# Definition of logarithm for universal vector extension

Let $S$ be a topological $\mathbb{Z}_p$-algebra and $R\to S$ a surjection (where $R$ has $p$ nilpotent) with topologically nilpotent kernel which has a PD structure.

We know that if $G/R$ is a $p$-divisible group, then we can lift $G$ to $S$, call this lift $H$, and let $EH$ be the universal vector extension of $G$. There is then supposed to be a logarithm map

$$\log_{EH}:EH(S)\to \text{Lie}_{EH}(S)[p^{-1}]=M(G)(S)[p^{-1}]$$

where $M(G)$ just means the Dieudonne module.

Could someone explain to me how this is defined or, perhaps, give me a reference (as 'accessible' as possible please :) ).

I know the classical picture of the logarithm for $p$-divisible groups over something like $\mathcal{O}_K$ (for $K$ a $p$-adic local field) in terms of a map $G(\mathcal{O}_{\mathbb{C}_K})\to \text{Lie}(G)\otimes_K \mathbb{C}_K$ but I don't really know how to interpret a lot of the objects in this definition for the generality I'm asking about.

Any help would be greatly appreciated!

EDIT: I found a reference for the fact I think I'm after, but am having trouble understanding it. Namely, it's essentially the content of 2.2.3 of Messing on page 88. I've been trying to figure out how one takes an element of $I\text{Lie}(U)$ and applies $\exp$ (as he's defined it) to it? To me, and I know this probably a silly statement, $\exp$ just defines a map $I\to 1+I$ in $S$.

I assume the intuition is that, I guess assuming our $EH$ is a formal Lie group, then we're essentially trying to define a map $(1+I)^n\to I^n$ (again, intuitively) which is just this $\log$ on the individual components, but I don't know how to make this truly rigorous (or if it's right).

So, any help interpreting this would be great!

EDIT EDIT: I now understand the construction in Messing for $p$-divisible groups, but still fail to see how one makes this work for something that is not $p$-divisible (like $EG$). I also don't see how this relates to the log map in the classic sense (i.e. the map $G(\mathcal{O}_C)\to \text{Lie}(G)\otimes C$ in Tate's paper).

Anything someone can say about any of this would be very helpful. Thanks!