Given a p-divisible group over $\mathbb{F}_p$, Grothendieck-Messing theory tells us that deforming the group to $\mathbb{Z}_p$ is the same as finding an admissible filtration of the Dieudonne-module evaluated on $\mathbb{Z}_p$. (I think the way this is done is that given the group on $\mathbb{F}_p$, we pick some lift to $\mathbb{Z}_p$ and "define" the Dieudonne-module to be the Lie algebra of the universal vector extension; this is independent of the lift. But the filtration one gets from the universal extension does depend on the lift, and in fact, determines it.)
I've been told that this sometimes applies in greater generality, i.e. deforming the group to an object not in the Crystalline site of $\mathbb{F}_p$. In particular, the space that represents the deformation functor. And this can be done because there is an integrable connection.
So my questions are:
- Where does this connection come from? Is it just the fact that the Dieudonne module is a crystal on the Crystalline site and every crystal is equipped with an integrable connection?
- (This is the main one) How does having the connection save the day?
Best Answer
The theory can be summarized as follows: First, to every $p$-divisible group $G$ over a scheme $S$ in characteristic $p$, you can attach a Dieudonne crystal $\mathbb{D}(G)$ over $S$. What this means is that, for any $S$-scheme $U$, and any divided power thickening of $U$--that is, a closed immersion of $\mathbb{Z}_p$-schemes $U\hookrightarrow T$ such that $U$ is cut out in $T$ by a nilpotent ideal equipped with divided powers (note that this is usually an additional structure)--we have a vector bundle $\mathbb{D}(G)\vert_T$ over $T$. Of course, you need all these vector bundles to patch together in a nice way.
It in fact has the structure of an $F$-crystal, but we don't need that here. There are many constructions of this crystal: First, using Grothendieck's idea of universal vector extensions, Messing (and then Mazur-Messing) built a crystal, which, however, is only defined on a smaller site, the so-called nilpotent crystalline site. The general construction is due to Berthelot-Breen-Messing.
When $S$ is smooth and admits a smooth lift $\widetilde{S}$ over $\mathbb{Z}_p$, giving such a crystal is equivalent to giving a vector bundle over $\widetilde{S}$ equipped with a (topologically quasi-nilpotent integrable) connection. The point is that the connection tells you how to differentiate sections of the vector bundle along vector fields, and hence lets you use Taylor series to identify the evaluations of the vector bundle along 'infinitesimally close' points of $\widetilde{S}$. To make sense of such a series, you need divided powers, and the 'topologically quasi-nilpotent' condition ensures that this series is always truncated at a finite level. In general, one has to work locally, and with divided power envelopes, but there exists a similar such description. See Theorem 6.6 of Berthelot-Ogus, 'Notes on crystalline cohomology'.
Once you have the crystal $\mathbb{D}(G)$, Messing showed that you can use it linearize the deformation theory of $p$-divisible groups. Namely, the restriction of $\mathbb{D}(G)$ to the Zariski site of $S$ has a natural (Hodge) filtration, given, as you point out, by the Lie algebra of the universal vector extension of $G$. Then, for any nilpotent divided power thickening (we need not just the ideal sheaf, but also the divided powers on it to be nilpotent) $U\hookrightarrow T$ in the crystalline site of $S$ over $\mathbb{Z}_p$, lifting $G\vert U$ over $T$ is equivalent to deforming the Hodge filtration on $\mathbb{D}(G)\vert_{U\hookrightarrow U}$ to a direct summand of $\mathbb{D}(G)\vert_{U\hookrightarrow T}$.
This specializes to the case you refer to in your question, by taking $S$ to be $\text{Spec }\mathbb{F}_p$ and $T$ to be $\text{Spec }\mathbb{Z}/p^n\mathbb{Z}$ (it is known (see 2.4.4 of de Jong's 'Crystalline Dieudonne theory...') that giving a $p$-divisible group over $\mathbb{Z}_p$ is equivalent to giving a compatible system of $p$-divisible groups over $\mathbb{Z}/p^n\mathbb{Z}$ as $n$ varies). The divided powers are the canonical ones: $p\mapsto\frac{p^n}{n!}$. Unfortunately, there is a hitch when $p=2$: these divided powers are not nilpotent, so Grothendieck-Messing theory is very delicate then. In fact, the statement as you have it is no longer true in this situation (one has to restrict to connected $p$-divisible groups).