Proposition 64.27.3 (03VG)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 64: The Trace Formula
Section 64.27: Fundamental groups
Proposition 64.27.3 (cite)

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Proposition 64.27.3. Let $X$ be an integral normal Noetherian scheme. Let $\overline y\to X$ be an algebraic geometric point lying over the generic point $\eta \in X$ . Then

$\pi _ x(X, \overline\eta ) = Gal(M/\kappa (\eta ))$

( $\kappa (\eta )$ , function field of $X$ ) where

$\kappa (\overline\eta )\supset M\supset \kappa (\eta ) = k(X)$

is the max sub-extension such that for every finite sub extension $M\supset L\supset \kappa (\eta )$ the normalization of $X$ in $L$ is finite étale over $X$ .

Proof. Omitted. $\square$

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