Group scheme - Imported Knowlpedia Content

Definition

Fix a base scheme $S$ . A group scheme over $S$ is an $S$ -scheme $G\to S$ equipped with morphisms of schemes over $S$

m:G\times_S G\longrightarrow G, \qquad e:S\longrightarrow G, \qquad i:G\longrightarrow G,

called multiplication, identity, and inverse, satisfying the usual group axioms. The product in the multiplication map is a fiber product over the base.

Functor-of-points viewpoint

For every $S$ -scheme $T$ , the set of $T$ -valued points

G(T)=\operatorname{Hom}_S(T,G)

is a group, functorially in $T$ . This viewpoint exposes the group law on points defined over every test scheme, including points carrying nilpotent or family-valued information that ordinary geometric points may miss.

Actions

A right action of $G$ on an $S$ -scheme $P$ is a morphism

P\times_S G\longrightarrow P

over $S$ satisfying the identity and associativity axioms. Such actions appear in the torsor condition.

Remarks