The Landweber exactness criterion determins if a given formal group law does arise as the formal group law defined by a weakly periodic cohomology theory.
Notice that since every formal group law over a ring is classified by a ring homomorphism where by Quillen’s theorem on MU|Quillen’s theorem] is the Lazard ring. So for every formal group one obtains a contravariant functor on topological spaces given by the assignment
where denotes the complex cobordism cohomology theory and where the tensor product is taken using the -module structure on induced by .
The point of Landweber-exactness is that if is Landweber exact (i.e. if the corresponding formal group law is) then this construction defines a cohomology theory .
Landweber criterion Let be a formal group law and a prime, the coefficient of in . If form a regular sequence for all and then is Landweber exact and hence gives a cohomology theory via the the formula above.
See at Landweber exact functor theorem
, , , , for all . The regularity conditions are trivial. Hence we know that is a cohomology theory.