The Stacks project

Remark 35.8.4. Let $f : T \to S$ be a morphism of schemes. Each of the morphisms of sites $f_{sites}$ listed in Topologies, Section 34.11 becomes a morphism of ringed sites. Namely, each of these morphisms of sites $f_{sites} : (\mathit{Sch}/T)_\tau \to (\mathit{Sch}/S)_{\tau '}$, or $f_{sites} : (\mathit{Sch}/S)_\tau \to S_{\tau '}$ is given by the continuous functor $S'/S \mapsto T \times _ S S'/S$. Hence, given $S'/S$ we let

\[ f_{sites}^\sharp : \mathcal{O}(S'/S) \longrightarrow f_{sites, *}\mathcal{O}(S'/S) = \mathcal{O}(S \times _ S S'/T) \]

be the usual map $\text{pr}_{S'}^\sharp : \mathcal{O}(S') \to \mathcal{O}(T \times _ S S')$. Similarly, the morphism $i_ f : \mathop{\mathit{Sh}}\nolimits (T_\tau ) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_\tau )$ for $\tau \in \{ Zar, {\acute{e}tale}\} $, see Topologies, Lemmas 34.3.12 and 34.4.12, becomes a morphism of ringed topoi because $i_ f^{-1}\mathcal{O} = \mathcal{O}$. Here are some special cases:

  1. The morphism of big sites $f_{big} : (\mathit{Sch}/X)_{fppf} \to (\mathit{Sch}/Y)_{fppf}$, becomes a morphism of ringed sites

    \[ (f_{big}, f_{big}^\sharp ) : ((\mathit{Sch}/X)_{fppf}, \mathcal{O}_ X) \longrightarrow ((\mathit{Sch}/Y)_{fppf}, \mathcal{O}_ Y) \]

    as in Modules on Sites, Definition 18.6.1. Similarly for the big syntomic, smooth, ├ętale and Zariski sites.

  2. The morphism of small sites $f_{small} : X_{\acute{e}tale}\to Y_{\acute{e}tale}$ becomes a morphism of ringed sites

    \[ (f_{small}, f_{small}^\sharp ) : (X_{\acute{e}tale}, \mathcal{O}_ X) \longrightarrow (Y_{\acute{e}tale}, \mathcal{O}_ Y) \]

    as in Modules on Sites, Definition 18.6.1. Similarly for the small Zariski site.


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Your email address will not be published.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 070R. Beware of the difference between the letter 'O' and the digit '0'.