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The Stacks project

Lemma 31.17.10. Let $\pi : X \to Y$ be a finite surjective morphism of schemes. Assume that $X$ is quasi-affine. If either

  1. $\pi $ is finite locally free, or

  2. $Y$ is an integral normal scheme

then $Y$ is quasi-affine.

Proof. Case (1) follows from a combination of Lemmas 31.17.6 and 31.17.5. In case (2) we first replace $X$ by an irreducible component of $X$ which dominates $Y$ (viewed as a reduced closed subscheme of $X$). Then we can apply Lemma 31.17.7. $\square$

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