Landweber exact functor theorem

Summary

In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal group law. The Landweber exact functor theorem (or LEFT for short) can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.

StatementEdit

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

  and  

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

 

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
 
is a homology theory on CW-complexes.

In particular, every formal group law F over a ring   yields a module over   since we get via F a ring morphism  .

RemarksEdit

  • 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).

ExamplesEdit

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

 

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(n) is not Landweber exact.

Modern reformulationEdit

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.

ReferencesEdit

  • Goerss, Paul. "Realizing families of Landweber exact homology theories" (PDF).
  • Hovey, Mark; Strickland, Neil P. (1999), "Morava K-theories and localisation", Memoirs of the American Mathematical Society, 139 (666), doi:10.1090/memo/0666, MR 1601906, archived from the original on 2004-12-07
  • Landweber, Peter S. (1976). "Homological properties of comodules over   and  ". American Journal of Mathematics. 98 (3): 591–610. doi:10.2307/2373808. JSTOR 2373808..
  • Lurie, Jacob (2010). "Chromatic Homotopy Theory. Lecture Notes".