Let $R$ be a ring of characteristic $p$. Let $G$ be the kernel of the natural map $\pi\_1^{et} (\operatorname{Spec} R[[x]] [1/x]) \to \pi\_1^{et}( \operatorname{Spec} R)$. $G$ has a natural map to $\prod\_{l\neq p} \mathbb Z\_l(1)$, coming from the etale coverings adjoining the $n$th roots of $x$ for $n$ prime to $p$. Is the kernel of this natural map always a pro-$p$ group?


For $R$ a field, this is standard. But I'm not sure when $R$ is not a field.



Etale fundamental group of punctured formal neighborhood
The answer is no.


Take $R=\mathbb F\_2[y]$ and consider the cover defined by the equation $t^3+yt+x=0$. By computing the discriminant, this cover is etale. The inertia group of this cover is a subgroup of $S\_3$. By reducing mod $(y)$, it contains a $3$-cycle. By reducing mod $y-1+x^2$, the equation factors into $(t^2-xt+1)(t+x)$, so the inertia group contains a $2$-cycle. Thus the inertia group is full $S\_3$.


So $S\_3$ is a quotient of $G$. But clearly if $G$ has the property I describe, every finite quotient of it must have a normal $2$-Sylow subgroup, which $S\_3$ does not.


