Commutative formal group law
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|
|1||Associativity||as formal power series|
|2||Identity element||for some power series . Thus,|
|3||Inverses||There exists a power series such that and|
Note that conditions (1)-(3) alone define formal group law (which is not necessarily commutative). Also, condition (3) is redundant.
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 from to , equals .|
|2||Identity element||terms of higher degree, so||For each , terms of higher degree (each further term is a product that involves at least one and one .|
|3||Inverse||There exists , a collection of formal power series in one variable, such that formally.||There exist , all formal power series in one variable, such that .|
Note that conditions (1)-(3) define a (not necessarily commutative) formal group law. Condition (3) is redundant, i.e., it can be deduced from (1) and (2).