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

This entry is about a section of the text

${\mathrm{Sch}}_{k}$ is copowered (= tensored) over $\mathrm{Set}$. We define the constant $k$-scheme on a set $E$ by

${E}_{k}:=E\otimes {\mathrm{Sp}}_{k}k=\coprod _{e\in E}{\mathrm{Sp}}_{k}k$E_k:=E\otimes Sp_k k=\coprod_{e\in E}Sp_k k

For a scheme $X$ we compute ${M}_{k}\left({E}_{k},E\right)=\mathrm{Set}\left({\mathrm{Sp}}_{k}k,X{\right)}^{E}=X\left(k{\right)}^{E}=\mathrm{Set}\left(E,X\left(k\right)\right)$ and see that there is an adjunction

$\left(\left(-{\right)}_{k}⊣\left(-\right)\left(k\right)\right):{\mathrm{Sch}}_{k}\to \mathrm{Set}$((-)_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 $k$-scheme is defined to be a directed colimit of $k$-spectra ${\mathrm{Sp}}_{k}{k}^{\prime }$ 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 ${\mathrm{Spf}}_{k}{k}^{\prime }$ of finite separable field-extensions ${k}^{\prime }$ of $k$.

Remark

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

1. $X$ is étale.

2. $X{\otimes }_{k}\mathrm{cl}\left(k\right)$ is constant.

3. $X{\otimes }_{k}{k}_{s}$ is constant.

where $\mathrm{cl}\left(k\right)$ denotes an algebraic closure of $k$, ${k}_{s}$ denotes the subextension of $\mathrm{cl}\left(k\right)$ consisting of all separable elements of $\mathrm{cl}\left(k\right)$ and ${\otimes }_{k}$ denotes skalar extension.

Proposition

Let $X$ be a $k$-formal scheme (resp. a locally algebraic scheme) then $X$ is étale iff the Frobenius morphism ${F}_{X}:X\to {X}^{\left(p\right)}$is a monomorphism (resp. an isomorphism).

Theorem

The functor

$\left\{\begin{array}{l}{\mathrm{Sch}}_{\mathrm{et}}\to \mathrm{Gal}\left({k}_{s}/k\right)-\mathrm{Mod}\\ X↦X\left({k}_{s}\right)\end{array}$\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 $\mathrm{Gal}\left({k}_{s}/s\right)-\mathrm{Mod}$ is an equivalence of categories.

Remark

The completion functor

$\left\{\begin{array}{l}{\mathrm{Sch}}_{\mathrm{et}}\to {\mathrm{fSch}}_{\mathrm{et}}\\ X↦\stackrel{^}{X}\end{array}$\begin{cases} Sch_{et}\to fSch_{et} \\ X\mapsto \hat X \end{cases}

is an equivalence of categories.