## 60.23 Applications

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

Proposition 60.23.1. In Situation 60.7.5. 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 60.15.1 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 60.9.5. Thus Lemma 60.21.2 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 60.23.2. The proof of Proposition 60.23.1 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 60.21.1. 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 60.15.1 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 60.23.1 holds is exactly the category of crystals in quasi-coherent modules.

In Situation 60.7.5. 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 60.23.1 allows us to construct a canonical map

60.23.2.1
$$\label{crystalline-equation-restriction} R\Gamma (\text{Cris}(X/S), \mathcal{F}) \longrightarrow R\Gamma (T, \mathcal{F}_ T \otimes _{\mathcal{O}_ T} \Omega ^*_{T/S, \delta })$$

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 60.12.3.

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