# Idea

The Landweber exactness criterion determins if a given formal group law does arise 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 $f:\mathrm{MP}\left(*\right)\to R$ where $\mathrm{MP}\left(*\right)$ is the Lazard ring. So for every formal group one obtains a contravariant functor on topological spaces given by the assignment

$X↦{A}_{f}^{n}\left(X\right):={\mathrm{MP}}^{n}\left(X\right){\otimes }_{\mathrm{MP}\left(*\right)}R\phantom{\rule{thinmathspace}{0ex}},$X \mapsto A_f^n(X) := MP^n(X) \otimes_{MP({*})} R \,,

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

The point of Lazard-exactness is that if $f$ is Lazard exact (i.e. if the corresponding formal group law is) then this construciton defines a cohomology theory ${A}^{•}\left(-\right)$.

# Definition

Landweber criterion Let $f\left(x,y\right)$ be a formal group law and $p$ a prime, ${v}_{i}$ the coefficient of ${x}^{{p}^{i}}$ in $\left[p{\right]}_{f}\left(x\right)=x{+}_{f}\cdots {+}_{\mathrm{fx}}$. If ${v}_{0},\dots ,{v}_{i}$ form a regular sequence for all $p$ and $i$ then $f\left(x,y\right)$ is Lazard exact and hence gives a cohomology theory via the the formula above.

Example. ${g}_{a}\left(x,y\right)=x+y$, $\left[p{\right]}_{a}\left(x\right)=\mathrm{px}$, ${v}_{0}=p$, ${v}_{i}=0$ for all $i\ge 1$; regularity condtions imply that the zero map $R/\left(p\right)\to R/\left(p\right)$ must be injective. The last statement implies that $R$ contains the rational numbers as a subring.

Note that ${\mathrm{HP}}^{*}\left(X,R\right)={\prod }_{k}{H}^{n+2k}\left(X,R\right)$ is a cohomology theory over any ring $R$.

Example. ${g}_{m}\left(x,y\right)=\mathrm{xy}$, $\left[p{\right]}_{m}\left(x\right)=\left(x+1{\right)}^{p}-1$, ${v}_{0}=p$, ${v}_{1}=1$, ${v}_{i}=0$ for all $i>1$. The regularity conditions are trivial. Hence we know that ${K}^{*}\left(X\right)={\mathrm{MP}}^{*}\left(X\right){\otimes }_{\mathrm{MP}\left(•\right)}ℤ$ is a cohomology theory.

# related entries

Created on September 14, 2009 17:20:59 by Urs Schreiber (195.37.209.182)