Landweber exact functor theorem

The coefficient ring of complex cobordism is ${\displaystyle MU_{*}(*)=MU_{*}\cong \mathbb {Z} [x_{1},x_{2},\dots ]}$, where the degree of ${\displaystyle x_{i}}$ is ${\displaystyle 2i}$. This is isomorphic to the graded Lazard ring ${\displaystyle {\mathcal {}}L_{*}}$. This means that giving a formal group law F (of degree ${\displaystyle -2}$) over a graded ring ${\displaystyle R_{*}}$ is equivalent to giving a graded ring morphism ${\displaystyle L_{*}\to R_{*}}$. Multiplication by an integer ${\displaystyle n>0}$ is defined inductively as a power series, by

${\displaystyle [n+1]^{F}x=F(x,[n]^{F}x)}$ and ${\displaystyle [1]^{F}x=x.}$

Let now F be a formal group law over a ring ${\displaystyle {\mathcal {}}R_{*}}$. Define for a topological space X

${\displaystyle E_{*}(X)=MU_{*}(X)\otimes _{MU_{*}}R_{*}}$

Here ${\displaystyle R_{*}}$ gets its ${\displaystyle MU_{*}}$-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 ${\displaystyle R_{*}}$ be flat over ${\displaystyle MU_{*}}$, 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 ${\displaystyle v_{1},v_{2},\dots \in MU_{*}}$ such that we have the following: Suppose that ${\displaystyle M_{*}}$ is a graded ${\displaystyle MU_{*}}$-module and the sequence ${\displaystyle (p,v_{1},v_{2},\dots ,v_{n})}$ is regular for ${\displaystyle M}$, for every p and n. Then

${\displaystyle E_{*}(X)=MU_{*}(X)\otimes _{MU_{*}}M_{*}}$

is a homology theory on CW-complexes.

In particular, every formal group law F over a ring ${\displaystyle R}$ yields a module over ${\displaystyle {\mathcal {}}MU_{*}}$ since we get via F a ring morphism ${\displaystyle MU_{*}\to R}$.

• There is also a version for Brown–Peterson cohomology BP. The spectrum BP is a direct summand of ${\displaystyle MU_{(p)}}$ with coefficients ${\displaystyle \mathbb {Z} _{(p)}[v_{1},v_{2},\dots ]}$. 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 ${\displaystyle BP_{*}}$ which are invariant under coaction of ${\displaystyle BP_{*}BP}$ are the ${\displaystyle I_{n}=(p,v_{1},\dots ,v_{n})}$. This allows to check flatness only against the ${\displaystyle BP_{*}/I_{n}}$ (see Landweber, 1976).
• The LEFT can be strengthened as follows: let ${\displaystyle {\mathcal {E}}_{*}}$ be the (homotopy) category of Landweber exact ${\displaystyle MU_{*}}$-modules and ${\displaystyle {\mathcal {E}}}$ the category of MU-module spectra M such that ${\displaystyle \pi _{*}M}$ is Landweber exact. Then the functor ${\displaystyle \pi _{*}\colon {\mathcal {E}}\to {\mathcal {E}}_{*}}$ is an equivalence of categories. The inverse functor (given by the LEFT) takes ${\displaystyle {\mathcal {}}MU_{*}}$-algebras to (homotopy) MU-algebra spectra (see Hovey, Strickland, 1999, Thm 2.7).

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 ${\displaystyle x+y+xy}$. The corresponding morphism ${\displaystyle MU_{*}\to K_{*}}$ is also known as the Todd genus. We have then an isomorphism

${\displaystyle K_{*}(X)=MU_{*}(X)\otimes _{MU_{*}}K_{*},}$

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 ${\displaystyle E(n)}$ and the Lubin–Tate spectra ${\displaystyle E_{n}}$.

While homology with rational coefficients ${\displaystyle H\mathbb {Q} }$ is Landweber exact, homology with integer coefficients ${\displaystyle H\mathbb {Z} }$ is not Landweber exact. Furthermore, Morava K-theory K(n) is not Landweber exact.

A module M over ${\displaystyle {\mathcal {}}MU_{*}}$ is the same as a quasi-coherent sheaf ${\displaystyle {\mathcal {F}}}$ over ${\displaystyle {\text{Spec }}L}$, where L is the Lazard ring. If ${\displaystyle M={\mathcal {}}MU_{*}(X)}$, then M has the extra datum of a ${\displaystyle {\mathcal {}}MU_{*}MU}$ coaction. A coaction on the ring level corresponds to that ${\displaystyle {\mathcal {F}}}$ is an equivariant sheaf with respect to an action of an affine group scheme G. It is a theorem of Quillen that ${\displaystyle G\cong \mathbb {Z} [b_{1},b_{2},\dots ]}$ and assigns to every ring R the group of power series

${\displaystyle g(t)=t+b_{1}t^{2}+b_{2}t^{3}+\cdots \in R[[t]]}$.

It acts on the set of formal group laws ${\displaystyle {\text{Spec }}L(R)}$ via

${\displaystyle F(x,y)\mapsto gF(g^{-1}x,g^{-1}y)}$.

These are just the coordinate changes of formal group laws. Therefore, one can identify the stack quotient ${\displaystyle {\text{Spec }}L//G}$ with the stack of (1-dimensional) formal groups ${\displaystyle {\mathcal {M}}_{fg}}$ and ${\displaystyle M=MU_{*}(X)}$ 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 ${\displaystyle {\mathcal {F}}}$ which is flat over ${\displaystyle {\mathcal {M}}_{fg}}$ in order that ${\displaystyle MU_{*}(X)\otimes _{MU_{*}}M}$ is a homology theory. The Landweber exactness theorem can then be interpreted as a flatness criterion for ${\displaystyle {\mathcal {M}}_{fg}}$ (see Lurie 2010).

While the LEFT is known to produce (homotopy) ring spectra out of ${\displaystyle {\mathcal {}}MU_{*}}$, it is a much more delicate question to understand when these spectra are actually ${\displaystyle E_{\infty }}$-ring spectra. As of 2010, the best progress was made by Jacob Lurie. If X is an algebraic stack and ${\displaystyle X\to {\mathcal {M}}_{fg}}$ 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 ${\displaystyle M_{p}(n)}$ (the stack of 1-dimensional p-divisible groups of height n) and the map ${\displaystyle X\to M_{p}(n)}$ is etale, then this presheaf can be refined to a sheaf of ${\displaystyle E_{\infty }}$-ring spectra (see Goerss). This theorem is important for the construction of topological modular forms.

• 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 ${\displaystyle MU_{*}(MU)}$ and ${\displaystyle BP_{*}(BP)}$". American Journal of Mathematics. 98 (3): 591–610. doi:10.2307/2373808. JSTOR 2373808..
• Lurie, Jacob (2010). "Chromatic Homotopy Theory. Lecture Notes".