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 \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 \mathcal{}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 \mathcal{}R_* gets its \mathcal{}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 \mathcal{}R_* is flat over \mathcal{}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,\cdots \in MU_* such that we have the following: Suppose that \mathcal{}M_* is a graded \mathcal{}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

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 ConnerFloyd 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 JohnsonWilson 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 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 \mathcal{}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.

References

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