The Stacks project

Lemma 73.8.8. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The functor

\[ u : (\textit{Spaces}/Y)_{ph} \longrightarrow (\textit{Spaces}/X)_{ph}, \quad V/Y \longmapsto V/X \]

is cocontinuous, and has a continuous right adjoint

\[ v : (\textit{Spaces}/X)_{ph} \longrightarrow (\textit{Spaces}/Y)_{ph}, \quad (U \to Y) \longmapsto (U \times _ X Y \to Y). \]

They induce the same morphism of topoi

\[ f_{big} : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{ph}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph}) \]

We have $f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)$. We have $f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times _ X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.

Proof. The functor $u$ is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 7.21.5 and 7.21.6 apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/T$ and $V/X$ we have $\mathop{\mathrm{Mor}}\nolimits _ X(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ Y(U, V \times _ X Y)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$. $\square$


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Your email address will not be published.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 0DFN. Beware of the difference between the letter 'O' and the digit '0'.