The questions are motivated by P. Scholze's answer in MO question https://mathoverflow.net/questions/132438/why-is-faltings-almost-purity-theorem-a-purity-theorem. Consider for every $m \in \mathbb{N}$ and $p$ prime the rings $$A_m = \mathbb{Z}_p[p^{1/p^m},T^{\pm 1/p^m}]$$
Each $A_m$ is a $R_m:= \mathbb{Z}_p[p^{1/p^m}]$-algebra given by canonical inclusion $$i_m:\mathbb{Z}_p[p^{1/p^m}] \to \mathbb{Z}_p[p^{1/p^m},T^{\pm 1/p^m}]$$
We can also endow the $R_m$ with transition maps $t_{mn}: R_m \to R_n$ for $m >n$ induced by iterations of the canonical inclusions:
$$t_{m,m+1}: \mathbb{Z}_p[p^{1/p^{m}},T^{\pm 1/p^{m}}] \to \mathbb{Z}_p[p^{1/p^{m+1}},T^{\pm 1/p^{m+1}}]$$
I have some questions about the properties of $i_m$ and $t_{mn}$.
Question I Why is $i_m$ is a smooth ring map? What might be the most elegant/ conventional/standard way to show smoothness of $i_m$ in this question?
One criterion for smoothness I'm familar with is that a morphism of schemes $f: X \to Y$ is smooth iff it is flat and all geometric fibers are smooth.
That's a very general criterion working for arbitrary schemes. Here $i_m$ induce a morphism $\operatorname{Spec} \ A_m \to \operatorname{Spec} R_m$ between affine schemes, so I hope that in this case there is a more "simple" way to show smoothness without passing explicitely to the $\operatorname{Spec}$ structure and for example not to talking about geometric fibers?
In my searching I found in Stacks this: https://stacks.math.columbia.edu/tag/00T1 They are working with the concept of cotangent complex and the criterion/definition Definition 10.136.1 requiring to check quasi-isomorpism looks not really more accesible than the criterion above.
Question II: Why the transition maps $t_{m+1,m}$ are etale after inverting $p$? ie when we invert $p$ in $R_m$ or localize $R_m$ at $p$ we obtain $(R_m)_p:=\mathbb{Z}_p[p^{ \pm 1/p^m},T^{\pm 1/p^m}]$, right?
So the question boils down why is the induced Frobenius $$(t_{m,m+1})_p: (R_{m})_p \to (R_{m+1})_p$$
etale? One criterion for etaleness (that I know) is flat and unramified.
On flatness I have no idea, on ramification we have to show that the discriminant ideal becomes unit ideal after interting $p$. The proplem is I haven't here an ideal how to calculate the discriminant ideal of $R_m$.