After describing the computation of ${\pi_* MU}$, 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 ${R}$ be a (commutative) ring. Recall that a formal group law (commutative and one-dimensional) is a power series ${f(x,y) \in R[[x,y]]}$ such that

1. ${f(x,y) = f(y,x)}$.
2. ${f(x, f(y,z)) = f(f(x,y), z)}$.
3. ${f(x,0) = f(0,x) = x}$.

It is automatic from this by a successive approximation argument that there exists an inverse power series ${i(x) \in R[[x]]}$ such that ${f(x, i(x)) = 0}$.

In particular, ${f}$ has the property that for any ${R}$-algebra ${S}$, the nilpotent elements of ${S}$ become an abelian group with addition given by ${f}$.

A key observation is that, given ${R}$, to specify a formal group law amounts to specifying a countable collection of elements ${c_{i,j}}$ to define the power series ${f(x,y) = \sum c_{i,j} x^i y^j}$. These ${c_{i,j}}$ are required to satisfy various polynomial constraints to ensure that the formal group identities hold. Consequently:

Theorem 1 There exists a universal ring ${L}$ together with a formal group law ${f_{univ}(x,y)}$ on ${L}$, such that any FGL ${f}$ on another ring ${R}$ determines a unique map ${L \rightarrow R}$ carrying ${f_{univ} \mapsto f}$. (more…)