Group scheme

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


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.