# 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 | as formal power series | If is the binary operation denoting multiplication, then is associative. |

2 | Identity element | for some power series . Thus, | The element is the identity element for multiplication. |

3 | Inverses | There exists a power series such that and . | 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 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 . |

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 . In other words, if we translate by 1, this is just multiplication. Now use associativity of multiplication | commutative formal group law |