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## Tag 07MR

Remark 51.24.7 (Specific boundedness). In Situation 51.7.5 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 51.21.4). These de Rham complexes are zero in all degrees $>e$. Hence (1) follows from Cohomology, Proposition 20.21.7. In case (2) we use the alternating Čech complex (see Remark 51.24.4) 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).

The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4679–4707 (see updates for more information).

\begin{remark}[Specific boundedness]
\label{remark-bounded-cohomology-over-point}
In Situation \ref{situation-global} 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.
\begin{enumerate}
\item If $\dim X = d$ and $X/S_0$ has embedding dimension $e$, then
$H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i > d + e$.
\item 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 $i > q + e$.
\end{enumerate}
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 \ref{lemma-compute-cohomology-crystal-smooth}).
These de Rham complexes are zero in all degrees $>e$. Hence (1)
follows from Cohomology, Proposition
\ref{cohomology-proposition-vanishing-Noetherian}.
In case (2) we use the alternating {\v C}ech complex (see
Remark \ref{remark-alternating-cech-complex}) 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).
\end{remark}

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