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Demazure, lectures on p-divisible groups, I.8, constant- and étale schemes

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Sch kSch_k is copowered (= tensored) over SetSet. We define the constant kk-scheme on a set EE by

E k:=ESp kk= eESp kkE_k:=E\otimes Sp_k k=\coprod_{e\in E}Sp_k k

For a scheme XX we compute M k(E k,E)=Set(Sp kk,X) E=X(k) E=Set(E,X(k))M_k(E_k,E)=Set(Sp_k k,X)^E=X(k)^E=Set(E,X(k)) and see that there is an adjunction

(() k()(k)):Sch kSet((-)_k\dashv (-)(k)):Sch_k\to Set

A constant formal scheme is defined to be a completion of constant scheme. The completion functor induces an equivalence between the category of constant schemes and the category of constant formal schemes.

An étale kk-scheme is defined to be a directed colimit of kk-spectra Sp kk Sp_k k^\prime of finite separable field-extensions k k^\prime of kk.

An étale formal kk-scheme is defined to be a directed colimit of formal kk-spectra Spf kk Spf_k k^\prime of finite separable field-extensions k k^\prime of kk.

Remark

Let XX be a kk-scheme or a formal kk-scheme. Then the following statements are equivalent:

  1. XX is étale.

  2. X kcl(k)X\otimes_k cl(k) is constant.

  3. X kk sX\otimes_k k_s is constant.

where cl(k)cl(k) denotes an algebraic closure of kk, k sk_s denotes the subextension of cl(k)cl(k) consisting of all separable elements of cl(k)cl(k) and k\otimes_k denotes skalar extension.

Proposition

Let XX be a kk-formal scheme (resp. a locally algebraic scheme) then XX is étale iff the Frobenius morphism F X:XX (p)F_X:X\to X^{(p)}is a monomorphism (resp. an isomorphism).

Theorem

(fundamental theorem of Galois theory)

The functor

{Sch etGal(k s/k)Mod XX(k s)\begin{cases} Sch_{et}\to Gal(k_s / k)- Mod \\ X\mapsto X(k_s) \end{cases}

from étale schemes to the category of Galois modules Gal(k s/s)ModGal(k_s/s)-Mod is an equivalence of categories.

Remark

The completion functor

{Sch etfSch et XX^\begin{cases} Sch_{et}\to fSch_{et} \\ X\mapsto \hat X \end{cases}

is an equivalence of categories.

Revised on June 7, 2012 19:42:00 by Stephan Alexander Spahn (79.227.183.2)