Suppose $k$ is a number field. I want to compute $H^\ast({\mathbb P}^n\_k,\mu\_l^{\otimes r})$ where $l,r\in {\mathbb N}$. I know that Milne has some computations, but he assumes throughout that his field is ${\mathbb C}$. I want to know if it differs when $k$ is number field.


Disclaimer: This is not a homework problem. However, I am just a starter when it comes to the etale world. Any hints/comments will be appreciated. Thanks.



etale cohomology of ${\mathbb P}^n_k$
Let $G:=\mathrm{Gal}(\bar{k}/k)$. There is a Hochschild-Serre spectral sequence
$$E^{pq}\_2=H^p(G, H^q(\mathbb{P}^n\_{\bar{k}}\,,\mu \_l^{\otimes r})\ \Rightarrow\ H^\*(\mathbb{P}^n\_{k}\,,\mu \_l^{\otimes r})\ .$$
The cohomology of $\mathbb{P}^n\_{\bar{k}}$ is well known, but the Galois cohomology is rather difficult to control, already for $H^1$ -- this is essentially class field theory.


