Lemma 54.5.2 (0BFT)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 54: Resolution of Surfaces
Section 54.5: Dominating by normalized blowups
Lemma 54.5.2 (cite)

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Lemma 54.5.2. In Definition 54.5.1 if $X$ is Nagata, then the normalized blowing up of $X$ at $x$ is normal, Nagata, and proper over $X$ .

Proof. The blowup morphism $X' \to X$ is proper (as $X$ is locally Noetherian we may apply Divisors, Lemma 31.32.13). Thus $X'$ is Nagata (Morphisms, Lemma 29.18.1). Therefore the normalization $X'' \to X'$ is finite (Morphisms, Lemma 29.54.11) and we conclude that $X'' \to X$ is proper as well (Morphisms, Lemmas 29.44.11 and 29.41.4). It follows that the normalized blowing up is a normal (Morphisms, Lemma 29.54.5) Nagata algebraic space. $\square$

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