Group scheme

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Definition

Abstract definition

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

Concrete definition

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

Operation name	Arity of operation	Operation description and notation
Multiplication or product	2	A binary operation $\mu :G\times _{S}G\to G$
Identity element (or neutral element)	0	A map $e:S\to G$.
Inverse map	1	A unary operation $\iota :G\to G$.

satisfying the following compatibility conditions:

Condition name	Arity of space from which the maps are equal	Condition description	Comments
Associativity	3	We have that $\mu \circ (\mu \times _{S}\operatorname {id} )=\mu \circ (\operatorname {id} \times _{S}\mu )$ as maps from $G\times _{S}G\times _{S}G$	Note: We are abusing notation somewhat and treating $G\times _{S}G\times _{S}G$ 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 $\operatorname {id} =\mu \circ (e\times _{S}\operatorname {id} )=\mu \circ \mu \circ (\operatorname {id} \times _{S}e)$ as maps from $G$ to $G$.	Note that we are using the canonical identification of $G$ with $G\times _{S}S$ and $S\times _{S}G$.
Inverse element	1	We have $\mu \circ (\operatorname {id} \times _{S}\iota )=\mu \circ (\iota \times _{S}\operatorname {id} )=e$ as maps from $G$ to $G$.	Note that the map $e$ is a map $S\to G$, but can be treated as a map $G\to G$ via pre-composition with the unique map from $G$ to $S$, on account of $S$ being a final object.