Lemma 32.11.3 (09NL)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 32: Limits of Schemes
Section 32.11: Characterizing affine schemes
Lemma 32.11.3 (cite)

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Lemma 32.11.3. Let $X$ be a scheme which is set theoretically the union of finitely many affine closed subschemes. Then $X$ is affine.

Proof. Let $Z_ i \subset X$, $i = 1, \ldots , n$ be affine closed subschemes such that $X = \bigcup Z_ i$ set theoretically. Then $\coprod Z_ i \to X$ is surjective and integral with affine source. Hence $X$ is affine by Proposition 32.11.2. $\square$

Comments (2)

Comment #3026 by Brian Lawrence on December 13, 2017 at 02:56

Suggested slogan: A set-theoretic union of finitely many affine closed subschemes is affine.

Comment #3140 by Johan on February 01, 2018 at 18:48

This slogan is ignored as it is the same as the statement of the lemma.

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