# The Stacks Project

## Tag 07LL

### 51.23. Applications

In this section we collect some applications of the material in the previous sections.

Proposition 51.23.1. In Situation 51.7.5. Let $\mathcal{F}$ be a crystal in quasi-coherent modules on $\text{Cris}(X/S)$. The truncation map of complexes $$(\mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^1_{X/S} \to \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^2_{X/S} \to \ldots) \longrightarrow \mathcal{F}[0],$$ while not a quasi-isomorphism, becomes a quasi-isomorphism after applying $Ru_{X/S, *}$. In fact, for any $i > 0$, we have $$Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0.$$

Proof. By Lemma 51.15.1 we get a de Rham complex as indicated in the lemma. We abbreviate $\mathcal{H} = \mathcal{F} \otimes \Omega^i_{X/S}$. Let $X' \subset X$ be an affine open subscheme which maps into an affine open subscheme $S' \subset S$. Then $$(Ru_{X/S, *}\mathcal{H})|_{X'_{Zar}} = Ru_{X'/S', *}(\mathcal{H}|_{\text{Cris}(X'/S')}),$$ see Lemma 51.9.5. Thus Lemma 51.21.2 shows that $Ru_{X/S, *}\mathcal{H}$ is a complex of sheaves on $X_{Zar}$ whose cohomology on any affine open is trivial. As $X$ has a basis for its topology consisting of affine opens this implies that $Ru_{X/S, *}\mathcal{H}$ is quasi-isomorphic to zero. $\square$

Remark 51.23.2. The proof of Proposition 51.23.1 shows that the conclusion $$Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0$$ for $i > 0$ is true for any $\mathcal{O}_{X/S}$-module $\mathcal{F}$ which satisfies conditions (1) and (2) of Proposition 51.21.1. This applies to the following non-crystals: $\Omega^i_{X/S}$ for all $i$, and any sheaf of the form $\underline{\mathcal{F}}$, where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module. In particular, it applies to the sheaf $\underline{\mathcal{O}_X} = \underline{\mathbf{G}_a}$. But note that we need something like Lemma 51.15.1 to produce a de Rham complex which requires $\mathcal{F}$ to be a crystal. Hence (currently) the collection of sheaves of modules for which the full statement of Proposition 51.23.1 holds is exactly the category of crystals in quasi-coherent modules.

In Situation 51.7.5. Let $\mathcal{F}$ be a crystal in quasi-coherent modules on $\text{Cris}(X/S)$. Let $(U, T, \delta)$ be an object of $\text{Cris}(X/S)$. Proposition 51.23.1 allows us to construct a canonical map $$\tag{51.23.2.1} R\Gamma(\text{Cris}(X/S), \mathcal{F}) \longrightarrow R\Gamma(T, \mathcal{F}_T \otimes_{\mathcal{O}_T} \Omega^*_{T/S, \delta})$$ Namely, we have $R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S), \mathcal{F} \otimes \Omega^*_{X/S})$, we can restrict global cohomology classes to $T$, and $\Omega_{X/S}$ restricts to $\Omega_{T/S, \delta}$ by Lemma 51.12.3.

The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4285–4384 (see updates for more information).

\section{Applications}
\label{section-applications}

\noindent
In this section we collect some applications of the material in
the previous sections.

\begin{proposition}
\label{proposition-compare-with-de-Rham}
In Situation \ref{situation-global}.
Let $\mathcal{F}$ be a crystal in quasi-coherent modules on
$\text{Cris}(X/S)$. The truncation map of complexes
$$(\mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^1_{X/S} \to \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^2_{X/S} \to \ldots) \longrightarrow \mathcal{F}[0],$$
while not a quasi-isomorphism, becomes a quasi-isomorphism after applying
$Ru_{X/S, *}$. In fact, for any $i > 0$, we have
$$Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0.$$
\end{proposition}

\begin{proof}
By Lemma \ref{lemma-automatic-connection} we get a de Rham complex
as indicated in the lemma. We abbreviate
$\mathcal{H} = \mathcal{F} \otimes \Omega^i_{X/S}$.
Let $X' \subset X$ be an affine open
subscheme which maps into an affine open subscheme $S' \subset S$.
Then
$$(Ru_{X/S, *}\mathcal{H})|_{X'_{Zar}} = Ru_{X'/S', *}(\mathcal{H}|_{\text{Cris}(X'/S')}),$$
see Lemma \ref{lemma-localize}. Thus
Lemma \ref{lemma-cohomology-is-zero}
shows that $Ru_{X/S, *}\mathcal{H}$ is a complex of sheaves on
$X_{Zar}$ whose cohomology on any affine open is trivial.
As $X$ has a basis for its topology consisting of affine opens
this implies that $Ru_{X/S, *}\mathcal{H}$ is quasi-isomorphic to zero.
\end{proof}

\begin{remark}
\label{remark-vanishing}
The proof of Proposition \ref{proposition-compare-with-de-Rham}
shows that the conclusion
$$Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0$$
for $i > 0$ is true for any $\mathcal{O}_{X/S}$-module
$\mathcal{F}$ which satisfies conditions (1) and (2) of
Proposition \ref{proposition-compute-cohomology}.
This applies to the following non-crystals:
$\Omega^i_{X/S}$ for all $i$, and any sheaf of the form
$\underline{\mathcal{F}}$, where $\mathcal{F}$ is a quasi-coherent
$\mathcal{O}_X$-module. In particular, it applies to the
sheaf $\underline{\mathcal{O}_X} = \underline{\mathbf{G}_a}$.
But note that we need something like Lemma \ref{lemma-automatic-connection}
to produce a de Rham complex which requires $\mathcal{F}$ to be a crystal.
Hence (currently) the collection of sheaves of modules for which the full
statement of Proposition \ref{proposition-compare-with-de-Rham} holds
is exactly the category of crystals in quasi-coherent modules.
\end{remark}

\noindent
In Situation \ref{situation-global}.
Let $\mathcal{F}$ be a crystal in quasi-coherent modules on
$\text{Cris}(X/S)$. Let $(U, T, \delta)$ be an object of
$\text{Cris}(X/S)$. Proposition \ref{proposition-compare-with-de-Rham}
allows us to construct a canonical map

\label{equation-restriction}
R\Gamma(\text{Cris}(X/S), \mathcal{F})
\longrightarrow
R\Gamma(T, \mathcal{F}_T \otimes_{\mathcal{O}_T} \Omega^*_{T/S, \delta})

Namely, we have $R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S), \mathcal{F} \otimes \Omega^*_{X/S})$,
we can restrict global cohomology classes to $T$, and $\Omega_{X/S}$
restricts to $\Omega_{T/S, \delta}$ by
Lemma \ref{lemma-module-of-differentials}.

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