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

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

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…)

The goal of the next few posts is to compute ${\pi_* MU}$:

Theorem 1 (Milnor) The complex cobordism ring ${\pi_* MU}$ is isomorphic to a polynomial ring ${\mathbb{Z}[c_1, c_2, \dots]}$ where each ${c_i}$ is in degree ${2i}$.

We are also going to work out what the image of the Hurewicz map is on indecomposables. The strategy will be to apply the Adams spectral sequence to ${MU}$, at each prime individually.

1. Change-of-rings theorem

In order to apply the ASS, we’re going to need the groups ${\mathrm{Ext}^{s,t}_{\mathcal{A}_p^{\vee}}(\mathbb{Z}/p, H_*(MU; \mathbb{Z}/p))}$ because the spectral sequence runs $\displaystyle E_2^{s,t} = \mathrm{Ext}^{s,t}_{\mathcal{A}_p^{\vee}}(\mathbb{Z}/p, H_*(MU; \mathbb{Z}/p)) \implies \widehat{\pi_{t-s}(MU)}.$

The ${\mathrm{Ext}}$ groups are computed in the category of (graded) comodules over ${\mathcal{A}_p^{\vee}}$.

In the previous post, we computed $\displaystyle H_*(MU; \mathbb{Z}/p) = P \otimes \mathbb{Z}/p[y_i]_{i + 1 \neq p^k},$

as a comodule over ${\mathcal{A}_p^{\vee}}$. In order to compute the ${\mathrm{Ext}}$ groups, we need a general machine. The idea is that ${P \otimes \mathbb{Z}[y_i]_{i + 1 \neq p^k}}$ is almost a coinduced comodule—if it were, the ${\mathrm{Ext}}$ groups would be trivial. It’s not, but the general “change-of-rings” machine will enable us to reduce the calculation of these ${\mathrm{Ext}}$ groups to the calculation of (much simpler) ${\mathrm{Ext}}$ groups over an exterior algebra. (more…)