Lemma 58.5.5. Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point. The functor

$F_{\overline{x}} : \textit{FÉt}_ X \longrightarrow \textit{Sets},\quad Y \longmapsto |Y_{\overline{x}}|$

defines a Galois category (Definition 58.3.6).

Proof. After identifying $\textit{FÉt}_{\overline{x}}$ with the category of finite sets (Example 58.5.1) we see that our functor $F_{\overline{x}}$ is nothing but the base change functor for the morphism $\overline{x} \to X$. Thus we see that $\textit{FÉt}_ X$ has finite limits and finite colimits and that $F_{\overline{x}}$ is exact by Lemma 58.5.2. We will also use that finite limits in $\textit{FÉt}_ X$ agree with the corresponding finite limits in the category of schemes over $X$, see Remark 58.5.3.

If $Y' \to Y$ is a monomorphism in $\textit{FÉt}_ X$ then we see that $Y' \to Y' \times _ Y Y'$ is an isomorphism, and hence $Y' \to Y$ is a monomorphism of schemes. It follows that $Y' \to Y$ is an open immersion (Étale Morphisms, Theorem 41.14.1). Since $Y'$ is finite over $X$ and $Y$ separated over $X$, the morphism $Y' \to Y$ is finite (Morphisms, Lemma 29.44.14), hence closed (Morphisms, Lemma 29.44.11), hence it is the inclusion of an open and closed subscheme of $Y$. It follows that $Y$ is a connected objects of the category $\textit{FÉt}_ X$ (as in Definition 58.3.6) if and only if $Y$ is connected as a scheme. Then it follows from Topology, Lemma 5.7.7 that $Y$ is a finite coproduct of its connected components both as a scheme and in the sense of Definition 58.3.6.

Let $Y \to Z$ be a morphism in $\textit{FÉt}_ X$ which induces a bijection $F_{\overline{x}}(Y) \to F_{\overline{x}}(Z)$. We have to show that $Y \to Z$ is an isomorphism. By the above we may assume $Z$ is connected. Since $Y \to Z$ is finite étale and hence finite locally free it suffices to show that $Y \to Z$ is finite locally free of degree $1$. This is true in a neighbourhood of any point of $Z$ lying over $\overline{x}$ and since $Z$ is connected and the degree is locally constant we conclude. $\square$

Comment #8806 by ZW on

Is the definition of "finite étale" here just finite and étale? Since finite flat is not the same as finite locally free, could you explain why, in the last paragraph, "since Y → Z is finite étale and hence finite locally free ..." without any noetherian assumption? Thanks.

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