Landweber exact functor theorem explained

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.

Statement

The coefficient ring of complex cobordism is

MU _{*(*)} = MU _{*} \cong \Z[x _{1,x} _{2,...]} , where the degree of x _{i} is 2 i . This is isomorphic to the graded Lazard ring l{}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 _{*\to} 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

l{}R _{*} . Define for a topological space XE _{*(X)} = MU _{*(X) ⊗} ^{MU _{*}} R _{*} Here R _{*} gets its MU _{*} -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 _{*} be flat over MU _{*} , 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,...} \in MU _{*} such that we have the following: Suppose that M _{*} is a graded MU _{*} -module and the sequence (p,v _{1,v} _{2,...,} v _{n)} is regular for M , for every p and n. Then E _{*(X)} = MU _{*(X) ⊗} ^{MU _{*}} M _{*}

is a homology theory on CW-complexes.

In particular, every formal group law F over a ring

R yields a module over l{}MU _{*} since we get via F a ring morphism MU _{*\to} R .

Remarks

• There is also a version for Brown–Peterson cohomology BP. The spectrum BP is a direct summand of

MU _{(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

BP _{*} which are invariant under coaction of BP _{*BP} are the I _{n} = (p,v _{1,...,} v _{n)} . This allows to check flatness only against the BP _{*/I} _{n} (see Landweber, 1976).

• The LEFT can be strengthened as follows: let

l{E} _{*} be the (homotopy) category of Landweber exact MU _{*} -modules and l{E} the category of MU-module spectra M such that \pi _{*M} is Landweber exact. Then the functor \pi _{*\colonl{E}\tol{E}} _{*} is an equivalence of categories. The inverse functor (given by the LEFT) takes l{}MU _{*} -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 + xy . The corresponding morphism MU _{*\to} K _{*} is also known as the Todd genus. We have then an isomorphismK _{*(X)} = MU _{*(X) ⊗} ^{MU _{*}} K _{*,} called the Conner–Floyd isomorphism.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

l{}MU _{*} is the same as a quasi-coherent sheaf l{F} over Spec L , where L is the Lazard ring. If M = l{}MU _{*(X)} , then M has the extra datum of a l{}MU _{*MU} coaction. A coaction on the ring level corresponds to that l{F} is an equivariant sheaf with respect to an action of an affine group scheme G. It is a theorem of Quillen that G \cong \Z[b _{1,} b _{2,...]} and assigns to every ring R the group of power series g(t) = t + b ^{3+ … \in} _{2 t} R[[t]] . It acts on the set of formal group laws Spec L(R) via F(x,y) \mapsto gF(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 l{M} _{fg} and M = MU _{*(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 l{F} which is flat over l{M} _{fg} in order that MU _{*(X) ⊗} ^{MU _{*}} M is a homology theory. The Landweber exactness theorem can then be interpreted as a flatness criterion for l{M} _{fg} (see Lurie 2010).

Refinements to

E _{infty} -ring spectra

While the LEFT is known to produce (homotopy) ring spectra out of

l{}MU _{*} , it is a much more delicate question to understand when these spectra are actually E _{infty} -ring spectra. As of 2010, the best progress was made by Jacob Lurie. If X is an algebraic stack and X\to l{M} _{fg} 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\to M _{p(n)} is etale, then this presheaf can be refined to a sheaf of E _{infty} -ring spectra (see Goerss). This theorem is important for the construction of topological modular forms.

See also

• Chromatic homotopy theory

References

• Web site: Paul. Goerss. Realizing families of Landweber exact homology theories.
  • Peter S.. Landweber. Peter Landweber. Homological properties of comodules over MU _{*(MU)} and BP _{*(BP)} . . 98 . 3. 1976. 591–610. 2373808. 10.2307/2373808. .
• Web site: Jacob. Lurie. Jacob Lurie. Chromatic Homotopy Theory. Lecture Notes . 2010.