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 over a graded ring is equivalent to giving a graded ring morphism. Multiplication by an integer is defined inductively as a power series, by Let now F be a formal group law over a ring. Define for a topological spaceX 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: In particular, every formal group law F over a ring yields a module over since we get via F a ring morphism.
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 .
The LEFT can be strengthened as follows: let be the 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 takes -algebras to MU-algebra spectra.
Examples
The archetypical and first known 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 called the Conner–Floyd isomorphism. While complex K-theory was constructed before by geometric means, many homology theories were first constructed via the Landweber exact functor theorem. This includes elliptic homology, the Johnson-Wilson theories and the Lubin-Tate spectra. While homology with rational coefficients is Landweber exact, homology with integer coefficients is not Landweber exact. Furthermore, Morava K-theory K is not Landweber exact.
Modern reformulation
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 sheafwith 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 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 .
While the LEFT is known to produce 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 ring spectra on X. If this map factors over and the map is etale, then this presheaf can be refined to a sheaf of -ring spectra. This theorem is important for the construction of topological modular forms.
