6
$\begingroup$

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?

$\endgroup$
5
  • 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. $\endgroup$ May 4, 2011 at 3:11
  • $\begingroup$ 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 $R$? $\endgroup$ May 4, 2011 at 3:37
  • 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 $\endgroup$ May 4, 2011 at 4:03
  • $\begingroup$ The precise result is Corollaire 3.3.6. $\endgroup$ May 4, 2011 at 4:12
  • $\begingroup$ Is it possible for you to move your responses to the 'Answer' section? $\endgroup$ May 4, 2011 at 4:19

2 Answers 2

12
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ 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)? $\endgroup$ May 3, 2011 at 23:53
  • $\begingroup$ The last line of the question has been modified. $\endgroup$ May 4, 2011 at 1:59
  • $\begingroup$ 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. $\endgroup$
    – JBorger
    May 4, 2011 at 2:13
  • $\begingroup$ My apologies. I meant the last line of my original posted question. $\endgroup$ May 4, 2011 at 2:42
7
$\begingroup$

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 $e< p-1$).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.