The Stacks project

Proposition 60.21.1. 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 60.18.2 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 60.18.2 and 60.18.3 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 26.24 and in particular Lemma 26.24.1. 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 30.2.2. Hence the transition maps of the inverse systems in Lemma 60.18.3 are surjective. We conclude that $R^ pg_*(\mathcal{F}|_\mathcal {C}) = 0$ for all $p \geq 1$ where $g$ is as in Lemma 60.18.3. The object $D$ of the category $\mathcal{C}^\wedge $ satisfies the assumption of Lemma 60.18.4 by Lemma 60.5.7 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 60.17.2. Thus we win. $\square$

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