# Formal group law

This is a variation of group|Find other variations of group | Read a survey article on varying group

## Definition

### One-dimensional formal group law

Let $R$ be a commutative unital ring. A one-dimensional formal group law on $R$ is a formal power series $F$ in two variables, denoted $x$ and $y$, such that:

Condition no. Name Description of condition Interpretation
1 Associativity $\! F(x,F(y,z)) = F(F(x,y),z)$ as formal power series If $F$ is the binary operation denoting multiplication, then $F$ is associative.
2 Identity element $\! F(x,y) = x + y + xyG(x,y)$ for some power series $G$. Thus, $F(x,0) = x, F(0,y) = y$ The element $0$ is the identity element for multiplication.
3 Inverses There exists a power series $m(x)$ such that $m(0) = 0$ and $F(x,m(x)) = 0$. Every element has an inverse for multiplication.

Condition (3) is redundant, i.e., it can be deduced from (1) and (2).

A one-dimensional commutative formal group law is a one-dimensional formal group law $F$ such that $F(x,y) = F(y,x)$. Two important examples of commutative formal group laws, that make sense for any ring, are the additive formal group law and the multiplicative formal group law.

### Higher-dimensional formal group law

Let $R$ be a commutative unital ring. A $n$-dimensional formal group law is a collection of $n$ formal power series $F_i$ involving $2n$ variables $(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n)$ satisfying a bunch of conditions.

Before stating the conditions, we introduce some shorthand. Consider $x = (x_1,x_2,\dots,x_n)$ and $y = (y_1,y_2,\dots,y_n)$. Then, $F(x,y)$ is the $n$-tuple $(F_1(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),F_2(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),\dots,F_n(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n))$.

Condition no. Name Description of condition in shorthand Description of condition in longhand
1 Associativity $\! F(x,F(y,z)) = F(F(x,y),z)$ For each $i$ from $1$ to $n$, $F_i(x_1,x_2,\dots,x_n,F_1(y_1,y_2,\dots,y_n,z_1,z_2,\dots,z_n),F_2(y_1,y_2,\dots,y_n,z_1,z_2,\dots,z_n),\dots,F_n(y_1,y_2,\dots,y_n,z_1,z_2,\dots,z_n))$ equals $F_i(F_1(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),F_2(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),F_n(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),z_1,z_2,\dots,z_n)$.
2 Identity element $\! F(x,y) = x + y +$ terms of higher degree, so $\! F(x,0) = F(0,x) = x$ For each $i$, $\! F_i(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n) = x_i + y_i +$ terms of higher degree (each further term is a product that involves at least one $x_j$ and one $y_k$.
3 Inverse There exists $m$, a collection of $n$ formal power series in one variable, such that $F(x,m(x)) = 0$ formally. There exist $m_i, 1 \le i \le n$, all formal power series in one variable, such that $\! F(x_1,x_2,\dots,x_n,m_1(x_1,x_2,\dots,x_n),m_2(x_1,x_2,\dots,x_n),\dots,m_n(x_1,x_2,\dots,x_n)) = 0$.

Condition (3) is redundant, i.e., it can be deduced from (1) and (2).

A commutative formal group law is a formal group law $F$ such that $F(x,y) = F(y,x)$. Two important examples of commutative formal group laws, that make sense for any ring, are the additive formal group law and the multiplicative formal group law.

## Interpretation as group

### For power series rings

A one-dimensional formal group law over a commutative unital ring $R$ gives a group structure on the maximal ideal $\langle t \rangle$ in the ring $R[[t]]$ of formal power series in one variable over $R$.

A one-dimensional formal group law can also be interpreted to give a group structure over the image of the maximal ideal $\langle t \rangle$ in any quotient ring of $R[[t]]$; i.e., a ring of the form $R[[t]]/(t^n) \cong R[t]/(t^n)$.

A $n$-dimensional formal group law over a commutative unital ring $R$ gives a group structure on the set of $n$-tuples of formal power series in one variable over $R$.

### For arbitrary algebras over $R$

Further information: formal group law functor from commutative algebras to groups

More generally, for any commutative $R$-algebra $S$, if $N$ is the set of nilpotent elements of $S$, then any $n$-dimensional formal group law $F$ over $S$ gives a group structure on the set $N^n$ of $n$-tuples over $N$. The formal group law thus gives a functor from the category of commutative $R$-algebras to the category of groups.

A particular case of this is when $R$ is a local ring and $M$ is its unique maximal ideal. In this case, we get what is called a $R$-standard group.

## Examples

### Examples of one-dimensional formal group laws

Name of law Expression for law Crude explanation for associativity Additional properties
additive formal group law $x + y$ addition is associative in the base ring commutative formal group law
multiplicative formal group law $x + y + xy$ rewrite as $(x + 1)(y + 1) - 1$. In other words, if we translate by 1, this is just multiplication. Now use associativity of multiplication commutative formal group law