Is completion of infinite degree extension of perfectoid fields perfectoid ? It is known that finite extension of perfectoid fields is also perftoid from tilting correspondence, but what about infinite cases ?
Infinite degree extension of perfectoid is not perfectoid because it is not always complete, but completion of it possibly be perfectoid. If you could find some counter example(infinite degree extension of perfectoid field $K$ whose completion is not perfectoid),I would be appreciated.
I'm assuming you mean infinite algebraic extensions, as otherwise there is no standard way of completing them.
Let $K$ be a perfectoid field, let $L$ be an infinite algebraic extension. Then $L$ admits a unique valuation extending that of $K$, and hence we can take the completion $\widehat L$. It is clearly complete and the valuation is nondiscrete (since the one on $K$ was). It remains to check the Frobenius is surjective on $O_{\widehat L}/p$. We have $O_{\widehat L}/p=O_L/p$. Write $L$ as a direct limit of finite extensions $L_i/K$. Then $O_L$ is the direct limit of $O_{L_i}$, and $O_L/p$ is the direct limit of $O_{L_i}/p$. Since each $L_i$ is perfectoid, the Frobenius is surjective on each $O_{L_i}/p$, and hence it is also so on the direct limit. We conclude $\widehat L$ is indeed perfectoid.
