Contents

# Contents

## Idea

The Lazard ring is a commutative ring which is

and by Quillen's theorem also

## Definition

The Lazard ring may be presented by generators $a_{i j}$ with $i,j \in \mathbb{N}$

$L = \mathbb{Z}[a_{i j}] / (relations\;1,2,3\;below)$

and relations as follows

1. $a_{i j} = a_{j i}$

2. $a_{10} = a_{01} = 1$; $\forall i \neq 1: a_{i 0} = 0$

3. the obvious associativity relations imposed by $\ell(x, \ell(y, z)) = \ell(\ell(x, y), z)$

where we write

$\ell(x,y) = \sum_{i,j} a_{i j} x^i y^j \in L[[x,y]].$

In other words, $\ell(x, y)$ becomes the universal 1-dimensional formal group law as a formal power series in two variables with coefficients in the Lazard ring, Theorem below.

## Properties

### As classifying ring for formal group laws

###### Theorem

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

review includes (Hopkins 99, theorem 2.3, theorem 2.5)

###### Remark

Passing to formal dual ring spectra, this says that $Spec(L)$ is something like the moduli space for formal groups. By Quillen's theorem on MU, the lift of $L$ to higher algebra is the E-infinity ring MU and the E-infinity ring spectrum $Spec(MU)$ is something like the derived moduli stack for formal group laws.

### Lazard’s theorem

Lazard's theorem states:

###### Theorem

The Lazard ring is isomorphic to a graded polynomial ring

$L \simeq \mathbb{Z}[t_1, t_2, \cdots]$

with the variable $t_i$ in degree $2 i$.

review includes (Hopkins 99, theorem 2.5, Lurie 10, lect 2, theorem 4)

### As the complex cobordism cohomology ring

By Quillen's theorem on MU the Lazard ring is the cohomology ring of complex cobordism cohomology theory.

###### Theorem

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

###### Remark

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

$E^n(X) := M P^n(X) \otimes_{M P({*})} R,$

for any $n\in\mathbb{Z}$. This construction could however break the left exactness condition. However, $E$ built this way will be left exact if the ring morphism $M P({*}) \to R$ is a flat morphism. This is the Landweber exactness condition (or maybe slightly stronger). See at Landweber exact functor theorem.