We are in the middle of proving an important result of Lazard:

Theorem 1The Lazard ring over which the universal formal group law is defined is a polynomial ring in variables of degree .

The fact that the Lazard ring is polynomial implies a number of results which are not a priori obvious: for instance, it shows that given a surjection of rings , then any formal group law on can be lifted to one over .

We began the proof of Lazard’s theorem last time: we produced a map

classifying the formal group law obtained from the additive one by the “change of coordinates” . We claimed that the map on indecomposables was injective, and that, in fact the image in the indecomposables of could be determined completely. I won’t get into the details of this (it was all in the previous post), because the purpose of this post is to prove a result to which we reduced last time.

Let be an abelian group. A **symmetric 2-cocycle** is a “polynomial” with the properties:

and

These symmetric 2-cocycles come up when one tries to classify formal group laws over the ring , as we saw last time: in fact, we can think of them as “deformations” of the additive formal group law.

The main lemma which we stated last time was the following:

Theorem 2 (Symmetric 2-cocycle lemma)A homogeneous symmetric 2-cocycle of degree is a multiple of where if is not a power of a prime, and if .

For a direct combinatorial proof of this theorem, see Lurie’s notes. I want to describe a longer homological proof, which is apparently due to Mike Hopkins and which appears in the COCTALOS notes. The strategy is to interpret these symmetric 2-cocycles as actual cocycles in a cobar complex computing an group. Then, the strategy is to compute this group independently.

This argument is somewhat longer than the combinatorial one, but it has the benefit (for me) of engaging with some homological algebra (which I need to learn more about), as well as potentially generalizing in other directions. (more…)