The coefficient ring of complex cobordism is , where the degree of is . This is isomorphic to the graded Lazard ring. This means that giving a formal group law F (of degree ) over a graded ring is equivalent to giving a graded ring morphism . Multiplication by an integer is defined inductively as a power series, by
Here gets its -algebra structure via F. The question is: is E a homology theory? It is obviously a homotopy invariant functor, which fulfills excision. The problem is that tensoring in general does not preserve exact sequences. One could demand that be flat over , but that would be too strong in practice. Peter Landweber found another criterion:
Theorem (Landweber exact functor theorem)
For every prime p, there are elements such that we have the following: Suppose that is a graded -module and the sequence is regular for , for every p and n. Then
In particular, every formal group law F over a ring yields a module over since we get via F a ring morphism .
There is also a version for Brown–Peterson cohomology BP. The spectrum BP is a direct summand of with coefficients . The statement of the LEFT stays true if one fixes a prime p and substitutes BP for MU.
The classical proof of the LEFT uses the Landweber–Morava invariant ideal theorem: the only prime ideals of which are invariant under coaction of are the . This allows to check flatness only against the (see Landweber, 1976).
The LEFT can be strengthened as follows: let be the (homotopy) category of Landweber exact -modules and the category of MU-module spectra M such that is Landweber exact. Then the functor is an equivalence of categories. The inverse functor (given by the LEFT) takes -algebras to (homotopy) MU-algebra spectra (see Hovey, Strickland, 1999, Thm 2.7).
The archetypical and first known (non-trivial) example is complex K-theory K. Complex K-theory is complex oriented and has as formal group law . The corresponding morphism is also known as the Todd genus. We have then an isomorphism
While homology with rational coefficients is Landweber exact, homology with integer coefficients is not Landweber exact. Furthermore, Morava K-theory K(n) is not Landweber exact.
A module M over is the same as a quasi-coherent sheaf over , where L is the Lazard ring. If , then M has the extra datum of a coaction. A coaction on the ring level corresponds to that is an equivariant sheaf with respect to an action of an affine group scheme G. It is a theorem of Quillen that and assigns to every ring R the group of power series
It acts on the set of formal group laws via
These are just the coordinate changes of formal group laws. Therefore, one can identify the stack quotient with the stack of (1-dimensional) formal groups and defines a quasi-coherent sheaf over this stack. Now it is quite easy to see that it suffices that M defines a quasi-coherent sheaf which is flat over in order that is a homology theory. The Landweber exactness theorem can then be interpreted as a flatness criterion for (see Lurie 2010).
Refinements to -ring spectraEdit
While the LEFT is known to produce (homotopy) ring spectra out of , it is a much more delicate question to understand when these spectra are actually -ring spectra. As of 2010, the best progress was made by Jacob Lurie. If X is an algebraic stack and a flat map of stacks, the discussion above shows that we get a presheaf of (homotopy) ring spectra on X. If this map factors over (the stack of 1-dimensional p-divisible groups of height n) and the map is etale, then this presheaf can be refined to a sheaf of -ring spectra (see Goerss). This theorem is important for the construction of topological modular forms.
Goerss, Paul. "Realizing families of Landweber exact homology theories" (PDF).