Remark 18.19.7. Localization and presheaves of modules; see Sites, Remark 7.25.10. 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

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

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.

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