Remark 60.24.7 (Specific boundedness). In Situation 60.7.5 let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Assume that $S_0$ has a unique point and that $X \to S_0$ is of finite presentation.

1. If $\dim X = d$ and $X/S_0$ has embedding dimension $e$, then $H^ i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i > d + e$.

2. If $X$ is separated and can be covered by $q$ affines, and $X/S_0$ has embedding dimension $e$, then $H^ i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i > q + e$.

Hints: In case (1) we can use that

$H^ i(\text{Cris}(X/S), \mathcal{F}) = H^ i(X_{Zar}, Ru_{X/S, *}\mathcal{F})$

and that $Ru_{X/S, *}\mathcal{F}$ is locally calculated by a de Rham complex constructed using an embedding of $X$ into a smooth scheme of dimension $e$ over $S$ (see Lemma 60.21.4). These de Rham complexes are zero in all degrees $> e$. Hence (1) follows from Cohomology, Proposition 20.20.7. In case (2) we use the alternating Čech complex (see Remark 60.24.4) to reduce to the case $X$ affine. In the affine case we prove the result using the de Rham complex associated to an embedding of $X$ into a smooth scheme of dimension $e$ over $S$ (it takes some work to construct such a thing).

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