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Tag 04IY

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

Lemma 18.19.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then there exists a commutative diagram $$ \xymatrix{ (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/V), \mathcal{O}_V) \ar[rd]_{(j_V, j_V^\sharp)} \ar[rr]_{(j, j^\sharp)} & & (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/U), \mathcal{O}_U) \ar[ld]^{(j_U, j_U^\sharp)} \\ & (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}) & } $$ of ringed topoi. Here $(j, j^\sharp)$ is the localization morphism associated to the object $V/U$ of the ringed site $(\mathcal{C}/V, \mathcal{O}_V)$.

Proof. The only thing to check is that $j_V^\sharp = j^\sharp \circ j^{-1}(j_U^\sharp)$, since everything else follows directly from Sites, Lemma 7.24.8 and Sites, Equation (7.24.8.1). We omit the verification of the equality. $\square$

    The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2211–2228 (see updates for more information).

    \begin{lemma}
    \label{lemma-relocalize}
    Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
    Let $f : V \to U$ be a morphism of $\mathcal{C}$.
    Then there exists a commutative diagram
    $$
    \xymatrix{
    (\Sh(\mathcal{C}/V), \mathcal{O}_V)
    \ar[rd]_{(j_V, j_V^\sharp)} \ar[rr]_{(j, j^\sharp)} & &
    (\Sh(\mathcal{C}/U), \mathcal{O}_U)
    \ar[ld]^{(j_U, j_U^\sharp)} \\
    & (\Sh(\mathcal{C}), \mathcal{O}) &
    }
    $$
    of ringed topoi. Here $(j, j^\sharp)$ is the localization morphism
    associated to the object $V/U$ of the ringed site
    $(\mathcal{C}/V, \mathcal{O}_V)$.
    \end{lemma}
    
    \begin{proof}
    The only thing to check is that
    $j_V^\sharp = j^\sharp \circ j^{-1}(j_U^\sharp)$,
    since everything else follows directly from
    Sites, Lemma \ref{sites-lemma-relocalize} and
    Sites, Equation (\ref{sites-equation-relocalize}).
    We omit the verification of the equality.
    \end{proof}

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