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}.
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.
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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