# Is the tangent space functor from PD formal groups to Lie algebras an equivalence?

The previous version of this question was rather badly broken, and I hope this version makes some sense.

There have been a few questions on MathOverflow about how much representation-theoretic information is lost when passing from a Lie group to its Lie algebra, e.g., away from the semisimple case, Lie algebras have many more representations. In the algebraic setting, there is an intermediate construction between an algebraic group and its Lie algebra, given by the formal group. One completes the algebraic group along the identity to get a formal scheme equipped with a group law, and one can pass from there to the tangent space to get the Lie algebra. In characteristic zero, the tangent space functor is an equivalence of categories from formal groups to Lie algebras, but in positive characteristic, formal groups form an honest intermediate category since the tangent space can lose a lot of information. For example, there is only one isomorphism class of one-dimensional Lie algebra, but one-dimensional formal groups have a rich arithmetic theory, with a moduli space stratified by positive integer heights. The completions at the identity of the additive group and the multiplicative group have very distinct formal group structures, and one way to explain the lack of isomorphism is by the presence of denominators in the usual logarithm and exponential power series.

It seems to me that in positive characteristic, there could be an intermediate construction between formal groups and Lie algebras, given by passing to PD rings and replacing the coordinate ring of the formal group with the divided power envelope of the identity section. If I'm not mistaken, this construction yields a group object in PD formal schemes.

Here is a bit of explanation for the uninitiated (see Berthelot-Ogus for more): PD rings are triples $(A,I,\gamma)$, where $A$ is a commutative ring, $I$ is an ideal, and $\gamma = \{ \gamma_n: I \to A \}_{n \geq 0}$ is a system of divided power operations. I think they arose when Grothendieck tried to get De Rham cohomology to give the expected answers for proper varieties in characteristic $p$, since the naïve definition tended to yield infinite dimensional spaces. There is a forgetful functor $(A,I,\gamma) \mapsto (A,I)$ from PD rings to ring-ideal pairs, and it has a left adjoint, called the divided power envelope. In characteristic zero, $\gamma$ is canonically given as $\gamma_n(x) = x^n/n!$, so both functors are equivalences in that case. The notion of PD ring can be sheafified and localizations have canonical PD structures, so one has notions of PD scheme and PD formal scheme.

Question: Do PD formal groups contain any more information than the underlying Lie algebra?

I have a suspicion that the answer is "no" and the answer to the title question is "yes". Vague word-association suggests that the divided power structure is exactly what one needs to get a formal logarithm, but maybe there is a more fundamental obstruction.

I was originally motivated by the question of how Gelfand-Kazhdan formal geometry would differ in charateristic $p$ if I switched between ordinary and PD structures (cf. David Jordan's question). Unfortunately, I was laboring under some misconceptions about formal completions, and I'm still a bit confused about the precise structure of the automorphism group of the completion (PD or ordinary) of a smooth variety at a point in characteristic $p$.

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Can you elaborate on how you obtain a formal group structure on the completion X^? I only know how this arises if X is a group scheme, or how to make the pair (X^, (X x X)^) into a groupoid object. – Tyler Lawson Dec 12 '09 at 13:25
"If X is a smooth variety, then the formal completion of X at a closed point x has a canonical formal group structure" -- this is a very strange assertion, as automorphisms of the formal completion of a point on a smooth variety do not preserve any formal group structure in general, neither do those of them that come from global automorphisms of the variety. – Leonid Positselski Dec 12 '09 at 13:28
This is rather embarrassing. You're both absolutely correct. In fact, I was wondering why the "canonical" structures in my examples weren't preserved under etale maps. I will try to turn this into an honest question later today. – S. Carnahan Dec 12 '09 at 22:00
I think your third paragraph, where you made the global construction, is correct. You might be able to make this an honest question by just striking out the second paragraph. – David Speyer Dec 14 '09 at 18:49
The formal completion of the tangent space at x has a canonical formal group structure, which is related to your formal completion by deformation to the normal cone. Of course this formal group is additive by construction. Maybe though you can rephrase as asking about recovering PD neighborhoods from analogues of solving Maurer-Cartan equations for some (restricted?) dgla. – David Ben-Zvi Dec 14 '09 at 18:59

It seems that this deals only with height $1$ neighbourhoods, i.e., divided power structures with $\gamma_p=0$. They are just rings in char. $p$ where the elements of the DP-ideal raised to the $p$'th power are zero. Their groups are the $p$-Lie algebras not the Lie algebras. – Torsten Ekedahl Jun 21 '10 at 20:32