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$Sch_k$ 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.
An étale $k$-scheme is defined to be a directed colimit of $k$-spectra $Sp_k k^\prime$ of finite separable field-extensions $k^\prime$ of $k$.
An étale formal $k$-scheme is defined to be a directed colimit of formal $k$-spectra $Spf_k k^\prime$ of finite separable field-extensions $k^\prime$ of $k$.
Let $X$ be a $k$-scheme or a formal $k$-scheme. Then the following statements are equivalent:
$X$ is étale.
$X\otimes_k cl(k)$ is constant.
$X\otimes_k k_s$ is constant.
where $cl(k)$ denotes an algebraic closure of $k$, $k_s$ denotes the subextension of $cl(k)$ consisting of all separable elements of $cl(k)$ and $\otimes_k$ denotes skalar extension.
Let $X$ be a $k$-formal scheme (resp. a locally algebraic scheme) then $X$ is étale iff the Frobenius morphism $F_X:X\to X^{(p)}$is a monomorphism (resp. an isomorphism).
(fundamental theorem of Galois theory)
The functor
from étale schemes to the category of Galois modules $Gal(k_s/s)-Mod$ is an equivalence of categories.
The completion functor
is an equivalence of categories.