In etale cohomology, for a variety $X$ defined over $\mathbb{Q}$ there is an isomorphism \begin{equation} H^n_{et}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_\ell(n)) \simeq H^n_{et}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_\ell)(n) \end{equation} which is in fact an isomorphism between representations of the absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Could anyone give a proof to it?
2026-02-22 23:29:47.1771802987
Twists of etale cohomology
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