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

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

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

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

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

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

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 $Sp_k k^\prime$ 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 $k$.

Remark

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

1. $X$ is étale.

2. $X\otimes_k cl(k)$ is constant.

3. $X\otimes_k k_s$ 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.

Proposition

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

Theorem

(fundamental theorem of Galois theory)

The functor

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

from étale schemes to the category of Galois modules $Gal(k_s/s)-Mod$ is an equivalence of categories.

Remark

The completion functor

$$\begin{cases} Sch_{et}\to fSch_{et} \\ X\mapsto \hat X \end{cases}$$

is an equivalence of categories.