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. 



Comparison of etale and singular cohomology for varieties over number fields
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).


