Remark 60.9.6 (Structure morphism). In Situation 60.7.5. Consider the closed subscheme $S_0 = V(\mathcal{I}) \subset S$. If we assume that $p$ is locally nilpotent on $S_0$ (which is always the case in practice) then we obtain a situation as in Definition 60.8.1 with $S_0$ instead of $X$. Hence we get a site $\text{Cris}(S_0/S)$. If $f : X \to S_0$ is the structure morphism of $X$ over $S$, then we get a commutative diagram of ringed topoi

$\xymatrix{ (X/S)_{\text{cris}} \ar[r]_{f_{\text{cris}}} \ar[d]_{u_{X/S}} & (S_0/S)_{\text{cris}} \ar[d]^{u_{S_0/S}} \\ \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \ar[r]^{f_{small}} & \mathop{\mathit{Sh}}\nolimits (S_{0, Zar}) \ar[rd] \\ & & \mathop{\mathit{Sh}}\nolimits (S_{Zar}) }$

see Remark 60.9.3. We think of the composition $(X/S)_{\text{cris}} \to \mathop{\mathit{Sh}}\nolimits (S_{Zar})$ as the structure morphism of the crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism

$\tau _{X/S} : (X/S)_{\text{cris}} \longrightarrow \mathop{\mathit{Sh}}\nolimits (S_{Zar})$

is defined as we can take the lower route through the diagram above.

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.

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 07IM. Beware of the difference between the letter 'O' and the digit '0'.