Proposition 33.38.12 (09NZ)—The Stacks project

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Part 2: Schemes
Chapter 33: Varieties
Section 33.38: One dimensional Noetherian schemes
Proposition 33.38.12 (cite)

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Proposition 33.38.12. Let $X$ be a Noetherian separated scheme of dimension $1$ . Then $X$ has an ample invertible sheaf.

Proof. Let $Z \subset X$ be the reduction of $X$ . By Lemma 33.38.10 the scheme $Z$ has an ample invertible sheaf. Thus by Lemma 33.38.11 there exists an invertible $\mathcal{O}_ X$ -module $\mathcal{L}$ on $X$ whose restriction to $Z$ is ample. Then $\mathcal{L}$ is ample by an application of Cohomology of Schemes, Lemma 30.17.5. $\square$

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