nLab Lazard ring




The Lazard ring is a commutative ring which is

and by Quillen's theorem also


The Lazard ring may be presented 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; i1:a i0=0\forall i \neq 1: a_{i 0} = 0

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

where we write

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

In other words, (x,y)\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.


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 ring homomorphism LSL \to S that sends \ell to gg.

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


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

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.

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.


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.


Last revised on July 16, 2023 at 16:51:09. See the history of this page for a list of all contributions to it.