4 Formal group laws

For the remainder of the talk, I wish to use a bit more algebraic geometry language. We assume $E$ is a commutative complex oriented ring spectrum. If $X$ is a CW complex, then $E^{2*}X_+$ 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 $E^{2*} X_+$ as a topological ring (or a pro-ring), with the topology given by

$$E^{2*} X_+ = \lim _{A \subseteq X \text{ finite}} E^{2*} A_+$$

For example,

$$E^{2*} \mathbb {CP}^\infty _+ = \lim _n E^{2*} \mathbb {CP}^n = E^{2*}[\! [u]\! ],$$

where the power series ring has the usual topology.

Since we are doing algebraic geometry, we are supposed to apply the functor $\operatorname{Spec}$ to rings. Actually, since we have topological rings, we should apply $\operatorname{Spf}$ instead, which remembers the topology, and end up with a formal scheme. If you don't know about $\operatorname{Spf}$, you can pretend it is $\operatorname{Spec}$ instead. The upshot is that the functor

$$X \mapsto X_E \equiv \operatorname{Spf}E^{2*} X_+$$

is a covariant functor in $X$. Moreover, if we restrict to spaces for which $E^*X_+$ is a free $E^*$-module, such as $\mathbb {CP}^n$ and complex oriented cohomology theories, this functor is symmetric monoidal by Künneth's theorem.

For $\mathbb {CP}^\infty$, we have

$$\mathbb {CP}^\infty _E = \operatorname{Spf}E^{2*}[\! [u]\! ].$$

This as an infinitesimal neighbourhood of $0 \in \mathbb {A}^1$ over $\operatorname{Spec}E^{2*}$, which we denote $\hat{\mathbb {A}}^1$. Our first conclusion is thus

Now recall that $X \mapsto X_E$ is symmetric monoidal. Moreover, $\mathbb {CP}^\infty$ has the structure of an abelian group (in the homotopy category), with the map $\otimes : \mathbb {CP}^\infty \times \mathbb {CP}^\infty \to \mathbb {CP}^\infty$ classifying the tensor product of line bundles. This turns $\mathbb {CP}^\infty _E$ into a (formal) group scheme.

Thus, if $E$ is complex orientable, then $\mathbb {CP}^\infty _E$ is a formal group. If it is complex oriented, then $\mathbb {CP}^\infty _E$ is given the structure of a formal group law. Different choices of complex orientations give different but isomorphic formal group laws.

Let us unwrap what it means to be a formal group law. Let $R$ be a ring. A formal group law is a map

$$\operatorname{Spf}R[\! [x]\! ] \times \operatorname{Spf}R[\! [x]\! ] \to \operatorname{Spf}R[\! [x]\! ]$$

satisfying certain properties. Undoing the $\operatorname{Spf}$ gives us a continuous map of $R$-algebras

$$R[\! [x]\! ] \to R[\! [x, y]\! ].$$

This is uniquely determined by the value of $x$. Call this $x +_F y$, which is a power series in $x$ and $y$. The property of being a commutative group is equivalent to the conditions

$$\begin{aligned} x +_F y & = y +_F x\\ x +_F 0 & = 0\\ (x +_F y) +_F z & = x +_F (y +_F z). \end{aligned}$$

An isomorphism of formal group laws is given by an automorphism of $R[\! [x]\! ]$ that sends one formal group law to the other. Again an automorphism $R[\! [x]\! ] \to R[\! [x]\! ]$ is uniquely specified by the image of $x$, say $f(x)$, and an isomorphism between $+_F$ and $+_G$ is an invertible $f$ such that

$$f(x +_F y) = f(x) +_G f(y).$$

A formal group is then a formal group law up to isomorphism.

Note that if $f: R \to S$ is a map of rings, then a formal group law over $R$ induces a formal group law over $S$ by applying $f$ 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 $\operatorname{Spec}R$ to $\operatorname{Spec}S$.

The key content of the theorem is the final line. The moduli space of formal group laws is an affine space!

Recall that $MU$ is the universal complex oriented cohomology theory, and a complex oriented cohomology theory has a canonical 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: