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

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Let kk be a field.

Definition

If EE is a group then the constant scheme? E kE_k on EE carries a group structure. Such an E kE_k is called constant kk-group.

Definition and Remark

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

()(k s):{etSch k Gal(k s/k)Set X X(k s)(-)(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 kk-schemes and that of sets equipped with a group action ofGal(k s/k)Gal(k_s/k).

Moreover this gives a functor

()(k s):{etGr k Gal(k s/k)Mod X X(k s)(-)(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 kk-groups and that of groups equipped with a group action of Gal(k s/k)Gal(k_s/k). Note that commutative Gal(k s/k)Gal(k_s/k)-groups are called Galois modules, too.

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

  • Michel Demazure, lectures on p-divisible groups web
Revised on May 27, 2012 13:25:32 by Stephan Alexander Spahn (79.227.168.80)