The Stacks project

Lemma 29.53.9. 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 $Y$. Every generic point of an irreducible component of $X'$ is the image of a generic point of an irreducible component of $Y$.

Proof. By Lemma 29.53.6 we may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine. Choose a finite affine open covering $Y = \bigcup \mathop{\mathrm{Spec}}(B_ i)$. Then $X' = \mathop{\mathrm{Spec}}(A')$ and the morphisms $\mathop{\mathrm{Spec}}(B_ i) \to Y \to X'$ jointly define an injective $A$-algebra map $A' \to \prod B_ i$. Thus the lemma follows from Algebra, Lemma 10.30.5. $\square$

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