Lemma 29.54.6 (0CDV)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.54: Normalization
Lemma 29.54.6 (cite)

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Lemma 29.54.6. Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. Let $Z_ i \subset X$ , $i \in I$ be the irreducible components of $X$ endowed with the reduced induced structure. Let $Z_ i^\nu \to Z_ i$ be the normalization. Then $\coprod _{i \in I} Z_ i^\nu \to X$ is the normalization of $X$ .

Proof. We may assume $X$ is reduced, see Lemma 29.54.2. Then the lemma follows either from the local description in Lemma 29.54.3 or from Lemma 29.54.5 part (3) because $\coprod Z_ i \to X$ is integral and locally birational (as $X$ is reduced and has locally finitely many irreducible components). $\square$

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