1
$\begingroup$

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) ?

$\endgroup$

1 Answer 1

4
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.