The Stacks project

Lemma 31.17.5. Let $\pi : X \to Y$ be a finite morphism of schemes. Assume $X$ is quasi-affine and there exists a norm of degree $d$ for $\pi $. Then $Y$ is quasi-affine.

Proof. By Properties, Lemma 28.27.1 we see that $\mathcal{O}_ X$ is an ample invertible sheaf on $X$. The proof of Lemma 31.17.4 shows that $\text{Norm}_\pi (\mathcal{O}_ X) = \mathcal{O}_ Y$ is an ample invertible $\mathcal{O}_ Y$-module. Hence Properties, Lemma 28.27.1 shows that $Y$ is quasi-affine. $\square$

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