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18.19. Localization of ringed sites

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$. We explain the counterparts of the results in Sites, Section 7.24 in this setting.

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 $$ (j_U, j_U^\sharp) : (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/U), \mathcal{O}_U) \longrightarrow (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}) $$ 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.

Definition 18.19.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$.

  1. The ringed site $(\mathcal{C}/U, \mathcal{O}_U)$ is called the localization of the ringed site $(\mathcal{C}, \mathcal{O})$ at the object $U$.
  2. The morphism of ringed topoi $(j_U, j_U^\sharp) : (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/U), \mathcal{O}_U) \to (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O})$ is called the localization morphism.
  3. The functor $j_{U*} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$ is called the direct image functor.
  4. For a sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{C}$ the sheaf $j_U^*\mathcal{F}$ is called the 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 $j_U^*(\mathcal{F})(X/U) = \mathcal{F}(X)$.
  5. The left adjoint $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$ of restriction is called extension by zero. It exists and is exact by Lemmas 18.19.2 and 18.19.3.

As in the topological case, see Sheaves, Section 6.31, 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.

Lemma 18.19.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$. 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 $$ \mathop{\rm Mor}\nolimits_{\textit{Mod}(\mathcal{O}_U)}(\mathcal{G}, j_U^*\mathcal{F}) = \mathop{\rm Mor}\nolimits_{\textit{Mod}(\mathcal{O})}(j_{U!}\mathcal{G}, \mathcal{F}) $$ for $\mathcal{F} \in \mathop{\rm Ob}\nolimits(\textit{Mod}(\mathcal{O}))$ and $\mathcal{G} \in \mathop{\rm Ob}\nolimits(\textit{Mod}(\mathcal{O}_U))$. Moreover, the extension by zero $j_{U!}\mathcal{G}$ of $\mathcal{G}$ is the sheaf associated to the presheaf $$ V \longmapsto \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) $$ with obvious restriction mappings and an obvious $\mathcal{O}$-module structure.

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 $\mathcal{C}/U$).

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 $$ \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) \longrightarrow \mathcal{F}(V) $$ 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 18.11.1).

Conversely, let $\beta : j_{U!}\mathcal{G} \to \mathcal{F}$ by a map of $\mathcal{O}$-modules. Recall from Sites, Section 7.24 that there exists an extension by the empty set $j^{Sh}_{U!} : \mathop{\textit{Sh}}\nolimits(\mathcal{C}/U) \to \mathop{\textit{Sh}}\nolimits(\mathcal{C})$ 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 $$ V \longmapsto \coprod\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) $$ 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.

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. $\square$

Lemma 18.19.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$. The functor $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$ is exact.

Proof. Since $j_{U!}$ is a left adjoint to $j_U^*$ we see that it is right exact (see Categories, Lemma 4.24.5 and Homology, Section 12.7). 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 18.19.2) is the map $$ \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}_1(V \xrightarrow{\varphi} U) \longrightarrow \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}_2(V \xrightarrow{\varphi} U) $$ which is injective by assumption. Since sheafification is exact by Lemma 18.11.2 we conclude $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2$ is injective and we win. $\square$

Lemma 18.19.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$. 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.

Proof. We already know that $j_{U!}$ is exact, see Lemma 18.19.3. Thus it suffices to show that $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$ reflects injections and surjections.

For every $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_U)$ the counit $j_U^*j_{U_!}\mathcal{G} \to \mathcal{G}$ is surjective (see construction in the proof of Lemma 18.19.2). If $\mathcal{G} \to \mathcal{G}'$ is a map of $\mathcal{O}_U$-modules with $j_{U!}\mathcal{G} \to j_{U!}\mathcal{G}'$ surjective, then $j_U^*j_{U!}\mathcal{G} \to j_U^*j_{U!}\mathcal{G}'$ is surjective (restriction is exact), hence $j_U^*j_{U!}\mathcal{G} \to \mathcal{G}'$ is surjective, hence $\mathcal{G} \to \mathcal{G}'$ is surjective. We conclude that $j_{U!}$ reflects surjections.

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 injective. Let $\mathcal{H}$ be the kernel of $a$. Then $j_{U!}\mathcal{H} = 0$ as $j_{U!}$ is exact. By the above the map $j^*_U j_{U!}\mathcal{H} \to \mathcal{H}$ is surjective. Hence $\mathcal{H} = 0$ as desired. $\square$

Lemma 18.19.5. 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 $$ \xymatrix{ (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/V), \mathcal{O}_V) \ar[rd]_{(j_V, j_V^\sharp)} \ar[rr]_{(j, j^\sharp)} & & (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/U), \mathcal{O}_U) \ar[ld]^{(j_U, j_U^\sharp)} \\ & (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}) & } $$ of ringed topoi. Here $(j, j^\sharp)$ is the localization morphism associated to the object $V/U$ of the ringed site $(\mathcal{C}/V, \mathcal{O}_V)$.

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 7.24.8 and Sites, Equation (7.24.8.1). We omit the verification of the equality. $\square$

Remark 18.19.6. In the situation of Lemma 18.19.2 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.

Remark 18.19.7. Localization and presheaves of modules; see Sites, Remark 7.24.10. 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 7.6.6). 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 18.19.2 we see that $j_{U!}\mathcal{G}$ is the presheaf $$ V \longmapsto \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) $$ In addition the functor $j_{U!}$ is exact (by Lemma 18.19.3 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.

Remark 18.19.8 (Map from lower shriek to pushforward). Let $U$ be an object of $\mathcal{C}$. For any abelian sheaf $\mathcal{G}$ on $\mathcal{C}/U$ there is a canonical map $$ c : j_{U!}\mathcal{G} \longrightarrow j_{U*}\mathcal{G} $$ Namely, this is the same thing as a map $j_U^{-1}j_{U!}\mathcal{G} \to \mathcal{G}$. Note that restriction commutes with sheafification. Thus we can use the presheaf of Lemma 18.19.2. Hence it suffices to define for $V/U$ a map $$ \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V) \longrightarrow \mathcal{G}(V) $$ compatible with restrictions. We simply take the map which is zero on all summands except for the one where $\varphi$ is the structure morphism $V \to U$ where we take $1$. Moreover, if $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}$ and $\mathcal{G}$ is an $\mathcal{O}_U$-module, then the displayed map above is a map of $\mathcal{O}$-modules.

    The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 1983–2337 (see updates for more information).

    \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}
    
    \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 $\textit{Mod}(\mathcal{O}_U)$
    the counit $j_U^*j_{U_!}\mathcal{G} \to \mathcal{G}$
    is surjective (see construction
    in the proof of Lemma \ref{lemma-extension-by-zero}).
    If $\mathcal{G} \to \mathcal{G}'$
    is a map of $\mathcal{O}_U$-modules with
    $j_{U!}\mathcal{G} \to j_{U!}\mathcal{G}'$ surjective,
    then $j_U^*j_{U!}\mathcal{G} \to j_U^*j_{U!}\mathcal{G}'$ is surjective
    (restriction is exact), hence
    $j_U^*j_{U!}\mathcal{G} \to \mathcal{G}'$ is surjective, hence
    $\mathcal{G} \to \mathcal{G}'$ is surjective.
    We conclude that $j_{U!}$ reflects surjections.
    
    \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 injective.
    Let $\mathcal{H}$ be the kernel of $a$.
    Then $j_{U!}\mathcal{H} = 0$ as $j_{U!}$ is exact.
    By the above the map $j^*_U j_{U!}\mathcal{H} \to \mathcal{H}$
    is surjective. 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{remark}[Map from lower shriek to pushforward]
    \label{remark-from-shriek-to-star}
    Let $U$ be an object of $\mathcal{C}$. For any abelian sheaf
    $\mathcal{G}$ on $\mathcal{C}/U$ there is a canonical map
    $$
    c : j_{U!}\mathcal{G} \longrightarrow j_{U*}\mathcal{G}
    $$
    Namely, this is the same thing as a map
    $j_U^{-1}j_{U!}\mathcal{G} \to \mathcal{G}$.
    Note that restriction commutes with sheafification.
    Thus we can use the presheaf of Lemma \ref{lemma-extension-by-zero}.
    Hence it suffices to define for $V/U$ a map
    $$
    \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V)
    \longrightarrow
    \mathcal{G}(V)
    $$
    compatible with restrictions. We simply take the map
    which is zero on all summands except for the one where $\varphi$
    is the structure morphism $V \to U$ where we take $1$.
    Moreover, if $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}$ and
    $\mathcal{G}$ is an $\mathcal{O}_U$-module, then
    the displayed map above is a map of $\mathcal{O}$-modules.
    \end{remark}

    Comments (2)

    Comment #2474 by Tanya Kaushal Srivastava (site) on April 4, 2017 a 6:25 pm UTC

    In the statement of Lemma 18.19.4, I think that maybe it should be the object $V/U$ of the ringed site $(C/U, \mathcal{O}_U)$, since the map is $f: V \rightarrow U$.

    Comment #2507 by Johan (site) on April 14, 2017 a 12:09 am UTC

    Thanks, fixed here.

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