# Completion of the stalk of a flat scheme over a complete local Noetherian ring

This is an exercise from Brian Conrad's 2006 problems on group schemes and p-divisible groups.

Let $$X$$ be a flat scheme, locally of finite type over a complete local Noetherian ring $$R$$ with residue field $$k$$. Let $$x\in X(R)$$ be a section.

If $$X$$ has smooth geometric closed fibre, prove that $$\mathcal O_{X,x_0}^\wedge$$ ($$x_0$$ the closed point of the section) is isomorphic to a formal power series ring over $$R$$ in finitely many variables.

This seems to me like a boosted form of Cohen's structure theorem, but instead of a coefficient field, it's a pretty nice ring.

Since the question is local on $$X$$, we can assume $$X = \operatorname{Spec}A$$ is affine. Then the section $$x$$ corresponds to a ring map $$A\to R$$ with kernel $$I$$ such that $$A/I\simeq R$$, and this is in fact an $$R$$-splitting, so $$A \simeq R\oplus I$$ as $$R$$-modules. Since $$R$$ is local, there is a unique maximal ideal lying over $$I$$, say $$\mathfrak m_0$$. Since $$R$$ is local, the map $$R\to A$$ is in fact faithfully flat. Since $$X_{\bar{k}}$$ is smooth, $$\mathcal O_{X_{\bar{k}},(x_0)_{\bar{k}}}^\wedge$$ is isomorphic to $$\bar{k}[[t_1,\dots,t_n]]$$ for some $$n$$. I want to show that this implies $$\mathcal O_{X,x_0}^\wedge\simeq R[[t_1,\dots,t_n]]$$.

As a module, $$A = R\oplus I$$, so it seems like the completion of $$A$$ at $$\mathfrak m_0$$ should be the sum of the completion of $$R$$ along $$\mathfrak m$$ (which is just $$R$$) and the completion of $$I$$ along $$\mathfrak m_0$$, which "feels like" a power series ring over $$R$$, but I don't know how to justify this.