nLab Landweber exactness

Redirected from "Landweber exactness criterion".
Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher algebra

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)=xy+x+yg_m(x,y)=xy+x+y, [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

Last revised on May 29, 2024 at 14:20:44. See the history of this page for a list of all contributions to it.