Lemma 38.11.4 (0B48)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.11: Extending properties from an open
Lemma 38.11.4 (cite)

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Lemma 38.11.4. Let $f : X \to S$ be a morphism of schemes which is flat and proper. Let $U \subset S$ be a dense open such that $X_ U \to U$ is finite. If also either $f$ is locally of finite presentation or $U \subset S$ is retrocompact, then $f$ is finite.

Proof. By Lemma 38.11.3 the fibres of $f$ have dimension zero. Hence $f$ is quasi-finite (Morphisms, Lemma 29.29.5) whence has finite fibres (Morphisms, Lemma 29.20.10). Hence $f$ is finite by More on Morphisms, Lemma 37.44.1. $\square$

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