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Tag 07MQ

Chapter 51: Crystalline Cohomology > Section 51.24: Some further results

Remark 51.24.6 (Boundedness). In the situation of Remark 51.24.1 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 51.24.5 (using Cohomology of Schemes, Lemma 29.4.5) we reduce to proving that $H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i \gg 0$ in the situation of Proposition 51.21.3 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 $i \gg 0$.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4659–4677 (see updates for more information).

    \begin{remark}[Boundedness]
    \label{remark-bounded-cohomology}
    In the situation of Remark \ref{remark-compute-direct-image}
    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 $i > i_0$.
    
    \medskip\noindent
    Hints: Arguing as in Remark \ref{remark-quasi-coherent} (using
    Cohomology of Schemes, Lemma
    \ref{coherent-lemma-quasi-coherence-higher-direct-images})
    we reduce to proving that $H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i \gg 0$
    in the situation of Proposition \ref{proposition-compute-cohomology-crystal}
    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 $i \gg 0$.
    \end{remark}

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