Consider a scheme $X$ over $\mathbb{Q}$. By the base change theorem for the étale cohomology, we have $$H^i_{ét}(X_{\bar{\mathbb{Q}}},\mathbb{Z}/\ell^n)\cong H^i_{ét}(X_{\mathbb{C}_p},\mathbb{Z}/\ell^n)$$ in a natural way and thus passing to the limit over $n$, we get an isomorphism of $\mathbb{Z}_\ell$-modues $$H^i_{ét}(X_{\bar{\mathbb{Q}}},\mathbb{Z}_\ell)\cong H^i_{ét}(X_{\mathbb{C}_p},\mathbb{Z}_\ell)$$ which yields after tensoring to $$(*) \quad H^i_{ét}(X_{\bar{\mathbb{Q}}},\mathbb{Q}_\ell)\cong H^i_{ét}(X_{\mathbb{C}_p},\mathbb{Q}_\ell).$$ On the vector space on the left hand side, there is a natural action of the absolute galois group $G_{\mathbb{Q}}$, where as on the righthand side the local absolute galois group acts, i.e. $G_{\mathbb{Q}_p}$. Since we have inclusion $G_{\mathbb{Q}_p}\rightarrow G_{\mathbb{Q}}$, it seems natural to me that $G_{\mathbb{Q}_p}$ acts on $H^i_{ét}(X_{\bar{\mathbb{Q}}},\mathbb{Q}_\ell)$. Does $G_{\mathbb{Q}}$ act on $H^i_{ét}(X_{\mathbb{C}_p},\mathbb{Q}_\ell)$ making $(*)$ an isomorphism of $G_{\mathbb{Q}}$-modules compatible with the $G_{\mathbb{Q}_p}$-action?



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