Lemma 28.27.2 (0BCK)—The Stacks project

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Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.27: Affine and quasi-affine schemes
Lemma 28.27.2 (cite)

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Lemma 28.27.2. Let $X$ be a quasi-affine scheme. For any quasi-compact immersion $i : X' \to X$ the scheme $X'$ is quasi-affine.

Proof. This can be proved directly without making use of the material on ample invertible sheaves; we urge the reader to do this on a napkin. Since $X$ is quasi-affine, we have that $\mathcal{O}_ X$ is ample by Lemma 28.27.1. Then $\mathcal{O}_{X'}$ is ample by Lemma 28.26.14. Then $X'$ is quasi-affine by Lemma 28.27.1. $\square$

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