The Stacks project

Remark 60.24.5 (Quasi-coherence). In the situation of Remark 60.24.1 assume that $S \to S'$ is quasi-compact and quasi-separated and that $X \to S_0$ is quasi-compact and quasi-separated. Then for a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$ the sheaves $R^ if_{\text{cris}, *}\mathcal{F}$ are locally quasi-coherent.

Hints: We have to show that the restrictions to $T'$ are quasi-coherent $\mathcal{O}_{T'}$-modules, where $(U', T', \delta ')$ is any object of $\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine. We use the formula (60.24.1.1), the fact that $T \to T'$ is quasi-compact and quasi-separated (as $T$ is affine over the base change of $T'$ by $S \to S'$), and Cohomology of Schemes, Lemma 30.4.5 to see that it suffices to show that the sheaves $R^ i\tau _{U/T, *}\mathcal{F}_ U$ are quasi-coherent. Note that $U \to T_0$ is also quasi-compact and quasi-separated, see Schemes, Lemmas 26.21.14 and 26.21.14.

This reduces us to proving that $R^ i\tau _{X/S, *}\mathcal{F}$ is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here $\tau _{X/S}$ is the structure morphism, see Remark 60.9.6. We may work locally on $S$, hence we may assume $S$ affine (see Lemma 60.9.5). Induction on the number of affines covering $X$ and Mayer-Vietoris (Remark 60.24.2) reduces the question to the case where $X$ is also affine (as in the proof of Cohomology of Schemes, Lemma 30.4.5). Say $X = \mathop{\mathrm{Spec}}(C)$ and $S = \mathop{\mathrm{Spec}}(A)$ so that $(A, I, \gamma )$ and $A \to C$ are as in Situation 60.5.1. Choose a polynomial algebra $P$ over $A$ and a surjection $P \to C$ as in Section 60.17. Let $(M, \nabla )$ be the module corresponding to $\mathcal{F}$, see Proposition 60.17.4. Applying Proposition 60.21.3 we see that $R\Gamma (\text{Cris}(X/S), \mathcal{F})$ is represented by $M \otimes _ D \Omega _ D^*$. Note that completion isn't necessary as $p$ is nilpotent in $A$! We have to show that this is compatible with taking principal opens in $S = \mathop{\mathrm{Spec}}(A)$. Suppose that $g \in A$. Then we conclude that similarly $R\Gamma (\text{Cris}(X_ g/S_ g), \mathcal{F})$ is computed by $M_ g \otimes _{D_ g} \Omega _{D_ g}^*$ (again this uses that $p$-adic completion isn't necessary). Hence we conclude because localization is an exact functor on $A$-modules.