\input{preamble} % OK, start here. % \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 :