# Group scheme

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This is a variation of group|Find other variations of group | Read a survey article on varying group

## 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.