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Landweber exactness

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 $f:\mathrm{MP}(*)\to R$ where $\mathrm{MP}(*)$ is the Lazard ring. So for every formal group one obtains a contravariant functor on topological spaces given by the assignment

$$X\mapsto A_f^n(X):={\mathrm{MP}}^n(X){\otimes }_{\mathrm{MP}(*)}R,$$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 $\mathrm{MP}(*)$ 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 $A^{•}(-)$.

Definition

Landweber criterion Let $f(x,y)$ be a formal group law and $p$ a prime, $v_i$ the coefficient of $x^{p^i}$ in $[p{]}_f(x)=x{+}_f\cdots {+}_{\mathrm{fx}}$. If $v_0,\dots ,v_i$ form a regular sequence for all $p$ and $i$ then $f(x,y)$ is Lazard exact and hence gives a cohomology theory via the the formula above.

Example. $g_a(x,y)=x+y$, $[p{]}_a(x)=\mathrm{px}$, $v_0=p$, $v_i=0$ for all $i\ge 1$; regularity condtions imply that the zero map $R/(p)\to R/(p)$ must be injective. The last statement implies that $R$ contains the rational numbers as a subring.

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

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

related entries

• A Survey of Elliptic Cohomology - formal groups and cohomology