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Tag 0931

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

Remark 18.19.7 (Map from lower shriek to pushforward). Let $U$ be an object of $\mathcal{C}$. For any abelian sheaf $\mathcal{G}$ on $\mathcal{C}/U$ there is a canonical map $$ c : j_{U!}\mathcal{G} \longrightarrow j_{U*}\mathcal{G} $$ Namely, this is the same thing as a map $j_U^{-1}j_{U!}\mathcal{G} \to \mathcal{G}$. Note that restriction commutes with sheafification. Thus we can use the presheaf of Lemma 18.19.2. Hence it suffices to define for $V/U$ a map $$ \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V) \longrightarrow \mathcal{G}(V) $$ compatible with restrictions. We simply take the map which is zero on all summands except for the one where $\varphi$ is the structure morphism $V \to U$ where we take $1$. Moreover, if $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}$ and $\mathcal{G}$ is an $\mathcal{O}_U$-module, then the displayed map above is a map of $\mathcal{O}$-modules.

    The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2264–2287 (see updates for more information).

    \begin{remark}[Map from lower shriek to pushforward]
    \label{remark-from-shriek-to-star}
    Let $U$ be an object of $\mathcal{C}$. For any abelian sheaf
    $\mathcal{G}$ on $\mathcal{C}/U$ there is a canonical map
    $$
    c : j_{U!}\mathcal{G} \longrightarrow j_{U*}\mathcal{G}
    $$
    Namely, this is the same thing as a map
    $j_U^{-1}j_{U!}\mathcal{G} \to \mathcal{G}$.
    Note that restriction commutes with sheafification.
    Thus we can use the presheaf of Lemma \ref{lemma-extension-by-zero}.
    Hence it suffices to define for $V/U$ a map
    $$
    \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V)
    \longrightarrow
    \mathcal{G}(V)
    $$
    compatible with restrictions. We simply take the map
    which is zero on all summands except for the one where $\varphi$
    is the structure morphism $V \to U$ where we take $1$.
    Moreover, if $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}$ and
    $\mathcal{G}$ is an $\mathcal{O}_U$-module, then
    the displayed map above is a map of $\mathcal{O}$-modules.
    \end{remark}

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