Let $K=\mathbf{Q}_p(\zeta_{p^{\infty}})$ and $C:=\widehat{\overline{\mathbf{Q}_P}}$. When I wrote an answer for a question a question answer, I have in my mind that
"$\widehat{K}$ is a perfectoid fields will imply for example $H^i(G_K, O_C)$ is almost zero for $i>0$"
as a result of some "almost purity for for perfectoid fields".
I then searched and find for example a lecture notes, putting Corollary 185 as a consequence of Theorem 183. I am fine to use Corollary 185 to conclude (as in my answer linked). But I didn't see how can I directly use the fact "$\widehat{K}$ is a perfectoid field + almost purity for for perfectoid fields" to conclude. More precisely,
Question: in a lecture notes, how does Theorem 183 imply Corollary 185?
Somehow Theorem 183 tells that "$L/\widehat{K}$ is almost finite etale", but in Corollary 185 one needs somehow "$L/K$ is almost finite etale."
Or similar Question: Would "$L/\widehat{K}$ being almost finite etale for any finite extension $L/\widehat{K}$" imply (removing the completion) "$M/K$ is almost finite etale for any finite extension $M/K$"?
Thank you for remarks, perhaps this is stupid and the answer is very direct.