nLab Demazure, lectures on p-divisible groups, II.2, constant and étale k-groups

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

Let $k$ be a field.

Definition

If $E$ is a group then the constant scheme? $E_k$ on $E$ carries a group structure. Such an $E_k$ is called constant $k$-group.

Definition and Remark

Let $k_s$ denote the field extension of $k$ consisting of separable elements. Let $\Pi:=Gal(k_s/k)$ denote the Galois group of this extension. Then the functor

$$(-)(k_s):\begin{cases} et Sch_k &\to& Gal(k_s/k)-Set \\ X &\to& X(k_s) \end{cases}$$

is an equivalence between the category of étale $k$-schemes and that of sets equipped with a group action of$Gal(k_s/k)$.

Moreover this gives a functor

$$(-)(k_s):\begin{cases} et Gr_k &\to& Gal(k_s/k)-Mod \\ X &\to& X(k_s) \end{cases}$$

which is an equivalence between the category of étale $k$-groups and that of groups equipped with a group action of $Gal(k_s/k)$. Note that commutative $Gal(k_s/k)$-groups are called Galois modules, too.

A $k$-group scheme is étale iff its coefficient extension $X\otimes_k k_s$ along $k\hookrightarrow k_s$ is a constant k-scheme.

• Michel Demazure, lectures on p-divisible groups web