p-Divisible Groups Revisited 1

I’ve posted about {p}-divisible groups all over the place over the past few years (see: here, here, and here). I’ll just do a quick recap here on the “classical setting” to remind you of what we know so far. This will kick-start a series on some more subtle aspects I’d like to discuss which are kind of scary at first.

Suppose {G} is a {p}-divisible group over {k}, a perfect field of characteristic {p>0}. We can be extremely explicit in classifying all such objects. Recall that {G} is just an injective limit of group schemes {G=\varinjlim G_\nu} where we have an exact sequence {0\rightarrow G_\nu \rightarrow G_{\nu+1}\stackrel{p^\nu}{\rightarrow} G_{\nu+1}} and there is a fixed integer {h} such that group schemes {G_{\nu}} are finite of rank {p^{\nu h}}.

As a corollary to the standard connected-étale sequence for group schemes we get a canonical decomposition called the connected-étale sequence:

\displaystyle 0\rightarrow G^0 \rightarrow G \rightarrow G^{et} \rightarrow 0

where {G^0} is connected and {G^{et}} is étale. Since {k} was assumed to be perfect, this sequence actually splits. Thus {G} is a semi-direct product of an étale {p}-divisible group and a connected {p}-divisible group. If you’ve seen the theory for finite, flat group schemes, then you’ll know that we usually decompose these two categories even further so that we get a piece that is connected with connected dual, connected with étale dual, étale with connected dual, and étale with étale dual.

The standard examples to keep in mind for these four categories are {\alpha_p}, {\mu_p}, {\mathbb{Z}/p}, and {\mathbb{Z}/\ell} for {\ell\neq p} respectively. When we restrict ourselves to {p}-divisible groups the last category can’t appear in the decomposition of {G_\nu} (since étale things are dimension 0, if something and its dual are both étale, then it would have to have height 0). I think it is not a priori clear, but the four category decomposition is a direct sum decomposition, and hence in this case we get that {G\simeq G^0\oplus G^{et}} giving us a really clear idea of what these things look like.

As usual we can describe étale group schemes in a nice way because they are just constant after base change. Thus the functor {G^{et}\mapsto G^{et}(\overline{k})} is an equivalence of categories between étale {p}-divisible groups and the category of inverse systems of {Gal(\overline{k}/k)}-sets of order {p^{\nu h}}. Thus, after sufficient base change, we get an abstract isomorphism with the constant group scheme {\prod \mathbb{Q}_p/\mathbb{Z}_p} for some product (for the {p}-divisible group case it will be a finite direct sum).

All we have left now is to describe the possibilities for {G^0}, but this is a classical result as well. There is an equivalence of categories between the category of divisible, commutative, formal Lie groups and connected {p}-divisible groups given simply by taking the colimit of the {p^n}-torsion {A\mapsto \varinjlim A[p^n]}. The canonical example to keep in mind is {\varinjlim \mathbb{G}_m[p^n]=\mu_{p^\infty}}. This is connected only because in characteristic {p} we have {(x^p-1)=(x-1)^p}, so {\mu_{p^n}=Spec(k[x]/(x-1)^{p^n})}. In any other characteristic this group scheme would be étale and totally disconnected.

This brings us to the first subtlety which can cause a lot of confusion because of the abuse of notation. A few times ago we talked about the fact that {E[p]} for an elliptic curve was either {\mathbb{Z}/p} or {0} depending on whether or not it was ordinary or supersingular (respectively). It is dangerous to write this, because here we mean {E} as a group (really {E(\overline{k})}) and {E[p]} the {p}-torsion in this group.

When talking about the {p}-divisible group {E[p^\infty]=\varinjlim E[p^n]} we are referring to {E/k} as a group scheme and {E[p^n]} as the (always!) non-trivial, finite, flat group scheme which is the kernel of the isogeny {p^n: E\rightarrow E}. The first way kills off the infinitesimal part so that we are just left with some nice reduced thing, and that’s why we can get {0}, because for a supersingular elliptic curve the group scheme {E[p^n]} is purely infinitesimal, i.e. has trivial étale part.

Recall also that we pointed out that {E[p]\simeq \mathbb{Z}/p} for an ordinary elliptic curve by using some flat cohomology trick. But this trick is only telling us that the reduced group is cyclic of order {p}, but it does not tell us the scheme structure. In fact, in this case {E[p^n]\simeq \mu_{p^n}\oplus \mathbb{Z}/p^n} giving us {E[p^\infty]\simeq \mu_{p^\infty}\oplus \mathbb{Q}_p/\mathbb{Z}_p}. So this is a word of warning that when working these things out you need to be very careful that you understand whether or not you are figuring out the full group scheme structure or just reduced part. It can be hard to tell sometimes.


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