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

Theorem 1 The Lazard ring {L} over which the universal formal group law is defined is a polynomial ring in variables {x_1, x_2, \dots, } of degree {2i}.

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 { A \twoheadrightarrow B}, then any formal group law on {B} can be lifted to one over {A}.

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

\displaystyle L \rightarrow \mathbb{Z}[b_1, b_2, \dots ], \quad \deg b_i = 2i,

classifying the formal group law obtained from the additive one {x+y} by the “change of coordinates” { \exp(x) = \sum b_i x^{i+1}}. We claimed that the map on indecomposables was injective, and that, in fact the image in the indecomposables of {\mathbb{Z}[b_1, b_2, \dots ]} 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 {A} be an abelian group. A symmetric 2-cocycle is a “polynomial” {P(x,y) \in A[x, y] = A \otimes_{\mathbb{Z}} \mathbb{Z}[x, y]} with the properties:

\displaystyle P(x, y) = P(y,x)


\displaystyle P(x, y+z) + P(y, z) = P(x,y) + P(x+y, z).

These symmetric 2-cocycles come up when one tries to classify formal group laws over the ring {\mathbb{Z} \oplus A}, 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 {n} is a multiple of {\frac{1}{d} ( ( x+y)^n - x^n - y^n )} where {d =1} if {n} is not a power of a prime, and {d = p} if {n = p^k}.

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 {\mathrm{Ext}} group. Then, the strategy is to compute this {\mathrm{Ext}} 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…)