The Stacks project

Lemma 60.18.1. In Situation 60.7.5. Let $\mathcal{F}$ be a locally quasi-coherent $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then we have

\[ H^ p((U, T, \delta ), \mathcal{F}) = 0 \]

for all $p > 0$ and all $(U, T, \delta )$ with $T$ or $U$ affine.

Proof. As $U \to T$ is a thickening we see that $U$ is affine if and only if $T$ is affine, see Limits, Lemma 32.11.1. Having said this, let us apply Cohomology on Sites, Lemma 21.10.9 to the collection $\mathcal{B}$ of affine objects $(U, T, \delta )$ and the collection $\text{Cov}$ of affine open coverings $\mathcal{U} = \{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\} $. The Čech complex ${\check C}^*(\mathcal{U}, \mathcal{F})$ for such a covering is simply the Čech complex of the quasi-coherent $\mathcal{O}_ T$-module $\mathcal{F}_ T$ (here we are using the assumption that $\mathcal{F}$ is locally quasi-coherent) with respect to the affine open covering $\{ T_ i \to T\} $ of the affine scheme $T$. Hence the Čech cohomology is zero by Cohomology of Schemes, Lemma 30.2.6 and 30.2.2. Thus the hypothesis of Cohomology on Sites, Lemma 21.10.9 are satisfied and we win. $\square$

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