The Stacks project

Lemma 60.25.6. In the situation above, assume that $X \to S_0$ is smooth of relative dimension $d$. Then $F_{X/S_0}$ is an iterated $\alpha _ p$-cover of degree $p^ d$. Hence Lemmas 60.25.4 and 60.25.5 apply to this situation. In particular, for any crystal in quasi-coherent modules $\mathcal{G}$ on $\text{Cris}(X^{(1)}/S)$ the map

\[ F_{X/S_0}^* : H^ i(\text{Cris}(X^{(1)}/S), \mathcal{G}) \longrightarrow H^ i(\text{Cris}(X/S), F_{X/S_0, \text{cris}}^*\mathcal{G}) \]

has kernel and cokernel annihilated by $p^{d(i + 1)}$.

Proof. It suffices to prove the first statement. To see this we may assume that $X$ is étale over $\mathbf{A}^ d_{S_0}$, see Morphisms, Lemma 29.36.20. Denote $\varphi : X \to \mathbf{A}^ d_{S_0}$ this étale morphism. In this case the relative Frobenius of $X/S_0$ fits into a diagram

\[ \xymatrix{ X \ar[d] \ar[r] & X^{(1)} \ar[d] \\ \mathbf{A}^ d_{S_0} \ar[r] & \mathbf{A}^ d_{S_0} } \]

where the lower horizontal arrow is the relative frobenius morphism of $\mathbf{A}^ d_{S_0}$ over $S_0$. This is the morphism which raises all the coordinates to the $p$th power, hence it is an iterated $\alpha _ p$-cover. The proof is finished by observing that the diagram is a fibre square, see Étale Morphisms, Lemma 41.14.3. $\square$

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