Landweber exact functor theorem

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Short description: Theorem relating to algebraic topology

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 [math]\displaystyle{ MU_*(*) = MU_* \cong \Z[x_1,x_2,\dots] }[/math], where the degree of [math]\displaystyle{ x_i }[/math] is [math]\displaystyle{ 2i }[/math]. This is isomorphic to the graded Lazard ring [math]\displaystyle{ \mathcal{}L_* }[/math]. This means that giving a formal group law F (of degree [math]\displaystyle{ -2 }[/math]) over a graded ring [math]\displaystyle{ R_* }[/math] is equivalent to giving a graded ring morphism [math]\displaystyle{ L_*\to R_* }[/math]. Multiplication by an integer [math]\displaystyle{ n\gt 0 }[/math] is defined inductively as a power series, by

[math]\displaystyle{ [n+1]^F x = F(x, [n]^F x) }[/math] and [math]\displaystyle{ [1]^F x = x. }[/math]

Let now F be a formal group law over a ring [math]\displaystyle{ \mathcal{}R_* }[/math]. Define for a topological space X

[math]\displaystyle{ E_*(X) = MU_*(X)\otimes_{MU_*}R_* }[/math]

Here [math]\displaystyle{ R_* }[/math] gets its [math]\displaystyle{ MU_* }[/math]-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 [math]\displaystyle{ R_* }[/math] be flat over [math]\displaystyle{ MU_* }[/math], 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 [math]\displaystyle{ v_1,v_2,\dots \in MU_* }[/math] such that we have the following: Suppose that [math]\displaystyle{ M_* }[/math] is a graded [math]\displaystyle{ MU_* }[/math]-module and the sequence [math]\displaystyle{ (p,v_1,v_2,\dots, v_n) }[/math] is regular for [math]\displaystyle{ M }[/math], for every p and n. Then
[math]\displaystyle{ E_*(X) = MU_*(X)\otimes_{MU_*}M_* }[/math]
is a homology theory on CW-complexes.

In particular, every formal group law F over a ring [math]\displaystyle{ R }[/math] yields a module over [math]\displaystyle{ \mathcal{}MU_* }[/math] since we get via F a ring morphism [math]\displaystyle{ MU_*\to R }[/math].

Remarks

  • There is also a version for Brown–Peterson cohomology BP. The spectrum BP is a direct summand of [math]\displaystyle{ MU_{(p)} }[/math] with coefficients [math]\displaystyle{ \Z_{(p)}[v_1,v_2,\dots] }[/math]. 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 [math]\displaystyle{ BP_* }[/math] which are invariant under coaction of [math]\displaystyle{ BP_*BP }[/math] are the [math]\displaystyle{ I_n = (p,v_1,\dots, v_n) }[/math]. This allows to check flatness only against the [math]\displaystyle{ BP_*/I_n }[/math] (see Landweber, 1976).
  • The LEFT can be strengthened as follows: let [math]\displaystyle{ \mathcal{E}_* }[/math] be the (homotopy) category of Landweber exact [math]\displaystyle{ MU_* }[/math]-modules and [math]\displaystyle{ \mathcal{E} }[/math] the category of MU-module spectra M such that [math]\displaystyle{ \pi_*M }[/math] is Landweber exact. Then the functor [math]\displaystyle{ \pi_*\colon\mathcal{E}\to \mathcal{E}_* }[/math] is an equivalence of categories. The inverse functor (given by the LEFT) takes [math]\displaystyle{ \mathcal{}MU_* }[/math]-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 [math]\displaystyle{ x+y+xy }[/math]. The corresponding morphism [math]\displaystyle{ MU_*\to K_* }[/math] is also known as the Todd genus. We have then an isomorphism

[math]\displaystyle{ K_*(X) = MU_*(X)\otimes_{MU_*}K_*, }[/math]

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 [math]\displaystyle{ E(n) }[/math] and the Lubin–Tate spectra [math]\displaystyle{ E_n }[/math].

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

Modern reformulation

A module M over [math]\displaystyle{ \mathcal{}MU_* }[/math] is the same as a quasi-coherent sheaf [math]\displaystyle{ \mathcal{F} }[/math] over [math]\displaystyle{ \text{Spec }L }[/math], where L is the Lazard ring. If [math]\displaystyle{ M = \mathcal{}MU_*(X) }[/math], then M has the extra datum of a [math]\displaystyle{ \mathcal{}MU_*MU }[/math] coaction. A coaction on the ring level corresponds to that [math]\displaystyle{ \mathcal{F} }[/math] is an equivariant sheaf with respect to an action of an affine group scheme G. It is a theorem of Quillen that [math]\displaystyle{ G \cong \Z[b_1, b_2,\dots] }[/math] and assigns to every ring R the group of power series

[math]\displaystyle{ g(t) = t+b_1t^2+b_2t^3+\cdots\in Rt }[/math].

It acts on the set of formal group laws [math]\displaystyle{ \text{Spec }L(R) }[/math] via

[math]\displaystyle{ F(x,y) \mapsto gF(g^{-1}x, g^{-1}y) }[/math].

These are just the coordinate changes of formal group laws. Therefore, one can identify the stack quotient [math]\displaystyle{ \text{Spec }L // G }[/math] with the stack of (1-dimensional) formal groups [math]\displaystyle{ \mathcal{M}_{fg} }[/math] and [math]\displaystyle{ M = MU_*(X) }[/math] 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 [math]\displaystyle{ \mathcal{F} }[/math] which is flat over [math]\displaystyle{ \mathcal{M}_{fg} }[/math] in order that [math]\displaystyle{ MU_*(X)\otimes_{MU_*}M }[/math] is a homology theory. The Landweber exactness theorem can then be interpreted as a flatness criterion for [math]\displaystyle{ \mathcal{M}_{fg} }[/math] (see Lurie 2010).

Refinements to [math]\displaystyle{ E_\infty }[/math]-ring spectra

While the LEFT is known to produce (homotopy) ring spectra out of [math]\displaystyle{ \mathcal{}MU_* }[/math], it is a much more delicate question to understand when these spectra are actually [math]\displaystyle{ E \infty }[/math]-ring spectra. As of 2010, the best progress was made by Jacob Lurie. If X is an algebraic stack and [math]\displaystyle{ X\to \mathcal{M}_{fg} }[/math] 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 [math]\displaystyle{ M_p(n) }[/math] (the stack of 1-dimensional p-divisible groups of height n) and the map [math]\displaystyle{ X\to M_p(n) }[/math] is etale, then this presheaf can be refined to a sheaf of [math]\displaystyle{ E_\infty }[/math]-ring spectra (see Goerss). This theorem is important for the construction of topological modular forms.

See also

References



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