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Tag 03EJ

Chapter 18: Modules on Sites > Section 18.19: Localization of ringed sites

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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