Remark 60.24.10 (Rlim). Let p be a prime number. Let (A, I, \gamma ) be a divided power ring with A an algebra over \mathbf{Z}_{(p)} with p nilpotent in A/I. Set S = \mathop{\mathrm{Spec}}(A) and S_0 = \mathop{\mathrm{Spec}}(A/I). Let X be a scheme over S_0 with p locally nilpotent on X. Let \mathcal{F} be any \mathcal{O}_{X/S}-module. For e \gg 0 we have (p^ e) \subset I is preserved by \gamma , see Divided Power Algebra, Lemma 23.4.5. Set S_ e = \mathop{\mathrm{Spec}}(A/p^ eA) for e \gg 0. Then \text{Cris}(X/S_ e) is a full subcategory of \text{Cris}(X/S) and we denote \mathcal{F}_ e the restriction of \mathcal{F} to \text{Cris}(X/S_ e). Then
Hints: Suffices to prove this for \mathcal{F} injective. In this case the sheaves \mathcal{F}_ e are injective modules too, the transition maps \Gamma (\mathcal{F}_{e + 1}) \to \Gamma (\mathcal{F}_ e) are surjective, and we have \Gamma (\mathcal{F}) = \mathop{\mathrm{lim}}\nolimits _ e \Gamma (\mathcal{F}_ e) because any object of \text{Cris}(X/S) is locally an object of one of the categories \text{Cris}(X/S_ e) by definition of \text{Cris}(X/S).
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