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 $k$.
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
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
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.
Let $X$ be a $k$-scheme or a formal $k$-scheme. Then the following statements are equivalent:
$X$ is étale.
$X \otimes_k \overline k$ is constant.
$X \otimes_k k_s$ 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.
$X$ is étale iff its skalar extension $X\otimes_k k_s$ is étale. And a $k_s$-scheme is étale iff it is constant.
The functor
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:
The functor
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:
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.
Michel Demazure, lectures on p-divisible groups, sections I.8 and II.2, web