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.
