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Tag 07MV

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

Remark 51.24.10 (Rlim). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ an algebra over $\mathbf{Z}_{(p)}$ with $p$ nilpotent in $A/I$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a scheme over $S_0$ with $p$ locally nilpotent on $X$. Let $\mathcal{F}$ be any $\mathcal{O}_{X/S}$-module. For $e \gg 0$ we have $(p^e) \subset I$ is preserved by $\gamma$, see Divided Power Algebra, Lemma 23.4.5. Set $S_e = \mathop{\rm Spec}(A/p^eA)$ for $e \gg 0$. Then $\text{Cris}(X/S_e)$ is a full subcategory of $\text{Cris}(X/S)$ and we denote $\mathcal{F}_e$ the restriction of $\mathcal{F}$ to $\text{Cris}(X/S_e)$. Then $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\mathop{\rm lim}\nolimits_e R\Gamma(\text{Cris}(X/S_e), \mathcal{F}_e) $$

Hints: Suffices to prove this for $\mathcal{F}$ injective. In this case the sheaves $\mathcal{F}_e$ are injective modules too, the transition maps $\Gamma(\mathcal{F}_{e + 1}) \to \Gamma(\mathcal{F}_e)$ are surjective, and we have $\Gamma(\mathcal{F}) = \mathop{\rm lim}\nolimits_e \Gamma(\mathcal{F}_e)$ because any object of $\text{Cris}(X/S)$ is locally an object of one of the categories $\text{Cris}(X/S_e)$ by definition of $\text{Cris}(X/S)$.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4813–4842 (see updates for more information).

    \begin{remark}[Rlim]
    \label{remark-rlim}
    Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
    ring with $A$ an algebra over $\mathbf{Z}_{(p)}$ with $p$ nilpotent
    in $A/I$. Set $S = \Spec(A)$ and $S_0 = \Spec(A/I)$.
    Let $X$ be a scheme over $S_0$ with $p$ locally
    nilpotent on $X$. Let $\mathcal{F}$ be any
    $\mathcal{O}_{X/S}$-module. For $e \gg 0$ we have $(p^e) \subset I$
    is preserved by $\gamma$, see
    Divided Power Algebra, Lemma \ref{dpa-lemma-extend-to-completion}.
    Set $S_e = \Spec(A/p^eA)$ for $e \gg 0$.
    Then $\text{Cris}(X/S_e)$ is a full subcategory of $\text{Cris}(X/S)$
    and we denote $\mathcal{F}_e$ the restriction of $\mathcal{F}$ to
    $\text{Cris}(X/S_e)$. Then
    $$
    R\Gamma(\text{Cris}(X/S), \mathcal{F}) =
    R\lim_e R\Gamma(\text{Cris}(X/S_e), \mathcal{F}_e)
    $$
    
    \medskip\noindent
    Hints: Suffices to prove this for $\mathcal{F}$ injective.
    In this case the sheaves $\mathcal{F}_e$ are injective
    modules too, the transition maps
    $\Gamma(\mathcal{F}_{e + 1}) \to \Gamma(\mathcal{F}_e)$ are
    surjective, and we have
    $\Gamma(\mathcal{F}) = \lim_e \Gamma(\mathcal{F}_e)$ because
    any object of $\text{Cris}(X/S)$ is locally an object of one
    of the categories $\text{Cris}(X/S_e)$ by definition of
    $\text{Cris}(X/S)$.
    \end{remark}

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