# On a Theorem of Fontaine

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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The answer is: this is true if $R$ is a complete DVR in mixed characteristic with absolute ramification index $e<p-1$ (where $p$ is the residue characteristic). This is the main result of Raynaud's (p,...,p) paper. –  Keerthi Madapusi Pera May 4 '11 at 3:11
Thank you for your response. I'm not familiar with Raynaud's main result, but I'm guessing from the title that he considers $p$-primary group schemes over such $R$? –  Saikat Biswas May 4 '11 at 3:37
Yes, he's considering finite flat group schemes over such $R$. What he shows is that, with this restriction on ramification, the functor sending a finite flat group scheme over $R$ to its generic fiber is fully faithful. When the order is prime-to-$p$, this is easy, because then the group scheme is necessarily etale. So the content of the result is for the $p$-primary part. Here's a link to the paper: archive.numdam.org/article/BSMF_1974__102__241_0.pdf –  Keerthi Madapusi Pera May 4 '11 at 4:03
The precise result is Corollaire 3.3.6. –  Keerthi Madapusi Pera May 4 '11 at 4:12
Splendid. Thank you. –  Saikat Biswas May 4 '11 at 4:13

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.

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Thank you for the nice counterexample. It helped clear a few other questions too that I had in mind. However, after posting my question, I realized that what I was really looking for are rings $R$ (not just those that are henselian local) for which Fontaine's result might hold. Even more specifically, can we expect it to hold if $R$ is the completion of the ring of integers of a number field at a prime ideal (i.e. a DVR)? –  Saikat Biswas May 3 '11 at 23:53
The last line of the question has been modified. –  Saikat Biswas May 4 '11 at 1:59
The $\mathbf{Z}_p[\zeta_p]$ is the completion of the ring of integers of $\mathbf{Q}(\zeta_p)$ at the unique prime ideal lying over $p$. So the example above is also a counterexample to modified question in your first comment. Regarding the edit to the original question, I think there's some confusion. I gave a counterexample to your question, and now you're asking the same question but with weaker hypotheses. So my example is still a counterexample. –  JBorger May 4 '11 at 2:13
My apologies. I meant the last line of my original posted question. –  Saikat Biswas May 4 '11 at 2:42

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 $e< p-1$).

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