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## 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 $MU_{*}(*)=MU_{*}\cong \mathbb {Z} [x_{1},x_{2},\dots ]$ , where the degree of $x_{i}$  is $2i$ . This is isomorphic to the graded Lazard ring ${\mathcal {}}L_{*}$ . 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 $^{F}x=x.$

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

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

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 $v_{1},v_{2},\dots \in MU_{*}$  such that we have the following: Suppose that $M_{*}$  is a graded $MU_{*}$ -module and the sequence $(p,v_{1},v_{2},\dots ,v_{n})$  is regular for $M$ , for every p and n. Then
$E_{*}(X)=MU_{*}(X)\otimes _{MU_{*}}M_{*}$
is a homology theory on CW-complexes.

In particular, every formal group law F over a ring $R$  yields a module over ${\mathcal {}}MU_{*}$  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 $\mathbb {Z} _{(p)}[v_{1},v_{2},\dots ]$ . 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 $BP_{*}$  which are invariant under coaction of $BP_{*}BP$  are the $I_{n}=(p,v_{1},\dots ,v_{n})$ . This allows to check flatness only against the $BP_{*}/I_{n}$  (see Landweber, 1976).
• The LEFT can be strengthened as follows: let ${\mathcal {E}}_{*}$  be the (homotopy) category of Landweber exact $MU_{*}$ -modules and ${\mathcal {E}}$  the category of MU-module spectra M such that $\pi _{*}M$  is Landweber exact. Then the functor $\pi _{*}\colon {\mathcal {E}}\to {\mathcal {E}}_{*}$  is an equivalence of categories. The inverse functor (given by the LEFT) takes ${\mathcal {}}MU_{*}$ -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 $x+y+xy$ . The corresponding morphism $MU_{*}\to K_{*}$  is also known as the Todd genus. We have then an isomorphism

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

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

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

## Modern reformulation

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

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

It acts on the set of formal group laws ${\text{Spec }}L(R)$  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 ${\text{Spec }}L//G$  with the stack of (1-dimensional) formal groups ${\mathcal {M}}_{fg}$  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 ${\mathcal {F}}$  which is flat over ${\mathcal {M}}_{fg}$  in order that $MU_{*}(X)\otimes _{MU_{*}}M$  is a homology theory. The Landweber exactness theorem can then be interpreted as a flatness criterion for ${\mathcal {M}}_{fg}$  (see Lurie 2010).

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

While the LEFT is known to produce (homotopy) ring spectra out of ${\mathcal {}}MU_{*}$ , 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 $X\to {\mathcal {M}}_{fg}$  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 $M_{p}(n)$  (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.