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Landweber exactness

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Cohomology

cohomology

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higher algebra

universal algebra

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Contents

Idea

The Landweber exactness criterion determines whether a given formal group law arises as the formal group law defined by a weakly periodic cohomology theory.

Notice that since every formal group law over a ring RR is classified by a ring homomorphism f:MP(*)Rf : MP({\ast}) \to R where by Quillen's theorem MP(*)MP({\ast}) is the Lazard ring. So for every formal group one obtains a contravariant functor on topological spaces given by the extension of scalars-assignment

XA f n(X):=MP n(X) MP(*)R, X \mapsto A_f^n(X) := MP^n(X) \otimes_{MP({*})} R \,,

where MP MP^\bullet denotes the complex cobordism cohomology theory and where the tensor product is taken using the RR-module structure on MP(*)MP({*}) induced by ff.

The point of Landweber-exactness is that if ff is Landweber exact (i.e. if the corresponding formal group law is) then this construction defines a cohomology theory A ()A^\bullet(-).

Definition

Proposition

Landweber criterion Let f(x,y)f(x,y) be a formal group law and pp a prime, v iv_i the coefficient of x p ix^{p^i} in

[p] f(x)x+ f+ fxpsummands. [p]_f(x) \coloneqq \underset{p\,\text{summands}}{\underbrace{x+_f\cdots+_f x}} \,.

If v 0,,v iv_0,\ldots,v_i form a regular sequence for all pp and ii then f(x,y)f(x,y) is Landweber exact and hence gives a cohomology theory via the the formula above.

See at Landweber exact functor theorem

Examples

Example

Let g a(x,y)=x+yg_a(x,y)=x+y be the formal additive group. Then [p] a(x)=px[p]_a(x)= p x and so v 0=pv_0=p, v i=0v_i=0 for all i1i\ge1. The regularity condtions imply that the zero map R/(p)R/(p)R/(p)\to R/(p) must be injective. This implies that RR contains the rational numbers as a subring.

Note that the ordinary cohomology HP *(X,R)= kH n+2k(X,R)HP^*(X,R)=\prod_k H^{n+2k}(X,R) is a cohomology theory over any ring RR.

Example

g m(x,y)=xyg_m(x,y)=xy, [p] m(x)=(x+1) p1[p]_m(x)=(x+1)^p-1, v 0=pv_0=p, v 1=1v_1=1, v i=0v_i=0 for all i>1i \gt 1. The regularity conditions are trivial. Hence we know that K *(X)=MP *(X) MP()K^*(X)=MP^*(X)\otimes_{MP(\bullet)} \mathbb{Z} is a cohomology theory.

References

Revised on June 6, 2017 14:01:57 by Urs Schreiber (88.77.226.246)