nLab etale scheme

Definition

Let $k$ be a field.

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

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

Properties

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

The category $Sch_k$ of $k$-schemes is copowered (= tensored) over $Set$. We define the constant $k$-scheme on a set $E$ by

$$E_k \coloneqq E \otimes Sp_k k = \coprod_{e\in E} Sp_k k$$

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

$$((-)_k \dashv (-)(k))\colon Sch_k \to Set$$

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 $X$ be a $k$-scheme or a formal $k$-scheme. Then the following statements are equivalent:

1. $X$ is étale.

2. $X \otimes_k \overline k$ is constant.

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

Proof

$X$ is étale iff its scalar extension $X\otimes_k k_s$ is étale. And a $k_s$-scheme is étale iff it is constant.

Proposition

The functor

$$\begin{cases} Sch_{et}\to Gal(k\hookrightarrow k_s)-Set \\ X\mapsto X(k_s) \end{cases}$$

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

$$\begin{cases} GrSch_{et}\to Gal(k\hookrightarrow k_s)-Mod \\ X\mapsto X(k_s) \end{cases}$$

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

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

Proposition

Let $X$ be a k-formal scheme resp. a locally algebraic scheme.

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

References

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