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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 $\Pi :=\mathrm{Gal}(k_s/k)$ denote the Galois group of this extension. Then the functor

$$(-)(k_s):\{\begin{array}{lll}\mathrm{et}{\mathrm{Sch}}_k& \to & \mathrm{Gal}(k_s/k)-\mathrm{Set}\\ X& \to & X(k_s)\end{array}$$(-)(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$\mathrm{Gal}(k_s/k)$.

Moreover this gives a functor

$$(-)(k_s):\{\begin{array}{lll}\mathrm{et}{\mathrm{Gr}}_k& \to & \mathrm{Gal}(k_s/k)-\mathrm{Mod}\\ X& \to & X(k_s)\end{array}$$(-)(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 $\mathrm{Gal}(k_s/k)$. Note that commutative $\mathrm{Gal}(k_s/k)$-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