Landweber exact functor theorem

Statement

The coefficient ring of complex cobordism is M U ∗ ( ∗ ) = M U ∗ ≅ Z [ x 1 , x 2 , … ] , where the degree of x i is 2i. This is isomorphic to the graded Lazard ring L ∗ . This means that giving a formal group law F (of degree −2) over a graded ring R ∗ is equivalent to giving a graded ring morphism L ∗ → R ∗ . Multiplication by an integer n >0 is defined inductively as a power series, by

[ n + 1 ] F x = F ( x , [ n ] F x ) and [ 1 ] F x = x .

Let now F be a formal group law over a ring R ∗ . Define for a topological space X

E ∗ ( X ) = M U ∗ ( X ) ⊗ M U ∗ R ∗

Here R ∗ gets its M U ∗ -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 R ∗ is flat over M U ∗ , 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 v 1 , v 2 , ⋯ ∈ M U ∗ such that we have the following: Suppose that M ∗ is a graded M U ∗ -module and the sequence ( p , v 1 , v 2 , … , v n ) is regular for M, for every p and n. Then E ∗ ( X ) = M U ∗ ( X ) ⊗ M U ∗ M ∗ is a homology theory on CW-complexes.

In particular, every formal group law F over a ring R yields a module over M U ∗ since we get via F a ring morphism M U ∗ → R .

Remarks

There is also a version for Brown–Peterson cohomology BP. The spectrum BP is a direct summand of M U ( p ) with coefficients Z ( p ) [ v 1 , v 2 , … ] . 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 B P ∗ which are invariant under coaction of B P ∗ B P are the I n = ( p , v 1 , … , v n ) . This allows to check flatness only against the B P ∗ / I n (see Landweber, 1976).

The LEFT can be strengthened as follows: let E ∗ be the (homotopy) category of Landweber exact M U ∗ -modules and E the category of MU-module spectra M such that π ∗ M is Landweber exact. Then the functor π ∗ : E → E ∗ is an equivalence of categories. The inverse functor (given by the LEFT) takes M U ∗ -algebras to (homotopy) MU-algebra spectra (see Hovey, Strickland, 1999, Thm 2.7).

Examples

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 x + y + x y . The corresponding morphism M U ∗ → K ∗ is also known as the Todd genus. We have then an isomorphism

K ∗ ( X ) = M U ∗ ( X ) ⊗ M U ∗ K ∗ ,

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 E ( n ) and the Lubin–Tate spectra E n .

While homology with rational coefficients H Q is Landweber exact, homology with integer coefficients H Z is not Landweber exact. Furthermore, Morava K-theory K(n) is not Landweber exact.

Modern reformulation

A module M over M U ∗ is the same as a quasi-coherent sheaf F over Spec  L , where L is the Lazard ring. If M = M U ∗ ( X ) , then M has the extra datum of a M U ∗ M U coaction. A coaction on the ring level corresponds to that F is an equivariant sheaf with respect to an action of an affine group scheme G. It is a theorem of Quillen that G ≅ Z [ b 1 , b 2 , … ] and assigns to every ring R the group of power series

g ( t ) = t + b 1 t 2 + b 2 t 3 + ⋯ ∈ R [ [ t ] ] .

It acts on the set of formal group laws Spec  L ( R ) via

F ( x , y ) ↦ g F ( g − 1 x , g − 1 y ) .

These are just the coordinate changes of formal group laws. Therefore, one can identify the stack quotient Spec  L / / G with the stack of (1-dimensional) formal groups M f g and M = M U ∗ ( X ) 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 F which is flat over M f g in order that M U ∗ ( X ) ⊗ M U ∗ M is a homology theory. The Landweber exactness theorem can then be interpreted as a flatness criterion for M f g (see Lurie 2010).

Refinements to E ∞ {\displaystyle E_{\infty }} -ring spectra

While the LEFT is known to produce (homotopy) ring spectra out of M U ∗ , it is a much more delicate question to understand when these spectra are actually E ∞ -ring spectra. As of 2010, the best progress was made by Jacob Lurie. If X is an algebraic stack and X → M f g 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 M p ( n ) (the stack of 1-dimensional p-divisible groups of height n) and the map X → M p ( n ) is etale, then this presheaf can be refined to a sheaf of E ∞ -ring spectra (see Goerss). This theorem is important for the construction of topological modular forms.