Group Scheme - Definition

Definition

A group scheme is a group object in a category of schemes that has fiber products and some final object S. That is, it is an S-scheme G equipped with one of the equivalent sets of data

• a triple of morphisms μ: G ×_S G → G, e: S → G, and ι: G → G, satisfying the usual compatibilities of groups (namely associativity of μ, identity, and inverse axioms)
• a functor from schemes over S to the category of groups, such that composition with the forgetful functor to sets is equivalent to the presheaf corresponding to G under the Yoneda embedding.

A homomorphism of group schemes is a map of schemes that respects multiplication. This can be precisely phrased either by saying that a map f satisfies the equation fμ = μ(f × f), or by saying that f is a natural transformation of functors from schemes to groups (rather than just sets).

A left action of a group scheme G on a scheme X is a morphism G ×_S X→ X that induces a left action of the group G(T) on the set X(T) for any S-scheme T. Right actions are defined similarly. Any group scheme admits natural left and right actions on its underlying scheme by multiplication and conjugation. Conjugation is an action by automorphisms, i.e., it commutes with the group structure, and this induces linear actions on naturally derived objects, such as its Lie algebra, and the algebra of left-invariant differential operators.

An S-group scheme G is commutative if the group G(T) is an abelian group for all S-schemes T. There are several other equivalent conditions, such as conjugation inducing a trivial action, or inversion map ι being a group scheme automorphism.