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Tag 07MJ

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

Remark 54.24.1 (Higher direct images). Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Let $$ \xymatrix{ X \ar[r]_f \ar[d] & X' \ar[d] \\ S_0 \ar[r] & S'_0 } $$ be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $X'$. Let $\mathcal{F}$ be an $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then $Rf_{\text{cris}, *}\mathcal{F}$ can be computed as follows.

Given an object $(U', T', \delta')$ of $\text{Cris}(X'/S')$ set $U = X \times_{X'} U' = f^{-1}(U')$ (an open subscheme of $X$). Denote $(T_0, T, \delta)$ the divided power scheme over $S$ such that $$ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } $$ is cartesian in the category of divided power schemes, see Lemma 54.7.4. There is an induced morphism $U \to T_0$ and we obtain a morphism $(U/T)_{\text{cris}} \to (X/S)_{\text{cris}}$, see Remark 54.9.3. Let $\mathcal{F}_U$ be the pullback of $\mathcal{F}$. Let $\tau_{U/T} : (U/T)_{\text{cris}} \to T_{Zar}$ be the structure morphism. Then we have \begin{equation} \tag{54.24.1.1} \left(Rf_{\text{cris}, *}\mathcal{F}\right)_{T'} = R(T \to T')_*\left(R\tau_{U/T, *} \mathcal{F}_U \right) \end{equation} where the left hand side is the restriction (see Section 54.10).

Hints: First, show that $\text{Cris}(U/T)$ is the localization (in the sense of Sites, Lemma 7.29.3) of $\text{Cris}(X/S)$ at the sheaf of sets $f_{\text{cris}}^{-1}h_{(U', T', \delta')}$. Next, reduce the statement to the case where $\mathcal{F}$ is an injective module and pushforward of modules using that the pullback of an injective $\mathcal{O}_{X/S}$-module is an injective $\mathcal{O}_{U/T}$-module on $\text{Cris}(U/T)$. Finally, check the result holds for plain pushforward.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4394–4444 (see updates for more information).

    \begin{remark}[Higher direct images]
    \label{remark-compute-direct-image}
    Let $p$ be a prime number. Let
    $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ be
    a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Let
    $$
    \xymatrix{
    X \ar[r]_f \ar[d] & X' \ar[d] \\
    S_0 \ar[r] & S'_0
    }
    $$
    be a commutative diagram of morphisms of schemes and assume $p$ is
    locally nilpotent on $X$ and $X'$. Let $\mathcal{F}$ be an
    $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then
    $Rf_{\text{cris}, *}\mathcal{F}$ can be computed as follows.
    
    \medskip\noindent
    Given an object $(U', T', \delta')$ of $\text{Cris}(X'/S')$ set
    $U = X \times_{X'} U' = f^{-1}(U')$ (an open subscheme of $X$). Denote
    $(T_0, T, \delta)$ the divided power scheme over $S$ such that
    $$
    \xymatrix{
    T \ar[r] \ar[d] & T' \ar[d] \\
    S \ar[r] & S'
    }
    $$
    is cartesian in the category of divided power schemes, see
    Lemma \ref{lemma-fibre-product}. There is an
    induced morphism $U \to T_0$ and we obtain a morphism
    $(U/T)_{\text{cris}} \to (X/S)_{\text{cris}}$, see
    Remark \ref{remark-functoriality-cris}.
    Let $\mathcal{F}_U$ be the pullback of $\mathcal{F}$.
    Let $\tau_{U/T} : (U/T)_{\text{cris}} \to T_{Zar}$ be the structure morphism.
    Then we have
    \begin{equation}
    \label{equation-identify-pushforward}
    \left(Rf_{\text{cris}, *}\mathcal{F}\right)_{T'} =
    R(T \to T')_*\left(R\tau_{U/T, *} \mathcal{F}_U \right)
    \end{equation}
    where the left hand side is the restriction (see
    Section \ref{section-sheaves}).
    
    \medskip\noindent
    Hints: First, show that $\text{Cris}(U/T)$ is the localization (in the sense
    of Sites, Lemma \ref{sites-lemma-localize-topos-site}) of $\text{Cris}(X/S)$
    at the sheaf of sets $f_{\text{cris}}^{-1}h_{(U', T', \delta')}$. Next, reduce
    the statement to the case where $\mathcal{F}$ is an injective module
    and pushforward of modules using that the pullback of an injective
    $\mathcal{O}_{X/S}$-module is an injective $\mathcal{O}_{U/T}$-module on
    $\text{Cris}(U/T)$. Finally, check the result holds for plain pushforward.
    \end{remark}

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