Let $S=Spec(R)$ where $R$ is a Henselian local ring with fraction field $K$. Let $G$ and $G'$ be finite, flat group schemes of odd order over $S$ with isomorphic generic fibers (over $Spec(K)$). Does this isomorphism extend to one over $S$?
When $R=\mathbb{Z}_p$, the answer is yes following Fontaine's discussion in his 1975 paper 'Groupes finis commutatifs sur les vecteurs de Witt' (where he works over Witt vectors of a perfect field. Mazur uses this result to prove Theorem I.4 in his Eisenstein Ideal paper). What happens when $R$ is any general local ring (not necessarily Henselian)? Are there rings other than $\mathbb{Z}_p$ for which the answer to the above question is also a yes?
[Math] On a Theorem of Fontaine
arithmetic-geometrygroup-schemesnt.number-theory
Best Answer
No. Here is a counterexample. Let $R=\mathbf{Z}_p[\zeta_p]$, where $p$ is a prime number and $\zeta_p$ is a primitive $p$-th root of unity. Let $G=\mu_p=\mathrm{Spec}(R[x]/(x^p-1))$ and let $G'$ be the constant group scheme $\mathbf{Z}/p\mathbf{Z}$. Then $G$ and $G'$ are not isomorphic because the special fiber of $G$ is connected but that of $G'$ isn't. But there is an isomorphism $G'\to G$ over the fraction field of $R$ given by $1\mapsto \zeta_p$.
A lot more is known about these things, but not so much more by me. There are a few people around here who could probably say a lot more.