Remark 60.8.7 (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 morphisms of ringed topoi

$\xymatrix{ (X/S)_{\text{CRIS}} \ar[r]_{f_{\text{CRIS}}} \ar[d]_{U_{X/S}} & (S_0/S)_{\text{CRIS}} \ar[d]^{U_{S_0/S}} \\ \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{Zar}) \ar[r]^{f_{big}} & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S_0)_{Zar}) \ar[rd] \\ & & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar}) }$

by Remark 60.8.5. We think of the composition $(X/S)_{\text{CRIS}} \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$ as the structure morphism of the big crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism

$(X/S)_{\text{CRIS}} \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$

is defined as we can take the lower route through the diagram above. Thus it is the morphism of topoi corresponding to the cocontinuous functor $\text{CRIS}(X/S) \to (\mathit{Sch}/S)_{Zar}$ given by the rule $(U, T, \delta )/S \mapsto U/S$, see Sites, Section 7.21.

Comment #3458 by ZY on

In the last line, shouldn't the cocontinuous functor be $(U, T, \delta)/S \mapsto U/S$ instead?

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