Lemma 69.16.8 (0D2V)—The Stacks project

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Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.16: Vanishing of cohomology
Lemma 69.16.8 (cite)

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Lemma 69.16.8. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ . Assume that for every coherent $\mathcal{O}_ X$ -module $\mathcal{F}$ we have $H^1(X, \mathcal{F}) = 0$ . Then $X$ is an affine scheme.

Proof. The assumption implies that $H^1(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_ X$ -module $\mathcal{F}$ by Lemmas 69.15.1 and 69.5.1. Then $X$ is affine by Proposition 69.16.7. $\square$

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