Lemma 29.54.2. Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. The normalization morphism $\nu $ factors through the reduction $X_{red}$ and $X^\nu \to X_{red}$ is the normalization of $X_{red}$.
Proof. Let $f : Y \to X$ be the morphism (29.54.0.1). We get a factorization $Y \to X_{red} \to X$ of $f$ from Schemes, Lemma 26.12.7. By Lemma 29.53.4 we obtain a canonical morphism $X^\nu \to X_{red}$ and that $X^\nu $ is the normalization of $X_{red}$ in $Y$. The lemma follows as $Y \to X_{red}$ is identical to the morphism (29.54.0.1) constructed for $X_{red}$. $\square$
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