Formal group law
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Contents
Definition
One-dimensional formal group law
Let be a commutative unital ring. A one-dimensional formal group law on
is a formal power series
in two variables, denoted
and
, such that:
Condition no. | Name | Description of condition | Interpretation |
---|---|---|---|
1 | Associativity | ![]() |
If ![]() ![]() |
2 | Identity element | ![]() ![]() ![]() |
The element ![]() |
3 | Inverses | There exists a power series ![]() ![]() ![]() |
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 such that
. 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 be a commutative unital ring. A
-dimensional formal group law is a collection of
formal power series
involving
variables
satisfying a bunch of conditions.
Before stating the conditions, we introduce some shorthand. Consider and
. Then,
is the
-tuple
.
Condition no. | Name | Description of condition in shorthand | Description of condition in longhand |
---|---|---|---|
1 | Associativity | ![]() |
For each ![]() ![]() ![]() ![]() ![]() |
2 | Identity element | ![]() ![]() |
For each ![]() ![]() ![]() ![]() |
3 | Inverse | There exists ![]() ![]() ![]() |
There exist ![]() ![]() |
Condition (3) is redundant, i.e., it can be deduced from (1) and (2).
A commutative formal group law is a formal group law such that
. 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 gives a group structure on the maximal ideal
in the ring
of formal power series in one variable over
.
A one-dimensional formal group law can also be interpreted to give a group structure over the image of the maximal ideal in any quotient ring of
; i.e., a ring of the form
.
A -dimensional formal group law over a commutative unital ring
gives a group structure on the set of
-tuples of formal power series in one variable over
.
For arbitrary algebras over 
Further information: formal group law functor from commutative algebras to groups
More generally, for any commutative -algebra
, if
is the set of nilpotent elements of
, then any
-dimensional formal group law
over
gives a group structure on the set
of
-tuples over
. The formal group law thus gives a functor from the category of commutative
-algebras to the category of groups.
A particular case of this is when is a local ring and
is its unique maximal ideal. In this case, we get what is called a
-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 | ![]() |
addition is associative in the base ring | commutative formal group law |
multiplicative formal group law | ![]() |
rewrite as ![]() |
commutative formal group law |