Étale cohomology of projective space I have some very basic question about étale cohomology. 
Namely I would like to compute the étale cohomology of of the projective space over the algebraic closure of $\mathbb F_q$ along with its Frobenius operation:

$$H^i(\mathbb P^n_{\mathbb F},\mathbb Z /l)$$

I would expect that it vanishes for $i&gt;2n$ or odd $i$ and is $\mathbb Z/l$ with Frobenius operation by multiplication by $q^{i/2}$ otherwise.
Using the Gysin sequence I can check, that the cohomology groups look as expected, however I don't know how to compute the operation of the Frobenius. 

So my questions are:

How does one compute the Frobenius operation on cohomology in this example?

What are general techniques to compute Frobenius action on $l$-adic or étale cohomology?
You could count points and use the Lefschetz fixed point formula.
You could count points and use the Lefschetz fixed point formula.