nLab Landweber exactness

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

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

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

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

Definition

Proposition

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

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

If $v_0,\ldots,v_i$ form a regular sequence for all $p$ and $i$ then $f(x,y)$ is Landweber exact and hence gives a cohomology theory via the the formula above.

See at Landweber exact functor theorem

Examples

Example

Let $g_a(x,y)=x+y$ be the formal additive group. Then $[p]_a(x)= p x$ and so $v_0=p$, $v_i=0$ for all $i\ge1$. The regularity condtions imply that the zero map $R/(p)\to R/(p)$ must be injective. This implies that $R$ contains the rational numbers as a subring.

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

Example

$g_m(x,y)=xy$, $[p]_m(x)=(x+1)^p-1$, $v_0=p$, $v_1=1$, $v_i=0$ for all $i \gt 1$. The regularity conditions are trivial. Hence we know that $K^*(X)=MP^*(X)\otimes_{MP(\bullet)} \mathbb{Z}$ is a cohomology theory.

Related entries

• Landweber exact functor theorem

• A Survey of Elliptic Cohomology - formal groups and cohomology

References

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010,

  Lecture 15 Flat modules over $\mathcal{M}_{FG}$ (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 16 The Landweber exact functor theorem (pdf)