The Stacks project

Proposition 57.11.3. Let $X$ be a normal integral scheme with function field $K$. Then the canonical map (57.10.6.1)

\[ \text{Gal}(K^{sep}/K) = \pi _1(\eta , \overline{\eta }) \longrightarrow \pi _1(X, \overline{\eta }) \]

is identified with the quotient map $\text{Gal}(K^{sep}/K) \to \text{Gal}(M/K)$ where $M \subset K^{sep}$ is the union of the finite subextensions $L$ such that $X$ is unramified in $L$.

Proof. The normal scheme $X$ is geometrically unibranch (Properties, Lemma 28.15.2). Hence Lemma 57.10.7 applies to $X$. Thus $\pi _1(\eta , \overline{\eta }) \to \pi _1(X, \overline{\eta })$ is surjective and top horizontal arrow of the commutative diagram

\[ \xymatrix{ \textit{FÉt}_ X \ar[r] \ar[d] \ar[rd]_ c & \textit{FÉt}_\eta \ar[d] \\ \textit{Finite-}\pi _1(X, \overline{\eta })\textit{-sets} \ar[r] & \textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-sets} } \]

is fully faithful. The left vertical arrow is the equivalence of Theorem 57.6.2 and the right vertical arrow is the equivalence of Lemma 57.6.3. The lower horizontal arrow is induced by the map of the proposition. By Lemmas 57.11.1 and 57.11.2 we see that the essential image of $c$ consists of $\text{Gal}(K^{sep}/K)\textit{-Sets}$ isomorphic to sets of the form

\[ S = \mathop{\mathrm{Hom}}\nolimits _ K(\prod \nolimits _{i = 1, \ldots , n} L_ i, K^{sep}) = \coprod \nolimits _{i = 1, \ldots , n} \mathop{\mathrm{Hom}}\nolimits _ K(L_ i, K^{sep}) \]

with $L_ i/K$ finite separable such that $X$ is unramified in $L_ i$. Thus if $M \subset K^{sep}$ is as in the statement of the lemma, then $\text{Gal}(K^{sep}/M)$ is exactly the subgroup of $\text{Gal}(K^{sep}/K)$ acting trivially on every object in the essential image of $c$. On the other hand, the essential image of $c$ is exactly the category of $S$ such that the $\text{Gal}(K^{sep}/K)$-action factors through the surjection $\text{Gal}(K^{sep}/K) \to \pi _1(X, \overline{\eta })$. We conclude that $\text{Gal}(K^{sep}/M)$ is the kernel. Hence $\text{Gal}(K^{sep}/M)$ is a normal subgroup, $M/K$ is Galois, and we have a short exact sequence

\[ 1 \to \text{Gal}(K^{sep}/M) \to \text{Gal}(K^{sep}/K) \to \text{Gal}(M/K) \to 1 \]

by Galois theory (Fields, Theorem 9.22.4 and Lemma 9.22.5). The proof is done. $\square$


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