# The Stacks Project

## Tag 03EJ

Remark 18.19.6. Localization and presheaves of modules; see Sites, Remark 7.24.9. 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.

The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2214–2262 (see updates for more information).

\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}

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