Lazard ring



The Lazard ring_ is a commutative ring which is

and by Quillen's theorem also


The Lazard ring can be presented as by generators a ija_{i j} with i,ji,j \in \mathbb{N}

L=[a ij]/(relations1,2,3below) L = \mathbb{Z}[a_{i j}] / (relations\;1,2,3\;below)

and relations as follows

  1. a ij=a jia_{i j} = a_{j i}

  2. a 10=a 01=1a_{10} = a_{01} = 1; i0:a i0=0\forall i \neq 0: a_{i 0} = 0

  3. the obvious associativity relation

the universal formal group law is the formal power series

(x,y)= i,ja ijx jy jL[[x,y]] \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.


As classifying ring for formal group laws

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

Lazard’s theorem

Lazard's theorem states:


The Lazard ring is isomorphic to a graded polynomial ring

L[t 1,t 2,] L \simeq \mathbb{Z}[t_1, t_2, \cdots]

with the variable t it_i in degree 2i2 i.

(e.g. Lurie 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.


Let MPM P denote the peridodic complex cobordism cohomology theory. Its cohomology ring MP(*)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,)(L,\ell).


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

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

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


Revised on November 25, 2013 06:22:56 by Zoran Škoda (