It is stated in Caruso  An introduction to $p$adic period rings (the remarks following equation (2)) that the $p$adic étale cohomology groups of an algebraic variety $X$ over a finite extension $K$ of $\mathbb{Q}_p$ are not $\mathbb{C}_p$admissible. In fact, $\mathbb{C}_p\otimes_{\mathbb{Q}_p}H_\text{ét}^r(X_{\overline{K}},\mathbb{Q}_p)$ is isomorphic to the graded module of $\mathbb{C}_p\otimes_K H_{\text{dR}}^r(X)$ for the de Rham filtration, but not to $\mathbb{C}_p\otimes_K H_{\text{dR}}^r(X)$ itself. Is there a reference for this result or a quick proof using a characterization of $\mathbb{C}_p$admissibility (at least to prove that they are not $\mathbb{C}_p$admissible) ?
1 Answer
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The characterization of $\mathbb C_p$admissibility in the notes is that the action of the inertia group factors through a finite quotient.
A standard calculation in étale cohomology shows that for $\mathbb P^1$, the action of the Galois group on $H^2( \mathbb P^1_{\overline{K}}, \mathbb Q_p)$ is by the inverse of the $p$adic cyclotomic character.
Since the $p$adic cyclotomic character does not have the finite inertia property, this shows that it fails for at least one variety.

$\begingroup$ This is exactly what I was looking for, thank you very much ! $\endgroup$ Commented Aug 1, 2022 at 15:37