Lemma 58.15.1 (0BUN)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.15: Homotopy exact sequence
Lemma 58.15.1 (cite)

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[Expose X, Proposition 1.2, p. 262, SGA1].

Lemma 58.15.1. Let $f : X \to S$ be a proper morphism of schemes. Let $X \to S' \to S$ be the Stein factorization of $f$ , see More on Morphisms, Theorem 37.53.5. If $f$ is of finite presentation, flat, with geometrically reduced fibres, then $S' \to S$ is finite étale.

Proof. This follows from Derived Categories of Schemes, Lemma 36.32.8 and the information contained in More on Morphisms, Theorem 37.53.5. $\square$

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