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