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## Tag 03DH

### 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.

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\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
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}

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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