The Stacks project

Remark 60.24.13 (Complete perfectness). Let $p$ be a prime number. Let $(A, I, \gamma )$ be a divided power ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. Let $X$ be a proper smooth scheme over $S_0$. Let $\mathcal{F}$ be a crystal in finite locally free quasi-coherent $\mathcal{O}_{X/S}$-modules. Then $R\Gamma (\text{Cris}(X/S), \mathcal{F})$ is a perfect object of $D(A)$.

Hints: We know that $K = R\Gamma (\text{Cris}(X/S), \mathcal{F})$ is the derived limit $K = R\mathop{\mathrm{lim}}\nolimits K_ e$ of the cohomologies over $A/p^ eA$, see Remark 60.24.10. Each $K_ e$ is a perfect complex of $D(A/p^ eA)$ by Remark 60.24.12. Since $A$ is $p$-adically complete the result follows from More on Algebra, Lemma 15.97.4.

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