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…)