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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.

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    $\begingroup$ The projective bundle formula gives that $H^{i}(\mathbf{P}^n,\mu^{\otimes r}_l)=\bigoplus_{0\leq k\leq n} H^{i-2k}(k,\mu^{\otimes r-k}_l)$ (note that $\mu^{\otimes r}_l$ makes sense for $r\in\mathbf Z$). $\endgroup$ Jun 3, 2014 at 20:22
  • $\begingroup$ Thanks. Is there a way to get $H^\ast({\mathbb P}^\infty_k)$ from $H^\ast({\mathbb P}^n_k)$, where $k$ is still a number field? $\endgroup$
    – us51
    Jun 4, 2014 at 19:14
  • $\begingroup$ @ Denis-Charles: How would you prove projective bundle formula for etale cohomology (over number field)? $\endgroup$
    – sms1
    Jun 6, 2014 at 2:20

1 Answer 1

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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.

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