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}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 04IZ

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?