The Stacks project

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

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

Comment #3140 by on

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

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