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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 $x$.

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    $\begingroup$ Please give precise definitions for $X_x$ and $Y_y$. $\endgroup$
    – Laurent Moret-Bailly
    Nov 19, 2014 at 8:59
  • $\begingroup$ the completion of the local ring $\mathcal{O}_{X,x}$, same for $Y$. $\endgroup$
    – prochet
    Nov 19, 2014 at 17:27

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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.

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  • $\begingroup$ Dear Laurent. This comment is just to be sure that I understand things properly. Compared to Grothendieck's criterion in (EGA IV, 17.5.3), what is missing in the statement given by prochet is the assumption that $Y$ is locally noetherian, isn't it? $\endgroup$
    – ACL
    Nov 20, 2014 at 9:31
  • $\begingroup$ Right. Without this assumption, taking completions (of local rings) may lead to strange things. $\endgroup$
    – Laurent Moret-Bailly
    Nov 20, 2014 at 10:43
  • $\begingroup$ And meanwhile, I have realized that the same example was given by Anton Geraschenko in answer to this question. $\endgroup$
    – Laurent Moret-Bailly
    Nov 20, 2014 at 10:48
  • $\begingroup$ and can we find an example if $Y=\mathbb{A}^{\mathbb{N}}$? $\endgroup$
    – prochet
    Dec 6, 2014 at 22:52
  • $\begingroup$ @prochet: as ACL notes, there is no such example if $Y$ is locally noetherian. $\endgroup$
    – Laurent Moret-Bailly
    Dec 7, 2014 at 8:36

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