Constant finite group scheme - Imported Knowlpedia Content

Definition

To let the abstract Galois group $G=\operatorname{Gal}(K/F)$ act on $\operatorname{Spec}K$ over $\operatorname{Spec}F$ , regard $G$ as a scheme: put one copy of the base at every group element, with multiplication dictated by the multiplication table of $G$ .

Let $G$ be a finite group and $S$ a scheme. The constant finite group scheme $G_S$ is

G_S:=\coprod_{g\in G}S,

with multiplication, identity, and inverse induced by those of $G$ . If $S=\operatorname{Spec}R$ , then

G_S\cong\operatorname{Spec}\!\left(\prod_{g\in G}R\right).

It is finite étale over $S$ , and its $T$ -points are the locally constant maps from $T$ to the finite set $G$ . It supplies the group scheme in the Galois extension as étale torsor construction.