let $X$ be a smooth, projective and geometrically integral $k$-scheme. the Brauer group of $X$ is defined by $Br(X)=H^2_{ét}(X, \mathbb{G}_m)$.
I'm searching for a proof of this Theorem: assume that $X$ as above and $X$ is $k$-rational, ie birational equivalent to some $\mathbb{P}^n_k$. then $Br(X)=Br(k)$.
obviously the problem can be splitted in two statements:
1) $X,Y$ smooth, projective and geometrically integral $k$-schemes which are birationally equivalent to each other. then $Br(X)=Br(Y)$.
2) $Br(\mathbb{P}^n_k)=Br(k)$
could anybody sketch these proofs or give a reference? additionally: do we need for 1) and 2) really every of the smooth, projective and geometrically integral conditions or can it be weakened?
Let me spell out how the computation for (2). We first note that we have the Kummer short exact sequence, which is $$ 0\rightarrow \mu_l \rightarrow \mathbb{G}_m\rightarrow \mathbb{G}_m\rightarrow 0.$$ The induced long exact sequence on etale cohomology is then $$ H^1(\mathbb{P}_k^n,\mathbb{G}_m) \rightarrow H^2(\mathbb{P}_k^n, \mu_l) \rightarrow H^2(\mathbb{P}_k^n,\mathbb{G}_m)\rightarrow H^2(\mathbb{P}_k^n,\mathbb{G}_m).$$ Note that $H^1(\mathbb{P}_k^n,\mathbb{G}_m)=\text{Pic}(\mathbb{P}^n_k)=\mathbb{Z}$. Then we want to compute $H^2(\mathbb{P}_k^n,\mu_l)$. For this we use the Hochschild-Serre spectral sequence. This then gives us $H^i(G_k, H^j(\mathbb{P}^n_{\bar{k}} ,\mu_l))$ $\Rightarrow$ $H^{i+j}(\mathbb{P}_{\bar{k}}^n, \mu_l)$. Hence we want to compute $H^0(\mathbb{P}_{\bar{k}}^n,\mu_l)$, $H^1(\mathbb{P}_{\bar{k}}^n,\mu_l)$ and $H^2(\mathbb{P}_{\bar{k}}^n,\mu_l)$, which are $\mu_l,0$ and $\mathbb{Z}/l$ respectively. Thus we have $$0 \rightarrow H^2(G_k,\mu_l)\rightarrow H^2(\mathbb{P}_k^n,\mu_l) \rightarrow H^0(G_k,\mathbb{Z}/l)\rightarrow 0.$$ Thus $H^2(G_k,\mu_l)\cong \text{coker}(\text{Pic}(\mathbb{P}_k^n)\rightarrow H^2(\mathbb{P}_k^n,\mu_l))[l]\cong Br(\mathbb{P}_k^n)[l].$ Then we note that $H^2(G_k,\mu_l)\cong Br(k)[l]$. Since this is true for all $l$, we see that in fact $Br(k)\cong Br(\mathbb{P}_k^n)$.