Landweber exact functor theorem

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 $$MU_*(*) = MU_* \cong \Z[x_1,x_2,\dots]$$, 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 $$[1]^F x = x.$$

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 $$\Z_{(p)}[v_1,v_2,\dots]$$. 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 $$G \cong \Z[b_1, b_2,\dots]$$ and assigns to every ring R the group of power series
 * $$g(t) = t+b_1t^2+b_2t^3+\cdots\in Rt$$.

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.