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…)