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Tag 07MR

Chapter 54: Crystalline Cohomology > Section 54.24: Some further results

Remark 54.24.7 (Specific boundedness). In Situation 54.7.5 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 54.21.4). 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 54.24.4) 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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