Lemma 89.5.2. In Definition 89.5.1 if $X$ is Nagata, then the normalized blowing up of $X$ at $x$ is a normal Nagata algebraic space proper over $X$.
Proof. The blowup morphism $X' \to X$ is proper (as $X$ is locally Noetherian we may apply Divisors on Spaces, Lemma 71.17.11). Thus $X'$ is Nagata (Morphisms of Spaces, Lemma 67.26.1). Therefore the normalization $X'' \to X'$ is finite (Morphisms of Spaces, Lemma 67.49.9) and we conclude that $X'' \to X$ is proper as well (Morphisms of Spaces, Lemmas 67.45.9 and 67.40.4). It follows that the normalized blowing up is a normal (Morphisms of Spaces, Lemma 67.49.8) Nagata algebraic space. $\square$
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