# Grothendieck-Messing theory for finite flat group schemes

Classical Grothendieck-Messing theory relates deformations of $p$-divisible groups to lifts of the Hodge filtration (if the ideal defining the nilpotent immersion is equipped with a PD-structure). If I understand it correctly, Faltings in his article "Group schemes with strict $\mathcal{O}$-action" proves a version of this result for finite locally free group schemes $G$ (and even for his variant with $\mathcal{O}$-action, but that is not important for my question), with $M(G)$ now being a filtered perfect complex rather than a filtered locally free module. Is there another reference for this result?

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Can you mention the reason that you seek another reference, or more specifically to what do you intend to apply the theory? (For some applications there are other classification theories which are easier to use.) Or are you learning the theory for its own sake? –  BCnrd Aug 13 '10 at 16:29
If you mean the results of Illusie, these are not quite what I need, as I am really interested in deformations with extra endomorphisms, and (as far as I see) Illusie only describes isomorphism classes of lifts but not their endomorphisms. Faltings' result in contrast appears to be some kind of categorical version, also describing endomorphisms. –  Peter Scholze Aug 13 '10 at 16:56
Dear Peter: Actually, I was wondering if your ultimate interest might be finite flat groups over a Dedekind base, in which case integral $p$-adic Hodge theory gives a nice technique amenable to doing calculations. What is the source of your interest in studying the deformation theory of such things? (You're right that to do it categorically with a general kind of base ring, Faltings' approach seems to be the only one out there. But I don't know if it's so suitable for doing computations.) –  BCnrd Aug 13 '10 at 17:30
My source of interest is roughly to relate deformation spaces of $p$-divisible groups to (versal) deformation spaces of their $p^n$-torsion, as is done (without extra structure) by Illusie. Hence I need something over a general base ring in a categorical way, and would be very happy if there was a clear treatment of the theorem out there in the literature... Thanks for your help! –  Peter Scholze Aug 13 '10 at 18:34
Dear Peter: presumably you require deformations to remain truncated BT-groups (e.g., $p$-power torsion levels are flat, and if $n = 1$ then Frob-kernel is flat). Ask Oort and/or Vasiu, since they're studied the question of when certain truncated BT-groups over $W(k)$ uniquely "extend" to $p$-divisible groups (satisfying conditions I don't recall), maybe assuming $k$ algebraically closed. Deformation spaces of $p$-divisible groups can be quite accessible via G-M theory, so why do you wish to study them via deformation theory of the torsion-levels? Sounds harder that way. –  BCnrd Aug 14 '10 at 2:09

Dear Peter, I will answer to the question in the comment since it seems this is you main interest. For the algebraicity of the $p^n$-torsion points of the universal deformation, I gave a proof of this to Matthias Strauch a few years ago. He included it in his article "Deformation spaces of one-dimensional formal groups and their cohomology", this is theorem 2.3.1 (the proof as written in Matthias article is for Lubin-Tate spaces but works in general without changing anything), see his webpage. You don't need deformation theory for finite flat group schemes for this...look at the proof there's a trick (due to Artin).
For Brian, you say "Do you mean there's a BT-group over a finitely presented algebra whose pullback to completion at some point is the universal formal deformation? If so, then I find that hard to believe". But in fact this is conjecturally true ! This would follow from the non-emptiness of Newton strata in unitary PEL type Shimura varieties at a split prime $p$. For example this is known for the deformation space of a principally polarized BT group thanks to the non-emptiness of Newton strata of Siegel modular varieties (I mean you deform not the BT group but the BT group together with its principal polarization).
Dear Laurent, thanks for your answer! I am aware of the arguments in Strauch's article, but as far as I see these do use Faltings' results (if the field $F$ is not unramified over $\mathbb{Q}_p$). More specifically, can you prove a similar algebraicity result for the $p^n$-torsion, with extra structure given by a general PEL datum (not necessarily unramified)? After some struggles, I think I can do this now (the details are yet to be written), but I don't think it is obvious. –  Peter Scholze Oct 8 '10 at 18:33
I think what you mean is: if $F$ is ramified this argument does not work directly since the deformation space is not formally smooth anymore. Am I right ? But it is not clear to me what is the deformation space in this case: do you put Kottwitz condition on the tangent space as usually stated (equality of two polynomials...) ? –  Laurent F. Oct 12 '10 at 9:05
My question concerns the fact that the argument in Strauch's article makes use of the fact (due to Faltings) that the deformation space of the $p$-divisible group with extra structure is also the versal deformation space of some truncated $p$-divisible group in the specific case at hand. For general PEL data, how do you prove such a thing (Yes, use Kottwitz condition as usually stated)? Or could one avoid reference to Faltings in that argument? That for $F$ ramified, the deformation space is not smooth anymore causes additional problems, which are not my principal concern... –  Peter Scholze Oct 12 '10 at 15:55