Demazure, lectures on p-divisible groups, II.2, constant and étale k-groups
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Let be a field.
If is a group then the constant scheme? on carries a group structure. Such an is called constant -group.
Definition and Remark
Let denote the field extension of consisting of separable elements. Let denote the Galois group of this extension. Then the functor
is an equivalence between the category of étale -schemes and that of sets equipped with a group action of.
Moreover this gives a functor
which is an equivalence between the category of étale -groups and that of groups equipped with a group action of . Note that commutative -groups are called Galois modules, too.
A -group scheme is étale iff its coefficient extension along is a constant k-scheme.
- Michel Demazure, lectures on p-divisible groups web