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Tag 03DJ

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

Lemma 18.19.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$. The functor $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$ is exact.

Proof. Since $j_{U!}$ is a left adjoint to $j_U^*$ we see that it is right exact (see Categories, Lemma 4.24.5 and Homology, Section 12.7). Hence it suffices to show that if $\mathcal{G}_1 \to \mathcal{G}_2$ is an injective map of $\mathcal{O}_U$-modules, then $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2$ is injective. The map on sections of presheaves over an object $V$ (as in Lemma 18.19.2) is the map $$ \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}_1(V \xrightarrow{\varphi} U) \longrightarrow \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}_2(V \xrightarrow{\varphi} U) $$ which is injective by assumption. Since sheafification is exact by Lemma 18.11.2 we conclude $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2$ is injective and we win. $\square$

    The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2137–2144 (see updates for more information).

    \begin{lemma}
    \label{lemma-extension-by-zero-exact}
    Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
    Let $U \in \Ob(\mathcal{C})$.
    The functor
    $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$
    is exact.
    \end{lemma}
    
    \begin{proof}
    Since $j_{U!}$ is a left adjoint to $j_U^*$ we see that it is right exact
    (see
    Categories, Lemma \ref{categories-lemma-exact-adjoint}
    and
    Homology, Section \ref{homology-section-functors}).
    Hence it suffices to show that if $\mathcal{G}_1 \to \mathcal{G}_2$
    is an injective map of $\mathcal{O}_U$-modules, then
    $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2$ is injective.
    The map on sections of presheaves over an object $V$
    (as in Lemma \ref{lemma-extension-by-zero}) is the map
    $$
    \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
    \mathcal{G}_1(V \xrightarrow{\varphi} U)
    \longrightarrow
    \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
    \mathcal{G}_2(V \xrightarrow{\varphi} U)
    $$
    which is injective by assumption. Since sheafification is exact by
    Lemma \ref{lemma-sheafification-exact}
    we conclude $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2$ is injective and
    we win.
    \end{proof}

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