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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1$\begingroup$ 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. $\endgroup$– Keerthi MadapusiCommented May 4, 2011 at 3:11
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$\begingroup$ 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 $R$? $\endgroup$– Saikat BiswasCommented May 4, 2011 at 3:37
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1$\begingroup$ 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 $\endgroup$– Keerthi MadapusiCommented May 4, 2011 at 4:03
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$\begingroup$ The precise result is Corollaire 3.3.6. $\endgroup$– Keerthi MadapusiCommented May 4, 2011 at 4:12
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$\begingroup$ Is it possible for you to move your responses to the 'Answer' section? $\endgroup$– Saikat BiswasCommented May 4, 2011 at 4:19
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2 Answers
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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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$\begingroup$ 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)? $\endgroup$ Commented May 3, 2011 at 23:53
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$\begingroup$ The last line of the question has been modified. $\endgroup$ Commented May 4, 2011 at 1:59
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$\mathbf{Z}_p[\zeta_p]$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. $\endgroup$– JBorgerCommented May 4, 2011 at 2:13 -
$\begingroup$ My apologies. I meant the last line of my original posted question. $\endgroup$ Commented May 4, 2011 at 2:42
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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 $e< p-1$).
answered May 4, 2011 at 4:40