I had a number of basic questions about $l$-adic cohomology about some examples that can be calculated.
- Is there calculations/interpretations of $l$-adic cohomology of fields? Let's say for a field as simple as the function field of the affine space.
- It is well-known that the compact support cohomology of $\mathbb{A}^n_k$ with coefficients in $\mathbb{Q}_l$ is trivial except at the top degree which is $\mathbb{Q}_l$? Is the same true for the non-compactly supported cohomology? (I think this might follow from the Poincare duality, but it seems Poincare duality works only over algebraically closed fields and one need to descend from the algebraic closure to the original field which might be possible ...)
- For a field $k$ what are the groups $H^i(\text{Spec}(k), \mu_l^{\otimes j})$? I know if $i=j$ by Bloch-Kato this is $K^M_i(k)/l$ but I am not sure about the other values. (BTW is Bloch-Kato true only when $l$ is an invertible prime or any invertible integer? I've only seen it stated for invertible primes but in the Wikipedia page it is stated for any invertible integer.)