Let $f:X\rightarrow Y$ a locally finitely presented map. Let $x\in X$ and $y=f(x)$.


We assume that the map on the level of fomal neighborhoods $X\_{x}\rightarrow Y\_{y}$ is formally smooth, can we find a étale neighborhood $S$ of $x$, such that $S\rightarrow Y$ is smooth at $x$.



Smoothness and smoothness over formal neighborhood
Let $k$ be a field and $Y=\mathrm{Spec}\,R$ where $R=\bigcup\_{n>0}k[[t^{1/n}]]$ is the ring of Puiseux series over $k$. Take $X=\mathrm{Spec}\,(R/tR)$, $f=$ the obvious embedding. The maximal ideal $m$ of $R$ satisfies $m=m^2$, and the same holds in $R/tR$, so both completions are equal to $k$, but of course $f$ is not smooth.


