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Tag 07MW

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

Remark 51.24.11 (Comparison). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $Y$ be a smooth scheme over $S$ and set $X = Y \times_S S_0$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Then

  1. $\gamma$ extends to a divided power structure on the ideal of $X$ in $Y$ so that $(X, Y, \gamma)$ is an object of $\text{Cris}(X/S)$,
  2. the restriction $\mathcal{F}_Y$ (see Section 51.10) comes endowed with a canonical integrable connection $\nabla : \mathcal{F}_Y \to \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S}$, and
  3. we have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(Y, \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega^\bullet_{Y/S}) $$ in $D(A)$.

Hints: See Divided Power Algebra, Lemma 23.4.2 for (1). See Lemma 51.15.1 for (2). For Part (3) note that there is a map, see (51.23.2.1). This map is an isomorphism when $X$ is affine, see Lemma 51.21.4. This shows that $Ru_{X/S, *}\mathcal{F}$ and $\mathcal{F}_Y \otimes \Omega^\bullet_{Y/S}$ are quasi-isomorphic as complexes on $Y_{Zar} = X_{Zar}$. Since $R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(X_{Zar}, Ru_{X/S, *}\mathcal{F})$ the result follows.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4844–4877 (see updates for more information).

    \begin{remark}[Comparison]
    \label{remark-comparison}
    Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
    ring with $p$ nilpotent in $A$. Set $S = \Spec(A)$ and
    $S_0 = \Spec(A/I)$. Let $Y$ be a smooth scheme over $S$ and set
    $X = Y \times_S S_0$. Let
    $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.
    Then
    \begin{enumerate}
    \item $\gamma$ extends to a divided power structure on the ideal
    of $X$ in $Y$ so that $(X, Y, \gamma)$ is an object of $\text{Cris}(X/S)$,
    \item the restriction $\mathcal{F}_Y$ (see Section \ref{section-sheaves})
    comes endowed with a canonical integrable connection
    $\nabla : \mathcal{F}_Y \to
    \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S}$, and
    \item we have
    $$
    R\Gamma(\text{Cris}(X/S), \mathcal{F}) =
    R\Gamma(Y, \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega^\bullet_{Y/S})
    $$
    in $D(A)$.
    \end{enumerate}
    Hints: See Divided Power Algebra, Lemma \ref{dpa-lemma-gamma-extends} for (1).
    See Lemma \ref{lemma-automatic-connection} for (2).
    For Part (3) note that there is a map, see
    (\ref{equation-restriction}). This map is an isomorphism when
    $X$ is affine, see
    Lemma \ref{lemma-compute-cohomology-crystal-smooth}.
    This shows that $Ru_{X/S, *}\mathcal{F}$ and
    $\mathcal{F}_Y \otimes \Omega^\bullet_{Y/S}$ are quasi-isomorphic
    as complexes on $Y_{Zar} = X_{Zar}$.
    Since $R\Gamma(\text{Cris}(X/S), \mathcal{F}) =
    R\Gamma(X_{Zar}, Ru_{X/S, *}\mathcal{F})$ the result follows.
    \end{remark}

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