Let $f: X\to S$ be a smooth morphism of schemes. Let $p$ be a section. Let $\hat{\mathscr{O}}_{X, p(S)}$ be the completion $X$ along $p(S)$. Then I think for every point $s\in S$, there exists an open neighborhood $ U=\operatorname{Spec} R$ of $s$, such that $\hat{\mathscr{O}}_{X, p(S)} (U)$ is a formal power series over $R$. How it should be proved? Or where can I find the reference? Thanks.
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7$\begingroup$ EGA IV$_4$, 16.9.9 and 17.2.1(c') (with $Y=S$ in the latter; i.e., $h = {\rm{id}}_S$ there). It's ultimately a quite concrete thing, building on the structure of $I/I^2$ that can be read off from the case of the zero-section of affine space over which the given situation is suitably etale. $\endgroup$– nfdc23Commented Apr 7, 2018 at 18:00
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$\begingroup$ nfdc23: Thank you. To apply 16.9.9, we need to know the section $p: S\to X$ is quasi-regular, it doesn't seem we can apply 17.2.1 to get it. Actually 17.2.1(c') doesn't exist. $\endgroup$– JJHCommented Apr 9, 2018 at 17:02
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1$\begingroup$ Oops, I made a typo: I should have written 17.12.1 rather than 17.2.1 (and part (c') there exists and addresses exactly the quasi-regular condition). $\endgroup$– nfdc23Commented Apr 10, 2018 at 2:23
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