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Tag 07MZ

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

Remark 54.24.14 (Complete comparison). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a Noetherian $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $Y$ be a proper smooth scheme over $S$ and set $X = Y \times_S S_0$. Let $\mathcal{F}$ be a finite type crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Then

  1. there exists a coherent $\mathcal{O}_Y$-module $\mathcal{F}_Y$ endowed with integrable connection $$ \nabla : \mathcal{F}_Y \longrightarrow \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S} $$ such that $\mathcal{F}_Y/p^e\mathcal{F}_Y$ is the module with connection over $A/p^eA$ found in Remark 54.24.11, and
  2. 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: The existence of $\mathcal{F}_Y$ is Grothendieck's existence theorem (insert future reference here). The isomorphism of cohomologies follows as both sides are computed as $R\mathop{\rm lim}\nolimits$ of the versions modulo $p^e$ (see Remark 54.24.10 for the left hand side; use the theorem on formal functions, see Cohomology of Schemes, Theorem 29.20.5 for the right hand side). Each of the versions modulo $p^e$ are isomorphic by Remark 54.24.11.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4971–5006 (see updates for more information).

    \begin{remark}[Complete comparison]
    \label{remark-complete-comparison}
    Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
    ring with $A$ a Noetherian $p$-adically complete ring and $p$ nilpotent
    in $A/I$. Set $S = \Spec(A)$ and
    $S_0 = \Spec(A/I)$. Let $Y$ be a proper smooth scheme over $S$ and set
    $X = Y \times_S S_0$. Let $\mathcal{F}$ be a finite type crystal in
    quasi-coherent $\mathcal{O}_{X/S}$-modules. Then
    \begin{enumerate}
    \item there exists a coherent $\mathcal{O}_Y$-module $\mathcal{F}_Y$
    endowed with integrable connection
    $$
    \nabla :
    \mathcal{F}_Y
    \longrightarrow
    \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S}
    $$
    such that $\mathcal{F}_Y/p^e\mathcal{F}_Y$ is the module with connection
    over $A/p^eA$ found in Remark \ref{remark-comparison}, 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: The existence of $\mathcal{F}_Y$ is Grothendieck's existence theorem
    (insert future reference here). The isomorphism of cohomologies follows
    as both sides are computed as $R\lim$ of the versions modulo $p^e$
    (see Remark \ref{remark-rlim} for the left hand side; use the theorem
    on formal functions, see
    Cohomology of Schemes, Theorem \ref{coherent-theorem-formal-functions}
    for the right hand side).
    Each of the versions modulo $p^e$ are isomorphic by
    Remark \ref{remark-comparison}.
    \end{remark}

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