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It's well-known, due to work of Fontaine (see e.g. this), that if $k$ is a perfect field of characteristic $p$, then one can classify $p$-divisible groups over $W(k)$ by Honda systems. Namely, these are pairs $(L,D)$ where $D$ is a Dieudonne-module over $k$ (free as a $W(k)$-module) and a sub $W(k)$-module $L$ such that composition $L\to W(k)/FW(k)$ induces an isomorphism $L/pL\to W(k)/FW(k)$.

I have always interpreted this in the following way. A $p$-divisible group $G$ over $\text{Spec}(W(k))$ is (by algebraizability) a $p$-divisible group over $\text{Spf}(W(k))$. One can interpret such data as a $p$-divisible group $G_0$ over $k$ and, thanks to Grothendieck-Messing, a submodule $L_n\subseteq D(G_0)(W_n(k))$ for each $n$, where $L_n$ is required to be free, and to be 'admissible' (in the sense that $L_n/p L_n$ is supposed to be $\omega_{G_0^\vee}$ (note that I'm using, to be consistent with Messing, covariant Dieudonne theory). Of course such data needs to be compatible in $n$. It seems to me that one recovers precisely the notion of Honda systems by taking this system and taking the limit. Thus, one seems to recover the classification of $p$-divisible groups over $W(k)$ from Grothendieck-Messing.

But, as far as I can tell, the approach that people usually take to the equality between Honda systems and $p$-divisible groups is via (the admittedly more concrete) approach of Fontaine using the logarithm map.

It seems strange to me that if one can recover this interpretation of the Honda system classification of $p$-divisible groups over $W(k)$ that it wouldn't be mentioned in the above article of B. Conrad. Also brief searches in google with the key words "Grothendieck-Messing" and "Honda systems" turn up nought.

Is there some subtlety that I'm missing, or does the above actually explain the classification of $p$-divisible groups over $W(k)$ by Honda systems.

Thanks!

EDIT: User nfdc23 states below that one does obtain this classification of $p$-divisible groups over $W(k)$, but it is not so obvious that it mathces, on the nose, Fontaine's classification. Namely, if $L\subseteq D(G_0)(W(k))$ is the admissible submodule from Grothendieck-Messing is it true that $L$ is the kernel logarithm map from Fontaine's work?

I will think on this, but if anyone has a quick answer, I'd love to hear it!

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    $\begingroup$ The Honda system formalism for (commutative, $p$-power order) finite flat group schemes $G$ over $W(k)$ and Dieudonne theory for $G_k$ set up from scratch by Fontaine in his Asterisque book can sometimes (as in the link you gave) be a simpler self-contained way to proceed than via G-M theory for $p$-divisible groups (+ Mazur-Messing, combined with Raynaud's theorem in [BBM, 3.3.1]). And perhaps B. Conrad (a graduate student when that link was written) didn't know G-M theory at that time. So doesn't seem strange that G-M isn't mentioned in that link. It depends on whether one needs agreement. $\endgroup$
    – nfdc23
    Commented Oct 8, 2017 at 20:09
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    $\begingroup$ @nfdc23 Dear nfdc23, that makes a lot of sense. Can you clarify what you mean by 'needs agreement'? Also, I hadn't realized that this was written by B. Conrad as a graduate student--the fact that B. Conrad didn't know Grothendieck-Messing at the time didn't even occur to me--I assumed that he was born knowing everything about $p$-divisible groups. :P I was more confused because not only was it not in that article, but I couldn't find a single reference that says you can deduce Honda theory from GM--I thought I might be missing some major subtlety. Thanks again! Best, Alex. $\endgroup$
    – SomeGuy
    Commented Oct 8, 2017 at 20:20
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    $\begingroup$ With these matters, it can be such a pain to match up two very distant-looking constructions that if one doesn't truly need the consistency (i.e., maybe one can get by purely from the G-M side abstractly, or from the Honda system side concretely, depending on the intended purpose) then it can be rather unappetizing to try to cook up a proof of agreement (maybe up to some Frobenius twist or duality or whatever). So unless one really needs to know the agreement, what is the incentive to sweat out making a proof? (Sure, it is nice to know...and even nicer if someone else grinds out that task.) $\endgroup$
    – nfdc23
    Commented Oct 8, 2017 at 20:30
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    $\begingroup$ Anyway, one has shown that G-M theory meshes well with classical Dieudonne theory (up to a Frobenius twist), which is the main conclusion of the Mazur-Messing book, and one knows how classical Dieudonne theory encodes the tangent space (all done in Fontaine's book), one does recover as you say a classification looking just like Honda systems. But does it really match Fontaine's log-style construction of $L$ on the nose (in terms of the submodule of $D(G_k)$ associated to $G$)? Perhaps for all purposes it doesn't matter, so nobody has the incentive to dig far down to check. $\endgroup$
    – nfdc23
    Commented Oct 8, 2017 at 20:37
  • $\begingroup$ @nfdc23 Dear nfcd23, I see. That makes a lot of sense. I guess I, somewhat naively, had the thought process "if the two classifications look the same, they're the same classification" which, of course, doesn't hold any real water. Do you think it's worth thinking about comparing the two? To be honest, my immediate motivation was that I was giving a talk on Breuil and Kisin's work, and wanted to explain what happens in the unramified case. I kind of wanted to just unify the classification in the literature (i.e. Fontaine's work for urnamified and Conrad's for low ramification) but realized $\endgroup$
    – SomeGuy
    Commented Oct 8, 2017 at 20:46

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