O}_X)$.
\begin{lemma}
\label{lemma-limits-colimits}
Let $(X, \mathcal{O}_X)$ be a ringed space.
\begin{enumerate}
\item All limits exist in $\textit{Mod}(\mathcal{O}_X)$.
Limits are the same as the corresponding limits of presheaves of
$\mathcal{O}_X$-modules (i.e., commute with taking
sections over opens).
\item All colimits exist in $\textit{Mod}(\mathcal{O}_X)$.
Colimits are the sheafification of the corresponding colimit in
the category of presheaves. Taking colimits commutes with taking
stalks.
\item Filtered colimits are exact.
\item Finite direct sums are the same as the corresponding
finite direct sums of presheaves of $\mathcal{O}_X$-modules.
\end{enumerate}
\end{lemma}
\begin{proof}
As $\textit{Mod}(\mathcal{O}_X)$ is abelian (Lemma \ref{lemma-abelian})
it has all finite limits and colimits
(Homology, Lemma \ref{homology-lemma-colimit-abelian-category}).
Thus the existence of limits and colimits and their description follows from
the existence of products and coproducts and their description
(see discussion above) and
Categories, Lemmas \ref{categories-lemma-limits-products-equalizers} and
\ref{categories-lemma-colimits-coproducts-coequalizers}.
Since sheafification commutes with taking stalks we see that
colimits commute with taking stalks. Part (3) signifies that given
a system $0 \to \mathcal{F}_i \to \mathcal{G}_i \to \mathcal{H}_i \to 0$
of exact sequences of $\mathcal{O}_X$-modules over a directed partially
ordered set $I$ the sequence
$0 \to \colim \mathcal{F}_i \to \colim \mathcal{G}_i \to
\colim \mathcal{H}_i \to 0$ is exact as well. Since we can check
exactness on stalks (Lemma \ref{lemma-abelian}) this follows from the case
of modules which is
Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}.
We omit the proof of (4).
\end{proof}
\noindent
The existence of limits and colimits
allows us to consider exactness properties of
functors defined on the category of $\mathcal{O}$-modules
in terms of limits and colimits, as in
Categories, Section \ref{categories-section-exact-functor}.
See Homology, Lemma \ref{homology-lemma-exact-functor} for a
description of exactness
properties in terms of short exact sequences.
\begin{lemma}
\label{lemma-exactness-pushforward-pullback}
Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
be a morphism of ringed spaces.
\begin{enumerate}
\item The functor
$f_* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_Y)$
is left exact. In fact it commutes with all limits.
\item The functor
$f^* : \textit{Mod}(\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$
is right exact. In fact it commutes with all colimits.
\item Pullback $f^{-1} : \textit{Ab}(Y) \to \textit{Ab}(X)$
on abelian sheaves is exact.
\end{enumerate}
\end{lemma}
\begin{proof}
Parts (1) and (2) hold because $(f^*, f_*)$ is an adjoint pair
of functors, see
Sheaves, Lemma \ref{sheaves-lemma-adjoint-pullback-pushforward-modules}
and
Categories, Section \ref{categories-section-adjoint}.
Part (3) holds because exactness can be checked on stalks
(Lemma \ref{lemma-abelian})
and the description of stalks of the pullback, see
Sheaves, Lemma \ref{sheaves-lemma-pullback-abelian-stalk}.
\end{proof}
\begin{lemma}
\label{lemma-j-shriek-exact}
Let $j : U \to X$ be an open immersion of topological spaces.
The functor $j_! : \textit{Ab}(U) \to \textit{Ab}(X)$
is exact.
\end{lemma}
\begin{proof}
Follows from the description of stalks
given in Sheaves, Lemma \ref{sheaves-lemma-j-shriek-abelian}.
\end{proof}
\begin{lemma}
\label{lemma-section-direct-sum-quasi-compact}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $I$ be a set. For $i \in I$, let $\mathcal{F}_i$
be a sheaf of $\mathcal{O}_X$-modules.
For $U \subset X$ quasi-compact open the map
$$
\bigoplus\nolimits_{i \in I} \mathcal{F}_i(U)
\longrightarrow
\left(\bigoplus\nolimits_{i \in I} \mathcal{F}_i\right)(U)
$$
is bijective.
\end{lemma}
\begin{proof}
If $s$ is an element of the right hand side, then
there exists an open covering $U = \bigcup_{j \in J} U_j$
such that $s|_{U_j}$ is a finite sum
$\sum_{i \in I_j} s_{ji}$ with $s_{ji} \in \mathcal{F}_i(U_j)$.
Because $U$ is quasi-compact we may assume that the
covering is finite, i.e., that $J$ is finite.
Then $I' = \bigcup_{j \in J} I_j$ is a finite subset of
$I$. Clearly, $s$ is a section of the subsheaf
$\bigoplus_{i \in I'} \mathcal{F}_i$. The result follows
from the fact that for a finite direct sum sheafification
is not needed, see Lemma \ref{lemma-limits-colimits} above.
\end{proof}
\section{Sections of sheaves of modules}
\label{section-sections}
\noindent
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
Let $s \in \Gamma(X, \mathcal{F}) = \mathcal{F}(X)$ be a
global section. There is a unique {\it map of $\mathcal{O}_X$-modules
$$
\mathcal{O}_X \longrightarrow \mathcal{F}, \ f \longmapsto fs
$$
associated to $s$}. The notation above signifies that a local
section $f$ of $\mathcal{O}_X$, i.e., a section $f$ over some open $U$,
is mapped to the multiplication of $f$ with the restriction of $s$ to
$U$. Conversely, any map $\varphi : \mathcal{O}_X \to \mathcal{F}$
gives rise to a section $s = \varphi(1)$ such that $\varphi$ is
the morphism associated to $s$.
\begin{definition}
\label{definition-globally-generated}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
We say that $\mathcal{F}$ is {\it generated by global
sections} if there exist a set $I$, and
global sections $s_i \in \Gamma(X, \mathcal{F})$, $i \in I$
such that the map
$$
\bigoplus\nolimits_{i \in I}
\mathcal{O}_X \longrightarrow \mathcal{F}
$$
which is the map associated to $s_i$ on the summand corresponding to $i$,
is surjective. In this case we say that the sections $s_i$
{\it generate} $\mathcal{F}$.
\end{definition}
\noindent
We often use the abuse of notation introduced in
Sheaves, Section \ref{sheaves-section-stalks} where, given a local
section $s$ of $\mathcal{F}$ defined in an open neighbourhood
of a point $x \in X$, we denote $s_x$, or even $s$ the image of $s$
in the stalk $\mathcal{F}_x$.
\begin{lemma}
\label{lemma-globally-generated}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
Let $I$ be a set. Let
$s_i \in \Gamma(X, \mathcal{F})$, $i \in I$
be global sections. The sections $s_i$ generate
$\mathcal{F}$ if and only if for all $x\in X$ the
elements $s_{i, x} \in \mathcal{F}_x$ generate
the $\mathcal{O}_{X, x}$-module $\mathcal{F}_x$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-tensor-product-globally-generated}
\begin{slogan}
The tensor product of globally generated sheaves of modules is
globally generated.
\end{slogan}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}_X$-modules.
If $\mathcal{F}$ and $\mathcal{G}$ are generated by global sections
then so is $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-generated-by-local-sections}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
Let $I$ be a set. Let $s_i$, $i \in I$ be a collection
of local sections of $\mathcal{F}$, i.e., $s_i \in \mathcal{F}(U_i)$
for some opens $U_i \subset X$. There exists a unique smallest
subsheaf of $\mathcal{O}_X$-modules $\mathcal{G}$ such
that each $s_i$ corresponds to a local section of
$\mathcal{G}$.
\end{lemma}
\begin{proof}
Consider the subpresheaf of $\mathcal{O}_X$-modules
defined by the rule
$$
U
\longmapsto
\{
\text{sums } \sum\nolimits_{i \in J} f_i (s_i|_U)
\text{ where } J \text{ is finite, }
U \subset U_i \text{ for } i\in J, \text{ and }
f_i \in \mathcal{O}_X(U)
\}
$$
Let $\mathcal{G}$ be the sheafification of this subpresheaf.
This is a subsheaf of $\mathcal{F}$ by
Sheaves, Lemma \ref{sheaves-lemma-characterize-epi-mono}.
Since all the finite sums clearly have to be in $\mathcal{G}$
this is the smallest subsheaf as desired.
\end{proof}
\begin{definition}
\label{definition-generated-by-local-sections}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
Given a set $I$, and
local sections $s_i$, $i \in I$ of $\mathcal{F}$
we say that the subsheaf $\mathcal{G}$ of
Lemma \ref{lemma-generated-by-local-sections}
above is the {\it subsheaf generated by the $s_i$}.
\end{definition}
\begin{lemma}
\label{lemma-generated-by-local-sections-stalk}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
Given a set $I$, and
local sections $s_i$, $i \in I$ of $\mathcal{F}$.
Let $\mathcal{G}$ be the subsheaf generated by the
$s_i$ and let $x\in X$.
Then $\mathcal{G}_x$ is the $\mathcal{O}_{X, x}$-submodule of
$\mathcal{F}_x$ generated by the elements $s_{i, x}$
for those $i$ such that $s_i$ is defined at $x$.
\end{lemma}
\begin{proof}
This is clear from the construction of $\mathcal{G}$
in the proof of Lemma \ref{lemma-generated-by-local-sections}.
\end{proof}
\section{Supports of modules and sections}
\label{section-support}
\begin{definition}
\label{definition-support}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
\begin{enumerate}
\item The {\it support of $\mathcal{F}$} is the set of
points $x \in X$ such that $\mathcal{F}_x \not = 0$.
\item We denote $\text{Supp}(\mathcal{F})$ the support of $\mathcal{F}$.
\item Let $s \in \Gamma(X, \mathcal{F})$ be a global section.
The {\it support of $s$} is the set of points $x \in X$
such that the image $s_x \in \mathcal{F}_x$ of $s$ is
not zero.
\end{enumerate}
\end{definition}
\noindent
Of course the support of a local section is then defined also
since a local section is a global section of the restriction of
$\mathcal{F}$.
\begin{lemma}
\label{lemma-support-section-closed}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
Let $U \subset X$ open.
\begin{enumerate}
\item The support of $s \in \mathcal{F}(U)$ is closed in $U$.
\item The support of $fs$ is contained in the intersections
of the supports of $f \in \mathcal{O}_X(U)$ and $s \in \mathcal{F}(U)$.
\item The support of $s + s'$ is contained in the union of
the supports of $s, s' \in \mathcal{F}(U)$.
\item The support of $\mathcal{F}$ is the union of the supports
of all local sections of $\mathcal{F}$.
\item If $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of
$\mathcal{O}_X$-modules, then the support of $\varphi(s)$ is
contained in the support of $s \in \mathcal{F}(U)$.
\end{enumerate}
\end{lemma}
\begin{proof}
This is true because if $s_x = 0$, then $s$ is zero
in an open neighbourhood of $x$ by definition of stalks.
Similarly for $f$. Details omitted.
\end{proof}
\noindent
In general the support of a sheaf of modules is not closed.
Namely, the sheaf could be an abelian sheaf on $\mathbf{R}$
(with the usual archimedean topology)
which is the direct sum of infinitely many nonzero skyscraper
sheaves each supported at a single point $p_i$ of $\mathbf{R}$.
Then the support would be the set of points $p_i$
which may not be closed.
\medskip\noindent
Another example is to consider the open immersion
$j : U = (0 , \infty) \to \mathbf{R} = X$, and the abelian sheaf
$j_!\underline{\mathbf{Z}}_U$. By Sheaves, Section
\ref{sheaves-section-open-immersions} the support of
this sheaf is exactly $U$.
\begin{lemma}
\label{lemma-support-sheaf-rings-closed}
Let $X$ be a topological space.
The support of a sheaf of rings is closed.
\end{lemma}
\begin{proof}
This is true because (according to our conventions)
a ring is $0$ if and only if
$1 = 0$, and hence the support of a sheaf of rings
is the support of the unit section.
\end{proof}
\section{Closed immersions and abelian sheaves}
\label{section-closed-immersions}
\noindent
Recall that we think of an abelian sheaf on a topological space $X$ as a
sheaf of $\underline{\mathbf{Z}}_X$-modules. Thus we may apply any results,
definitions for sheaves of modules to abelian sheaves.
\begin{lemma}
\label{lemma-i-star-exact}
Let $X$ be a topological space. Let $Z \subset X$ be a closed subset.
Denote $i : Z \to X$ the inclusion map. The functor
$$
i_* : \textit{Ab}(Z) \longrightarrow \textit{Ab}(X)
$$
is exact, fully faithful, with essential image exactly those
abelian sheaves whose support is contained in $Z$. The functor $i^{-1}$
is a left inverse to $i_*$.
\end{lemma}
\begin{proof}
Exactness follows from the description of
stalks in Sheaves, Lemma \ref{sheaves-lemma-stalks-closed-pushforward}
and Lemma \ref{lemma-abelian}. The rest was shown in
Sheaves, Lemma \ref{sheaves-lemma-equivalence-categories-closed-abelian}.
\end{proof}
\noindent
Let $\mathcal{F}$ be a sheaf on $X$. There is a canonical subsheaf
of $\mathcal{F}$ which consists of exactly those sections whose
support is contained in $Z$. Here is the exact statement.
\begin{lemma}
\label{lemma-sections-support-in-closed}
Let $X$ be a topological space. Let $Z \subset X$ be a closed subset.
Let $\mathcal{F}$ be a sheaf on $X$. For $U \subset X$ open set
$$
\Gamma(U, \mathcal{H}_Z(\mathcal{F})) =
\{s \in \mathcal{F}(U) \mid
\text{ the support of }s\text{ is contained in }Z \cap U\}
$$
Then $\mathcal{H}_Z(\mathcal{F})$ is an abelian subsheaf of
$\mathcal{F}$. It is the largest abelian subsheaf of
$\mathcal{F}$ whose support is contained in $Z$. The construction
$\mathcal{F} \mapsto \mathcal{H}_Z(\mathcal{F})$ is functorial
in the abelian sheaf $\mathcal{F}$.
\end{lemma}
\begin{proof}
This follows from Lemma \ref{lemma-support-section-closed}.
\end{proof}
\noindent
This seems like a good opportunity to show that the functor
$i_*$ has a right adjoint on abelian sheaves.
\begin{lemma}
\label{lemma-i-star-right-adjoint}
Let $i : Z \to X$ be the inclusion of a closed subset into the
topological space $X$.
Denote\footnote{This is likely nonstandard notation.}
$i^! : \textit{Ab}(X) \to \textit{Ab}(Z)$ the functor
$\mathcal{F} \mapsto i^{-1}\mathcal{H}_Z(\mathcal{F})$.
Then $i^!$ is a right adjoint to $i_*$, in a formula
$$
\Mor_{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{F})
=
\Mor_{\textit{Ab}(Z)}(\mathcal{G}, i^!\mathcal{F}).
$$
In particular $i_*$ commutes with arbitrary colimits.
\end{lemma}
\begin{proof}
Note that $i_*i^!\mathcal{F} = \mathcal{H}_Z(\mathcal{F})$.
Since $i_*$ is fully faithful we are reduced to showing that
$$
\Mor_{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{F})
=
\Mor_{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{H}_Z(\mathcal{F})).
$$
This follows since the support of the image via any homomorphism
of a section of $i_*\mathcal{G}$ is contained in $Z$, see
Lemma \ref{lemma-support-section-closed}.
\end{proof}
\begin{remark}
\label{remark-i-star-right-adjoint}
In Sheaves, Remark \ref{sheaves-remark-i-star-not-exact}
we showed that $i_*$ as a functor
on the categories of sheaves of sets
does not have a right adjoint simply because
it is not exact. However, it is very close to being
true, in fact, the functor $i_*$ is exact on sheaves
of pointed sets, sections with support in $Z$ can
be defined for sheaves of pointed sets, and $i^!$
makes sense and is a right adjoint to $i_*$.
\end{remark}
\section{A canonical exact sequence}
\label{section-canonical-exact-sequence}
\noindent
We give this exact sequence its own section.
\begin{lemma}
\label{lemma-canonical-exact-sequence}
Let $X$ be a topological space.
Let $U \subset X$ be an open subset with complement $Z \subset X$.
Denote $j : U \to X$ the open immersion and
$i : Z \to X$ the closed immersion.
For any sheaf of abelian groups $\mathcal{F}$ on $X$
the adjunction mappings $j_{!}j^*\mathcal{F} \to \mathcal{F}$ and
$\mathcal{F} \to i_*i^*\mathcal{F}$ give a short exact
sequence
$$
0 \to j_{!}j^*\mathcal{F} \to \mathcal{F} \to i_*i^*\mathcal{F} \to 0
$$
of sheaves of abelian groups. For any morphism
$\varphi : \mathcal{F} \to \mathcal{G}$ of abelian sheaves on $X$
we obtain a morphism of short exact sequences
$$
\xymatrix{
0 \ar[r] &
j_{!}j^*\mathcal{F} \ar[r] \ar[d] &
\mathcal{F} \ar[r] \ar[d] &
i_*i^*\mathcal{F} \ar[r] \ar[d] &
0 \\
0 \ar[r] &
j_{!}j^*\mathcal{G} \ar[r] &
\mathcal{G} \ar[r] &
i_*i^*\mathcal{G} \ar[r] &
0
}
$$
\end{lemma}
\begin{proof}
The functoriality of the short exact sequence is
immediate from the naturality of the adjunction mappings.
We may check exactness on stalks (Lemma \ref{lemma-abelian}).
For a description of the stalks in question see
Sheaves, Lemmas \ref{sheaves-lemma-j-shriek-abelian}
and \ref{sheaves-lemma-stalks-closed-pushforward}.
\end{proof}
\section{Modules locally generated by sections}
\label{section-locally-generated}
\noindent
Let $(X, \mathcal{O}_X)$ be a ringed space.
In this and the following section we will often restrict
sheaves to open subspaces $U \subset X$, see
Sheaves, Section \ref{sheaves-section-open-immersions}.
In particular, we will often denote the open subspace
by $(U, \mathcal{O}_U)$ instead of the more correct
notation $(U, \mathcal{O}_X|_U)$, see
Sheaves, Definition \ref{sheaves-definition-restriction}.
\medskip\noindent
Consider the open immersion
$j : U = (0 , \infty) \to \mathbf{R} = X$, and the abelian sheaf
$j_!\underline{\mathbf{Z}}_U$. By Sheaves, Section
\ref{sheaves-section-open-immersions} the stalk of
$j_!\underline{\mathbf{Z}}_U$ at $x = 0$ is $0$. In fact the
sections of this sheaf over any open interval containing $0$
are $0$. Thus there is no open neighbourhood of the point
$0$ over which the sheaf can be generated by sections.
\begin{definition}
\label{definition-locally-generated}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
We say that $\mathcal{F}$ is {\it locally generated by sections}
if for every $x \in X$ there exists an open
neighbourhood $U$ such that $\mathcal{F}|_U$
is globally generated as a sheaf of $\mathcal{O}_U$-modules.
\end{definition}
\noindent
In other words there exists a set $I$ and for
each $i$ a section $s_i \in \mathcal{F}(U)$ such
that the associated map
$$
\bigoplus\nolimits_{i \in I} \mathcal{O}_U
\longrightarrow
\mathcal{F}|_U
$$
is surjective.
\begin{lemma}
\label{lemma-pullback-locally-generated}
Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
be a morphism of ringed spaces.
The pullback $f^*\mathcal{G}$ is locally generated by sections
if $\mathcal{G}$ is locally generated by sections.
\end{lemma}
\begin{proof}
Given an open subspace $V$ of $Y$ we may
consider the commutative diagram of ringed spaces
$$
\xymatrix{
(f^{-1}V, \mathcal{O}_{f^{-1}V}) \ar[r]_{j'} \ar[d]_{f'} &
(X, \mathcal{O}_X) \ar[d]^f \\
(V, \mathcal{O}_V) \ar[r]^j &
(Y, \mathcal{O}_Y)
}
$$
We know that $f^*\mathcal{G}|_{f^{-1}V} \cong (f')^*(\mathcal{G}|_V)$,
see Sheaves, Lemma \ref{sheaves-lemma-push-pull-composition-modules}.
Thus we may assume that $\mathcal{G}$ is globally generated.
\medskip\noindent
We have seen that $f^*$ commutes with all colimits,
and is right exact, see Lemma \ref{lemma-exactness-pushforward-pullback}.
Thus if we have a surjection
$$
\bigoplus\nolimits_{i \in I}
\mathcal{O}_Y
\to
\mathcal{G}
\to
0
$$
then upon applying $f^*$ we obtain the surjection
$$
\bigoplus\nolimits_{i \in I}
\mathcal{O}_X
\to
f^*\mathcal{G}
\to
0.
$$
This implies the lemma.
\end{proof}
\section{Modules of finite type}
\label{section-finite-type}
\begin{definition}
\label{definition-finite-type}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
We say that $\mathcal{F}$ is of {\it finite type}
if for every $x \in X$ there exists an open
neighbourhood $U$ such that $\mathcal{F}|_U$
is generated by finitely many sections.
\end{definition}
\begin{lemma}
\label{lemma-pullback-finite-type}
Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
be a morphism of ringed spaces.
The pullback $f^*\mathcal{G}$ of a finite type
$\mathcal{O}_Y$-module is a finite type $\mathcal{O}_X$-module.
\end{lemma}
\begin{proof}
Arguing as in the proof of Lemma \ref{lemma-pullback-locally-generated}
we may assume $\mathcal{G}$ is globally generated by finitely
many sections.
We have seen that $f^*$ commutes with all colimits,
and is right exact, see Lemma \ref{lemma-exactness-pushforward-pullback}.
Thus if we have a surjection
$$
\bigoplus\nolimits_{i = 1, \ldots, n}
\mathcal{O}_Y
\to
\mathcal{G}
\to
0
$$
then upon applying $f^*$ we obtain the surjection
$$
\bigoplus\nolimits_{i = 1, \ldots, n}
\mathcal{O}_X
\to
f^*\mathcal{G}
\to
0.
$$
This implies the lemma.
\end{proof}
\begin{lemma}
\label{lemma-extension-finite-type}
Let $X$ be a ringed space.
The image of a morphism of $\mathcal{O}_X$-modules of finite
type is of finite type.
Let
$0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$
be a short exact sequence of $\mathcal{O}_X$-modules.
If $\mathcal{F}_1$ and $\mathcal{F}_3$ are of finite type,
so is $\mathcal{F}_2$.
\end{lemma}
\begin{proof}
The statement on images is trivial.
The statement on short exact sequences comes from the
fact that sections of $\mathcal{F}_3$ locally lift to sections
of $\mathcal{F}_2$ and the corresponding result in
the category of modules over a ring (applied to the stalks
for example).
\end{proof}
\begin{lemma}
\label{lemma-finite-type-surjective-on-stalk}
Let $X$ be a ringed space.
Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism
of $\mathcal{O}_X$-modules.
Let $x \in X$. Assume $\mathcal{F}$ of finite type and
the map on stalks
$\varphi_x : \mathcal{G}_x \to \mathcal{F}_x$ surjective.
Then there exists an open neighbourhood
$x \in U \subset X$ such that $\varphi|_U$ is surjective.
\end{lemma}
\begin{proof}
Choose an open neighbourhood $U \subset X$ of $x$ such that $\mathcal{F}$ is
generated by $s_1, \ldots, s_n \in \mathcal{F}(U)$ over $U$.
By assumption of surjectivity of $\varphi_x$,
after shrinking $U$ we may assume that $s_i = \varphi(t_i)$
for some $t_i \in \mathcal{G}(U)$.
Then $U$ works.
\end{proof}
\begin{lemma}
\label{lemma-finite-type-stalk-zero}
Let $X$ be a ringed space.
Let $\mathcal{F}$ be an $\mathcal{O}_X$-module.
Let $x \in X$.
Assume $\mathcal{F}$ of finite type and $\mathcal{F}_x = 0$.
Then there exists an open neighbourhood
$x \in U \subset X$ such that $\mathcal{F}|_U$ is zero.
\end{lemma}
\begin{proof}
This is a special case of
Lemma \ref{lemma-finite-type-surjective-on-stalk}
applied to the morphism $0 \to \mathcal{F}$.
\end{proof}
\begin{lemma}
\label{lemma-support-finite-type-closed}
\begin{slogan}
Over any ringed space, sheaves of modules of finite type have closed support.
\end{slogan}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
If $\mathcal{F}$ is of finite type then support of $\mathcal{F}$ is closed.
\end{lemma}
\begin{proof}
This is a reformulation of Lemma \ref{lemma-finite-type-stalk-zero}.
\end{proof}
\begin{lemma}
\label{lemma-finite-type-quasi-compact-colimit}
Let $X$ be a ringed space. Let $I$ be a partially ordered set and
let $(\mathcal{F}_i, f_{ii'})$ be a system over $I$ consisting of sheaves
of $\mathcal{O}_X$-modules (see
Categories, Section \ref{categories-section-posets-limits}).
Let $\mathcal{F} = \colim \mathcal{F}_i$ be the colimit. Assume
(a) $I$ is directed,
(b) $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module, and
(c) $X$ is quasi-compact. Then there exists an $i$ such that
$\mathcal{F}_i \to \mathcal{F}$ is surjective.
If the transition maps $f_{ii'}$ are injective
then we conclude that $\mathcal{F} = \mathcal{F}_i$ for some $i \in I$.
\end{lemma}
\begin{proof}
Let $x \in X$. There exists an open neighbourhood
$U \subset X$ of $x$ and finitely many sections
$s_j \in \mathcal{F}(U)$, $j = 1, \ldots, m$ such that
$s_1, \ldots, s_m$ generate $\mathcal{F}$ as $\mathcal{O}_U$-module.
After possibly shrinking $U$ to a smaller open neighbourhood of $x$
we may assume that each $s_j$ comes from a section of $\mathcal{F}_i$
for some $i \in I$.
Hence, since $X$ is quasi-compact we can find a finite open
covering $X = \bigcup_{j = 1, \ldots, m} U_j$, and for each $j$
an index $i_j$ and finitely many sections $s_{jl} \in \mathcal{F}_{i_j}(U_j)$
whose images generate the restriction of $\mathcal{F}$ to
$U_j$. Clearly, the lemma holds for any index $i \in I$ which
is $\geq$ all $i_j$.
\end{proof}
\begin{lemma}
\label{lemma-set-isomorphism-classes-finite-type-modules}
Let $X$ be a ringed space.
There exists a set of $\mathcal{O}_X$-modules
$\{\mathcal{F}_i\}_{i \in I}$ of finite type
such that each finite type $\mathcal{O}_X$-module
on $X$ is isomorphic to exactly one of the $\mathcal{F}_i$.
\end{lemma}
\begin{proof}
For each open covering $\mathcal{U} : X = \bigcup U_j$ consider the
sheaves of $\mathcal{O}_X$-modules $\mathcal{F}$ such that each
restriction $\mathcal{F}|_{U_j}$ is a quotient of
$\mathcal{O}_{U_j}^{\oplus r}$ for some $r_j \geq 0$.
These are parametrized by subsheaves
$\mathcal{K}_i \subset \mathcal{O}_{U_j}^{\oplus r_j}$ and glueing
data
$$
\varphi_{jj'} :
\mathcal{O}_{U_j \cap U_{j'}}^{\oplus r_j}/
(\mathcal{K}_j|_{U_j \cap U_{j'}})
\longrightarrow
\mathcal{O}_{U_j \cap U_{j'}}^{\oplus r_{j'}}/
(\mathcal{K}_{j'}|_{U_j \cap U_{j'}})
$$
see Sheaves, Section \ref{sheaves-section-glueing-sheaves}.
Note that the collection of all glueing data forms a set.
The collection of all coverings $\mathcal{U} : X = \bigcup_{j \in J} U_i$
where $J \to \mathcal{P}(X)$, $j \mapsto U_j$ is injective forms a set as
well. Hence the collection of all sheaves of $\mathcal{O}_X$-modules
gotten from glueing quotients as above forms a set $\mathcal{I}$.
By definition every finite type $\mathcal{O}_X$-module
is isomorphic to an element of $\mathcal{I}$. Choosing an
element out of each isomorphism class inside $\mathcal{I}$
gives the desired set of sheaves (uses axiom of choice).
\end{proof}