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Tag 07MY

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

Remark 54.24.13 (Complete perfectness). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a proper smooth scheme over $S_0$. Let $\mathcal{F}$ be a crystal in finite locally free quasi-coherent $\mathcal{O}_{X/S}$-modules. Then $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is a perfect object of $D(A)$.

Hints: We know that $K = R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is the derived limit $K = R\mathop{\rm lim}\nolimits K_e$ of the cohomologies over $A/p^eA$, see Remark 54.24.10. Each $K_e$ is a perfect complex of $D(A/p^eA)$ by Remark 54.24.12. Since $A$ is $p$-adically complete the result follows from More on Algebra, Lemma 15.83.4.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4950–4969 (see updates for more information).

    \begin{remark}[Complete perfectness]
    \label{remark-complete-perfect}
    Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
    ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set
    $S = \Spec(A)$ and $S_0 = \Spec(A/I)$. Let $X$ be a proper
    smooth scheme over $S_0$. Let $\mathcal{F}$ be a crystal in
    finite locally free quasi-coherent $\mathcal{O}_{X/S}$-modules.
    Then $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is a
    perfect object of $D(A)$.
    
    \medskip\noindent
    Hints: We know that $K = R\Gamma(\text{Cris}(X/S), \mathcal{F})$
    is the derived limit $K = R\lim K_e$ of the cohomologies over $A/p^eA$,
    see Remark \ref{remark-rlim}.
    Each $K_e$ is a perfect complex of $D(A/p^eA)$ by
    Remark \ref{remark-perfect}.
    Since $A$ is $p$-adically complete the result
    follows from
    More on Algebra, Lemma \ref{more-algebra-lemma-Rlim-perfect-gives-complete}.
    \end{remark}

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