The Stacks Project


Tag 07MP

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

Remark 54.24.5 (Quasi-coherence). In the situation of Remark 54.24.1 assume that $S \to S'$ is quasi-compact and quasi-separated and that $X \to S_0$ is quasi-compact and quasi-separated. Then for a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$ the sheaves $R^if_{\text{cris}, *}\mathcal{F}$ are locally quasi-coherent.

Hints: We have to show that the restrictions to $T'$ are quasi-coherent $\mathcal{O}_{T'}$-modules, where $(U', T', \delta')$ is any object of $\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine. We use the formula (54.24.1.1), the fact that $T \to T'$ is quasi-compact and quasi-separated (as $T$ is affine over the base change of $T'$ by $S \to S'$), and Cohomology of Schemes, Lemma 29.4.5 to see that it suffices to show that the sheaves $R^i\tau_{U/T, *}\mathcal{F}_U$ are quasi-coherent. Note that $U \to T_0$ is also quasi-compact and quasi-separated, see Schemes, Lemmas 25.21.15 and 25.21.15.

This reduces us to proving that $R^i\tau_{X/S, *}\mathcal{F}$ is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here $\tau_{X/S}$ is the structure morphism, see Remark 54.9.6. We may work locally on $S$, hence we may assume $S$ affine (see Lemma 54.9.5). Induction on the number of affines covering $X$ and Mayer-Vietoris (Remark 54.24.2) reduces the question to the case where $X$ is also affine (as in the proof of Cohomology of Schemes, Lemma 29.4.5). Say $X = \mathop{\rm Spec}(C)$ and $S = \mathop{\rm Spec}(A)$ so that $(A, I, \gamma)$ and $A \to C$ are as in Situation 54.5.1. Choose a polynomial algebra $P$ over $A$ and a surjection $P \to C$ as in Section 54.17. Let $(M, \nabla)$ be the module corresponding to $\mathcal{F}$, see Proposition 54.17.4. Applying Proposition 54.21.3 we see that $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is represented by $M \otimes_D \Omega_D^*$. Note that completion isn't necessary as $p$ is nilpotent in $A$! We have to show that this is compatible with taking principal opens in $S = \mathop{\rm Spec}(A)$. Suppose that $g \in A$. Then we conclude that similarly $R\Gamma(\text{Cris}(X_g/S_g), \mathcal{F})$ is computed by $M_g \otimes_{D_g} \Omega_{D_g}^*$ (again this uses that $p$-adic completion isn't necessary). Hence we conclude because localization is an exact functor on $A$-modules.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4605–4657 (see updates for more information).

    \begin{remark}[Quasi-coherence]
    \label{remark-quasi-coherent}
    In the situation of Remark \ref{remark-compute-direct-image}
    assume that $S \to S'$ is quasi-compact and quasi-separated and
    that $X \to S_0$ is quasi-compact and quasi-separated. Then for a crystal
    in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$
    the sheaves $R^if_{\text{cris}, *}\mathcal{F}$ are locally quasi-coherent.
    
    \medskip\noindent
    Hints: We have to show that the restrictions to $T'$ are quasi-coherent
    $\mathcal{O}_{T'}$-modules, where $(U', T', \delta')$ is any object of
    $\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine.
    We use the formula (\ref{equation-identify-pushforward}),
    the fact that $T \to T'$ is quasi-compact and quasi-separated (as $T$
    is affine over the base change of $T'$ by $S \to S'$), and
    Cohomology of Schemes, Lemma
    \ref{coherent-lemma-quasi-coherence-higher-direct-images}
    to see that it suffices to show that the sheaves
    $R^i\tau_{U/T, *}\mathcal{F}_U$ are quasi-coherent.
    Note that $U \to T_0$ is also quasi-compact and quasi-separated, see
    Schemes, Lemmas \ref{schemes-lemma-quasi-compact-permanence} and
    \ref{schemes-lemma-quasi-compact-permanence}.
    
    \medskip\noindent
    This reduces us to proving that $R^i\tau_{X/S, *}\mathcal{F}$
    is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here
    $\tau_{X/S}$ is the structure morphism, see
    Remark \ref{remark-structure-morphism}.
    We may work locally on $S$, hence we may assume $S$ affine
    (see Lemma \ref{lemma-localize}). Induction on the number
    of affines covering $X$ and Mayer-Vietoris
    (Remark \ref{remark-mayer-vietoris}) reduces the question to
    the case where $X$ is also affine (as in the proof of
    Cohomology of Schemes, Lemma
    \ref{coherent-lemma-quasi-coherence-higher-direct-images}).
    Say $X = \Spec(C)$ and $S = \Spec(A)$ so that $(A, I, \gamma)$ and
    $A \to C$ are as
    in Situation \ref{situation-affine}. Choose a polynomial algebra
    $P$ over $A$ and a surjection $P \to C$ as in
    Section \ref{section-quasi-coherent-crystals}.
    Let $(M, \nabla)$ be the module corresponding to $\mathcal{F}$, see
    Proposition \ref{proposition-crystals-on-affine}.
    Applying 
    Proposition \ref{proposition-compute-cohomology-crystal}
    we see that $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is represented by
    $M \otimes_D \Omega_D^*$. Note that completion isn't necessary
    as $p$ is nilpotent in $A$! We have to show that this is compatible
    with taking principal opens in $S = \Spec(A)$. Suppose that $g \in A$.
    Then we conclude that similarly $R\Gamma(\text{Cris}(X_g/S_g), \mathcal{F})$
    is computed by $M_g \otimes_{D_g} \Omega_{D_g}^*$ (again this uses that
    $p$-adic completion isn't necessary). Hence we conclude because localization
    is an exact functor on $A$-modules.
    \end{remark}

    Comments (0)

    There are no comments yet for this tag.

    There are also 2 comments on Section 54.24: Crystalline Cohomology.

    Add a comment on tag 07MP

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?