In [1], Propisition 6.1.9(2), it said that if $R$ is a perfectoid ring such that $pR^\circ$ is closed in $R^\circ$ (this includes the case if $R$ is of character $p$, or if $p$ is invertible in $R$, in particular if $R$ is a perfectoid $K$-algebra, $K$ a field), then the Frobenius $\Phi:R^\circ/p\to R^\circ/p$ is surjective (it is easy to see that this is equivalent to $R^{\flat\circ}\to R^\circ/p$ is surjective because $R^{\flat\circ}=\varprojlim R^\circ/p$). In [1], Remark 6.1.10 it said that probably $pR^\circ$ is not closed in general.


However, in the proof of [1], Lemma 6.2.7, it makes the extensive use of the surjectivity of thie map $R^{\flat\circ}\to R^\circ/p$ (say, by choosing $R^+$ to be $R^\circ$), ~~and all subsequent development of the theory relies on this lemma~~.


So my problem is: is this map surjective in general (i.e. when $pR^\circ$ is not closed)? I noticed that in the video of Scholze's fall 2014 course he said that this is surjective without giving details. Is it wrong, or still open, or I missed something obvious? Thanks in advance.


[1] Peter Scholze's lectures on $p$-adic geometry, fall 2014



Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$?
[Probably this question is no longer interesting to the author. But since I faced the same problem while trying to learn basics of perfectoid spaces I decided to write down an argument here]


We start with a perfectoid ring $R$ and a pseudo-uniformizer $\varpi$ s.t. $\varphi:R^{\circ}/\varpi \to R^{\circ}/\varpi^p$ is surjective. This means that for any element $x\in R^{\circ}$ there is a pair of power-bounded elements $x\_0, y\_1$ s.t. $x=x\_0^p+\varpi^py\_1$. Apply the same procedure to $y\_1$ to get $x\_1,y\_2\in R^{\circ}$ s.t. 
$$
x=x\_0^p+\varpi^p(x\_1^p+\varpi^py\_2)=x\_0^p+\varpi^px\_1^p+\varpi^{2p}y\_2.
$$


Keep going to obtain $x\_1,\dots, x\_n, y\_{n+1}\in R^{\circ}$ s.t.
$$
x=x\_0^p+\varpi^px\_1^p+\varpi^{2p}x\_2^p+\dots+\varpi^{np}x\_n^p+\varpi^{(n+1)p}y\_{n+1}.
$$
This implies that 
$$
x-(x\_0^p+\varpi^px\_1^p+\varpi^{2p}x\_2^p+\dots+\varpi^{np}x\_n^p)\in \varpi^{(n+1)p}R^{\circ}.
$$
Now note that since $R$ is a uniform Tate ring, so the induced topology on $R^{\circ}$ is just $\varpi$-adic topology and $R^{\circ}$ is complete with respect to this topology. Therefore, we can conclude that
$$
x=\sum\_{i=0}^{\infty}\varpi^{ip}x\_i^p.
$$
The last thing to observe is that 
$$
x-(\sum\_{i=0}^{\infty} \varpi^{i}x\_i)^p= \sum\_M \varpi^M \Big(\sum\_N \sum\_{k\_0+\dots k\_N=p, k\_N\geq 1, k\_n\neq p, k\_1+2k\_2+\dots+Nk\_N=M} \prod\_{t=1}^N \frac{p!}{k\_0!\dots k\_N!}x\_t^{k\_t}\Big)\in pR^{\circ}.
$$
So we are done.


[To rigorously prove that this sum makes sense and lies inside $pR^{\circ}$ note that each sum in parentheses is actually finite and each of $\frac{p!}{k\_0!\dots k\_N!}$ is divisible by $p$ in $\mathbf Z$ (under the assumption that $k\_N\geq 1$ and $k\_N\neq p$). Let's say that $\frac{p!}{k\_0!\dots k\_N!}=pc\_{k\_0,\dots, k\_N}$ for some integer $c\_{k\_0,\dots, k\_N}$. Then we have 
$$
\sum\_M \varpi^M \Big(\sum\_N \sum\_{k\_0+\dots k\_N=p, k\_N\geq 1, k\_n\neq p, k\_1+2k\_2+\dots+Nk\_N=M} \prod\_{t=1}^N \frac{p!}{k\_0!\dots k\_N!}x\_t^{k\_t}\Big)=
$$
$$
x-(\sum\_{i=0}^{\infty} \varpi^{i}x\_i)^p=p\bigg(\sum\_M \varpi^M \Big(\sum\_N \sum\_{k\_0+\dots k\_N=p, k\_N\geq 1, k\_n\neq p, k\_1+2k\_2+\dots+Nk\_N=M} \prod\_{t=1}^N c\_{k\_0,\dots, k\_N}x\_t^{k\_t}\Big)\bigg) \in pR^{\circ}.]
$$


