The Stacks project

Lemma 18.19.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. 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

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O}_ U)}(\mathcal{G}, j_ U^*\mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}(j_{U!}\mathcal{G}, \mathcal{F}) \]

for $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}))$ and $\mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}_ U))$. Moreover, the extension by zero $j_{U!}\mathcal{G}$ of $\mathcal{G}$ is the sheaf associated to the presheaf

\[ V \longmapsto \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \]

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

\[ \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \longrightarrow \mathcal{F}(V) \]

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 7.25 that there exists an extension by the empty set $j^{Sh}_{U!} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ 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

\[ V \longmapsto \coprod \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \]

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. $\square$

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