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

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

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

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