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.
Comments (0)