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. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 60.25: Pulling back along purely inseparable maps

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07QB. Beware of the difference between the letter 'O' and the digit '0'.