Landweber exact functor theorem

Theorem relating to algebraic topology

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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 $MU_{*}(*)=MU_{*}\cong \mathbb {Z}$ , 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

^{F}x=F(x,^{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 $\mathbb {Z} _{(p)}$ . 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 $G\cong \mathbb {Z}$ 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]

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.