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Let $X$ be a smooth and proper scheme over $Spec(\mathbb{Z}_p)$. Let $l$ be a prime number coprime to $p$. Then the proper base change theorem gives me an isomorphism $$H^r_{et}(X\times_{\mathbb{Z}_p}\bar{\mathbb{Q}}_p,\mathbb{Z}/l)\cong H^r_{et}(X\times_{\mathbb{Z}_p}\bar{\mathbb{F}}_p,\mathbb{Z}/l)$$

The left-hand side of this isomorphism has an action of $G_{\mathbb{Q}_p}$, the absolute Galois group of $\mathbb{Q}_p$ while the right-hand side has an action of $G_{\mathbb{F}_p}$ the absolute Galois group of $\mathbb{F}_p$. I am wondering how these two actions are related through the above isomorphism.

The most obvious thing one can do is to say that this isomorphism if $G_{\mathbb{Q}_p}$-equivariant if I give the right-hand side the $G_{\mathbb{Q}_p}$-action coming from the surjective morphism $G_{\mathbb{Q}_p}\to G_{\mathbb{F}_p}$ (which comes from identifying $G_{\mathbb{F}_p}$ with the Galois group of the maximal unramified extension of $\mathbb{Q}_p$).

This is probably overly naive but maybe this is at least true in some cases. Anyway the literature on Galois action on etale cohomology groups is extremely vast and I am unable to find an answer to such an elementary question. I would be grateful to anyone either answering this question or suggesting a readable reference that discusses this kind of questions.

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  • $\begingroup$ Your guess is completely correct (and it works with $\mathbf{Z}_\ell$ or $\mathbf{Q}_\ell$ coefficients too). $\endgroup$ – David Loeffler May 13 '16 at 6:33
  • $\begingroup$ Thanks a lot David ! Do you know a reference for this ? It's probably obvious if one really understands the proper base change theorem. Anyway, if you can provide a reference, I'd be happy to accept this as an answer. $\endgroup$ – Geoffroy May 13 '16 at 10:43
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Here is an answer that's basically transcribed from a comment by Keerthi Madapusi Pera on this MO question of mine (where I was asking what happens when the properness assumption is removed).

Let $f: X \to \operatorname{Spec} \mathbf{Z}_p$ be the structure map. Then we can form the sheaf $R^i f_* \mathbf{Z}_\ell$ on $\operatorname{Spec} \mathbf{Z}_p$, and since $f$ is a smooth proper map by hypothesis, the proper base-change theorem tells us that $R^i f_* \mathbf{Z}_\ell$ is a locally constant sheaf. The fibre of this sheaf at a generic point $\overline{x}$ is $H^i(X_{\overline{\mathbf{Q}}_p}, \mathbf{Z}_\ell)$; and since it is locally constant, the action of Galois on the generic fibre must factor through $\pi_1(\operatorname{Spec} \mathbf{Z}_p, \overline{x})$, which is the Galois group of the maximal unramified extension of $\mathbf{Q}_p$.

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