nLab
Lazard ring

Contents

Idea

The Lazard ring_ is a commutative ring which is

and by Quillen's theorem also

Definition

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 1-dimensional 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.

Properties

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:

Theorem

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.

Theorem

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).

Remark

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.

References

Revised on May 18, 2014 09:52:31 by Urs Schreiber (82.113.99.63)