I have a question about Faltings' paper "Crystalline cohomology and p-adic Galois representations". Suppose $R$ is a smooth $\mathbb{Z}_p$-algebra, of relative dimension $1$, such that there is an etale map $\mathbb{Z}_{p}[T,T^{-1}]\to R$.

By $\bar{R}$ we denote the maximal extension of $R$ which is etale in characteristic zero. That is if $R$ is geometrically integral we take the maximal field extension of its fraction-field such that the normalisation of $R[1/p]$ in this field is unramified over $R[1/p]$. Then $\bar{R}$ is the normalisation of R in this field. In general $R\otimes_{\mathbb{Z}_p}\bar{\mathbb{Q}}_p $ is a product of integral domains, and $\bar{R}$ is the product of the corresponding normalisations.

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

doesprovide an etale extension over $\mathbf{Q}_p$. So you give inconsistent descriptions of what you want. Why not focus on $R$ a domain, since your initial $R$ of interest is a product of domains anyway, so the domain case should be all that matters. $\endgroup$11more comments