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etale scheme

Étale schemes

Definition

Let kk be a field.

An étale kk-scheme is defined to be a directed colimit of kk-spectra Sp kkSp_k k' of finite separable field-extensions kk' of kk.

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

Properties

We give a characterization of étale kk-schemes and étale formal kk-schemes in terms of constant schemes?:

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

E kESp kk= eESp kkE_k \coloneqq 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))\colon 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.

Remark

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

  1. XX is étale.

  2. X kk¯X \otimes_k \overline k is constant.

  3. X kk sX \otimes_k k_s is constant. where k¯\overline k denotes an algebraic closure of kk, k sk_s denotes the subextension of k¯\overline k consisting of all separable elements of k¯\overline k and k\otimes_k denotes skalar extension.

Proof

XX is étale iff its skalar extension X kk sX\otimes_k k_s is étale. And a k sk_s-scheme is étale iff it is constant.

Proposition

The functor

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

from the category of étale schemes to the category of sets equipped with an action of the absolute Galois group is an equivalence of categories.

This statement is an instance of the main theorem of Grothendieck's Galois theory in the classical case of fields.

Since this functor preserves products we have the analogue statement for group schemes:

Definition

The functor

{GrSch etGal(kk s)Mod XX(k s)\begin{cases} GrSch_{et}\to Gal(k\hookrightarrow k_s)-Mod \\ X\mapsto X(k_s) \end{cases}

from the category of étale group schemes? to the category of Galois modules of the absolute Galois group of kk is an equivalence of categories.

If now the characteristic of kk is a prime number pp there is a relation of étale formal schemes resp. étale group schemes and the Frobenius morphism:

Proposition

Let XX be a k-formal scheme resp. a locally algebraic scheme.

Then XX is étale iff the Frobenius morphism F:XX (p)F:X\to X^{(p)} is a monomorphism resp. an isomorphism.

References

Michel Demazure, lectures on p-divisible groups, sections I.8 and II.2, web

Revised on June 7, 2012 21:10:57 by Stephan Alexander Spahn (79.227.183.2)