# Idea

The Lazard ring is

and at the same time

# Details

The Lazard ring can be presented as by generators ${a}_{ij}$ with $i,j\in ℕ$

$L=ℤ\left[{a}_{ij}\right]/\left(\mathrm{relations}1-3\mathrm{below}\right)$L = \mathbb{Z}[a_{i j}] / (relations 1-3 below)

and relatins as follows

1. ${a}_{ij}={a}_{ji}$

2. ${a}_{10}={a}_{01}=1$; $\forall i\ne 0:{a}_{i0}=0$

3. the obvious associativity relation

the universal formal group law is the formal power series

$\ell \left(x,y\right)=\sum _{i,j}{a}_{ij}{x}^{j}{y}^{j}\in L\left[\left[x,y\right]\right]$\ell(x,y) = \sum_{i,j} a_{i j} x^j y^j \in L[[x,y]]

in two variables with coefficients in the Lazard ring.

For any ring $S$ with formal group law $g\left(x,y\right)\in S\left[\left[x,y\right]\right]$ there is a unique morphism $L\to S$ that sends $\ell$ to $g$.

We now describe Quillen’s theorem on how the Lazard ring is the cohomlogy ring of peridodic complex cobrdism theory over the point.

Theorem (Quillen) Let $MP$ denote the peridodic complex cobordism cohomology theory. Its cohomology ring $MP\left(*\right)$ over the point together with its formal group law is naturally isomorphic to the universal Lazard ring with its formal group law $\left(L,\ell \right)$.

This can be used to make a cohomology theory out of a formal group law $\left(R,f\left(x,y\right)\right)$. Namely, one can use the classifying map $MP\left(*\right)\to R$ to build the tensor product

${E}^{n}\left(X\right):=M{P}^{n}\left(X\right){\otimes }_{MP\left(*\right)}R,$E^n(X) := M P^n(X) \otimes_{M P({*})} R,

for any $n\in ℤ$. This construction could however break the left exactness condition. However, $E$ built this way will be left exact if the ring morphism $MP\left(*\right)\to R$ is a flat morphism. This is the Landweber exactness condition (or maybe slightly stronger).

# related entries

for some context see

or this blog entry

# References:

Daniel Quillen: On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. Volume 75, Number 6 (1969), 1293-1298.