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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

${\mathrm{Sch}}_k$ 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 $M_k(E_k,E)=\mathrm{Set}({\mathrm{Sp}}_{k}k,X{)}^E=X(k{)}^E=\mathrm{Set}(E,X(k))$ and see that there is an adjunction

$$((-{)}_k⊣(-)(k)):{\mathrm{Sch}}_k\to \mathrm{Set}$$((-)_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. $X{\otimes }_k\mathrm{cl}(k)$ is constant.

3. $X{\otimes }_k{k}_s$ is constant.

where $\mathrm{cl}(k)$ denotes an algebraic closure of $k$, $k_s$ denotes the subextension of $\mathrm{cl}(k)$ consisting of all separable elements of $\mathrm{cl}(k)$ 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

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

Remark

The completion functor

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

is an equivalence of categories.