# 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 $$R$$. 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 $$u(c_{ij}) \in R’$$)… … we can get from $$F(x,y)$$ to any other polynomial $$F'(x,y)$$. #### 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. 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.

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.

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]]