Section 60.23 (07LL): 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

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.

The Stacks project

60.23 Applications

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