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\begin{document}
\title{Modules on Sites}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
In this document we work out basic notions of sheaves of modules on
ringed topoi or ringed sites. We first work out some basic facts on
abelian sheaves. After this we introduce ringed sites and ringed topoi.
We work through some of the very basic notions on (pre)sheaves of
$\mathcal{O}$-modules, analogous to the material on (pre)sheaves of
$\mathcal{O}$-modules in the chapter on sheaves on spaces.
Having done this, we duplicate much of the discussion in the chapter on
sheaves of modules (see Modules, Section \ref{modules-section-introduction}).
Basic references are \cite{FAC}, \cite{EGA} and \cite{SGA4}.
\section{Abelian presheaves}
\label{section-abelian-pre-sheaves}
\noindent
Let $\mathcal{C}$ be a category.
Abelian presheaves were introduced in
Sites, Sections \ref{sites-section-presheaves}
and \ref{sites-section-sheaves} and discussed a bit more
in Sites, Section \ref{sites-section-sheaves-algebraic-structures}.
We will follow the convention of this last reference, in that we think
of an abelian presheaf as a presheaf of sets endowed with addition rules
on all sets of sections compatible with the restriction mappings.
Recall that the category of abelian presheaves on $\mathcal{C}$
is denoted $\textit{PAb}(\mathcal{C})$.
\medskip\noindent
The category $\textit{PAb}(\mathcal{C})$ is abelian as defined in
Homology, Definition \ref{homology-definition-abelian-category}.
Given a map of presheaves $\varphi : \mathcal{G}_1 \to \mathcal{G}_2$
the kernel of $\varphi$ is the abelian presheaf
$U \mapsto \Ker(\mathcal{G}_1(U) \to \mathcal{G}_2(U))$ and
the cokernel of $\varphi$ is the presheaf
$U \mapsto \Coker(\mathcal{G}_1(U) \to \mathcal{G}_2(U))$.
Since the category of abelian groups is abelian it follows that
$\Coim = \Im$ because this holds over each $U$.
A sequence of abelian presheaves
$$
\mathcal{G}_1 \longrightarrow
\mathcal{G}_2 \longrightarrow
\mathcal{G}_3
$$
is exact if and only if
$\mathcal{G}_1(U) \to \mathcal{G}_2(U) \to \mathcal{G}_3(U)$
is an exact sequence of abelian groups for all $U \in \Ob(\mathcal{C})$.
We leave the verifications to the reader.
\begin{lemma}
\label{lemma-limits-colimits-abelian-presheaves}
Let $\mathcal{C}$ be a category.
\begin{enumerate}
\item All limits and colimits exist in $\textit{PAb}(\mathcal{C})$.
\item All limits and colimits commute with taking sections over objects of
$\mathcal{C}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\mathcal{I} \to \textit{PAb}(\mathcal{C})$, $i \mapsto \mathcal{F}_i$
be a diagram. We can simply define abelian presheaves
$L$ and $C$ by the rules
$$
L : U \longmapsto \lim_i \mathcal{F}_i(U)
$$
and
$$
C : U \longmapsto \colim_i \mathcal{F}_i(U).
$$
It is clear that there are maps of abelian presheaves $L \to \mathcal{F}_i$
and $\mathcal{F}_i \to C$, by using the corresponding maps on groups of
sections over each $U$. It is straightforward to check that $L$ and $C$ endowed
with these maps are the limit and colimit of the diagram in
$\textit{PAb}(\mathcal{C})$. This proves (1) and (2). Details omitted.
\end{proof}
\section{Abelian sheaves}
\label{section-abelian-sheaves}
\noindent
Let $\mathcal{C}$ be a site.
The category of abelian sheaves on $\mathcal{C}$ is denoted
$\textit{Ab}(\mathcal{C})$. It is the full subcategory of
$\textit{PAb}(\mathcal{C})$ consisting of those abelian presheaves
whose underlying presheaves of sets are sheaves.
Properties ($\alpha$) -- ($\zeta$) of
Sites, Section \ref{sites-section-sheaves-algebraic-structures}
hold, see
Sites,
Proposition \ref{sites-proposition-functoriality-algebraic-structures-topoi}.
In particular the inclusion functor
$\textit{Ab}(\mathcal{C}) \to \textit{PAb}(\mathcal{C})$
has a left adjoint, namely the sheafification functor
$\mathcal{G} \mapsto \mathcal{G}^\#$.
\medskip\noindent
We suggest the reader prove the lemma on a piece of scratch paper rather
than reading the proof.
\begin{lemma}
\label{lemma-abelian-abelian}
Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{F} \to \mathcal{G}$
be a morphism of abelian sheaves on $\mathcal{C}$.
\begin{enumerate}
\item The category $\textit{Ab}(\mathcal{C})$ is an abelian category.
\item The kernel $\Ker(\varphi)$ of $\varphi$ is the same as the
kernel of $\varphi$ as a morphism of presheaves.
\item The morphism $\varphi$ is injective
(Homology, Definition \ref{homology-definition-injective-surjective})
if and only if $\varphi$ is injective as a map of presheaves
(Sites, Definition \ref{sites-definition-presheaves-injective-surjective}),
if and only if $\varphi$ is injective as a map of sheaves
(Sites, Definition \ref{sites-definition-sheaves-injective-surjective}).
\item The cokernel $\Coker(\varphi)$ of $\varphi$ is the sheafification
of the cokernel of $\varphi$ as a morphism of presheaves.
\item The morphism $\varphi$ is surjective
(Homology, Definition \ref{homology-definition-injective-surjective})
if and only if $\varphi$ is surjective as a map of sheaves
(Sites, Definition \ref{sites-definition-sheaves-injective-surjective}).
\item A complex of abelian sheaves
$$
\mathcal{F} \to \mathcal{G} \to \mathcal{H}
$$
is exact at $\mathcal{G}$ if and only if for all
$U \in \Ob(\mathcal{C})$ and all $s \in \mathcal{G}(U)$
mapping to zero in $\mathcal{H}(U)$ there exists a covering
$\{U_i \to U\}_{i \in I}$ in $\mathcal{C}$ such that each
$s|_{U_i}$ is in the image of $\mathcal{F}(U_i) \to \mathcal{G}(U_i)$.
\end{enumerate}
\end{lemma}
\begin{proof}
We claim that Homology, Lemma \ref{homology-lemma-adjoint-get-abelian}
applies to the categories $\mathcal{A} = \textit{Ab}(\mathcal{C})$
and $\mathcal{B} = \textit{PAb}(\mathcal{C})$, and the functors
$a : \mathcal{A} \to \mathcal{B}$ (inclusion), and
$b : \mathcal{B} \to \mathcal{A}$ (sheafification).
Let us check the assumptions of
Homology, Lemma \ref{homology-lemma-adjoint-get-abelian}.
Assumption (1) is that $\mathcal{A}$, $\mathcal{B}$ are additive categories,
$a$, $b$ are additive functors, and $a$ is right adjoint to $b$.
The first two statements are clear and adjointness is
Sites, Section \ref{sites-section-sheaves-algebraic-structures} ($\epsilon$).
Assumption (2) says that $\textit{PAb}(\mathcal{C})$ is abelian
which we saw in Section \ref{section-abelian-pre-sheaves} and
that sheafification is left exact, which is
Sites, Section \ref{sites-section-sheaves-algebraic-structures} ($\zeta$).
The final assumption is that $ba \cong \text{id}_\mathcal{A}$ which is
Sites, Section \ref{sites-section-sheaves-algebraic-structures} ($\delta$).
Hence Homology, Lemma \ref{homology-lemma-adjoint-get-abelian}
applies and we conclude that $\textit{Ab}(\mathcal{C})$ is abelian.
\medskip\noindent
In the proof of Homology, Lemma \ref{homology-lemma-adjoint-get-abelian}
it is shown that $\Ker(\varphi)$ and $\Coker(\varphi)$
are equal to the sheafification of the kernel and cokernel of $\varphi$
as a morphism of abelian presheaves. This proves (4). Since the kernel
is a equalizer (i.e., a limit) and since sheafification commutes with
finite limits, we conclude that (2) holds.
\medskip\noindent
Statement (2) implies (3). Statement (4) implies (5) by our description
of sheafification. The characterization of exactness in (6) follows from
(2) and (5), and the fact that the sequence is exact if and only if
$\Im(\mathcal{F} \to \mathcal{G}) =
\Ker(\mathcal{G} \to \mathcal{H})$.
\end{proof}
\noindent
Another way to say part (6) of the lemma is that a
sequence of abelian sheaves
$$
\mathcal{F}_1 \longrightarrow
\mathcal{F}_2 \longrightarrow
\mathcal{F}_3
$$
is exact if and only if the sheafification of
$U \mapsto \Im(\mathcal{F}_1(U) \to \mathcal{F}_2(U))$
is equal to the kernel of $\mathcal{F}_2 \to \mathcal{F}_3$.
\begin{lemma}
\label{lemma-limits-colimits-abelian-sheaves}
Let $\mathcal{C}$ be a site.
\begin{enumerate}
\item All limits and colimits exist in $\textit{Ab}(\mathcal{C})$.
\item Limits are the same as the corresponding limits of abelian presheaves
over $\mathcal{C}$ (i.e., commute with taking sections over objects of
$\mathcal{C}$).
\item Finite direct sums are the same as the corresponding finite direct sums
in the category of abelian pre-sheaves over $\mathcal{C}$.
\item A colimit is the sheafification of the corresponding colimit in
the category of abelian presheaves.
\item Filtered colimits are exact.
\end{enumerate}
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-limits-colimits-abelian-presheaves} limits and colimits
of abelian presheaves exist, and are described by taking limits and colimits
on the level of sections over objects.
\medskip\noindent
Let $\mathcal{I} \to \textit{Ab}(\mathcal{C})$, $i \mapsto \mathcal{F}_i$
be a diagram. Let $\lim_i \mathcal{F}_i$ be the limit of the diagram
as an abelian presheaf. By Sites, Lemma \ref{sites-lemma-limit-sheaf}
this is an abelian sheaf. Then it is quite easy to see that
$\lim_i \mathcal{F}_i$ is the limit of the diagram in
$\textit{Ab}(\mathcal{C})$. This proves limits exist and (2) holds.
\medskip\noindent
By Categories, Lemma \ref{categories-lemma-adjoint-exact}, and because
sheafification is left adjoint to the inclusion functor we see that
$\colim_i \mathcal{F}$ exists and is the sheafification of the colimit
in $\textit{PAb}(\mathcal{C})$. This proves colimits exist and (4) holds.
\medskip\noindent
Finite direct sums are the same thing as finite products in any abelian
category. Hence (3) follows from (2).
\medskip\noindent
Proof of (5). The statement means that given a system
$0 \to \mathcal{F}_i \to \mathcal{G}_i \to \mathcal{H}_i \to 0$
of exact sequences of abelian sheaves over a directed 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. A formal argument using
Homology, Lemma \ref{homology-lemma-check-exactness} and the
definition of colimits shows that the sequence
$\colim \mathcal{F}_i \to \colim \mathcal{G}_i \to \colim \mathcal{H}_i \to 0$
is exact. Note that $\colim \mathcal{F}_i \to \colim \mathcal{G}_i$
is the sheafification of the map of presheaf colimits which is
injective as each of the maps $\mathcal{F}_i \to \mathcal{G}_i$ is
injective. Since sheafification is exact we conclude.
\end{proof}
\section{Free abelian presheaves}
\label{section-free-abelian-presheaf}
\noindent
In order to prepare notation for the following definition, let us agree
to denote the free abelian group on a set $S$ as\footnote{In other chapters
the notation $\mathbf{Z}[S]$ sometimes indicates the polynomial ring over
$\mathbf{Z}$ on $S$.}
$\mathbf{Z}[S] = \bigoplus_{s \in S} \mathbf{Z}$. It is characterized
by the property
$$
\Mor_{\textit{Ab}}(\mathbf{Z}[S], A)
=
\Mor_{\textit{Sets}}(S, A)
$$
In other words the construction $S \mapsto \mathbf{Z}[S]$ is a left adjoint
to the forgetful functor $\textit{Ab} \to \textit{Sets}$.
\begin{definition}
\label{definition-free-abelian-presheaf-on}
Let $\mathcal{C}$ be a category. Let $\mathcal{G}$ be a presheaf of sets.
The {\it free abelian presheaf} $\mathbf{Z}_\mathcal{G}$ on $\mathcal{G}$
is the abelian presheaf defined by the rule
$$
U \longmapsto \mathbf{Z}[\mathcal{G}(U)].
$$
In the special case $\mathcal{G} = h_X$ of a representable presheaf
associated to an object $X$ of $\mathcal{C}$
we use the notation $\mathbf{Z}_X = \mathbf{Z}_{h_X}$. In other words
$$
\mathbf{Z}_X(U) = \mathbf{Z}[\Mor_\mathcal{C}(U, X)].
$$
\end{definition}
\noindent
This construction is clearly functorial in the presheaf $\mathcal{G}$.
In fact it is adjoint to the forgetful functor
$\textit{PAb}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$.
Here is the precise statement.
\begin{lemma}
\label{lemma-obvious-adjointness}
Let $\mathcal{C}$ be a category.
Let $\mathcal{G}$, $\mathcal{F}$ be a presheaves of sets.
Let $\mathcal{A}$ be an abelian presheaf.
Let $U$ be an object of $\mathcal{C}$. Then
we have
\begin{align*}
\Mor_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F})
& =
\mathcal{F}(U), \\
\Mor_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_\mathcal{G}, \mathcal{A})
& =
\Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{G}, \mathcal{A}), \\
\Mor_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_U, \mathcal{A})
& =
\mathcal{A}(U).
\end{align*}
All of these equalities are functorial.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-coproduct-sum-free-abelian-presheaf}
Let $\mathcal{C}$ be a category.
Let $I$ be a set. For each $i \in I$ let
$\mathcal{G}_i$ be a presheaf of sets.
Then
$$
\mathbf{Z}_{\coprod_i \mathcal{G}_i}
=
\bigoplus\nolimits_{i \in I} \mathbf{Z}_{\mathcal{G}_i}
$$
in $\textit{PAb}(\mathcal{C})$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\section{Free abelian sheaves}
\label{section-free-abelian-sheaf}
\noindent
Here is the notion of a free abelian sheaf on a sheaf of sets.
\begin{definition}
\label{definition-free-abelian-sheaf-on}
Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a presheaf of sets.
The {\it free abelian sheaf} $\mathbf{Z}_\mathcal{G}^\#$
on $\mathcal{G}$ is the abelian sheaf $\mathbf{Z}_\mathcal{G}^\#$
which is the sheafification of the free abelian presheaf on $\mathcal{G}$.
In the special case $\mathcal{G} = h_X$ of a representable presheaf
associated to an object $X$ of $\mathcal{C}$
we use the notation $\mathbf{Z}_X^\#$.
\end{definition}
\noindent
This construction is clearly functorial in the presheaf $\mathcal{G}$.
In fact it provides an adjoint to the forgetful functor
$\textit{Ab}(\mathcal{C}) \to \Sh(\mathcal{C})$.
Here is the precise statement.
\begin{lemma}
\label{lemma-obvious-adjointness-sheaves}
Let $\mathcal{C}$ be a site.
Let $\mathcal{G}$, $\mathcal{F}$ be a sheaves of sets.
Let $\mathcal{A}$ be an abelian sheaf.
Let $U$ be an object of $\mathcal{C}$. Then
we have
\begin{align*}
\Mor_{\Sh(\mathcal{C})}(h_U^\#, \mathcal{F})
& =
\mathcal{F}(U), \\
\Mor_{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_\mathcal{G}^\#,
\mathcal{A})
& =
\Mor_{\Sh(\mathcal{C})}(\mathcal{G}, \mathcal{A}), \\
\Mor_{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_U^\#, \mathcal{A})
& =
\mathcal{A}(U).
\end{align*}
All of these equalities are functorial.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-may-sheafify-before-abelianize}
Let $\mathcal{C}$ be a site.
Let $\mathcal{G}$ be a presheaf of sets.
Then $\mathbf{Z}_\mathcal{G}^\# = (\mathbf{Z}_{\mathcal{G}^\#})^\#$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\section{Ringed sites}
\label{section-ringed-sites}
\noindent
In this chapter we mainly work with sheaves of modules on a ringed site.
Hence we need to define this notion.
\begin{definition}
\label{definition-ringed-site}
Ringed sites.
\begin{enumerate}
\item A {\it ringed site} is a pair $(\mathcal{C}, \mathcal{O})$
where $\mathcal{C}$ is a site and $\mathcal{O}$ is a sheaf of rings
on $\mathcal{C}$. The sheaf $\mathcal{O}$ is called the
{\it structure sheaf} of the ringed site.
\item Let $(\mathcal{C}, \mathcal{O})$, $(\mathcal{C}', \mathcal{O}')$ be ringed
sites. A {\it morphism of ringed sites}
$$
(f, f^\sharp) :
(\mathcal{C}, \mathcal{O})
\longrightarrow
(\mathcal{C}', \mathcal{O}')
$$
is given by a morphism of sites $f : \mathcal{C} \to \mathcal{C}'$
(see Sites, Definition \ref{sites-definition-morphism-sites})
together with a map of sheaves of rings
$f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$, which by adjunction
is the same thing as a map of sheaves of rings
$f^\sharp : \mathcal{O}' \to f_*\mathcal{O}$.
\item Let
$(f, f^\sharp) :
(\mathcal{C}_1, \mathcal{O}_1) \to (\mathcal{C}_2, \mathcal{O}_2)$ and
$(g, g^\sharp) :
(\mathcal{C}_2, \mathcal{O}_2) \to (\mathcal{C}_3, \mathcal{O}_3)$
be morphisms of ringed sites. Then we define
the {\it composition of morphisms of ringed sites}
by the rule
$$
(g, g^\sharp) \circ (f, f^\sharp) = (g \circ f, f^\sharp \circ g^\sharp).
$$
Here we use composition of morphisms of sites defined in
Sites, Definition \ref{sites-definition-composition-morphisms-sites}
and $f^\sharp \circ g^\sharp$ indicates the morphism of sheaves of
rings
$$
\mathcal{O}_3 \xrightarrow{g^\sharp} g_*\mathcal{O}_2
\xrightarrow{g_*f^\sharp} g_*f_*\mathcal{O}_1 = (g \circ f)_*\mathcal{O}_1
$$
\end{enumerate}
\end{definition}
\section{Ringed topoi}
\label{section-ringed-topoi}
\noindent
A ringed topos is just a ringed site, except that the notion of
a morphism of ringed topoi is different from the notion of a morphism
of ringed sites.
\begin{definition}
\label{definition-ringed-topos}
Ringed topoi.
\begin{enumerate}
\item A {\it ringed topos} is a pair
$(\Sh(\mathcal{C}), \mathcal{O})$
where $\mathcal{C}$ is a site and $\mathcal{O}$ is a sheaf of rings
on $\mathcal{C}$. The sheaf $\mathcal{O}$ is called the
{\it structure sheaf} of the ringed topos.
\item Let $(\Sh(\mathcal{C}), \mathcal{O})$,
$(\Sh(\mathcal{C}'), \mathcal{O}')$ be ringed topoi.
A {\it morphism of ringed topoi}
$$
(f, f^\sharp) :
(\Sh(\mathcal{C}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{C}'), \mathcal{O}')
$$
is given by a morphism of topoi $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$
(see Sites, Definition \ref{sites-definition-topos})
together with a map of sheaves of rings
$f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$, which by adjunction
is the same thing as a map of sheaves of rings
$f^\sharp : \mathcal{O}' \to f_*\mathcal{O}$.
\item Let
$(f, f^\sharp) :
(\Sh(\mathcal{C}_1), \mathcal{O}_1)
\to (\Sh(\mathcal{C}_2), \mathcal{O}_2)$ and
$(g, g^\sharp) :
(\Sh(\mathcal{C}_2), \mathcal{O}_2) \to
(\Sh(\mathcal{C}_3), \mathcal{O}_3)$
be morphisms of ringed topoi. Then we define
the {\it composition of morphisms of ringed topoi}
by the rule
$$
(g, g^\sharp) \circ (f, f^\sharp) = (g \circ f, f^\sharp \circ g^\sharp).
$$
Here we use composition of morphisms of topoi defined in
Sites, Definition \ref{sites-definition-topos}
and $f^\sharp \circ g^\sharp$ indicates the morphism of sheaves of
rings
$$
\mathcal{O}_3 \xrightarrow{g^\sharp} g_*\mathcal{O}_2
\xrightarrow{g_*f^\sharp} g_*f_*\mathcal{O}_1 = (g \circ f)_*\mathcal{O}_1
$$
\end{enumerate}
\end{definition}
\noindent
Every morphism of ringed topoi is the composition of an equivalence
of ringed topoi with a morphism of ringed topoi associated to a morphism
of ringed sites. Here is the precise statement.
\begin{lemma}
\label{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}
Let $(f, f^\sharp) :
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
\to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
be a morphism of ringed topoi. There exists a factorization
$$
\xymatrix{
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
\ar[rr]_{(f, f^\sharp)}
\ar[d]_{(g, g^\sharp)}
& &
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \ar[d]^{(e, e^\sharp)}
\\
(\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'})
\ar[rr]^{(h, h^\sharp)} & &
(\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'})
}
$$
where
\begin{enumerate}
\item $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$
is an equivalence of topoi induced by a special cocontinuous functor
$\mathcal{C} \to \mathcal{C}'$ (see
Sites, Definition \ref{sites-definition-special-cocontinuous-functor}),
\item $e : \Sh(\mathcal{D}) \to \Sh(\mathcal{D}')$
is an equivalence of topoi induced by a special cocontinuous functor
$\mathcal{D} \to \mathcal{D}'$ (see
Sites, Definition \ref{sites-definition-special-cocontinuous-functor}),
\item $\mathcal{O}_{\mathcal{C}'} = g_*\mathcal{O}_\mathcal{C}$
and $g^\sharp$ is the obvious map,
\item $\mathcal{O}_{\mathcal{D}'} = e_*\mathcal{O}_\mathcal{D}$
and $e^\sharp$ is the obvious map,
\item the sites $\mathcal{C}'$ and $\mathcal{D}'$ have final objects
and fibre products (i.e., all finite limits),
\item $h$ is a morphism of sites induced by a continuous functor
$u : \mathcal{D}' \to \mathcal{C}'$ which commutes with all finite limits
(i.e., it satisfies the assumptions of
Sites, Proposition \ref{sites-proposition-get-morphism}), and
\item given any set of sheaves $\mathcal{F}_i$ (resp.\ $\mathcal{G}_j$)
on $\mathcal{C}$ (resp.\ $\mathcal{D}$) we may assume each of these is
a representable sheaf on $\mathcal{C}'$ (resp.\ $\mathcal{D}'$).
\end{enumerate}
Moreover, if $(f, f^\sharp)$ is an equivalence of ringed topoi,
then we can choose the diagram such that
$\mathcal{C}' = \mathcal{D}'$,
$\mathcal{O}_{\mathcal{C}'} = \mathcal{O}_{\mathcal{D}'}$
and $(h, h^\sharp)$ is the identity.
\end{lemma}
\begin{proof}
This follows from
Sites, Lemma \ref{sites-lemma-morphism-topoi-comes-from-morphism-sites},
and
Sites, Remarks
\ref{sites-remark-morphism-topoi-comes-from-morphism-sites} and
\ref{sites-remark-equivalence-topoi-comes-from-morphism-sites}.
You just have to carry along the sheaves of rings. Some details omitted.
\end{proof}
\section{2-morphisms of ringed topoi}
\label{section-2-category}
\noindent
This is a brief section concerning the notion of a $2$-morphism
of ringed topoi.
\begin{definition}
\label{definition-2-morphism-ringed-topoi}
Let
$f, g :
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
\to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
be two morphisms of ringed topoi. A {\it 2-morphism from $f$ to $g$}
is given by a transformation of functors $t : f_* \to g_*$ such that
$$
\xymatrix{
& \mathcal{O}_\mathcal{D}
\ar[ld]_{f^\sharp}
\ar[rd]^{g^\sharp} \\
f_*\mathcal{O}_\mathcal{C} \ar[rr]^t & &
g_*\mathcal{O}_\mathcal{C}
}
$$
is commutative.
\end{definition}
\noindent
Pictorially we sometimes represent $t$ as follows:
$$
\xymatrix{
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
\rrtwocell^f_g{t}
&
&
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})
}
$$
As in
Sites, Section \ref{sites-section-2-category}
giving a 2-morphism $t : f_* \to g_*$ is equivalent to giving
$t : g^{-1} \to f^{-1}$ (usually denoted by the same symbol)
such that the diagram
$$
\xymatrix{
f^{-1}\mathcal{O}_\mathcal{D}
\ar[rd]_{f^\sharp} & &
g^{-1}\mathcal{O}_\mathcal{D} \ar[ll]^t \ar[ld]^{g^\sharp} \\
& \mathcal{O}_\mathcal{C}
}
$$
is commutative. As in
Sites, Section \ref{sites-section-2-category}
the axioms of a strict 2-category hold with horizontal and
vertical compositions defined as explained in loc.\ cit.
\section{Presheaves of modules}
\label{section-presheaves-modules}
\noindent
Let $\mathcal{C}$ be a category.
Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$.
At this point we have not yet defined a presheaf of $\mathcal{O}$-modules.
Thus we do so right now.
\begin{definition}
\label{definition-presheaf-modules}
Let $\mathcal{C}$ be a category, and
let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$.
\begin{enumerate}
\item A {\it presheaf of $\mathcal{O}$-modules}
is given by an abelian presheaf $\mathcal{F}$ together with a
map of presheaves of sets
$$
\mathcal{O} \times \mathcal{F} \longrightarrow \mathcal{F}
$$
such that for every object $U$ of $\mathcal{C}$ the map
$\mathcal{O}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$
defines the structure of an $\mathcal{O}(U)$-module
structure on the abelian group $\mathcal{F}(U)$.
\item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$
of presheaves of $\mathcal{O}$-modules} is a morphism of abelian presheaves
$\varphi : \mathcal{F} \to \mathcal{G}$ such that
the diagram
$$
\xymatrix{
\mathcal{O} \times \mathcal{F} \ar[r] \ar[d]_{\text{id} \times \varphi} &
\mathcal{F} \ar[d]^{\varphi} \\
\mathcal{O} \times \mathcal{G} \ar[r] &
\mathcal{G}
}
$$
commutes.
\item The set of $\mathcal{O}$-module morphisms as above is
denoted $\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})$.
\item The category of presheaves of $\mathcal{O}$-modules is denoted
$\textit{PMod}(\mathcal{O})$.
\end{enumerate}
\end{definition}
\noindent
Suppose that $\mathcal{O}_1 \to \mathcal{O}_2$ is a
morphism of presheaves of rings on the category $\mathcal{C}$. In this case,
if $\mathcal{F}$ is a presheaf of $\mathcal{O}_2$-modules
then we can think of $\mathcal{F}$ as a presheaf of
$\mathcal{O}_1$-modules by using the composition
$$
\mathcal{O}_1 \times \mathcal{F}
\to
\mathcal{O}_2 \times \mathcal{F}
\to
\mathcal{F}.
$$
We sometimes denote this by $\mathcal{F}_{\mathcal{O}_1}$
to indicate the restriction of rings. We call this
the {\it restriction of $\mathcal{F}$}. We obtain the
restriction functor
$$
\textit{PMod}(\mathcal{O}_2)
\longrightarrow
\textit{PMod}(\mathcal{O}_1)
$$
\medskip\noindent
On the other hand, given a presheaf of $\mathcal{O}_1$-modules
$\mathcal{G}$
we can construct a presheaf of $\mathcal{O}_2$-modules
$\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}$
by the rule
$$
U \longmapsto
\left(\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}\right)(U)
=
\mathcal{O}_2(U) \otimes_{\mathcal{O}_1(U)} \mathcal{G}(U)
$$
where $U \in \Ob(\mathcal{C})$, with obvious restriction mappings.
The index $p$ stands for ``presheaf'' and not ``point''.
This presheaf is called the tensor product presheaf. We obtain
the {\it change of rings} functor
$$
\textit{PMod}(\mathcal{O}_1)
\longrightarrow
\textit{PMod}(\mathcal{O}_2)
$$
\begin{lemma}
\label{lemma-adjointness-tensor-restrict-presheaves}
With $\mathcal{C}$, $\mathcal{O}_1 \to \mathcal{O}_2$, $\mathcal{F}$ and
$\mathcal{G}$ as above there exists a canonical bijection
$$
\Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1})
=
\Hom_{\mathcal{O}_2}(
\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G},
\mathcal{F}
)
$$
In other words, the restriction and change of rings functors defined
above are adjoint to each other.
\end{lemma}
\begin{proof}
This follows from the fact that for a ring map
$A \to B$ the restriction functor and the change
of ring functor are adjoint to each other.
\end{proof}
\section{Sheaves of modules}
\label{section-sheaves-modules}
\begin{definition}
\label{definition-sheaf-modules}
Let $\mathcal{C}$ be a site.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$.
\begin{enumerate}
\item A {\it sheaf of $\mathcal{O}$-modules} is a presheaf
of $\mathcal{O}$-modules $\mathcal{F}$,
see Definition \ref{definition-presheaf-modules},
such that the underlying presheaf of abelian groups $\mathcal{F}$
is a sheaf.
\item A {\it morphism of sheaves of $\mathcal{O}$-modules}
is a morphism of presheaves of $\mathcal{O}$-modules.
\item Given sheaves of $\mathcal{O}$-modules
$\mathcal{F}$ and $\mathcal{G}$ we denote
$\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})$
the set of morphism of sheaves of $\mathcal{O}$-modules.
\item The category of sheaves of $\mathcal{O}$-modules
is denoted $\textit{Mod}(\mathcal{O})$.
\end{enumerate}
\end{definition}
\noindent
This definition kind of makes sense even if $\mathcal{O}$ is just a
presheaf of rings, although we do not know any examples where
this is useful, and we will avoid using the terminology
``sheaves of $\mathcal{O}$-modules''
in case $\mathcal{O}$ is not a sheaf of rings.
\section{Sheafification of presheaves of modules}
\label{section-sheafification-presheaves-modules}
\begin{lemma}
\label{lemma-sheafification-presheaf-modules}
Let $\mathcal{C}$ be a site.
Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$.
Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules.
Let $\mathcal{O}^\#$ be the sheafification of $\mathcal{O}$ as a presheaf
of rings, see Sites, Section \ref{sites-section-sheaves-algebraic-structures}.
Let $\mathcal{F}^\#$ be the sheafification of $\mathcal{F}$
as a presheaf of abelian groups. There exists a unique map of
sheaves of sets
$$
\mathcal{O}^\# \times \mathcal{F}^\#
\longrightarrow
\mathcal{F}^\#
$$
which makes the diagram
$$
\xymatrix{
\mathcal{O} \times \mathcal{F} \ar[r] \ar[d] &
\mathcal{F} \ar[d] \\
\mathcal{O}^\# \times \mathcal{F}^\# \ar[r] &
\mathcal{F}^\#
}
$$
commute and which makes $\mathcal{F}^\#$ into a sheaf
of $\mathcal{O}^\#$-modules. In addition, if $\mathcal{G}$
is a sheaf of $\mathcal{O}^\#$-modules, then any morphism
of presheaves of $\mathcal{O}$-modules $\mathcal{F} \to \mathcal{G}$
(into the restriction of $\mathcal{G}$ to a $\mathcal{O}$-module)
factors uniquely as $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$
where $\mathcal{F}^\# \to \mathcal{G}$ is a morphism of
$\mathcal{O}^\#$-modules.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\noindent
This actually means that the functor
$i : \textit{Mod}(\mathcal{O}^\#) \to \textit{PMod}(\mathcal{O})$
(combining restriction and including sheaves into presheaves)
and the sheafification functor of the lemma
${}^\# : \textit{PMod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}^\#)$
are adjoint. In a formula
$$
\Mor_{\textit{PMod}(\mathcal{O})}(\mathcal{F}, i\mathcal{G})
=
\Mor_{\textit{Mod}(\mathcal{O}^\#)}(\mathcal{F}^\#, \mathcal{G})
$$
An important case happens when $\mathcal{O}$ is already a sheaf of
rings. In this case the formula reads
$$
\Mor_{\textit{PMod}(\mathcal{O})}(\mathcal{F}, i\mathcal{G})
=
\Mor_{\textit{Mod}(\mathcal{O})}(\mathcal{F}^\#, \mathcal{G})
$$
because $\mathcal{O} = \mathcal{O}^\#$ in this case.
\begin{lemma}
\label{lemma-sheafification-exact}
Let $\mathcal{C}$ be a site.
Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$
The sheafification functor
$$
\textit{PMod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}^\#), \quad
\mathcal{F} \longmapsto \mathcal{F}^\#
$$
is exact.
\end{lemma}
\begin{proof}
This is true because it holds for sheafification
$\textit{PAb}(\mathcal{C}) \to \textit{Ab}(\mathcal{C})$.
See the discussion in Section \ref{section-abelian-sheaves}.
\end{proof}
\noindent
Let $\mathcal{C}$ be a site.
Let $\mathcal{O}_1 \to \mathcal{O}_2$ be
a morphism of sheaves of rings on $\mathcal{C}$.
In Section \ref{section-presheaves-modules}
we defined a restriction functor
and a change of rings functor on presheaves of modules
associated to this situation.
\medskip\noindent
If $\mathcal{F}$ is a sheaf of $\mathcal{O}_2$-modules
then the restriction $\mathcal{F}_{\mathcal{O}_1}$
of $\mathcal{F}$ is clearly a sheaf
of $\mathcal{O}_1$-modules. We obtain the restriction functor
$$
\textit{Mod}(\mathcal{O}_2)
\longrightarrow
\textit{Mod}(\mathcal{O}_1)
$$
\medskip\noindent
On the other hand, given a sheaf of $\mathcal{O}_1$-modules
$\mathcal{G}$ the presheaf of $\mathcal{O}_2$-modules
$\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}$
is in general not a sheaf. Hence we define the
{\it tensor product sheaf}
$\mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}$
by the formula
$$
\mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}
=
(\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G})^\#
$$
as the sheafification of our construction for presheaves.
We obtain the {\it change of rings} functor
$$
\textit{Mod}(\mathcal{O}_1)
\longrightarrow
\textit{Mod}(\mathcal{O}_2)
$$
\begin{lemma}
\label{lemma-adjointness-tensor-restrict}
With $X$, $\mathcal{O}_1$, $\mathcal{O}_2$, $\mathcal{F}$ and
$\mathcal{G}$ as above there exists a canonical bijection
$$
\Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1})
=
\Hom_{\mathcal{O}_2}(
\mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G},
\mathcal{F}
)
$$
In other words, the restriction and change of rings functors
are adjoint to each other.
\end{lemma}
\begin{proof}
This follows from
Lemma \ref{lemma-adjointness-tensor-restrict-presheaves}
and the fact that
$\Hom_{\mathcal{O}_2}(
\mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G},
\mathcal{F}
)
=
\Hom_{\mathcal{O}_2}(
\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G},
\mathcal{F}
)$
because $\mathcal{F}$ is a sheaf.
\end{proof}
\begin{lemma}
\label{lemma-epimorphism-modules}
Let $\mathcal{C}$ be a site.
Let $\mathcal{O} \to \mathcal{O}'$ be an epimorphism of sheaves of rings.
Let $\mathcal{G}_1, \mathcal{G}_2$ be $\mathcal{O}'$-modules.
Then
$$
\Hom_{\mathcal{O}'}(\mathcal{G}_1, \mathcal{G}_2) =
\Hom_\mathcal{O}(\mathcal{G}_1, \mathcal{G}_2).
$$
In other words, the restriction functor
$\textit{Mod}(\mathcal{O}') \to \textit{Mod}(\mathcal{O})$ is fully faithful.
\end{lemma}
\begin{proof}
This is the sheaf version of
Algebra, Lemma \ref{algebra-lemma-epimorphism-modules}
and is proved in exactly the same way.
\end{proof}
\section{Morphisms of topoi and sheaves of modules}
\label{section-sheaves-modules-functorial}
\noindent
All of this material is completely straightforward.
We formulate everything in the case of morphisms of topoi,
but of course the results also hold in the case of morphisms of sites.
\begin{lemma}
\label{lemma-pushforward-module}
Let $\mathcal{C}$, $\mathcal{D}$ be sites.
Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$
be a morphism of topoi.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules.
There is a natural map of sheaves of sets
$$
f_*\mathcal{O} \times f_*\mathcal{F}
\longrightarrow
f_*\mathcal{F}
$$
which turns $f_*\mathcal{F}$ into a sheaf of $f_*\mathcal{O}$-modules.
This construction is functorial in $\mathcal{F}$.
\end{lemma}
\begin{proof}
Denote $\mu : \mathcal{O} \times \mathcal{F} \to \mathcal{F}$ the
multiplication map. Recall that $f_*$ (on sheaves of sets) is left exact
and hence commutes with products. Hence $f_*\mu$ is a map as
indicated. This proves the lemma.
\end{proof}
\begin{lemma}
\label{lemma-pullback-module}
Let $\mathcal{C}$, $\mathcal{D}$ be sites.
Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$
be a morphism of topoi.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{D}$.
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules.
There is a natural map of sheaves of sets
$$
f^{-1}\mathcal{O} \times f^{-1}\mathcal{G}
\longrightarrow
f^{-1}\mathcal{G}
$$
which turns $f^{-1}\mathcal{G}$ into a sheaf of $f^{-1}\mathcal{O}$-modules.
This construction is functorial in $\mathcal{G}$.
\end{lemma}
\begin{proof}
Denote $\mu : \mathcal{O} \times \mathcal{G} \to \mathcal{G}$ the
multiplication map. Recall that $f^{-1}$ (on sheaves of sets) is exact
and hence commutes with products. Hence $f^{-1}\mu$ is a map as
indicated. This proves the lemma.
\end{proof}
\begin{lemma}
\label{lemma-adjoint-push-pull-modules}
Let $\mathcal{C}$, $\mathcal{D}$ be sites.
Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$
be a morphism of topoi.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{D}$.
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules.
Let $\mathcal{F}$ be a sheaf of $f^{-1}\mathcal{O}$-modules.
Then
$$
\Mor_{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F})
=
\Mor_{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).
$$
Here we use
Lemmas \ref{lemma-pullback-module}
and \ref{lemma-pushforward-module}, and we think of
$f_*\mathcal{F}$ as an $\mathcal{O}$-module by restriction via
$\mathcal{O} \to f_*f^{-1}\mathcal{O}$.
\end{lemma}
\begin{proof}
First we note that we have
$$
\Mor_{\textit{Ab}(\mathcal{C})}(f^{-1}\mathcal{G}, \mathcal{F})
=
\Mor_{\textit{Ab}(\mathcal{D})}(\mathcal{G}, f_*\mathcal{F}).
$$
by Sites,
Proposition \ref{sites-proposition-functoriality-algebraic-structures-topoi}.
Suppose that $\alpha : f^{-1}\mathcal{G} \to \mathcal{F}$ and
$\beta : \mathcal{G} \to f_*\mathcal{F}$ are morphisms of abelian
sheaves which correspond via the formula above. We have to show that
$\alpha$ is $f^{-1}\mathcal{O}$-linear if and only if $\beta$
is $\mathcal{O}$-linear. For example, suppose $\alpha$ is
$f^{-1}\mathcal{O}$-linear, then clearly $f_*\alpha$ is
$f_*f^{-1}\mathcal{O}$-linear, and hence (as restriction is a functor)
is $\mathcal{O}$-linear. Hence it suffices to prove that the
adjunction map $\mathcal{G} \to f_*f^{-1}\mathcal{G}$ is
$\mathcal{O}$-linear. Using that both $f_*$ and $f^{-1}$ commute
with products (on sheaves of sets) this comes down to showing that
$$
\xymatrix{
\mathcal{O} \times \mathcal{G} \ar[r] \ar[d] &
f_*f^{-1}(\mathcal{O} \times \mathcal{G}) \ar[d] \\
\mathcal{G} \ar[r] & f_*f^{-1}\mathcal{G}
}
$$
is commutative. This holds because the adjunction mapping
$\text{id}_{\Sh(\mathcal{D})} \to f_*f^{-1}$ is a
transformation of functors. We omit the proof of the implication
$\beta$ linear $\Rightarrow$ $\alpha$ linear.
\end{proof}
\begin{lemma}
\label{lemma-adjoint-pull-push-modules}
Let $\mathcal{C}$, $\mathcal{D}$ be sites.
Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$
be a morphism of topoi.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules.
Let $\mathcal{G}$ be a sheaf of $f_*\mathcal{O}$-modules.
Then
$$
\Mor_{\textit{Mod}(\mathcal{O})}(
\mathcal{O} \otimes_{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F})
=
\Mor_{\textit{Mod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).
$$
Here we use
Lemmas \ref{lemma-pullback-module}
and \ref{lemma-pushforward-module}, and we use
the canonical map $f^{-1}f_*\mathcal{O} \to \mathcal{O}$
in the definition of the tensor product.
\end{lemma}
\begin{proof}
Note that we have
$$
\Mor_{\textit{Mod}(\mathcal{O})}(
\mathcal{O} \otimes_{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F})
=
\Mor_{\textit{Mod}(f^{-1}f_*\mathcal{O})}(
f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}f_*\mathcal{O}})
$$
by Lemma \ref{lemma-adjointness-tensor-restrict}. Hence the result follows
from Lemma \ref{lemma-adjoint-push-pull-modules}.
\end{proof}
\section{Morphisms of ringed topoi and modules}
\label{section-functoriality-modules}
\noindent
We have now introduced enough notation so that we are able to
define the pullback and pushforward of modules along a morphism
of ringed topoi.
\begin{definition}
\label{definition-pushforward}
Let
$(f, f^\sharp) :
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
\to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
be a morphism of ringed topoi or ringed sites.
\begin{enumerate}
\item Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{C}$-modules.
We define the {\it pushforward} of $\mathcal{F}$ as the
sheaf of $\mathcal{O}_\mathcal{D}$-modules which as a sheaf
of abelian groups equals $f_*\mathcal{F}$ and with
module structure given by the restriction
via $f^\sharp : \mathcal{O}_\mathcal{D} \to f_*\mathcal{O}_\mathcal{C}$
of the module structure
$$
f_*\mathcal{O}_\mathcal{C} \times f_*\mathcal{F}
\longrightarrow
f_*\mathcal{F}
$$
from Lemma \ref{lemma-pushforward-module}.
\item Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal{D}$-modules.
We define the {\it pullback} $f^*\mathcal{G}$ to be the
sheaf of $\mathcal{O}_\mathcal{C}$-modules defined by the formula
$$
f^*\mathcal{G}
=
\mathcal{O}_\mathcal{C} \otimes_{f^{-1}\mathcal{O}_\mathcal{D}}
f^{-1}\mathcal{G}
$$
where the ring map
$f^{-1}\mathcal{O}_\mathcal{D} \to \mathcal{O}_\mathcal{C}$
is $f^\sharp$, and where the module
structure is given by Lemma \ref{lemma-pullback-module}.
\end{enumerate}
\end{definition}
\noindent
Thus we have defined functors
\begin{eqnarray*}
f_* : \textit{Mod}(\mathcal{O}_\mathcal{C})
& \longrightarrow &
\textit{Mod}(\mathcal{O}_\mathcal{D}) \\
f^* : \textit{Mod}(\mathcal{O}_\mathcal{D})
& \longrightarrow &
\textit{Mod}(\mathcal{O}_\mathcal{C})
\end{eqnarray*}
The final result on these functors is that they are indeed
adjoint as expected.
\begin{lemma}
\label{lemma-adjoint-pullback-pushforward-modules}
Let
$(f, f^\sharp) :
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
\to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
be a morphism of ringed topoi or ringed sites.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{C}$-modules.
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal{D}$-modules.
There is a canonical bijection
$$
\Hom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{G}, \mathcal{F})
=
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, f_*\mathcal{F}).
$$
In other words: the functor $f^*$ is the left adjoint to
$f_*$.
\end{lemma}
\begin{proof}
This follows from the work we did before:
\begin{eqnarray*}
\Hom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{G}, \mathcal{F})
& = &
\Mor_{\textit{Mod}(\mathcal{O}_\mathcal{C})}(
\mathcal{O}_\mathcal{C}
\otimes_{f^{-1}\mathcal{O}_\mathcal{D}} f^{-1}\mathcal{G},
\mathcal{F}) \\
& = &
\Mor_{\textit{Mod}(f^{-1}\mathcal{O}_\mathcal{D})}(
f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}\mathcal{O}_\mathcal{D}}) \\
& = &
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, f_*\mathcal{F}).
\end{eqnarray*}
Here we use Lemmas \ref{lemma-adjointness-tensor-restrict}
and \ref{lemma-adjoint-push-pull-modules}.
\end{proof}
\begin{lemma}
\label{lemma-push-pull-composition-modules}
$(f, f^\sharp) :
(\Sh(\mathcal{C}_1), \mathcal{O}_1)
\to (\Sh(\mathcal{C}_2), \mathcal{O}_2)$ and
$(g, g^\sharp) :
(\Sh(\mathcal{C}_2), \mathcal{O}_2) \to
(\Sh(\mathcal{C}_3), \mathcal{O}_3)$
be morphisms of ringed topoi.
There are canonical isomorphisms of functors
$(g \circ f)_* \cong g_* \circ f_*$ and
$(g \circ f)^* \cong f^* \circ g^*$.
\end{lemma}
\begin{proof}
This is clear from the definitions.
\end{proof}
\section{The abelian category of sheaves of modules}
\label{section-kernels}
\noindent
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos.
Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}$-modules, see
Sheaves, Definition \ref{sheaves-definition-sheaf-modules}.
Let $\varphi, \psi : \mathcal{F} \to \mathcal{G}$
be morphisms of sheaves of $\mathcal{O}$-modules.
We define $\varphi + \psi : \mathcal{F} \to \mathcal{G}$
to be the sum of $\varphi$ and $\psi$ as morphisms of abelian sheaves.
This is clearly again a map of $\mathcal{O}$-modules.
It is also clear that composition of maps of
$\mathcal{O}$-modules is bilinear with respect to this
addition. Thus $\textit{Mod}(\mathcal{O})$ is a pre-additive
category, see Homology, Definition \ref{homology-definition-preadditive}.
\medskip\noindent
We will denote $0$ the sheaf of $\mathcal{O}$-modules
which has constant value $\{0\}$ for all objects $U$ of $\mathcal{C}$.
Clearly this is both a final and an initial object of
$\textit{Mod}(\mathcal{O})$. Given a morphism
of $\mathcal{O}$-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 object $U$.
\medskip\noindent
Moreover, given a pair
$\mathcal{F}$, $\mathcal{G}$ of sheaves of $\mathcal{O}$-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 $\textit{Mod}(\mathcal{O})$
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 $\mathcal{O}$-modules. We may define $\Ker(\varphi)$
to be the kernel of $\varphi$ as a map of abelian sheaves.
By Section \ref{section-abelian-sheaves} this is 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 objects $U$ of $\mathcal{C}$. It is easy to see that this is indeed
a kernel in the category of $\mathcal{O}$-modules. In other words,
a morphism $\alpha : \mathcal{H} \to \mathcal{F}$ factors
through $\Ker(\varphi)$ if and only if $\varphi \circ \alpha = 0$.
\medskip\noindent
Similarly, we define $\Coker(\varphi)$ as the cokernel of
$\varphi$ as a map of abelian sheaves. There is a unique
multiplication map
$$
\mathcal{O} \times \Coker(\varphi) \longrightarrow \Coker(\varphi)
$$
such that the map $\mathcal{G} \to \Coker(\varphi)$ becomes a
morphism of $\mathcal{O}$-modules (verification omitted).
The map $\mathcal{G} \to \Coker(\varphi)$ is surjective
(as a map of sheaves of sets, see Section \ref{section-abelian-sheaves}).
To show that $\Coker(\varphi)$ is a cokernel in
$\textit{Mod}(\mathcal{O})$, note that if
$\beta : \mathcal{G} \to \mathcal{H}$ is a morphism of $\mathcal{O}$-modules
such that $\beta \circ \varphi$ is zero, then you get for every
object $U$ of $\mathcal{C}$ 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 $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos.
The category $\textit{Mod}(\mathcal{O})$ is an abelian category.
The forgetful functor
$\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$
is exact, hence kernels, cokernels and exactness of
$\mathcal{O}$-modules, correspond to the corresponding notions
for abelian sheaves.
\end{lemma}
\begin{proof}
Above we have seen that $\textit{Mod}(\mathcal{O})$ is an additive
category, with kernels and cokernels
and that $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$
preserves kernels and cokernels.
By Homology, Definition \ref{homology-definition-abelian-category}
we have to show that image and coimage agree. This is clear
because it is true in $\textit{Ab}(\mathcal{C})$. The lemma follows.
\end{proof}
\begin{lemma}
\label{lemma-limits-colimits}
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos.
All limits and colimits exist in $\textit{Mod}(\mathcal{O})$
and the forgetful functor
$\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$
commutes with them. Moreover, filtered colimits are exact.
\end{lemma}
\begin{proof}
The final statement follows from the first as filtered colimits are
exact in $\textit{Ab}(\mathcal{C})$ by
Lemma \ref{lemma-limits-colimits-abelian-sheaves}.
Let $\mathcal{I} \to \textit{Mod}(\mathcal{C})$, $i \mapsto \mathcal{F}_i$
be a diagram. Let $\lim_i \mathcal{F}_i$ be the limit of the diagram
in $\textit{Ab}(\mathcal{C})$. By the description of this limit in
Lemma \ref{lemma-limits-colimits-abelian-sheaves} we see immediately that
there exists a multiplication
$$
\mathcal{O} \times \lim_i \mathcal{F}_i
\longrightarrow
\lim_i \mathcal{F}_i
$$
which turns $\lim_i \mathcal{F}_i$ into a sheaf of
$\mathcal{O}$-modules. It is easy to see that this is the
limit of the diagram in $\textit{Mod}(\mathcal{C})$. Let
$\colim_i \mathcal{F}_i$ be the colimit of the diagram
in $\textit{PAb}(\mathcal{C})$. By the description of this colimit
in the proof of Lemma \ref{lemma-limits-colimits-abelian-presheaves}
we see immediately that there exists a multiplication
$$
\mathcal{O} \times \colim_i \mathcal{F}_i
\longrightarrow
\colim_i \mathcal{F}_i
$$
which turns $\colim_i \mathcal{F}_i$ into a presheaf of
$\mathcal{O}$-modules. Applying sheafification we get a
sheaf of $\mathcal{O}$-modules $(\colim_i \mathcal{F}_i)^\#$,
see Lemma \ref{lemma-sheafification-presheaf-modules}.
It is easy to see that $(\colim_i \mathcal{F}_i)^\#$
is the colimit of the diagram in $\textit{Mod}(\mathcal{O})$, and
by Lemma \ref{lemma-limits-colimits-abelian-sheaves}
forgetting the $\mathcal{O}$-module structure is
the colimit in $\textit{Ab}(\mathcal{C})$.
\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 $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
\to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
be a morphism of ringed topoi.
\begin{enumerate}
\item The functor $f_*$ is left exact. In fact it commutes with
all limits.
\item The functor $f^*$ is right exact. In fact it commutes
with all colimits.
\end{enumerate}
\end{lemma}
\begin{proof}
This is true because $(f^*, f_*)$ is an adjoint pair
of functors, see
Lemma \ref{lemma-adjoint-pullback-pushforward-modules}.
See Categories, Section \ref{categories-section-adjoint}.
\end{proof}
\begin{lemma}
\label{lemma-check-exactness-stalks}
Let $\mathcal{C}$ be a site. If $\{p_i\}_{i \in I}$ is a conservative
family of points, then we may check exactness of a sequence of abelian
sheaves on the stalks at the points $p_i$, $i \in I$.
If $\mathcal{C}$ has enough points, then
exactness of a sequence of abelian sheaves may
be checked on stalks.
\end{lemma}
\begin{proof}
This is immediate from
Sites, Lemma \ref{sites-lemma-exactness-stalks}.
\end{proof}
\section{Exactness of pushforward}
\label{section-pushforward}
\noindent
Some technical lemmas concerning exactness properties of pushforward.
\begin{lemma}
\label{lemma-reflect-surjections}
Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be
a morphism of topoi. The following are equivalent:
\begin{enumerate}
\item $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is surjective for
all $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, and
\item $f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$
reflects surjections.
\end{enumerate}
In this case the functor
$f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$
is faithful.
\end{lemma}
\begin{proof}
Assume (1). Suppose that $a : \mathcal{F} \to \mathcal{F}'$
is a map of abelian sheaves on $\mathcal{C}$ such that $f_*a$ is surjective.
As $f^{-1}$ is exact this implies that
$f^{-1}f_*a : f^{-1}f_*\mathcal{F} \to f^{-1}f_*\mathcal{F}'$
is surjective. Combined with (1) this implies that $a$ is surjective.
This means that (2) holds.
\medskip\noindent
Assume (2). Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$.
We have to show that the map $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is
surjective. By (2) it suffices to show that
$f_*f^{-1}f_*\mathcal{F} \to f_*\mathcal{F}$ is surjective.
And this is true because there is a canonical map
$f_*\mathcal{F} \to f_*f^{-1}f_*\mathcal{F}$ which is a one-sided inverse.
\medskip\noindent
We omit the proof of the final assertion.
\end{proof}
\begin{lemma}
\label{lemma-exactness}
Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be
a morphism of topoi. Assume at least one of the following properties
holds
\begin{enumerate}
\item $f_*$ transforms surjections of sheaves of sets into surjections,
\item $f_*$ transforms surjections of abelian sheaves into surjections,
\item $f_*$ commutes with coequalizers on sheaves of sets,
\item $f_*$ commutes with pushouts on sheaves of sets,
\end{enumerate}
Then $f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$
is exact.
\end{lemma}
\begin{proof}
Since $f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$
is a right adjoint we already know that it transforms a short exact sequence
$0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$
of abelian sheaves on $\mathcal{C}$ into an exact sequence
$$
0 \to f_*\mathcal{F}_1 \to f_*\mathcal{F}_2 \to f_*\mathcal{F}_3
$$
see
Categories, Sections \ref{categories-section-exact-functor} and
\ref{categories-section-adjoint}
and
Homology, Section \ref{homology-section-functors}. Hence it suffices to
prove that the map $f_*\mathcal{F}_2 \to f_*\mathcal{F}_3$ is surjective.
If (1), (2) holds, then this is clear from the definitions. By
Sites, Lemma \ref{sites-lemma-exactness-properties}
we see that either (3) or (4) formally implies (1), hence in these cases
we are done also.
\end{proof}
\begin{lemma}
\label{lemma-morphism-ringed-sites-almost-cocontinuous}
Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites
associated to the continuous functor $u : \mathcal{C} \to \mathcal{D}$.
Assume $u$ is almost cocontinuous. Then
\begin{enumerate}
\item $f_* : \textit{Ab}(\mathcal{D}) \to \textit{Ab}(\mathcal{C})$ is exact.
\item if $f^\sharp : f^{-1}\mathcal{O}_\mathcal{C} \to \mathcal{O}_\mathcal{D}$
is given so that $f$ becomes a morphism of ringed sites, then
$f_* : \textit{Mod}(\mathcal{O}_\mathcal{D}) \to
\textit{Mod}(\mathcal{O}_\mathcal{C})$ is exact.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (2) follows from part (1) by
Lemma \ref{lemma-limits-colimits}.
Part (1) follows from
Sites, Lemmas
\ref{sites-lemma-morphism-of-sites-almost-cocontinuous} and
\ref{sites-lemma-exactness-properties}.
\end{proof}
\section{Exactness of lower shriek}
\label{section-exactness-lower-shriek}
\noindent
Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between sites.
Assume that
\begin{enumerate}
\item[(a)] $u$ is cocontinuous, and
\item[(b)] $u$ is continuous.
\end{enumerate}
Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the
morphism of topoi associated with $u$, see
Sites, Lemma \ref{sites-lemma-cocontinuous-morphism-topoi}.
Recall that $g^{-1} = u^p$, i.e., $g^{-1}$ is given by the simple formula
$(g^{-1}\mathcal{G})(U) = \mathcal{G}(u(U))$, see
Sites, Lemma \ref{sites-lemma-when-shriek}.
We would like to show that
$g^{-1} : \textit{Ab}(\mathcal{D}) \to \textit{Ab}(\mathcal{C})$
has a left adjoint $g_!$. By
Sites, Lemma \ref{sites-lemma-when-shriek}
the functor $g^{Sh}_! = (u_p\ )^\#$ is a left adjoint on sheaves of sets.
Moreover, we know that $g^{Sh}_!\mathcal{F}$ is the sheaf
associated to the presheaf
$$
V \longmapsto \colim_{V \to u(U)} \mathcal{F}(U)
$$
where the colimit is over $(\mathcal{I}_V^u)^{opp}$ and is taken in the
category of sets. Hence the following definition is natural.
\begin{definition}
\label{definition-g-shriek}
With $u : \mathcal{C} \to \mathcal{D}$ satisfying (a), (b) above.
For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we define
{\it $g_{p!}\mathcal{F}$} as the presheaf
$$
V \longmapsto \colim_{V \to u(U)} \mathcal{F}(U)
$$
with colimits over $(\mathcal{I}_V^u)^{opp}$ taken in $\textit{Ab}$. For
$\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we set
{\it $g_!\mathcal{F} = (g_{p!}\mathcal{F})^\#$}.
\end{definition}
\noindent
The reason for being so explicit with this is that the functors
$g^{Sh}_!$ and $g_!$ are different. Whenever we use both
we have to be careful to make the distinction clear.
\begin{lemma}
\label{lemma-g-shriek-adjoint}
The functor $g_{p!}$ is a left adjoint to the functor $u^p$.
The functor $g_!$ is a left adjoint to the functor $g^{-1}$.
In other words the formulas
\begin{align*}
\Mor_{\textit{PAb}(\mathcal{C})}(\mathcal{F}, u^p\mathcal{G})
& =
\Mor_{\textit{PAb}(\mathcal{D})}(g_{p!}\mathcal{F}, \mathcal{G}), \\
\Mor_{\textit{Ab}(\mathcal{C})}(\mathcal{F}, g^{-1}\mathcal{G})
& =
\Mor_{\textit{Ab}(\mathcal{D})}(g_!\mathcal{F}, \mathcal{G})
\end{align*}
hold bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
\end{lemma}
\begin{proof}
The second formula follows formally from the first, since if
$\mathcal{F}$ and $\mathcal{G}$ are abelian sheaves then
\begin{align*}
\Mor_{\textit{Ab}(\mathcal{C})}(\mathcal{F}, g^{-1}\mathcal{G})
& =
\Mor_{\textit{PAb}(\mathcal{D})}(g_{p!}\mathcal{F}, \mathcal{G}) \\
& =
\Mor_{\textit{Ab}(\mathcal{D})}(g_!\mathcal{F}, \mathcal{G})
\end{align*}
by the universal property of sheafification.
\medskip\noindent
To prove the first formula, let $\mathcal{F}$, $\mathcal{G}$ be abelian
presheaves. To prove the lemma we will construct maps from the group on the
left to the group on the right and omit the verification that these are
mutually inverse.
\medskip\noindent
Note that there is a canonical map of abelian presheaves
$\mathcal{F} \to u^pg_{p!}\mathcal{F}$ which on sections over $U$ is the
natural map
$\mathcal{F}(U) \to \colim_{u(U) \to u(U')} \mathcal{F}(U')$, see
Sites, Lemma \ref{sites-lemma-recover}.
Given a map $\alpha : g_{p!}\mathcal{F} \to \mathcal{G}$
we get $u^p\alpha : u^pg_{p!}\mathcal{F} \to u^p\mathcal{G}$.
which we can precompose by the map $\mathcal{F} \to u^pg_{p!}\mathcal{F}$.
\medskip\noindent
Note that there is a canonical map of abelian presheaves
$g_{p!}u^p\mathcal{G} \to \mathcal{G}$ which on sections over
$V$ is the natural map
$\colim_{V \to u(U)} \mathcal{G}(u(U)) \to \mathcal{G}(V)$.
It maps a section $s \in u(U)$ in the summand corresponding to
$t : V \to u(U)$ to $t^*s \in \mathcal{G}(V)$.
Hence, given a map $\beta : \mathcal{F} \to u^p\mathcal{G}$
we get a map $g_{p!}\beta : g_{p!}\mathcal{F} \to g_{p!}u^p\mathcal{G}$
which we can postcompose with the map $g_{p!}u^p\mathcal{G} \to \mathcal{G}$
above.
\end{proof}
\begin{lemma}
\label{lemma-exactness-lower-shriek}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites.
Let $u : \mathcal{C} \to \mathcal{D}$ be a functor.
Assume that
\begin{enumerate}
\item[(a)] $u$ is cocontinuous,
\item[(b)] $u$ is continuous, and
\item[(c)] fibre products and equalizers exist in $\mathcal{C}$ and
$u$ commutes with them.
\end{enumerate}
In this case the functor
$g_! : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$
is exact.
\end{lemma}
\begin{proof}
Compare with
Sites, Lemma \ref{sites-lemma-preserve-equalizers}.
Assume (a), (b), and (c).
We already know that $g_!$ is right exact as it is a left adjoint, see
Categories, Lemma \ref{categories-lemma-exact-adjoint} and
Homology, Section \ref{homology-section-functors}.
We have $g_! = (g_{p!}\ )^\#$. We have to show that
$g_!$ transforms injective maps of abelian sheaves into injective maps
of abelian presheaves.
Recall that sheafification of abelian presheaves is exact, see
Lemma \ref{lemma-limits-colimits-abelian-sheaves}.
Thus it suffices to show that $g_{p!}$ transforms injective maps of
abelian presheaves into injective maps of abelian presheaves.
To do this it suffices that colimits over the categories
$(\mathcal{I}_V^u)^{opp}$ of
Sites, Section \ref{sites-section-functoriality-PSh}
transform injective maps between diagrams into injections.
This follows from
Sites, Lemma \ref{sites-lemma-almost-directed}
and
Algebra, Lemma \ref{algebra-lemma-almost-directed-colimit-exact}.
\end{proof}
\begin{lemma}
\label{lemma-back-and-forth}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites.
Let $u : \mathcal{C} \to \mathcal{D}$ be a functor.
Assume that
\begin{enumerate}
\item[(a)] $u$ is cocontinuous,
\item[(b)] $u$ is continuous, and
\item[(c)] $u$ is fully faithful.
\end{enumerate}
For $g_!, g^{-1}, g_*$ as above the canonical maps
$\mathcal{F} \to g^{-1}g_!\mathcal{F}$ and
$g^{-1}g_*\mathcal{F} \to \mathcal{F}$
are isomorphisms
for all abelian sheaves $\mathcal{F}$ on $\mathcal{C}$.
\end{lemma}
\begin{proof}
The map $g^{-1}g_*\mathcal{F} \to \mathcal{F}$ is an isomorphism
by Sites, Lemma \ref{sites-lemma-back-and-forth} and the fact that
pullback and pushforward of abelian sheaves agrees with
pullback and pushforward on the underlying sheaves of sets.
\medskip\noindent
Pick $U \in \Ob(\mathcal{C})$. We will show that
$g^{-1}g_!\mathcal{F}(U) = \mathcal{F}(U)$. First, note that
$g^{-1}g_!\mathcal{F}(U) = g_!\mathcal{F}(u(U))$. Hence it suffices
to show that $g_!\mathcal{F}(u(U)) = \mathcal{F}(U)$.
We know that $g_!\mathcal{F}$ is the (abelian) sheaf associated
to the presheaf $g_{p!}\mathcal{F}$ which is defined by the rule
$$
V \longmapsto \colim_{V \to u(U')} \mathcal{F}(U')
$$
with colimit taken in $\textit{Ab}$. If $V = u(U)$, then, as $u$ is
fully faithful this colimit is over $U \to U'$. Hence we conclude
that $g_{p!}\mathcal{F}(u(U) = \mathcal{F}(U)$.
Since $u$ is cocontinuous and continuous any covering of $u(U)$ in
$\mathcal{D}$ can be refined by a covering (!) $\{u(U_i) \to u(U)\}$
of $\mathcal{D}$ where $\{U_i \to U\}$ is a covering in $\mathcal{C}$.
This implies that $(g_{p!}\mathcal{F})^+(u(U)) = \mathcal{F}(U)$ also,
since in the colimit defining the value of $(g_{p!}\mathcal{F})^+$
on $u(U)$ we may restrict to the cofinal system of coverings
$\{u(U_i) \to u(U)\}$ as above. Hence we see that
$(g_{p!}\mathcal{F})^+(u(U)) = \mathcal{F}(U)$ for all objects $U$
of $\mathcal{C}$ as well. Repeating this argument one more time
gives the equality $(g_{p!}\mathcal{F})^\#(u(U)) = \mathcal{F}(U)$
for all objects $U$ of $\mathcal{C}$. This produces the desired
equality $g^{-1}g_!\mathcal{F} = \mathcal{F}$.
\end{proof}
\begin{remark}
\label{remark-no-extension}
In general the functor $g_!$ cannot be extended to categories of modules
in case $g$ is (part of) a morphism of ringed topoi. Namely, given any
ring map $A \to B$ the functor $M \mapsto B \otimes_A M$ has a right adjoint
(restriction) but not in general a left adjoint (because its existence
would imply that $A \to B$ is flat). We will see in
Section \ref{section-localize}
below that it is possible to define $j_!$ on sheaves of modules
in the case of a localization of sites.
We will discuss this in greater generality in
Section \ref{section-lower-shriek-modules} below.
\end{remark}
\begin{lemma}
\label{lemma-have-left-adjoint}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let
$g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the morphism of topoi
associated to a continuous and cocontinuous functor
$u : \mathcal{C} \to \mathcal{D}$.
\begin{enumerate}
\item If $u$ has a left adjoint $w$, then $g_!$ agrees with $g_!^{\Sh}$
on underlying sheaves of sets and $g_!$ is exact.
\item If in addition $w$ is cocontinuous, then $g_! = h^{-1}$ and
$g^{-1} = h_*$ where
$h : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ is the morphism of topoi
associated to $w$.
\end{enumerate}
\end{lemma}
\begin{proof}
This Lemma is the analogue of
Sites, Lemma \ref{sites-lemma-have-left-adjoint}.
From Sites, Lemma \ref{sites-lemma-adjoint-functors} we see that the categories
$\mathcal{I}_V^u$ have an initial object. Thus the underlying set of a
colimit of a system of abelian groups over $(\mathcal{I}_V^u)^{opp}$
is the colimit of the underlying sets. Whence the agreement
of $g_!^{\Sh}$ and $g_!$ by our construction of $g_!$ in
Definition \ref{definition-g-shriek}.
The exactness and (2) follow immediately from the corresponding statements of
Sites, Lemma \ref{sites-lemma-have-left-adjoint}.
\end{proof}
\section{Global types of modules}
\label{section-global}
\begin{definition}
\label{definition-global}
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules.
\begin{enumerate}
\item We say $\mathcal{F}$ is a {\it free $\mathcal{O}$-module}
if $\mathcal{F}$ is isomorphic as an $\mathcal{O}$-module
to a sheaf of the form $\bigoplus_{i \in I} \mathcal{O}$.
\item We say $\mathcal{F}$ is {\it finite free} if
$\mathcal{F}$ is isomorphic as an $\mathcal{O}$-module
to a sheaf of the form $\bigoplus_{i \in I} \mathcal{O}$
with a finite index set $I$.
\item We say $\mathcal{F}$ is {\it generated by global sections}
if there exists a surjection
$$
\bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{F}
$$
from a free $\mathcal{O}$-module onto $\mathcal{F}$.
\item Given $r \geq 0$ we say $\mathcal{F}$ is
{\it generated by $r$ global sections} if there exists a surjection
$\mathcal{O}^{\oplus r} \to \mathcal{F}$.
\item We say $\mathcal{F}$ is {\it generated by finitely many global sections}
if it is generated by $r$ global sections for some $r \geq 0$.
\item We say $\mathcal{F}$ has a {\it global presentation}
if there exists an exact sequence
$$
\bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow
\bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow
\mathcal{F} \longrightarrow 0
$$
of $\mathcal{O}$-modules.
\item We say $\mathcal{F}$ has a {\it global finite presentation}
if there exists an exact sequence
$$
\bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow
\bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow
\mathcal{F} \longrightarrow 0
$$
of $\mathcal{O}$-modules with $I$ and $J$ finite sets.
\end{enumerate}
\end{definition}
\noindent
Note that for any set $I$ the direct sum
$\bigoplus_{i \in I} \mathcal{O}$ exists
(Lemma \ref{lemma-limits-colimits})
and is the sheafification of the presheaf
$U \mapsto \bigoplus_{i \in I} \mathcal{O}(U)$.
This module is called the {\it free $\mathcal{O}$-module on the set $I$}.
\begin{lemma}
\label{lemma-global-pullback}
Let
$(f, f^\sharp) :
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})
\to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
be a morphism of ringed topoi.
Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal{D}$-module.
\begin{enumerate}
\item If $\mathcal{F}$ is free then $f^*\mathcal{F}$ is free.
\item If $\mathcal{F}$ is finite free then $f^*\mathcal{F}$ is finite free.
\item If $\mathcal{F}$ is generated by global sections
then $f^*\mathcal{F}$ is generated by global sections.
\item Given $r \geq 0$ if $\mathcal{F}$ is generated by $r$ global
sections, then $f^*\mathcal{F}$ is generated by $r$ global sections.
\item If $\mathcal{F}$ is generated by finitely many global sections
then $f^*\mathcal{F}$ is generated by finitely many global sections.
\item If $\mathcal{F}$ has a global presentation then
$f^*\mathcal{F}$ has a global presentation.
\item If $\mathcal{F}$ has a finite global presentation
then $f^*\mathcal{F}$ has a finite global presentation.
\end{enumerate}
\end{lemma}
\begin{proof}
This is true because $f^*$ commutes with arbitrary colimits
(Lemma \ref{lemma-exactness-pushforward-pullback})
and $f^*\mathcal{O}_\mathcal{D} = \mathcal{O}_\mathcal{C}$.
\end{proof}
\section{Intrinsic properties of modules}
\label{section-intrinsic}
\noindent
Let $\mathcal{P}$ be a property of sheaves of modules on ringed topoi.
We say $\mathcal{P}$ is an {\it intrinsic property} if we have
$\mathcal{P}(\mathcal{F}) \Leftrightarrow \mathcal{P}(f^*\mathcal{F})$
whenever $(f, f^\sharp) :
(\Sh(\mathcal{C}'), \mathcal{O}')
\to
(\Sh(\mathcal{C}), \mathcal{O})$
is an equivalence of ringed topoi.
For example, the property of being free is intrinsic. Indeed, the free
$\mathcal{O}$-module on the set $I$ is characterized by the property
that
$$
\Mor_{\textit{Mod}(\mathcal{O})}(
\bigoplus\nolimits_{i \in I} \mathcal{O},
\mathcal{F})
=
\prod\nolimits_{i \in I} \Mor_{\Sh(\mathcal{C})}(\{*\},
\mathcal{F})
$$
for a variable $\mathcal{F}$ in $\textit{Mod}(\mathcal{O})$.
Alternatively, we can also use Lemma \ref{lemma-global-pullback}
to see that being free is intrinsic. In fact, each of the properties
defined in Definition \ref{definition-global} is intrinsic for the
same reason.
How will we go about defining other intrinsic properties of
$\mathcal{O}$-modules?
\medskip\noindent
The upshot of
Lemma \ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}
is the following: Suppose you want to define
an intrinsic property $\mathcal{P}$ of an $\mathcal{O}$-module on a
topos. Then you can proceed as follows:
\begin{enumerate}
\item Given any site $\mathcal{C}$, any sheaf of rings $\mathcal{O}$
on $\mathcal{C}$ and any $\mathcal{O}$-module $\mathcal{F}$
define the corresponding
property $\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})$.
\item For any pair of sites $\mathcal{C}$, $\mathcal{C}'$, any
special cocontinuous functor $u : \mathcal{C} \to \mathcal{C}'$,
any sheaf of rings $\mathcal{O}$ on $\mathcal{C}$ any
$\mathcal{O}$-module $\mathcal{F}$, show that
$$
\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})
\Leftrightarrow
\mathcal{P}(\mathcal{C}', g_*\mathcal{O}, g_*\mathcal{F})
$$
where $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$
is the equivalence of topoi associated to $u$.
\end{enumerate}
In this case, given any ringed topos $(\Sh(\mathcal{C}), \mathcal{O})$
and any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ we simply say that
$\mathcal{F}$ has property $\mathcal{P}$ if
$\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})$ is true.
And Lemma \ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}
combined with (2) above guarantees that this is well defined.
\medskip\noindent
Moreover, the same
Lemma \ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}
also guarantees that if in addition
\begin{enumerate}
\item[(3)] For any morphism of ringed sites
$(f, f^\sharp) :
(\mathcal{C}, \mathcal{O}_\mathcal{C})
\to
(\mathcal{D}, \mathcal{O}_\mathcal{D})$
such that $f$ is given by a functor
$u : \mathcal{D} \to \mathcal{C}$ satisfying the
assumptions of Sites, Proposition \ref{sites-proposition-get-morphism},
and any $\mathcal{O}_\mathcal{D}$-module $\mathcal{G}$
we have
$$
\mathcal{P}(\mathcal{D}, \mathcal{O}_\mathcal{D}, \mathcal{F})
\Rightarrow
\mathcal{P}(\mathcal{C}, \mathcal{O}_\mathcal{C}, f^*\mathcal{F})
$$
\end{enumerate}
then it is true that $\mathcal{P}$ is preserved under pullback
of modules w.r.t.\ arbitrary morphisms of ringed topoi.
\medskip\noindent
We will use this method in the following sections to see
that:
locally free,
locally generated by sections,
locally generated by $r$ sections,
finite type,
finite presentation,
quasi-coherent, and
coherent
are intrinsic properties of modules.
\medskip\noindent
Perhaps a more satisfying method would be to find an intrinsic definition
of these notions, rather than the laborious process sketched here.
On the other hand, in many geometric situations where we want to apply
these definitions we are given a definite ringed site, and a definite
sheaf of modules, and it is nice to have a definition already adapted to
this language.
\section{Localization of ringed sites}
\label{section-localize}
\noindent
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
Let $U \in \Ob(\mathcal{C})$.
We explain the counterparts of the results in
Sites, Section \ref{sites-section-localize}
in this setting.
\medskip\noindent
Denote
$\mathcal{O}_U = j_U^{-1}\mathcal{O}$ the restriction of $\mathcal{O}$
to the site $\mathcal{C}/U$. It is described by the simple
rule $\mathcal{O}_U(V/U) = \mathcal{O}(V)$. With this notation
the localization morphism $j_U$ becomes a morphism of ringed topoi
$$
(j_U, j_U^\sharp) :
(\Sh(\mathcal{C}/U), \mathcal{O}_U)
\longrightarrow
(\Sh(\mathcal{C}), \mathcal{O})
$$
namely, we take $j_U^\sharp : j_U^{-1}\mathcal{O} \to \mathcal{O}_U$
the identity map.
Moreover, we obtain the following descriptions for pushforward
and pullback of modules.
\begin{definition}
\label{definition-localize-ringed-site}
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
Let $U \in \Ob(\mathcal{C})$.
\begin{enumerate}
\item The ringed site $(\mathcal{C}/U, \mathcal{O}_U)$ is called the
{\it localization of the ringed site $(\mathcal{C}, \mathcal{O})$
at the object $U$}.
\item The morphism of ringed topoi
$(j_U, j_U^\sharp) :
(\Sh(\mathcal{C}/U), \mathcal{O}_U)
\to
(\Sh(\mathcal{C}), \mathcal{O})$
is called the {\it localization morphism}.
\item The functor
$j_{U*} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$
is called the {\it direct image functor}.
\item For a sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{C}$
the sheaf $j_U^*\mathcal{F}$ is called the
{\it restriction of $\mathcal{F}$ to $\mathcal{C}/U$}.
We will sometimes denote it by
$\mathcal{F}|_{\mathcal{C}/U}$ or even $\mathcal{F}|_U$.
It is described by the simple rule $j_U^*(\mathcal{F})(X/U) = \mathcal{F}(X)$.
\item The left adjoint
$j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$
of restriction is called {\it extension by zero}. It exists and is
exact by
Lemmas \ref{lemma-extension-by-zero} and
\ref{lemma-extension-by-zero-exact}.
\end{enumerate}
\end{definition}
\noindent
As in the topological case, see
Sheaves, Section \ref{sheaves-section-open-immersions},
the extension by zero $j_{U!}$ functor is different from
extension by the empty set $j_{U!}$ defined on sheaves of sets.
Here is the lemma defining extension by zero.
\begin{lemma}
\label{lemma-extension-by-zero}
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
Let $U \in \Ob(\mathcal{C})$.
The restriction functor
$j_U^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_U)$
has a left adjoint
$j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$.
So
$$
\Mor_{\textit{Mod}(\mathcal{O}_U)}(\mathcal{G}, j_U^*\mathcal{F})
=
\Mor_{\textit{Mod}(\mathcal{O})}(j_{U!}\mathcal{G}, \mathcal{F})
$$
for $\mathcal{F} \in \Ob(\textit{Mod}(\mathcal{O}))$
and $\mathcal{G} \in \Ob(\textit{Mod}(\mathcal{O}_U))$.
Moreover, the extension by zero $j_{U!}\mathcal{G}$ of $\mathcal{G}$
is the sheaf associated to the presheaf
$$
V
\longmapsto
\bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
\mathcal{G}(V \xrightarrow{\varphi} U)
$$
with obvious restriction mappings and an obvious $\mathcal{O}$-module
structure.
\end{lemma}
\begin{proof}
The $\mathcal{O}$-module structure on the presheaf is defined as
follows. If $f \in \mathcal{O}(V)$ and
$s \in \mathcal{G}(V \xrightarrow{\varphi} U)$, then
we define $f \cdot s = fs$ where
$f \in \mathcal{O}_U(\varphi : V \to U) = \mathcal{O}(V)$
(because $\mathcal{O}_U$ is the restriction of $\mathcal{O}$ to
$\mathcal{C}/U$).
\medskip\noindent
Similarly, let $\alpha : \mathcal{G} \to \mathcal{F}|_U$ be a
morphism of $\mathcal{O}_U$-modules. In this case we can define
a map from the presheaf of the lemma into $\mathcal{F}$ by mapping
$$
\bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
\mathcal{G}(V \xrightarrow{\varphi} U)
\longrightarrow
\mathcal{F}(V)
$$
by the rule that $s \in \mathcal{G}(V \xrightarrow{\varphi} U)$
maps to $\alpha(s) \in \mathcal{F}(V)$. It is clear that this is
$\mathcal{O}$-linear, and hence induces a morphism of
$\mathcal{O}$-modules $\alpha' : j_{U!}\mathcal{G} \to \mathcal{F}$
by the properties of sheafification of modules
(Lemma \ref{lemma-sheafification-presheaf-modules}).
\medskip\noindent
Conversely, let $\beta : j_{U!}\mathcal{G} \to \mathcal{F}$
by a map of $\mathcal{O}$-modules.
Recall from Sites, Section \ref{sites-section-localize}
that there exists an extension by the empty set
$j^{Sh}_{U!} : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})$
on sheaves of sets which is left adjoint to $j_U^{-1}$.
Moreover, $j^{Sh}_{U!}\mathcal{G}$ is the sheaf associated to the presheaf
$$
V
\longmapsto
\coprod\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
\mathcal{G}(V \xrightarrow{\varphi} U)
$$
Hence there is a natural map
$j^{Sh}_{U!}\mathcal{G} \to j_{U!}\mathcal{G}$ of sheaves of sets.
Hence precomposing $\beta$ by this map we get a map of sheaves of sets
$j^{Sh}_{U!}\mathcal{G} \to \mathcal{F}$ which by adjunction corresponds
to a map of sheaves of sets $\beta' : \mathcal{G} \to \mathcal{F}|_U$.
We claim that $\beta'$ is $\mathcal{O}_U$-linear. Namely, suppose
that $\varphi : V \to U$ is an object of $\mathcal{C}/U$ and that
$s, s' \in \mathcal{G}(\varphi : V \to U)$, and
$f \in \mathcal{O}(V) = \mathcal{O}_U(\varphi : V \to U)$.
Then by the discussion above we see that
$\beta'(s + s')$, resp.\ $\beta'(fs)$ in $\mathcal{F}|_U(\varphi : V \to U)$
correspond to $\beta(s + s')$, resp.\ $\beta(fs)$ in
$\mathcal{F}(V)$. Since $\beta$ is a homomorphism we conclude.
\medskip\noindent
To conclude the proof of the lemma we have to show that the constructions
$\alpha \mapsto \alpha'$ and $\beta \mapsto \beta'$ are mutually inverse.
We omit the verifications.
\end{proof}
\noindent
Note that we have in the situation of
Definition \ref{definition-localize-ringed-site} we have
\begin{equation}
\label{equation-map-lower-shriek-OU-into-module}
\Hom_\mathcal{O}(j_{U!}\mathcal{O}_U, \mathcal{F}) =
\Hom_{\mathcal{O}_U}(\mathcal{O}_U, j_U^*\mathcal{F}) =
\mathcal{F}(U)
\end{equation}
for every $\mathcal{O}$-module $\mathcal{F}$. Namely, the first equality
holds by the adjointness of $j_{U!}$ and $j_U^*$ and the second because
$\Hom_{\mathcal{O}_U}(\mathcal{O}_U, j_U^*\mathcal{F}) =
j_U^*\mathcal{F}(U/U) = \mathcal{F}|_U(U/U) = \mathcal{F}(U)$.
\begin{lemma}
\label{lemma-extension-by-zero-exact}
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
Let $U \in \Ob(\mathcal{C})$.
The functor
$j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$
is exact.
\end{lemma}
\begin{proof}
Since $j_{U!}$ is a left adjoint to $j_U^*$ we see that it is right exact
(see
Categories, Lemma \ref{categories-lemma-exact-adjoint}
and
Homology, Section \ref{homology-section-functors}).
Hence it suffices to show that if $\mathcal{G}_1 \to \mathcal{G}_2$
is an injective map of $\mathcal{O}_U$-modules, then
$j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2$ is injective.
The map on sections of presheaves over an object $V$
(as in Lemma \ref{lemma-extension-by-zero}) is the map
$$
\bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
\mathcal{G}_1(V \xrightarrow{\varphi} U)
\longrightarrow
\bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
\mathcal{G}_2(V \xrightarrow{\varphi} U)
$$
which is injective by assumption. Since sheafification is exact by
Lemma \ref{lemma-sheafification-exact}
we conclude $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2$ is injective and
we win.
\end{proof}
\begin{lemma}
\label{lemma-j-shriek-reflects-exactness}
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
Let $U \in \Ob(\mathcal{C})$. A complex of $\mathcal{O}_U$-modules
$\mathcal{G}_1 \to \mathcal{G}_2 \to \mathcal{G}_3$ is exact
if and only if
$j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2 \to j_{U!}\mathcal{G}_3$
is exact as a sequence of $\mathcal{O}$-modules.
\end{lemma}
\begin{proof}
We already know that $j_{U!}$ is exact, see
Lemma \ref{lemma-extension-by-zero-exact}.
Thus it suffices to show that
$j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$
reflects injections and surjections.
\medskip\noindent
For every $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_U)$
we have the unit $\mathcal{G} \to j_U^*j_{U!}\mathcal{G}$
of the adjunction. We claim this map is an injection of sheaves.
Namely, looking at the construction of Lemma \ref{lemma-extension-by-zero}
we see that this map is the sheafification of the rule sending the object
$V/U$ of $\mathcal{C}/U$ to the injective map
$$
\mathcal{G}(V/U) \longrightarrow
\bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
\mathcal{G}(V \xrightarrow{\varphi} U)
$$
given by the inclusion of the summand corresponding to the structure
morphism $V \to U$. Since sheafification is exact the claim follows.
Some details omitted.
\medskip\noindent
If $\mathcal{G} \to \mathcal{G}'$ is a map of $\mathcal{O}_U$-modules with
$j_{U!}\mathcal{G} \to j_{U!}\mathcal{G}'$ injective,
then $j_U^*j_{U!}\mathcal{G} \to j_U^*j_{U!}\mathcal{G}'$ is injective
(restriction is exact), hence
$\mathcal{G} \to j_U^*j_{U!}\mathcal{G}'$ is injective, hence
$\mathcal{G} \to \mathcal{G}'$ is injective.
We conclude that $j_{U!}$ reflects injections.
\medskip\noindent
Let $a : \mathcal{G} \to \mathcal{G}'$ be a map of $\mathcal{O}_U$-modules
such that $j_{U!}\mathcal{G} \to j_{U!}\mathcal{G}'$ is surjective.
Let $\mathcal{H}$ be the cokernel of $a$.
Then $j_{U!}\mathcal{H} = 0$ as $j_{U!}$ is exact.
By the above the map $\mathcal{H} \to j^*_U j_{U!}\mathcal{H}$
is injective. Hence $\mathcal{H} = 0$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-relocalize}
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
Let $f : V \to U$ be a morphism of $\mathcal{C}$.
Then there exists a commutative diagram
$$
\xymatrix{
(\Sh(\mathcal{C}/V), \mathcal{O}_V)
\ar[rd]_{(j_V, j_V^\sharp)} \ar[rr]_{(j, j^\sharp)} & &
(\Sh(\mathcal{C}/U), \mathcal{O}_U)
\ar[ld]^{(j_U, j_U^\sharp)} \\
& (\Sh(\mathcal{C}), \mathcal{O}) &
}
$$
of ringed topoi. Here $(j, j^\sharp)$ is the localization morphism
associated to the object $V/U$ of the ringed site
$(\mathcal{C}/V, \mathcal{O}_V)$.
\end{lemma}
\begin{proof}
The only thing to check is that
$j_V^\sharp = j^\sharp \circ j^{-1}(j_U^\sharp)$,
since everything else follows directly from
Sites, Lemma \ref{sites-lemma-relocalize} and
Sites, Equation (\ref{sites-equation-relocalize}).
We omit the verification of the equality.
\end{proof}
\begin{remark}
\label{remark-localize-shriek-equal}
In the situation of Lemma \ref{lemma-extension-by-zero}
the diagram
$$
\xymatrix{
\textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} &
\textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[d]^{forget} \\
\textit{Ab}(\mathcal{C}/U) \ar[r]^{j^{Ab}_{U!}} &
\textit{Ab}(\mathcal{C})
}
$$
commutes. This is clear from the explicit description of the functor
$j_{U!}$ in the lemma.
\end{remark}
\begin{remark}
\label{remark-localize-presheaves}
Localization and presheaves of modules; see
Sites, Remark \ref{sites-remark-localize-presheaves}.
Let $\mathcal{C}$ be a category.
Let $\mathcal{O}$ be a presheaf of rings.
Let $U$ be an object of $\mathcal{C}$.
Strictly speaking the functors $j_U^*$, $j_{U*}$ and $j_{U!}$
have not been defined for presheaves of $\mathcal{O}$-modules.
But of course, we can think of a presheaf as a sheaf for the
chaotic topology on $\mathcal{C}$ (see
Sites, Examples \ref{sites-example-indiscrete}).
Hence we also obtain a functor
$$
j_U^* :
\textit{PMod}(\mathcal{O})
\longrightarrow
\textit{PMod}(\mathcal{O}_U)
$$
and functors
$$
j_{U*}, j_{U!} :
\textit{PMod}(\mathcal{O}_U)
\longrightarrow
\textit{PMod}(\mathcal{O})
$$
which are right, left adjoint to $j_U^*$. Inspecting the proof of
Lemma \ref{lemma-extension-by-zero} we see that $j_{U!}\mathcal{G}$
is the presheaf
$$
V \longmapsto
\bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
\mathcal{G}(V \xrightarrow{\varphi} U)
$$
In addition the functor $j_{U!}$ is exact (by
Lemma \ref{lemma-extension-by-zero-exact} in the
case of the discrete topologies). Moreover, if $\mathcal{C}$
is actually a site, and $\mathcal{O}$ is actually a sheaf of rings,
then the diagram
$$
\xymatrix{
\textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} &
\textit{Mod}(\mathcal{O}) \\
\textit{PMod}(\mathcal{O}_U) \ar[r]^{j_{U!}} &
\textit{PMod}(\mathcal{O}) \ar[u]_{(\ )^\#}
}
$$
commutes.
\end{remark}
\begin{lemma}
\label{lemma-restrict-back}
Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$.
Assume that every $X$ in $\mathcal{C}$ has at most
one morphism to $U$. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}/U$.
The canonical maps $\mathcal{F} \to j_U^{-1}j_{U!}\mathcal{F}$
and $j_U^{-1}j_{U*}\mathcal{F} \to \mathcal{F}$ are
isomorphisms.
\end{lemma}
\begin{proof}
This is a special case of Lemma \ref{lemma-back-and-forth}
because the assumption on $U$ is equivalent to the fully faithfulness
of the localization functor $\mathcal{C}/U \to \mathcal{C}$.
\end{proof}
\section{Localization of morphisms of ringed sites}
\label{section-localize-morphisms}
\noindent
This section is the analogue of
Sites, Section \ref{sites-section-localize-morphisms}.
\begin{lemma}
\label{lemma-localize-morphism-ringed-sites}
Let
$(f, f^\sharp) :
(\mathcal{C}, \mathcal{O})
\longrightarrow
(\mathcal{D}, \mathcal{O}')$
be a morphism of ringed sites where $f$ is given by the continuous
functor $u : \mathcal{D} \to \mathcal{C}$.
Let $V$ be an object of $\mathcal{D}$ and set $U = u(V)$.
Then there is a canonical map of sheaves of rings $(f')^\sharp$
such that the diagram of
Sites, Lemma \ref{sites-lemma-localize-morphism}
is turned into a commutative diagram of ringed topoi
$$
\xymatrix{
(\Sh(\mathcal{C}/U), \mathcal{O}_U)
\ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f', (f')^\sharp)} & &
(\Sh(\mathcal{C}), \mathcal{O})
\ar[d]^{(f, f^\sharp)} \\
(\Sh(\mathcal{D}/V), \mathcal{O}'_V)
\ar[rr]^{(j_V, j_V^\sharp)} & &
(\Sh(\mathcal{D}), \mathcal{O}').
}
$$
Moreover, in this situation we have $f'_*j_U^{-1} = j_V^{-1}f_*$
and $f'_*j_U^* = j_V^*f_*$.
\end{lemma}
\begin{proof}
Just take $(f')^\sharp$ to be
$$
(f')^{-1}\mathcal{O}'_V =
(f')^{-1}j_V^{-1}\mathcal{O}' =
j_U^{-1}f^{-1}\mathcal{O}' \xrightarrow{j_U^{-1}f^\sharp}
j_U^{-1}\mathcal{O} = \mathcal{O}_U
$$
and everything else follows from
Sites, Lemma \ref{sites-lemma-localize-morphism}.
(Note that $j^{-1} = j^*$ on sheaves of modules if $j$ is a localization
morphism, hence the first equality of functors implies the second.)
\end{proof}
\begin{lemma}
\label{lemma-relocalize-morphism-ringed-sites}
Let
$(f, f^\sharp) :
(\mathcal{C}, \mathcal{O})
\longrightarrow
(\mathcal{D}, \mathcal{O}')$
be a morphism of ringed sites where $f$ is given by the continuous
functor $u : \mathcal{D} \to \mathcal{C}$.
Let $V \in \Ob(\mathcal{D})$, $U \in \Ob(\mathcal{C})$
and $c : U \to u(V)$ a morphism of $\mathcal{C}$.
There exists a commutative diagram of ringed topoi
$$
\xymatrix{
(\Sh(\mathcal{C}/U), \mathcal{O}_U)
\ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f_c, f_c^\sharp)} & &
(\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\
(\Sh(\mathcal{D}/V), \mathcal{O}'_V)
\ar[rr]^{(j_V, j_V^\sharp)} & &
(\Sh(\mathcal{D}), \mathcal{O}').
}
$$
The morphism $(f_c, f_c^\sharp)$
is equal to the composition of the morphism
$$
(f', (f')^\sharp) :
(\Sh(\mathcal{C}/u(V)), \mathcal{O}_{u(V)})
\longrightarrow
(\Sh(\mathcal{D}/V), \mathcal{O}'_V)
$$
of
Lemma \ref{lemma-localize-morphism-ringed-sites}
and the morphism
$$
(j, j^\sharp) :
(\Sh(\mathcal{C}/U), \mathcal{O}_U)
\to
(\Sh(\mathcal{C}/u(V)), \mathcal{O}_{u(V)})
$$
of
Lemma \ref{lemma-relocalize}.
Given any morphisms $b : V' \to V$, $a : U' \to U$ and
$c' : U' \to u(V')$ such that
$$
\xymatrix{
U' \ar[r]_-{c'} \ar[d]_a & u(V') \ar[d]^{u(b)} \\
U \ar[r]^-c & u(V)
}
$$
commutes, then the following diagram of ringed topoi
$$
\xymatrix{
(\Sh(\mathcal{C}/U'), \mathcal{O}_{U'})
\ar[rr]_{(j_{U'/U}, j_{U'/U}^\sharp)} \ar[d]_{(f_{c'}, f_{c'}^\sharp)} & &
(\Sh(\mathcal{C}/U), \mathcal{O}_U)
\ar[d]^{(f_c, f_c^\sharp)} \\
(\Sh(\mathcal{D}/V'), \mathcal{O}'_{V'})
\ar[rr]^{(j_{V'/V}, j_{V'/V}^\sharp)} & &
(\Sh(\mathcal{D}/V), \mathcal{O}'_{V'})
}
$$
commutes.
\end{lemma}
\begin{proof}
On the level of morphisms of topoi this is
Sites, Lemma \ref{sites-lemma-relocalize-morphism}.
To check that the diagrams commute as morphisms of ringed topoi use
Lemmas \ref{lemma-relocalize} and
\ref{lemma-localize-morphism-ringed-sites}
exactly as in the proof of
Sites, Lemma \ref{sites-lemma-relocalize-morphism}.
\end{proof}
\section{Localization of ringed topoi}
\label{section-localize-ringed-topoi}
\noindent
This section is the analogue of
Sites, Section \ref{sites-section-localize-topoi}
in the setting of ringed topoi.
\begin{lemma}
\label{lemma-localize-ringed-topos}
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos.
Let $\mathcal{F} \in \Sh(\mathcal{C})$ be a sheaf.
For a sheaf $\mathcal{H}$ on $\mathcal{C}$ denote
$\mathcal{H}_\mathcal{F}$ the sheaf $\mathcal{H} \times \mathcal{F}$
seen as an object of the category $\Sh(\mathcal{C})/\mathcal{F}$.
The pair
$(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})$
is a ringed topos and there is a canonical morphism of ringed topoi
$$
(j_\mathcal{F}, j_\mathcal{F}^\sharp) :
(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})
\longrightarrow
(\Sh(\mathcal{C}), \mathcal{O})
$$
which is a localization as in
Section \ref{section-localize}
such that
\begin{enumerate}
\item the functor $j_\mathcal{F}^{-1}$ is the functor
$\mathcal{H} \mapsto \mathcal{H}_\mathcal{F}$,
\item the functor $j_\mathcal{F}^*$ is the functor
$\mathcal{H} \mapsto \mathcal{H}_\mathcal{F}$,
\item the functor $j_{\mathcal{F}!}$ on sheaves of sets is the forgetful
functor $\mathcal{G}/\mathcal{F} \mapsto \mathcal{G}$,
\item the functor $j_{\mathcal{F}!}$ on sheaves of modules associates
to the $\mathcal{O}_\mathcal{F}$-module
$\varphi : \mathcal{G} \to \mathcal{F}$ the $\mathcal{O}$-module
which is the sheafification of the presheaf
$$
V \longmapsto
\bigoplus\nolimits_{s \in \mathcal{F}(V)}
\{\sigma \in \mathcal{G}(V) \mid \varphi(\sigma) = s \}
$$
for $V \in \Ob(\mathcal{C})$.
\end{enumerate}
\end{lemma}
\begin{proof}
By
Sites, Lemma \ref{sites-lemma-localize-topos}
we see that $\Sh(\mathcal{C})/\mathcal{F}$ is a topos
and that (1) and (3) are true. In particular this shows that
$j_\mathcal{F}^{-1}\mathcal{O} = \mathcal{O}_\mathcal{F}$
and shows that $\mathcal{O}_\mathcal{F}$ is a sheaf of rings.
Thus we may choose the map $j_\mathcal{F}^\sharp$ to be the identity,
in particular we see that (2) is true.
Moreover, the proof of
Sites, Lemma \ref{sites-lemma-localize-topos}
shows that we may assume $\mathcal{C}$ is a site with all finite limits
and a subcanonical topology and that $\mathcal{F} = h_U$ for some object
$U$ of $\mathcal{C}$.
Then (4) follows from the description of $j_{\mathcal{F}!}$ in
Lemma \ref{lemma-extension-by-zero}.
Alternatively one could show directly that the functor described
in (4) is a left adjoint to $j_\mathcal{F}^*$.
\end{proof}
\begin{definition}
\label{definition-localize-ringed-topos}
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos.
Let $\mathcal{F} \in \Sh(\mathcal{C})$.
\begin{enumerate}
\item The ringed topos
$(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})$
is called the
{\it localization of the ringed topos
$(\Sh(\mathcal{C}), \mathcal{O})$ at $\mathcal{F}$}.
\item The morphism of ringed topoi
$(j_\mathcal{F}, j_\mathcal{F}^\sharp) :
(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})
\to
(\Sh(\mathcal{C}), \mathcal{O})$ of
Lemma \ref{lemma-localize-ringed-topos}
is called the {\it localization morphism}.
\end{enumerate}
\end{definition}
\noindent
We continue the tradition, established in the chapter on sites, that we
check the localization constructions on topoi are compatible with the
constructions of localization on sites, whenever this makes sense.
\begin{lemma}
\label{lemma-localize-compare}
With
$(\Sh(\mathcal{C}), \mathcal{O})$ and
$\mathcal{F} \in \Sh(\mathcal{C})$ as in
Lemma \ref{lemma-localize-ringed-topos}.
If $\mathcal{F} = h_U^\#$ for some object $U$ of $\mathcal{C}$
then via the identification
$\Sh(\mathcal{C}/U) = \Sh(\mathcal{C})/h_U^\#$ of
Sites, Lemma \ref{sites-lemma-essential-image-j-shriek}
we have
\begin{enumerate}
\item canonically $\mathcal{O}_U = \mathcal{O}_\mathcal{F}$, and
\item with these identifications
we have $(j_\mathcal{F}, j_\mathcal{F}^\sharp) = (j_U, j_U^\sharp)$.
\end{enumerate}
\end{lemma}
\begin{proof}
The assertion for underlying topoi is
Sites, Lemma \ref{sites-lemma-localize-compare}.
Note that $\mathcal{O}_U$ is the restriction of $\mathcal{O}$
which by
Sites, Lemma \ref{sites-lemma-compute-j-shriek-restrict}
corresponds to $\mathcal{O} \times h_U^\#$ under the equivalence of
Sites, Lemma \ref{sites-lemma-essential-image-j-shriek}.
By definition of $\mathcal{O}_\mathcal{F}$ we get (1).
What's left is to prove that $j_\mathcal{F}^\sharp = j_U^\sharp$
under this identification. We omit the verification.
\end{proof}
\noindent
Localization is functorial in the following two ways:
We can ``relocalize'' a localization (see
Lemma \ref{lemma-relocalize-ringed-topos})
or we can given a morphism of ringed topoi, localize upstairs at
the inverse image of a sheaf downstairs and get a commutative
diagram of ringed topoi (see
Lemma \ref{lemma-localize-morphism-ringed-topoi}).
\begin{lemma}
\label{lemma-relocalize-ringed-topos}
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos.
If $s : \mathcal{G} \to \mathcal{F}$ is a morphism of sheaves
on $\mathcal{C}$ then there exists a natural commutative diagram of
morphisms of ringed topoi
$$
\xymatrix{
(\Sh(\mathcal{C})/\mathcal{G}, \mathcal{O}_\mathcal{G})
\ar[rd]_{(j_\mathcal{G}, j_\mathcal{G}^\sharp)} \ar[rr]_{(j, j^\sharp)} & &
(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})
\ar[ld]^{(j_\mathcal{F}, j_\mathcal{F}^\sharp)} \\
& (\Sh(\mathcal{C}), \mathcal{O}) &
}
$$
where $(j, j^\sharp)$ is the localization morphism of the ringed topos
$(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})$
at the object $\mathcal{G}/\mathcal{F}$.
\end{lemma}
\begin{proof}
All assertions follow from
Sites, Lemma \ref{sites-lemma-relocalize-topos}
except the assertion that
$j_\mathcal{G}^\sharp = j^\sharp \circ j^{-1}(j_\mathcal{F}^\sharp)$.
We omit the verification.
\end{proof}
\begin{lemma}
\label{lemma-relocalize-compare}
With $(\Sh(\mathcal{C}), \mathcal{O})$,
$s : \mathcal{G} \to \mathcal{F}$ as in
Lemma \ref{lemma-relocalize-ringed-topos}.
If there exist a morphism $f : V \to U$ of $\mathcal{C}$
such that $\mathcal{G} = h_V^\#$ and $\mathcal{F} = h_U^\#$
and $s$ is induced by $f$, then the
diagrams of
Lemma \ref{lemma-relocalize}
and
Lemma \ref{lemma-relocalize-ringed-topos}
agree via the identifications
$(j_\mathcal{F}, j_\mathcal{F}^\sharp) = (j_U, j_U^\sharp)$
and
$(j_\mathcal{G}, j_\mathcal{G}^\sharp) = (j_V, j_V^\sharp)$
of
Lemma \ref{lemma-localize-compare}.
\end{lemma}
\begin{proof}
All assertions follow from
Sites, Lemma \ref{sites-lemma-relocalize-compare}
except for the assertion that the two maps $j^\sharp$
agree. This holds since in both cases the map
$j^\sharp$ is simply the identity. Some details omitted.
\end{proof}
\section{Localization of morphisms of ringed topoi}
\label{section-localize-morphisms-ringed-topoi}
\noindent
This section is the analogue of
Sites, Section \ref{sites-section-localize-morphisms-topoi}.
\begin{lemma}
\label{lemma-localize-morphism-ringed-topoi}
Let
$$
f :
(\Sh(\mathcal{C}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{D}), \mathcal{O}')
$$
be a morphism of ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$.
Set $\mathcal{F} = f^{-1}\mathcal{G}$.
Then there exists a commutative diagram of ringed topoi
$$
\xymatrix{
(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})
\ar[rr]_{(j_\mathcal{F}, j_\mathcal{F}^\sharp)}
\ar[d]_{(f', (f')^\sharp)} & &
(\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\
(\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G})
\ar[rr]^{(j_\mathcal{G}, j_\mathcal{G}^\sharp)} & &
(\Sh(\mathcal{D}), \mathcal{O}')
}
$$
We have $f'_*j_\mathcal{F}^{-1} = j_\mathcal{G}^{-1}f_*$
and $f'_*j_\mathcal{F}^* = j_\mathcal{G}^*f_*$. Moreover, the
morphism $f'$ is characterized by the rule
$$
(f')^{-1}(\mathcal{H} \xrightarrow{\varphi} \mathcal{G})
=
(f^{-1}\mathcal{H} \xrightarrow{f^{-1}\varphi} \mathcal{F}).
$$
\end{lemma}
\begin{proof}
By
Sites, Lemma \ref{sites-lemma-localize-morphism-topoi}
we have the diagram of underlying topoi, the
equality $f'_*j_\mathcal{F}^{-1} = j_\mathcal{G}^{-1}f_*$, and
the description of $(f')^{-1}$.
To define $(f')^\sharp$ we use the map
$$
(f')^\sharp :