
> 
> Let $X$ be a smooth variety of a field $k$. Then is 
>  $$H\_{et}^i(X, \mathbb{Q}) = 0$$
>  for all $i > 0$?
> 
> 
> 


* The result is true for $i=1$. This follows from the same argument given for $\mathbb{Z}$-coefficients given here: [Etale cohomology with coefficients in the integers](https://mathoverflow.net/questions/84414/etale-cohomology-with-coefficients-in-the-integers). Note that one needs to at the very least assume that $X$ is normal; otherwise $H\_{et}^1(X, \mathbb{Q})$ can be non-trivial.
* The result is true if $X = \mathrm{Spec}(k)$. This follows from standard properties of Galois cohomology. Note that the analogous result is not true with $\mathbb{Z}$-coefficients; $H^2(k, \mathbb{Z})$ is often non-trivial.
* If it helps, I mostly care about the case $i=2$.



Etale cohomology with coefficients in $\mathbb{Q}$
The following is surely expressing whatever is in the core non-formal aspect of Joe Berner's answer (which is above my pay grade); it is offered as an alternative version of the same ideas.


Let $X$ be a normal noetherian scheme. We'll show the higher etale cohomology with coefficients in any flat $\mathbf{Z}$-module $M$ is torsion, and hence vanishes when $M$ is a $\mathbf{Q}$-vector space. Via the exact sequence
$$0 \rightarrow M \rightarrow \mathbf{Q} \otimes\_{\mathbf{Z}} M \rightarrow T \rightarrow 0$$
with $T$ visibly torsion, we just have to show the higher etale cohomology of the middle term vanishes; i.e., we are back to the original question of vanishing of higher etale cohomology with coefficients in a $\mathbf{Q}$-vector space $V$.


We can assume $X$ is connected, so it is irreducible (as $X$ is noetherian and normal); let $\eta$ be its unique generic point. For any quasi-compact etale $E \rightarrow X$, the scheme $E$ is normal and noetherian, so its connected components are all *irreducible* and as such have their unique generic point over $\eta$ (due to flatness of $E$ over $X$). Thus, $E \mapsto E\_{\eta}$ does not lose any information about connected components, so for the natural map $j:\eta \rightarrow X$ we see that the natural map of etale sheaves $V\_X \rightarrow j\_{\ast}(V\_{\eta})$ induces an isomorphism upon evaluating on any such $E$, whence it is an isomorphism of sheaves. Thus, our task is the same as proving the vanishing of the higher etale cohomology on $X$ of $j\_{\ast}(V\_{\eta})$. 


Consider the Leray spectral sequence
$$E\_2^{n,m} = {\rm{H}}^n(X, {\rm{R}}^mj\_{\ast}(V\_{\eta})) \Rightarrow {\rm{H}}^{n+m}(\eta,V\_{\eta}).$$
The sheaf ${\rm{R}}^m j\_{\ast}(V\_{\eta}$ is the etale sheafification of the presheaf on $X\_{\rm{et}}$ given by $U \mapsto {\rm{H}}^m(U\_{\eta},V)$ (with quasi-compact etale $U$ over $X$). The latter vanishes for $m>0$ since it is Galois cohomology (for the finitely many points of $U\_{\eta}$) and $V$ is a $\mathbf{Q}$-vector space. Hence, $E\_2^{n,m}=0$ for $m>0$, so
$${\rm{H}}^n(X, j\_{\ast}(V\_{\eta})) \simeq {\rm{H}}^n(\eta,V\_{\eta})$$
for all $n \ge 0$. But the right side vanishes for $n>0$ for the same Galois-cohomological reasons, so we are done.


