I have a question for Falings' paper "crystalline cohomology and p-adic galois representation" Suppose $R$ is a ring such that there is an etale map $\mathbb{Z}_{p}[T,T^{-1}]\to R$.

By $\bar{R}$ we denote the maixmal extension of $R$ which is etale in characteristic zero.

The paper stat that the Frobenius map on $\bar{R}/p\bar{R}$ is surjective. I wonder why, is there any reference for the proof? Thank you!