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Tag 08P4

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

Remark 18.19.5. In the situation of Lemma 18.19.2 the diagram $$ \xymatrix{ \textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}/U) \ar[r]^{j^{Ab}_{U!}} & \textit{Ab}(\mathcal{C}) } $$ commutes. This is clear from the explicit description of the functor $j_{U!}$ in the lemma.

    The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2198–2212 (see updates for more information).

    \begin{remark}
    \label{remark-localize-shriek-equal}
    In the situation of Lemma \ref{lemma-extension-by-zero}
    the diagram
    $$
    \xymatrix{
    \textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} &
    \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[d]^{forget} \\
    \textit{Ab}(\mathcal{C}/U) \ar[r]^{j^{Ab}_{U!}} &
    \textit{Ab}(\mathcal{C})
    }
    $$
    commutes. This is clear from the explicit description of the functor
    $j_{U!}$ in the lemma.
    \end{remark}

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