The Stacks project

Lemma 7.31.3. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Let $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$, $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and $s : \mathcal{F} \to f^{-1}\mathcal{G}$ a morphism of sheaves. There exists a commutative diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \ar[r]_{j_\mathcal {F}} \ar[d]_{f_ s} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G} \ar[r]^{j_\mathcal {G}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}). } \]

We have $f_ s = f' \circ j_{\mathcal{F}/f^{-1}\mathcal{G}}$ where $f' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/f^{-1}\mathcal{G} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{F}$ is as in Lemma 7.31.1 and $j_{\mathcal{F}/f^{-1}\mathcal{G}} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/f^{-1}\mathcal{G}$ is as in Lemma 7.30.6. The functor $(f_ s)^{-1}$ is described by the rule

\[ (f_ s)^{-1}(\mathcal{H} \xrightarrow {\varphi } \mathcal{G}) = (f^{-1}\mathcal{H} \times _{f^{-1}\varphi , f^{-1}\mathcal{G}, s} \mathcal{F} \rightarrow \mathcal{F}). \]

Finally, given any morphisms $b : \mathcal{G}' \to \mathcal{G}$, $a : \mathcal{F}' \to \mathcal{F}$ and $s' : \mathcal{F}' \to f^{-1}\mathcal{G}'$ such that

\[ \xymatrix{ \mathcal{F}' \ar[r]_-{s'} \ar[d]_ a & f^{-1}\mathcal{G}' \ar[d]^{f^{-1}b} \\ \mathcal{F} \ar[r]^-s & f^{-1}\mathcal{G} } \]

commutes, then the diagram

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}' \ar[r]_{j_{\mathcal{F}'/\mathcal{F}}} \ar[d]_{f_{s'}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \ar[d]^{f_ s} \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}' \ar[r]^{j_{\mathcal{G}'/\mathcal{G}}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}. } \]


Proof. The commutativity of the first square follows from the commutativity of the diagram in Lemma 7.30.6 and the commutativity of the diagram in Lemma 7.31.1. The description of $f_ s^{-1}$ follows on combining the descriptions of $(f')^{-1}$ in Lemma 7.31.1 with the description of $(j_{\mathcal{F}/f^{-1}\mathcal{G}})^{-1}$ in Lemma 7.30.6. The commutativity of the last square then follows from the equality

\[ f^{-1}\mathcal{H} \times _{f^{-1}\mathcal{G}, s} \mathcal{F} \times _\mathcal {F} \mathcal{F}' = f^{-1}(\mathcal{H} \times _\mathcal {G} \mathcal{G}') \times _{f^{-1}\mathcal{G}', s'} \mathcal{F}' \]

which is formal. $\square$

Comments (0)

Post a comment

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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

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 input field. As a reminder, this is tag 04IV. Beware of the difference between the letter 'O' and the digit '0'.