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…)