The goal of the next few posts is to compute :

Theorem 1 (Milnor)The complex cobordism ring is isomorphic to a polynomial ring where each is in degree .

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 , at each prime individually.

**1. Change-of-rings theorem**

In order to apply the ASS, we’re going to need the groups because the spectral sequence runs

The groups are computed in the category of (graded) comodules over .

In the previous post, we computed

as a comodule over . In order to compute the groups, we need a general machine. The idea is that is *almost* a coinduced comodule—if it were, the groups would be trivial. It’s not, but the general “change-of-rings” machine will enable us to reduce the calculation of these groups to the calculation of (much simpler) groups over an exterior algebra. (more…)