The Stacks project

58.2 Schemes étale over a point

In this section we describe schemes étale over the spectrum of a field. Before we state the result we introduce the category of $G$-sets for a topological group $G$.

Definition 58.2.1. Let $G$ be a topological group. A $G$-set, sometimes called a discrete $G$-set, is a set $X$ endowed with a left action $a : G \times X \to X$ such that $a$ is continuous when $X$ is given the discrete topology and $G \times X$ the product topology. A morphism of $G$-sets $f : X \to Y$ is simply any $G$-equivariant map from $X$ to $Y$. The category of $G$-sets is denoted $G\textit{-Sets}$.

The condition that $a : G \times X \to X$ is continuous signifies simply that the stabilizer of any $x \in X$ is open in $G$. If $G$ is an abstract group $G$ (i.e., a group but not a topological group) then this agrees with our preceding definition (see for example Sites, Example 7.6.5) provided we endow $G$ with the discrete topology.

Recall that if $L/K$ is an infinite Galois extension then the Galois group $G = \text{Gal}(L/K)$ comes endowed with a canonical topology, see Fields, Section 9.22.

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. $\square$

Remark 58.2.3. Under the correspondence of Lemma 58.2.2, the coverings in the small étale site $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$ of $K$ correspond to surjective families of maps in $G\textit{-Sets}$.

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