Lemma 29.13.7 (01SP)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.13: Quasi-affine morphisms
Lemma 29.13.7 (cite)

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Lemma 29.13.7. Let $S$ be a scheme. Let $X$ be an affine scheme. A morphism $f : X \to S$ is quasi-affine if and only if it is quasi-compact. In particular any morphism from an affine scheme to a quasi-separated scheme is quasi-affine.

Proof. Let $V \subset S$ be an affine open. Then $f^{-1}(V)$ is an open subscheme of the affine scheme $X$ , hence quasi-affine if and only if it is quasi-compact. This proves the first assertion. The quasi-compactness of any $f : X \to S$ where $X$ is affine and $S$ quasi-separated follows from Schemes, Lemma 26.21.14 applied to $X \to S \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ . $\square$

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