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Tag 07JN

Chapter 54: Crystalline Cohomology > Section 54.21: Cohomology in the affine case

Proposition 54.21.1. With notations as above assume that

  1. $\mathcal{F}$ is locally quasi-coherent, and
  2. for any morphism $(U, T, \delta) \to (U', T', \delta')$ of $\text{Cris}(X/S)$ where $f : T \to T'$ is a closed immersion the map $c_f : f^*\mathcal{F}_{T'} \to \mathcal{F}_T$ is surjective.

Then the complex $$ M(0) \to M(1) \to M(2) \to \ldots $$ computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

Proof. Using assumption (1) and Lemma 54.18.2 we see that $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is isomorphic to $R\Gamma(\mathcal{C}, \mathcal{F})$. Note that the categories $\mathcal{C}$ used in Lemmas 54.18.2 and 54.18.3 agree. Let $f : T \to T'$ be a closed immersion as in (2). Surjectivity of $c_f : f^*\mathcal{F}_{T'} \to \mathcal{F}_T$ is equivalent to surjectivity of $\mathcal{F}_{T'} \to f_*\mathcal{F}_T$. Hence, if $\mathcal{F}$ satisfies (1) and (2), then we obtain a short exact sequence $$ 0 \to \mathcal{K} \to \mathcal{F}_{T'} \to f_*\mathcal{F}_T \to 0 $$ of quasi-coherent $\mathcal{O}_{T'}$-modules on $T'$, see Schemes, Section 25.24 and in particular Lemma 25.24.1. Thus, if $T'$ is affine, then we conclude that the restriction map $\mathcal{F}(U', T', \delta') \to \mathcal{F}(U, T, \delta)$ is surjective by the vanishing of $H^1(T', \mathcal{K})$, see Cohomology of Schemes, Lemma 29.2.2. Hence the transition maps of the inverse systems in Lemma 54.18.3 are surjective. We conclude that that $R^pg_*(\mathcal{F}|_\mathcal{C}) = 0$ for all $p \geq 1$ where $g$ is as in Lemma 54.18.3. The object $D$ of the category $\mathcal{C}^\wedge$ satisfies the assumption of Lemma 54.18.4 by Lemma 54.5.7 with $$ D \times \ldots \times D = D(n) $$ in $\mathcal{C}$ because $D(n)$ is the $n + 1$-fold coproduct of $D$ in $\text{Cris}^\wedge(C/A)$, see Lemma 54.17.2. Thus we win. $\square$

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 3866–3880 (see updates for more information).

    \begin{proposition}
    \label{proposition-compute-cohomology}
    With notations as above assume that
    \begin{enumerate}
    \item $\mathcal{F}$ is locally quasi-coherent, and
    \item for any morphism $(U, T, \delta) \to (U', T', \delta')$
    of $\text{Cris}(X/S)$ where $f : T \to T'$ is a closed immersion
    the map $c_f : f^*\mathcal{F}_{T'} \to \mathcal{F}_T$ is surjective.
    \end{enumerate}
    Then the complex
    $$
    M(0) \to M(1) \to M(2) \to \ldots
    $$
    computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.
    \end{proposition}
    
    \begin{proof}
    Using assumption (1) and Lemma \ref{lemma-compare} we see that
    $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is isomorphic to
    $R\Gamma(\mathcal{C}, \mathcal{F})$. Note that the categories
    $\mathcal{C}$ used in Lemmas \ref{lemma-compare} and \ref{lemma-complete}
    agree. Let $f : T \to T'$ be a closed immersion as in (2). Surjectivity
    of $c_f : f^*\mathcal{F}_{T'} \to \mathcal{F}_T$ is equivalent to
    surjectivity of $\mathcal{F}_{T'} \to f_*\mathcal{F}_T$. Hence, if
    $\mathcal{F}$ satisfies (1) and (2), then we obtain a short exact sequence
    $$
    0 \to \mathcal{K} \to \mathcal{F}_{T'} \to f_*\mathcal{F}_T \to 0
    $$
    of quasi-coherent $\mathcal{O}_{T'}$-modules on $T'$, see
    Schemes, Section \ref{schemes-section-quasi-coherent} and in particular
    Lemma \ref{schemes-lemma-push-forward-quasi-coherent}.
    Thus, if $T'$ is affine, then we conclude that the restriction map
    $\mathcal{F}(U', T', \delta') \to \mathcal{F}(U, T, \delta)$
    is surjective by the vanishing of $H^1(T', \mathcal{K})$, see
    Cohomology of Schemes, Lemma
    \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.
    Hence the transition maps of the inverse systems in Lemma \ref{lemma-complete}
    are surjective. We conclude that
    that $R^pg_*(\mathcal{F}|_\mathcal{C}) = 0$ for all $p \geq 1$
    where $g$ is as in Lemma \ref{lemma-complete}.
    The object $D$ of the category $\mathcal{C}^\wedge$
    satisfies the assumption of Lemma \ref{lemma-category-with-covering} by
    Lemma \ref{lemma-generator-completion}
    with
    $$
    D \times \ldots \times D = D(n)
    $$
    in $\mathcal{C}$ because $D(n)$ is the $n + 1$-fold coproduct of
    $D$ in $\text{Cris}^\wedge(C/A)$, see Lemma \ref{lemma-property-Dn}.
    Thus we win.
    \end{proof}

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