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The usual theory of ($p$-typical) Witt vectors gives an equivalence of categories $ \{ \text{strict $p$-rings} \} \to \{\text{perfect }\mathbb{F}_p \text{ algebras} \}$. Here a strict $p$-ring is a $\mathbb{Z}_p$ algebras $A$ that is $p$-torsion free, $p$-adically complete and such that $A/p$ is perfect. The functor is given by $A \mapsto A/p$ and the inverse functor is given by the Witt vectors $R \mapsto W(R)$.

Now we can generalize this to the ramified case, we consider now $\mathbb{Z}[p^{1/p^n}]$-algebras $A$ such that $A/p^{1/p^n}$ is perfect (and $A$ is $p$-torsionfree and $p$-adically complete). Then there is an equivalence of categories $ \{ \text{strict $p^{1/p^n}$-rings} \} \to \{\text{perfect }\mathbb{F}_p \text{ algebras} \}$. One direction is given by $A \mapsto A/p^{1/p^n}$ and the functor in the reverse direction is given by $R \mapsto W(R) \otimes_{\mathbb{Z}_p} \mathbb{Z}[p^{1/p^n}]$.

My question is as follows: Can this be generalized to include the case $n=\infty$?. More precisely, consider the category $\mathcal{C}$ of $p$-adically complete and $p$-torsion free $\mathbb{Z}_p[p^{1/p^{\infty}}]^{\wedge}$ algebras $A$ such that $A/p^{1/p^{\infty}}$ is perfect. Is it then true that the reduction mod $p^{1/p^{\infty}}$ functor fully faithfull? The problem is defining a Teichmuller lift $[-]:A/p^{1/p^{\infty}} \to A$ (the usual construction is not well defined).

Note that if we would have a Teichmuller lift then we could construct a map $W(A/p^{1/p^{\infty}}) \hat{\otimes}_{\mathbb{Z}_p} \mathbb{Z}_p[p^{1/p^{\infty}}]^{\wedge} \to A$, which is what I am actually interested in.

PS: I am happy to assume that $A$ is in fact integral perfectoid. I know that in this case, tilting is the best way to go to perfect rings in characteristic $p$, but that doesn't seem to help me. I am trying to construct a section to the map $A/p \to A/p^{1/p^{\infty}}$, which is closely related to the above. A counterexample would be very interesting as well.

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    $\begingroup$ I won't be able help you answer this question, but may I ask where you learned these things? It sounds very interesting to me. $\endgroup$ – Lukas Heger Mar 2 '18 at 22:08
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    $\begingroup$ Two comments: First, this may be too advanced for you to get a response on MSE. You might wait a week or two and reask on Math Overflow. Second, is there any particular reason why you choose the $n$-th roots of $p$ rather than any other infinitely ramified tower? If you took the infinitely ramified cyclotomic extension, you would have something Galois... $\endgroup$ – Lubin Mar 3 '18 at 0:48
  • $\begingroup$ I do not particularly care about the specific tower of ramified extensions, as long as it gives me something perfectoid. So I guess the question could be generalized to that setting. $\endgroup$ – Pol van Hoften Mar 3 '18 at 18:06

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