Lemma 58.2.2. 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

is an equivalence of categories.

Lemma 58.2.2. 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

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

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 29.36.7). 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. $\square$

Your email address will not be published.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.

## Comments (2)

Comment #3388 by Dario on

Comment #3456 by Johan on