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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.

## Statement

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 ${\displaystyle MU_{*}\to R}$ .

## Remarks

• 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).

## 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 ${\displaystyle x+y+xy}$ . 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.

## Modern reformulation

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).

## Refinements to ${\displaystyle E_{\infty }}$-ring spectra

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.

## References

• 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".