After describing the computation of , I’d now like to handle the remaining half of the machinery that goes into Quillen’s theorem: the structure of the universal formal group law.

Let be a (commutative) ring. Recall that a **formal group law** (commutative and one-dimensional) is a power series such that

- .
- .
- .

It is automatic from this by a successive approximation argument that there exists an inverse power series such that .

In particular, has the property that for any -algebra , the nilpotent elements of become an abelian group with addition given by .

A key observation is that, given , to specify a formal group law amounts to specifying a countable collection of elements to define the power series . These are required to satisfy various polynomial constraints to ensure that the formal group identities hold. Consequently:

Theorem 1There exists a universal ring together with a formal group law on , such that any FGL on another ring determines a unique map carrying . (more…)