# Group scheme

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## Definition

### Abstract definition

A **group scheme** over is a group object in a category of schemes with fiber products and final object .

### Concrete definition

A **group scheme** over is a -scheme (in a category of schemes that admits fiber products over ) equipped with the following operations:

Operation name | Arity of operation | Operation description and notation |
---|---|---|

Multiplication or product | 2 | A binary operation |

Identity element (or neutral element) | 0 | A map . |

Inverse map | 1 | A unary operation . |

satisfying the following compatibility conditions:

Condition name | Arity of space from which the maps are equal | Condition description | Comments |
---|---|---|---|

Associativity | 3 | We have that as maps from | Note: We are abusing notation somewhat and treating as well defined, but the rigorous way of doing it is to actually use the associativity isomorphism to go between the two versions of it. |

Identity element (or neutral element) | 1 | We have as maps from to . | Note that we are using the canonical identification of with and . |

Inverse element | 1 | We have as maps from to . | Note that the map is a map , but can be treated as a map via pre-composition with the unique map from to , on account of being a final object. |