Grothendieck-Messing theory for finite flat group schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:36:06Z http://mathoverflow.net/feeds/question/35502 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35502/grothendieck-messing-theory-for-finite-flat-group-schemes Grothendieck-Messing theory for finite flat group schemes Peter Scholze 2010-08-13T15:37:21Z 2010-10-04T18:04:34Z <p>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?</p> http://mathoverflow.net/questions/35502/grothendieck-messing-theory-for-finite-flat-group-schemes/41044#41044 Answer by Laurent F. for Grothendieck-Messing theory for finite flat group schemes Laurent F. 2010-10-04T18:04:34Z 2010-10-04T18:04:34Z <p>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).</p> <p>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). </p>