This is the third in the series of posts intended to work through the proof of Lazard’s theorem, that the Lazard ring classifying the universal formal group law is actually a polynomial ring on a countable set of generators. In the first post, we reduced the result to an elementary but tricky “symmetric 2-cocycle lemma.” In the previous post, we proved most of the symmetric 2-cocycle lemma, except in characteristic zero. The case of characteristic zero is not harder than the cases we handled (it’s easier), but in this post we’ll complete the proof of that case by exhibiting a very direct construction of logarithms in characteristic zero. Next, I’ll describe an application in Lazard’s original paper, on “approximate” formal group laws.
After this, I’m going to try to move back to topology, and describe the proof of Quillen’s theorem on the formal group law of complex cobordism. The purely algebraic calculations of the past couple of posts will be necessary, though.
1. Formal group laws in characteristic zero
The last step missing in the proof of Lazard’s theorem was the claim that the map
classifying the formal group law obtained from the additive one by “change-of-coordinates” by the exponential series is an isomorphism mod torsion. In other words, we have an isomorphism
In fact, we didn’t really need this: we could have proved the homological 2-cocycle lemma in all cases, instead of just the finite field case, and it would have been easier. But I’d like to emphasize that the result is really something elementary here. In fact, what it is saying is that to give a formal group law over a -algebra is equivalent to giving a choice of series .
Definition 1 An exponential for a formal group law is a power series such that
The inverse power series is called the logarithm.
That is, a logarithm is an isomorphism of with the additive formal group law.
So another way of phrasing this result is that:
Proposition 2 A formal group law over a -algebra has a unique logarithm (i.e., is uniquely isomorphic to the additive one). (more…)