It is stated in [Caruso - An introduction to $p$-adic period rings](https://arxiv.org/abs/1908.08424) (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) ?



$p$-adic étale cohomology groups are not $\mathbb{C}_p$-admissible
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.


