# The Stacks Project

## Tag 03DI

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. $\square$

The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2049–2075 (see updates for more information).

\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
$j_U^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_U)$
$j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$.
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}

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