The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 53.2.2. Let $K$ be a field. Let $K^{sep}$ be a separable closure of $K$. Consider the profinite group $G = \text{Gal}(K^{sep}/K)$. The functor

\[ \begin{matrix} \text{schemes étale over }K & \longrightarrow & G\textit{-Sets} \\ X/K & \longmapsto & \mathop{Mor}\nolimits _{\mathop{\mathrm{Spec}}(K)}(\mathop{\mathrm{Spec}}(K^{sep}), X) \end{matrix} \]

is an equivalence of categories.

Proof. A scheme $X$ over $K$ is étale over $K$ if and only if $X \cong \coprod _{i\in I} \mathop{\mathrm{Spec}}(K_ i)$ with each $K_ i$ a finite separable extension of $K$ (Morphisms, Lemma 28.34.7). The functor of the lemma associates to $X$ the $G$-set

\[ \coprod \nolimits _ i \mathop{\mathrm{Hom}}\nolimits _ K(K_ i, K^{sep}) \]

with its natural left $G$-action. Each element has an open stabilizer by definition of the topology on $G$. Conversely, any $G$-set $S$ is a disjoint union of its orbits. Say $S = \coprod S_ i$. Pick $s_ i \in S_ i$ and denote $G_ i \subset G$ its open stabilizer. By Galois theory (Fields, Theorem 9.22.4) the fields $(K^{sep})^{G_ i}$ are finite separable field extensions of $K$, and hence the scheme

\[ \coprod \nolimits _ i \mathop{\mathrm{Spec}}((K^{sep})^{G_ i}) \]

is étale over $K$. This gives an inverse to the functor of the lemma. Some details omitted. $\square$


Comments (2)

Comment #3388 by Dario on

Typo: Let a separable closure...missing be


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03QR. Beware of the difference between the letter 'O' and the digit '0'.