TITLE: Etale cohomology with coefficients in $\mathbb{Q}$
QUESTION [9 upvotes]: 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. 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 $i=2$.

REPLY [2 votes]: You can find this for normal schemes in Deninger, A proper base change theorem for non-torsion sheaves in étale cohomology, Journal of Pure and Applied Algebra 50 (1988), 231–235. http://www.sciencedirect.com/science/article/pii/0022404988901028 (2.1) (proof by reduction to Galois cohomology using the Leray spectral sequence)