The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

53.7 Galois covers of connected schemes

Let $X$ be a connected scheme with geometric point $\overline{x}$. Since $F_{\overline{x}} : \textit{FÉt}_ X \to \textit{Sets}$ is a Galois category (Lemma 53.5.5) the material in Section 53.3 applies. In this section we explicity transfer some of the terminology and results to the setting of schemes and finite étale morphisms.

We will say a finite étale morphism $Y \to X$ is a Galois cover if $Y$ defines a Galois object of $\textit{FÉt}_ X$. For a finite étale morphism $Y \to X$ with $G = \text{Aut}(X/Y)$ the following are equivalent

  1. $Y$ is a Galois cover of $X$,

  2. $Y$ is connected and $|G|$ is equal to the degree of $Y \to X$,

  3. $Y$ is connected and $G$ acts transitively on $F_{\overline{x}}(Y)$, and

  4. $Y$ is connected and $G$ acts simply transitively on $F_{\overline{x}}(Y)$, and

This follows immediately from the discussion in Section 53.3.

For any finite étale morphism $f : Y \to X$ with $Y$ connected, there is a finite étale Galois cover $Y' \to X$ which dominates $Y$ (Lemma 53.3.8).

The Galois objects of $\textit{FÉt}_ X$ correspond, via the equivalence

\[ F_{\overline{x}} : \textit{FÉt}_ X \to \textit{Finite-}\pi _1(X, \overline{x})\textit{-Sets} \]

of Theorem 53.6.2, with the finite $\pi _1(X, \overline{x})\textit{-Sets}$ of the form $G = \pi _1(X, \overline{x})/H$ where $H$ is a normal open subgroup. Equivalently, if $G$ is a finite group and $\pi _1(X, \overline{x}) \to G$ is a continuous surjection, then $G$ viewed as a $\pi _1(X, \overline{x})$-set corresponds to a Galois covering.

If $Y_ i \to X$, $i = 1, 2$ are finite étale Galois covers with Galois groups $G_ i$, then there exists a finite étale Galois cover $Y \to X$ whose Galois group is a subgroup of $G_1 \times G_2$. Namely, take the corresponding continuous homomorphisms $\pi _1(X, \overline{x}) \to G_ i$ and let $G$ be the image of the induced continuous homomorphism $\pi _1(X, \overline{x}) \to G_1 \times G_2$.


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