# 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 $i>2n$ 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?

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Thanks Theo for polishing this post. BTW maybe it would be useful if someone created an "étale" tag, I can't because I lack reputation. –  Jan Aug 31 '11 at 17:24
I don't yet see how this helps. We only know about the sum of all frobenius actions then. It could for example a still happen that the frobenius acts on $H^2$ by $q^{10}$ but on $H^{20}$ by $q$. –  Jan Oct 27 '11 at 18:31