Complex oriented cohomology theoriesFormal group laws

4 Formal group laws

For the remainder of the talk, I wish to use a bit more algebraic geometry language. We assume EE is a commutative complex oriented ring spectrum. If XX is a CW complex, then E2X+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 E2X+E^{2*} X_+ as a topological ring (or a pro-ring), with the topology given by

E2X+=limAX finiteE2A+ E^{2*} X_+ = \lim _{A \subseteq X \text{ finite}} E^{2*} A_+

For example,

E2CP+=limnE2CPn=E2[ ⁣[u] ⁣], 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 Spec\operatorname{Spec} to rings. Actually, since we have topological rings, we should apply Spf\operatorname{Spf} instead, which remembers the topology, and end up with a formal scheme. If you don't know about Spf\operatorname{Spf}, you can pretend it is Spec\operatorname{Spec} instead. The upshot is that the functor

XXESpfE2X+ X \mapsto X_E \equiv \operatorname{Spf}E^{2*} X_+

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

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

CPE=SpfE2[ ⁣[u] ⁣]. \mathbb {CP}^\infty _E = \operatorname{Spf}E^{2*}[\! [u]\! ].

This as an infinitesimal neighbourhood of 0A10 \in \mathbb {A}^1 over SpecE2\operatorname{Spec}E^{2*}, which we denote A^1\hat{\mathbb {A}}^1. Our first conclusion is thus

Lemma 4.1

A complex orientation of EE gives an isomorphism

CPE=A^1. \mathbb {CP}^\infty _E = \hat{\mathbb {A}}^1.

The fact that EE is complex orientable tells us CPE\mathbb {CP}^\infty _E is abstractly isomorphic to A^1\hat{\mathbb {A}}^1, and a complex orientation is a choice of isomorphism.

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

Definition 4.2

A formal group is a commutative formal group scheme whose underlying scheme is (locally) isomorphic A^1\hat{\mathbb {A}}^1.

A formal group law is a commutative group scheme where the underlying scheme is equipped with an isomorphism with A^1\hat{\mathbb {A}}^1.

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 A^1\hat{\mathbb {A}}^1, or else they are extremely boring).

Thus, if EE is complex orientable, then CPE\mathbb {CP}^\infty _E is a formal group. If it is complex oriented, then CPE\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 RR be a ring. A formal group law is a map

SpfR[ ⁣[x] ⁣]×SpfR[ ⁣[x] ⁣]SpfR[ ⁣[x] ⁣] \operatorname{Spf}R[\! [x]\! ] \times \operatorname{Spf}R[\! [x]\! ] \to \operatorname{Spf}R[\! [x]\! ]

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

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

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

x+Fy=y+Fxx+F0=0(x+Fy)+Fz=x+F(y+Fz). \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] ⁣]R[\! [x]\! ] that sends one formal group law to the other. Again an automorphism R[ ⁣[x] ⁣]R[ ⁣[x] ⁣]R[\! [x]\! ] \to R[\! [x]\! ] is uniquely specified by the image of xx, say f(x)f(x), and an isomorphism between +F+_F and +G+_G is an invertible ff such that

f(x+Fy)=f(x)+Gf(y). 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:RSf: R \to S is a map of rings, then a formal group law over RR induces a formal group law over SS by applying ff 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 SpecR\operatorname{Spec}R to SpecS\operatorname{Spec}S.

Theorem 4.3 (Lazard)

There is a universal formal group law. That is, there is a ring LL with a formal group law G^L\hat{\mathbb {G}}_L on LL such that for any other formal group law G^\hat{\mathbb {G}} on a ring RR, there is a unique map f:LRf: L \to R such that G^=fG^L\hat{\mathbb {G}} = f^* \hat{\mathbb {G}}_L. Moreover, LZ[1,2,3,]L \cong \mathbb {Z}[\ell _1, \ell _2, \ell _3, \ldots ].

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

Recall that MUMU is the universal complex oriented cohomology theory, and a complex oriented cohomology theory has a canonical formal group law.

Theorem 4.4 (Quillen)

CPMU\mathbb {CP}^\infty _{MU} 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:

Example 4.5

If EE is complex oriented and FF is any ring spectrum, then EFE \wedge F is also complex oriented, and the formal group law on EFE \wedge F is pulled back from that of EE.

Thus, if both EE and FF are complex oriented, then we get two complex orientations of EFE \wedge F, 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 E=HZE = H\mathbb {Z} and F=KUF = KU. The respective formal group laws are G^a\hat{\mathbb {G}}_a and G^m\hat{\mathbb {G}}_m. We then know that π(HZKU)\pi _*(H\mathbb {Z}\wedge KU) is a ring on which G^a\hat{\mathbb {G}}_a and G^m\hat{\mathbb {G}}_m are isomorphic. Standard theory of formal group laws tells us this is possible only if the ring is rational (since one has height \infty and the other has height 11). So HKU=π(HZKU)H_*KU = \pi _*(H\mathbb {Z}\wedge KU) is rational.

Now consider the rationalization map KUKUQKU \to KU_\mathbb {Q}. Since HKUH_* KU is already rational, this map is an isomorphism on integral homology. However, πKU\pi _* KU 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.