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I've read a bit about Dieudonné modules, mainly from Fontaine's "Groupes p-divisibles sur les corps locaux" and Demazure's "Lectures on $p$-divisible groups". I am familiar with the main correspondences when working over a perfect field.

  1. I am trying to learn more, but a lot of references I find are about crystals. I am interested in a not-too-technical reference or answer elaborating on the relation between Dieudonné modules and crystals. As far as I understand (correct me if wrong) Dieudonne crystals over say $\text{Spec}(k)$ with $k$ a perfect field of characteristic $p$, are essentially the usual Dieudonné modules?

  2. As a related (and somewhat vague) question, if we fix a perfect ring $R$ (over $k$ or over $k=\mathbb F_p$ if you prefer), and I am interested in $p$-divisible groups over $S=\text{Spec}(R)$, then is this "closer" to Dieudonné theory over $k$ or to Dieudonné crystals over $S=\text{Spec}(R)$?

Apologies for the vagueness, but I can't find many references for Dieudonné modules over non-fields. Thanks!

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    $\begingroup$ Did you look inside Grothendieck's "Groupes de Barsotti–Tate et cristaux de Dieudonne"? He deals with perfect schemes in section IV 4. Is that what you want in question 2? $\endgroup$ Apr 3, 2017 at 10:14
  • $\begingroup$ Thanks for the reference, @PiotrAchinger. Sorry for the delay in responding - I just finished reading up to and including that section. Unfortunately, it doesn't seem to address my second question (or maybe I am misunderstanding). However, it was a nice and gentle introduction to crystals/crystalline site - thank you. $\endgroup$
    – aytio
    Apr 9, 2017 at 4:10

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