The Stacks project

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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