Lemma 33.38.6 (09ND)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.38: One dimensional Noetherian schemes
Lemma 33.38.6 (cite)

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Lemma 33.38.6. Let $X$ be a Noetherian integral separated scheme of dimension $1$ . Then $X$ has an ample invertible sheaf.

Proof. Choose an affine open covering $X = U_1 \cup \ldots \cup U_ n$ . Since $X$ is Noetherian, each of the sets $X \setminus U_ i$ is finite. Thus by Lemma 33.38.4 we can find a pair $(\mathcal{L}_ i, s_ i)$ consisting of a globally generated invertible sheaf $\mathcal{L}_ i$ and a global section $s_ i$ such that $U_ i = X_{s_ i}$ . We conclude that $X$ has an ample invertible sheaf by Lemma 33.38.5. $\square$

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