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

$R\Gamma (\text{Cris}(X/S), \mathcal{F}) = R\mathop{\mathrm{lim}}\nolimits _ e R\Gamma (\text{Cris}(X/S_ e), \mathcal{F}_ e)$

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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