I have some questions on an answer to this MO thread presented by ulrich on the statement that for field $k$ of positive characteristic the map $Br(k) \to Br(\mathbb{P}^n)$ is an isomorphism.
we collect this exact sequences:
this commutative square we obtain by exactness of Kummer sequence $1 \to \mu_r \to \mathbb{G}_m \stackrel{r}{\to} \mathbb{G}_m \to 1$ and functorility of $H$ is induced by canonical $\mathbb{P}^n_{\overline{k}}\to \mathbb{P}^n$ (we abbreviate $\mathbb{P}^n_k:=\mathbb{P}^n)$)
$$ \require{AMScd} \begin{CD} H^1(\mathbb{P}^n,\mathbb{G}_m) @>{d}>> H^2(\mathbb{P}^n, \mu_r) @>{e} >> Br(\mathbb{P}^n)[r]=Ker(r:Br(\mathbb{P}^n) \to Br(\mathbb{P}^n)) @>{} >> 0\\ @VaVV @VbVV @VcVV @VVV\\ H^1(\mathbb{P}^n_{\overline{k}},\mathbb{G}_m) @>{d}>>H^2(\mathbb{P}^n_{\overline{k}}, \mu_r) @>{f}>> Br(\mathbb{P}^n_{\overline{k}})[r] @>{} >> 0; \end{CD} $$
$\tag{SQ}$
in addition, using Hochschild-Serre spectral sequence with $H^i(G_k, H^j(\mathbb{P}^n_{\bar{k}} ,\mu_l))$ we obtain
$$0 \to H^2(Gal(\bar{k}/k),\mu_r) \stackrel{i}{\to} H^2(\mathbb{P}_k^n, \mu_r) \to H^0(Gal(\bar{k}/k), \mathbb{Z}/r) \to 0 \tag{HS}$$
next, we observe that the composition $b \circ d$ is surjective
$$\mathbb{Z} = Pic(\mathbb{P}^n) \stackrel{d}{\to} H^2(\mathbb{P}^n, \mu_r) \stackrel{b}{\to} H^2(\mathbb{P}_{\bar{k}}^n, \mu_r) = \mathbb{Z}/r \tag{S}$$
I have two questions:
1) why is the last composition (S) surjective
2) assume that we now that 1) is true, ie the last map is surjective, how we concretly can conclude from this and the other sequences (SQ), (HS) and (S) above that $e \circ i:Br(k) \to Br(\mathbb{P}^n)$ is an isomorphism?
from (S) and (SQ) we conclude that the composition $f \circ b \circ d$ is surjective. what do we know about the vertical map $c$? could somebody explain in detail why the desired map is an isomorphism?
the background of that all is this former question where I asked about: ulrich's answer occured in a link and the given answer by curious math guy seems to use the same argument, which I still not understand.