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?
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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 such local ring (not necessarily Henselian)? |
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On a Theorem of FontaineLet $S=Spec(R)$ 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 such ring?
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