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The answer is yes (in fact the result also holds over separably closed fields). You can find this statement in Corollary 7.5.21 of: Poonen - Rational points on varieties. Poonen gives a sketch of a …

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 $\mathbb{Z}$ …

An explicit simple counter-example is the following: just take the quaternion algebra $(x,y)$ over $k(x,y)$, where $k$ is a field with $\mathrm{char}(k) \neq 2$. This is non-zero on any open subset of …

Let $k$ be an algebraically closed field of characteristic $p \geq 0$. Let $X$ be a smooth Fano variety over $k$ and let $\ell \neq p$ be a prime. Is the natural morphism $\mathrm{Pic}(X) \otimes …

By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$. Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite f …

With regards to Q1, I believe one can easily prove that the Mumford-Tate conjecture holds whenever all the cohomology is generated by algebraic cycles. This occurs for example for quadric hypersurface …