Understanding the Lazard Ring

When I define a polynomial, I am simply handing you an indexed collection of coefficients.

A polynomial with two variables, \(x, y\) and coefficients \(c\), is of the form:

\(F(x, y) = \sum\limits_{ij} c_{ij} x^i y^j\)

The coefficients of a polynomial form a ring. In other words, the coefficients \(c_{ij}\) are members of a coefficient ring \(R\). When we say \(F\) is over \(R\), we mean that \(F\) has coefficients in \(R\).

Screenshot from 2015-01-03 16:13:31

Example: The polynomial
\(F(x,y) = 7 + 5xy^2 + 2x^3\) can be written out as
\(F(x,y) =7x^0y^0 + 5x^1y^2 + 2x^3y^0\) such that
\(c_{00} = 7\), \(c_{12} = 5\), \(c_{30} = 2\), and the rest of \(c_{ij} = 0\).

Alright, now let’s change the coefficients; reassign \(c_{00} = 4\), \(c_{78} = 3\), and all other \(c_{ij} = 0\).

Out pops a very different polynomial \(P(x,y) = 4 + 3x^7y^8\).

In other words, by altering the coefficients \(c_{ij}\) of \(F(x,y)\) via a ring homomorphism \(u: R \to R’\) (from the coefficient ring \(c_{ij} \in R\) to a coefficient ring \(u(c_{ij}) \in R’\))…

Screenshot from 2015-01-03 16:15:03

… we can get from \(F(x,y)\) to any other polynomial \(F'(x,y)\).

Screenshot from 2015-01-03 16:17:59

What’s a group-y polynomial?

Intuitively, a polynomial is “group-y” if there’s a constraint on our coefficients that forces the polynomial to satisfy the laws of a commutative group.

Concretely, a group-y polynomial is an operation of the form \(F(x,y) = \sum\limits_{ij}c_{ij}x^iy^j\) such that

  1. commutativity: \(F(x,y) = F(y,x)\)
  2. identity: \(F(x, 0) = x = F(0, x)\)
  3. associativity: \(F(F(x,y), z) – F(x, F(y,z)) = 0\)

We can make sure that our polynomial satisfies these constraints! How? We mod out our coefficient ring \(c_{ij}\) by the ideal \(I\) — generated by the relations among \(c_{ij}\) imposed by these constraints.

If you’d like to see the explicit relations, I wrote a cry for help post on stack overflow.

The ring of coefficients that results is called the Lazard ring \(L = \mathbb{Z}[c_{ij}]/I\).

It’s important to note here that group-y polynomials are morphisms out of the Lazard ring, not elements of the Lazard ring (i.e., that an assignment of values to each of the \(c_{ij}\) describes a group-y polynomial, but the ring of the \(c_{ij}\) itself is just a polynomial ring).

In other words, group-y polynomials \(f(x,y)\) are morphisms out of the Lazard ring, not elements of the Lazard ring.

Screen Shot 2015-03-04 at 2.28.45 PM

More formally: for any ring \(R\) with group-y polynomial \(f(x,y) \in R[[x,y]]\) there is a unique morphism \(L \to R\) that sends \(\ell \mapsto f\).

\(L \to R \simeq F_R\)

(where \(F_R\) denotes a group-y polynomial with coefficients in \(R\))

This makes sense. If it doesn’t, then scroll up a bit! As we saw above, a change of base ring corresponds to a new group-y polynomial.

Grading the Lazard Ring

As we’ve noted, the Lazard ring \(L = \mathbb{Z}[c_{ij}]/I\) is the quotient of a polynomial ring on the \(c_{ij}\) by some relations.

Lazard proved that it is also a polynomial ring (no relations) on a different set of generators. More specifically, \(\alpha\) is a graded ring isomorphism:

\(\mathbb{Z}[c_{ij}]/I \xrightarrow{\alpha} \mathbb{Z}[t_1, t_2 …]\)

(where the degree of \(t_i\) is \(2i\)).

Lurie talks about this a bit (Theorem 4, Lecture 2: The Lazard Ring), but I have yet to understand the proof myself.

Thanks to Alex Mennen for deriving constraints the associativity condition puts on our coefficients; thanks to Qiaochu Yuan and Josh Grochow for kindly explaining some basic details and mechanics of the Lazard ring.

For your ventures ahead…

In this post, I have committed two semantic sins in the name of pedagogy. Namely, sins of oversimplification which I’ll attempt to rectify s.t. you aren’t hopelessly confused by the literature:

  1. group-y polynomial = “1-dimensional abelian formal group law”
  2. polynomial = “formal power series”

Conventionally, a “polynomial” is a special case of a formal power series (in which we expect that our variables evaluate to a number – useful if we care about convergence).

polynomials \(\subset\) formal power series

The polynomial ring \(R[x]\) is the ring of all polynomials (in two variables) over a given coefficient ring \(R\).
The ring of formal power series \(R[[x]]\) is the ring of all formal power series (in two variables) over a given coefficient ring \(R\).

polynomial ring \(\subset\) ring of formal power series
R[x] \(\subset\) R[[x]]


Groupes de Lie formels à un paramètre
Groupes analytiques en caractéristique 0
Groupes de Lie algébriques (travaux de Chavelley)
Formal Group Laws – Ravenel