Formal group law

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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