4 Formal group laws
For the remainder of the talk, I wish to use a bit more algebraic geometry language. We assume is a commutative complex oriented ring spectrum. If is a CW complex, then is a ring. We restrict to the evenly graded parts so that it is actually commutative. If one is concerned, one can take into account all degrees and forgo the algebro-geometric language, but doing so gains us nothing (this works out because the spaces we care about only have even cells).
To be honest mathematicians, we should consider as a topological ring (or a pro-ring), with the topology given by
where the power series ring has the usual topology.
Since we are doing algebraic geometry, we are supposed to apply the functor to rings. Actually, since we have topological rings, we should apply instead, which remembers the topology, and end up with a formal scheme. If you don't know about , you can pretend it is instead. The upshot is that the functor
is a covariant functor in . Moreover, if we restrict to spaces for which is a free -module, such as and complex oriented cohomology theories, this functor is symmetric monoidal by Künneth's theorem.
For , we have
This as an infinitesimal neighbourhood of over , which we denote . Our first conclusion is thus
A complex orientation of gives an isomorphism
The fact that is complex orientable tells us is abstractly isomorphic to , and a complex orientation is a choice of isomorphism.
Now recall that is symmetric monoidal. Moreover, has the structure of an abelian group (in the homotopy category), with the map classifying the tensor product of line bundles. This turns into a (formal) group scheme.
A formal group is a commutative formal group scheme whose underlying scheme is (locally) isomorphic .
A formal group law is a commutative group scheme where the underlying scheme is equipped with an isomorphism with .
By convention, an isomorphism of formal group laws is an isomorphism of the underlying formal groups (that is, it is not required to act as the identity on , or else they are extremely boring).
Let us unwrap what it means to be a formal group law. Let be a ring. A formal group law is a map
satisfying certain properties. Undoing the gives us a continuous map of -algebras
This is uniquely determined by the value of . Call this , which is a power series in and . The property of being a commutative group is equivalent to the conditions
An isomorphism of formal group laws is given by an automorphism of that sends one formal group law to the other. Again an automorphism is uniquely specified by the image of , say , and an isomorphism between and is an invertible such that
A formal group is then a formal group law up to isomorphism.
Note that if is a map of rings, then a formal group law over induces a formal group law over by applying to the coefficients of the power series. Since we are thinking in terms of schemes, we call this “pulling back” the formal group law from to .
There is a universal formal group law. That is, there is a ring with a formal group law on such that for any other formal group law on a ring , there is a unique map such that . Moreover, .
Recall that is the universal complex oriented cohomology theory, and a complex oriented cohomology theory has a canonical formal group law.
is the universal formal group law.
In general, the formal group of a complex orientable cohomology theory captures a lot of important information about the theory. It also allows us to perform some nice computations:
If is complex oriented and is any ring spectrum, then is also complex oriented, and the formal group law on is pulled back from that of .
Thus, if both and are complex oriented, then we get two complex orientations of , hence two formal group laws. However, since the formal group of a complex orientable cohomology theory is well-defined, these two formal group laws must be isomorphic.
For example, take and . The respective formal group laws are and . We then know that is a ring on which and are isomorphic. Standard theory of formal group laws tells us this is possible only if the ring is rational (since one has height and the other has height ). So is rational.
Now consider the rationalization map . Since is already rational, this map is an isomorphism on integral homology. However, is definitely not rational, so the map is not an isomorphism on homotopy groups. Thus the cofiber of this map has trivial integral homology, but is non-contractible.