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Tag 07LL

54.23. Applications

In this section we collect some applications of the material in the previous sections.

Proposition 54.23.1. In Situation 54.7.5. Let $\mathcal{F}$ be a crystal in quasi-coherent modules on $\text{Cris}(X/S)$. The truncation map of complexes $$ (\mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^1_{X/S} \to \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^2_{X/S} \to \ldots) \longrightarrow \mathcal{F}[0], $$ while not a quasi-isomorphism, becomes a quasi-isomorphism after applying $Ru_{X/S, *}$. In fact, for any $i > 0$, we have $$ Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0. $$

Proof. By Lemma 54.15.1 we get a de Rham complex as indicated in the lemma. We abbreviate $\mathcal{H} = \mathcal{F} \otimes \Omega^i_{X/S}$. Let $X' \subset X$ be an affine open subscheme which maps into an affine open subscheme $S' \subset S$. Then $$ (Ru_{X/S, *}\mathcal{H})|_{X'_{Zar}} = Ru_{X'/S', *}(\mathcal{H}|_{\text{Cris}(X'/S')}), $$ see Lemma 54.9.5. Thus Lemma 54.21.2 shows that $Ru_{X/S, *}\mathcal{H}$ is a complex of sheaves on $X_{Zar}$ whose cohomology on any affine open is trivial. As $X$ has a basis for its topology consisting of affine opens this implies that $Ru_{X/S, *}\mathcal{H}$ is quasi-isomorphic to zero. $\square$

Remark 54.23.2. The proof of Proposition 54.23.1 shows that the conclusion $$ Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0 $$ for $i > 0$ is true for any $\mathcal{O}_{X/S}$-module $\mathcal{F}$ which satisfies conditions (1) and (2) of Proposition 54.21.1. This applies to the following non-crystals: $\Omega^i_{X/S}$ for all $i$, and any sheaf of the form $\underline{\mathcal{F}}$, where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module. In particular, it applies to the sheaf $\underline{\mathcal{O}_X} = \underline{\mathbf{G}_a}$. But note that we need something like Lemma 54.15.1 to produce a de Rham complex which requires $\mathcal{F}$ to be a crystal. Hence (currently) the collection of sheaves of modules for which the full statement of Proposition 54.23.1 holds is exactly the category of crystals in quasi-coherent modules.

In Situation 54.7.5. Let $\mathcal{F}$ be a crystal in quasi-coherent modules on $\text{Cris}(X/S)$. Let $(U, T, \delta)$ be an object of $\text{Cris}(X/S)$. Proposition 54.23.1 allows us to construct a canonical map \begin{equation} \tag{54.23.2.1} R\Gamma(\text{Cris}(X/S), \mathcal{F}) \longrightarrow R\Gamma(T, \mathcal{F}_T \otimes_{\mathcal{O}_T} \Omega^*_{T/S, \delta}) \end{equation} Namely, we have $R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S), \mathcal{F} \otimes \Omega^*_{X/S})$, we can restrict global cohomology classes to $T$, and $\Omega_{X/S}$ restricts to $\Omega_{T/S, \delta}$ by Lemma 54.12.3.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4285–4384 (see updates for more information).

    \section{Applications}
    \label{section-applications}
    
    \noindent
    In this section we collect some applications of the material in
    the previous sections.
    
    \begin{proposition}
    \label{proposition-compare-with-de-Rham}
    In Situation \ref{situation-global}.
    Let $\mathcal{F}$ be a crystal in quasi-coherent modules on
    $\text{Cris}(X/S)$. The truncation map of complexes
    $$
    (\mathcal{F} \to
    \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^1_{X/S} \to
    \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^2_{X/S} \to \ldots)
    \longrightarrow \mathcal{F}[0],
    $$
    while not a quasi-isomorphism, becomes a quasi-isomorphism after applying
    $Ru_{X/S, *}$. In fact, for any $i > 0$, we have 
    $$
    Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0.
    $$
    \end{proposition}
    
    \begin{proof}
    By Lemma \ref{lemma-automatic-connection} we get a de Rham complex
    as indicated in the lemma. We abbreviate
    $\mathcal{H} = \mathcal{F} \otimes \Omega^i_{X/S}$.
    Let $X' \subset X$ be an affine open
    subscheme which maps into an affine open subscheme $S' \subset S$.
    Then
    $$
    (Ru_{X/S, *}\mathcal{H})|_{X'_{Zar}} =
    Ru_{X'/S', *}(\mathcal{H}|_{\text{Cris}(X'/S')}),
    $$
    see Lemma \ref{lemma-localize}. Thus
    Lemma \ref{lemma-cohomology-is-zero}
    shows that $Ru_{X/S, *}\mathcal{H}$ is a complex of sheaves on
    $X_{Zar}$ whose cohomology on any affine open is trivial.
    As $X$ has a basis for its topology consisting of affine opens
    this implies that $Ru_{X/S, *}\mathcal{H}$ is quasi-isomorphic to zero.
    \end{proof}
    
    \begin{remark}
    \label{remark-vanishing}
    The proof of Proposition \ref{proposition-compare-with-de-Rham}
    shows that the conclusion
    $$
    Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0
    $$
    for $i > 0$ is true for any $\mathcal{O}_{X/S}$-module
    $\mathcal{F}$ which satisfies conditions (1) and (2) of
    Proposition \ref{proposition-compute-cohomology}.
    This applies to the following non-crystals:
    $\Omega^i_{X/S}$ for all $i$, and any sheaf of the form
    $\underline{\mathcal{F}}$, where $\mathcal{F}$ is a quasi-coherent
    $\mathcal{O}_X$-module. In particular, it applies to the
    sheaf $\underline{\mathcal{O}_X} = \underline{\mathbf{G}_a}$.
    But note that we need something like Lemma \ref{lemma-automatic-connection}
    to produce a de Rham complex which requires $\mathcal{F}$ to be a crystal.
    Hence (currently) the collection of sheaves of modules for which the full
    statement of Proposition \ref{proposition-compare-with-de-Rham} holds
    is exactly the category of crystals in quasi-coherent modules.
    \end{remark}
    
    \noindent
    In Situation \ref{situation-global}.
    Let $\mathcal{F}$ be a crystal in quasi-coherent modules on
    $\text{Cris}(X/S)$. Let $(U, T, \delta)$ be an object of
    $\text{Cris}(X/S)$. Proposition \ref{proposition-compare-with-de-Rham}
    allows us to construct a canonical map
    \begin{equation}
    \label{equation-restriction}
    R\Gamma(\text{Cris}(X/S), \mathcal{F})
    \longrightarrow
    R\Gamma(T, \mathcal{F}_T \otimes_{\mathcal{O}_T} \Omega^*_{T/S, \delta})
    \end{equation}
    Namely, we have $R\Gamma(\text{Cris}(X/S), \mathcal{F}) =
    R\Gamma(\text{Cris}(X/S), \mathcal{F} \otimes \Omega^*_{X/S})$,
    we can restrict global cohomology classes to $T$, and $\Omega_{X/S}$
    restricts to $\Omega_{T/S, \delta}$ by
    Lemma \ref{lemma-module-of-differentials}.

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