Comparison of etale and singular cohomology for varieties over number fields

Question

Whilst reading Hartshorne's appendix C I came across the comparison theorem for etale cohomology and singular cohomology:

Let $X$ be a smooth projective variety over a number field $K$ and $\ell$ a prime number. Fix an embedding of $K$ into $\mathbb C$. Then there is a "natural" isomorphism of $\mathbb C$-vector space $$\mathrm H^n(X_{\bar K,et},\mathbb Q_\ell)\otimes \mathbb C \cong \mathrm H^n(X(\mathbb C),\mathbb C).$$

Do we have the slightly stronger "natural" isomorphism of $\mathbb Q_\ell$-vector spaces

$$\mathrm H^n(X_{\bar K,et},\mathbb Q_\ell) \cong \mathrm H^n(X(\mathbb C),\mathbb Q)\otimes \mathbb Q_\ell?$$

I would appreciate references to the literature.

Answer 1

Yes. In fact what Artin proves in SGA4 exp XI thm 4.4 is that étale cohomology and singular cohomology agree for smooth schemes over $\mathbb{C}$ with finite coefficients. The statement you want will follow from this by taking inverse limits to get to $\mathbb{Z}_\ell$ and then extending scalars to $\mathbb{Q}_\ell$. If you don't feel like looking at SGA, you can find treatments of this in the books by Freitag-Kiehl, Milne,…

Added (in response to comment). The isomorphism $H_{et}^*(X_{\bar K}, \mathbb{Q}_\ell)\cong H_{et}^*(X_{\mathbb{C}}, \mathbb{Q}_\ell)$ follows from the smooth base change theorem (cf. Milne, Etale cohomology, p 231 cor 4.3).