TITLE: On a Theorem of Fontaine
QUESTION [5 upvotes]: 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?

REPLY [7 votes]: The answer to the edited question is 'yes' if $R$ is a complete DVR in mixed characteristic with absolute ramification index $e< p-1$ (where $p$ is the residue characteristic). This is one of the main results of Raynaud's (p,...,p) paper. He shows in Corollaire 3.3.6 that the functor sending a finite flat group scheme of $p$-power order over R to its generic fiber is fully faithful (when $e< p-1$).