Remark 60.24.6 (Boundedness). In the situation of Remark 60.24.1 assume that $S \to S'$ is quasi-compact and quasi-separated and that $X \to S_0$ is of finite type and quasi-separated. Then there exists an integer $i_0$ such that for any crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$ we have $R^ if_{\text{cris}, *}\mathcal{F} = 0$ for all $i > i_0$.

Hints: Arguing as in Remark 60.24.5 (using Cohomology of Schemes, Lemma 30.4.5) we reduce to proving that $H^ i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i \gg 0$ in the situation of Proposition 60.21.3 when $C$ is a finite type algebra over $A$. This is clear as we can choose a finite polynomial algebra and we see that $\Omega ^ i_ D = 0$ for $i \gg 0$.

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