The Stacks project

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. $\square$

Comments (2)

Comment #3388 by Dario on

Typo: Let a separable closure...missing be

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