# The Stacks Project

## Tag 07JN

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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