Lemma 29.53.8. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X' \to X$ be the normalization of $X$ in $Y$. If $Y$ is reduced, so is $X'$.

**Proof.**
This follows from the fact that a subring of a reduced ring is reduced. Some details omitted.
$\square$

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