## Tag `04IY`

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

Lemma 18.19.4. 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 7.24.7 and Sites, Equation (7.24.7.1). 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 2170–2187 (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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