# The Stacks Project

## Tag 04IZ

### 18.20. Localization of morphisms of ringed sites

This section is the analogue of Sites, Section 7.27.

Lemma 18.20.1. Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$ and set $U = u(V)$. Then there is a canonical map of sheaves of rings $(f')^\sharp$ such that the diagram of Sites, Lemma 7.27.1 is turned into a commutative diagram of ringed topoi $$\xymatrix{ (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/U), \mathcal{O}_U) \ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f', (f')^\sharp)} & & (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\mathop{\textit{Sh}}\nolimits(\mathcal{D}/V), \mathcal{O}'_V) \ar[rr]^{(j_V, j_V^\sharp)} & & (\mathop{\textit{Sh}}\nolimits(\mathcal{D}), \mathcal{O}'). }$$ Moreover, in this situation we have $f'_*j_U^{-1} = j_V^{-1}f_*$ and $f'_*j_U^* = j_V^*f_*$.

Proof. Just take $(f')^\sharp$ to be $$(f')^{-1}\mathcal{O}'_V = (f')^{-1}j_V^{-1}\mathcal{O}' = j_U^{-1}f^{-1}\mathcal{O}' \xrightarrow{j_U^{-1}f^\sharp} j_U^{-1}\mathcal{O} = \mathcal{O}_U$$ and everything else follows from Sites, Lemma 7.27.1. (Note that $j^{-1} = j^*$ on sheaves of modules if $j$ is a localization morphism, hence the first equality of functors implies the second.) $\square$

Lemma 18.20.2. Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \mathop{\rm Ob}\nolimits(\mathcal{D})$, $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$ and $c : U \to u(V)$ a morphism of $\mathcal{C}$. There exists a commutative diagram of ringed topoi $$\xymatrix{ (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/U), \mathcal{O}_U) \ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f_c, f_c^\sharp)} & & (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\mathop{\textit{Sh}}\nolimits(\mathcal{D}/V), \mathcal{O}'_V) \ar[rr]^{(j_V, j_V^\sharp)} & & (\mathop{\textit{Sh}}\nolimits(\mathcal{D}), \mathcal{O}'). }$$ The morphism $(f_c, f_c^\sharp)$ is equal to the composition of the morphism $$(f', (f')^\sharp) : (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/u(V)), \mathcal{O}_{u(V)}) \longrightarrow (\mathop{\textit{Sh}}\nolimits(\mathcal{D}/V), \mathcal{O}'_V)$$ of Lemma 18.20.1 and the morphism $$(j, j^\sharp) : (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/U), \mathcal{O}_U) \to (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/u(V)), \mathcal{O}_{u(V)})$$ of Lemma 18.19.4. Given any morphisms $b : V' \to V$, $a : U' \to U$ and $c' : U' \to u(V')$ such that $$\xymatrix{ U' \ar[r]_-{c'} \ar[d]_a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-c & u(V) }$$ commutes, then the following diagram of ringed topoi $$\xymatrix{ (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/U'), \mathcal{O}_{U'}) \ar[rr]_{(j_{U'/U}, j_{U'/U}^\sharp)} \ar[d]_{(f_{c'}, f_{c'}^\sharp)} & & (\mathop{\textit{Sh}}\nolimits(\mathcal{C}/U), \mathcal{O}_U) \ar[d]^{(f_c, f_c^\sharp)} \\ (\mathop{\textit{Sh}}\nolimits(\mathcal{D}/V'), \mathcal{O}'_{V'}) \ar[rr]^{(j_{V'/V}, j_{V'/V}^\sharp)} & & (\mathop{\textit{Sh}}\nolimits(\mathcal{D}/V), \mathcal{O}'_{V'}) }$$ commutes.

Proof. On the level of morphisms of topoi this is Sites, Lemma 7.27.3. To check that the diagrams commute as morphisms of ringed topoi use Lemmas 18.19.4 and 18.20.1 exactly as in the proof of Sites, Lemma 7.27.3. $\square$

The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2297–2435 (see updates for more information).

\section{Localization of morphisms of ringed sites}
\label{section-localize-morphisms}

\noindent
This section is the analogue of
Sites, Section \ref{sites-section-localize-morphisms}.

\begin{lemma}
\label{lemma-localize-morphism-ringed-sites}
Let
$(f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$
be a morphism of ringed sites where $f$ is given by the continuous
functor $u : \mathcal{D} \to \mathcal{C}$.
Let $V$ be an object of $\mathcal{D}$ and set $U = u(V)$.
Then there is a canonical map of sheaves of rings $(f')^\sharp$
such that the diagram of
Sites, Lemma \ref{sites-lemma-localize-morphism}
is turned into a commutative diagram of ringed topoi
$$\xymatrix{ (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f', (f')^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}/V), \mathcal{O}'_V) \ar[rr]^{(j_V, j_V^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}'). }$$
Moreover, in this situation we have $f'_*j_U^{-1} = j_V^{-1}f_*$
and $f'_*j_U^* = j_V^*f_*$.
\end{lemma}

\begin{proof}
Just take $(f')^\sharp$ to be
$$(f')^{-1}\mathcal{O}'_V = (f')^{-1}j_V^{-1}\mathcal{O}' = j_U^{-1}f^{-1}\mathcal{O}' \xrightarrow{j_U^{-1}f^\sharp} j_U^{-1}\mathcal{O} = \mathcal{O}_U$$
and everything else follows from
Sites, Lemma \ref{sites-lemma-localize-morphism}.
(Note that $j^{-1} = j^*$ on sheaves of modules if $j$ is a localization
morphism, hence the first equality of functors implies the second.)
\end{proof}

\begin{lemma}
\label{lemma-relocalize-morphism-ringed-sites}
Let
$(f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$
be a morphism of ringed sites where $f$ is given by the continuous
functor $u : \mathcal{D} \to \mathcal{C}$.
Let $V \in \Ob(\mathcal{D})$, $U \in \Ob(\mathcal{C})$
and $c : U \to u(V)$ a morphism of $\mathcal{C}$.
There exists a commutative diagram of ringed topoi
$$\xymatrix{ (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f_c, f_c^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}/V), \mathcal{O}'_V) \ar[rr]^{(j_V, j_V^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}'). }$$
The morphism $(f_c, f_c^\sharp)$
is equal to the composition of the morphism
$$(f', (f')^\sharp) : (\Sh(\mathcal{C}/u(V)), \mathcal{O}_{u(V)}) \longrightarrow (\Sh(\mathcal{D}/V), \mathcal{O}'_V)$$
of
Lemma \ref{lemma-localize-morphism-ringed-sites}
and the morphism
$$(j, j^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}/u(V)), \mathcal{O}_{u(V)})$$
of
Lemma \ref{lemma-relocalize}.
Given any morphisms $b : V' \to V$, $a : U' \to U$ and
$c' : U' \to u(V')$ such that
$$\xymatrix{ U' \ar[r]_-{c'} \ar[d]_a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-c & u(V) }$$
commutes, then the following diagram of ringed topoi
$$\xymatrix{ (\Sh(\mathcal{C}/U'), \mathcal{O}_{U'}) \ar[rr]_{(j_{U'/U}, j_{U'/U}^\sharp)} \ar[d]_{(f_{c'}, f_{c'}^\sharp)} & & (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[d]^{(f_c, f_c^\sharp)} \\ (\Sh(\mathcal{D}/V'), \mathcal{O}'_{V'}) \ar[rr]^{(j_{V'/V}, j_{V'/V}^\sharp)} & & (\Sh(\mathcal{D}/V), \mathcal{O}'_{V'}) }$$
commutes.
\end{lemma}

\begin{proof}
On the level of morphisms of topoi this is
Sites, Lemma \ref{sites-lemma-relocalize-morphism}.
To check that the diagrams commute as morphisms of ringed topoi use
Lemmas \ref{lemma-relocalize} and
\ref{lemma-localize-morphism-ringed-sites}
exactly as in the proof of
Sites, Lemma \ref{sites-lemma-relocalize-morphism}.
\end{proof}

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