The Stacks project

Lemma 33.38.10. Let $X$ be a Noetherian reduced separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf.

Proof. Let $Z_ i$, $i = 1, \ldots , n$ be the irreducible components of $X$. We view these as reduced closed subschemes of $X$. By Lemma 33.38.6 there exist ample invertible sheaves $\mathcal{L}_ i$ on $Z_ i$. Set $T = \bigcup _{i \not= j} Z_ i \cap Z_ j$. As $X$ is Noetherian of dimension $1$, the set $T$ is finite and consists of closed points of $X$. For each $i$ we may, possibly after replacing $\mathcal{L}_ i$ by a power, choose $s_ i \in \Gamma (Z_ i, \mathcal{L}_ i)$ such that $(Z_ i)_{s_ i}$ is affine and contains $T \cap Z_ i$, see Properties, Lemma 28.29.6.

By Lemma 33.38.8 we can find an invertible sheaf $\mathcal{L}$ on $X$ and $s \in \Gamma (X, \mathcal{L})$ such that $(\mathcal{L}, s)|_{Z_ i} = (\mathcal{L}_ i, s_ i)$. Observe that $X_ s$ contains $T$ and is set theoretically equal to the affine closed subschemes $(Z_ i)_{s_ i}$. Thus it is affine by Limits, Lemma 32.11.3. To finish the proof, it suffices to find for every $x \in X$, $x \not\in T$ an integer $m > 0$ and a section $t \in \Gamma (X, \mathcal{L}^{\otimes m})$ such that $X_ t$ is affine and $x \in X_ t$. Since $x \not\in T$ we see that $x \in Z_ i$ for some unique $i$, say $i = 1$. Let $Z \subset X$ be the reduced closed subscheme whose underlying topological space is $Z_2 \cup \ldots \cup Z_ n$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal sheaf of $Z$. Denote that $\mathcal{I}_1 \subset \mathcal{O}_{Z_1}$ the inverse image of this ideal sheaf under the inclusion morphism $Z_1 \to X$. Observe that

\[ \Gamma (X, \mathcal{I}\mathcal{L}^{\otimes m}) = \Gamma (Z_1, \mathcal{I}_1 \mathcal{L}_1^{\otimes m}) \]

see Remark 33.38.9. Thus it suffices to find $m > 0$ and $t \in \Gamma (Z_1, \mathcal{I}_1 \mathcal{L}_1^{\otimes m})$ with $x \in (Z_1)_ t$ affine. Since $\mathcal{L}_1$ is ample and since $x$ is not in $Z_1 \cap T = V(\mathcal{I}_1)$ we can find a section $t_1 \in \Gamma (Z_1, \mathcal{I}_1 \mathcal{L}_1^{\otimes m_1})$ with $x \in (Z_1)_{t_1}$, see Properties, Proposition 28.26.13. Since $\mathcal{L}_1$ is ample we can find a section $t_2 \in \Gamma (Z_1, \mathcal{L}_1^{\otimes m_2})$ with $x \in (Z_1)_{t_2}$ and $(Z_1)_{t_2}$ affine, see Properties, Definition 28.26.1. Set $m = m_1 + m_2$ and $t = t_1 t_2$. Then $t \in \Gamma (Z_1, \mathcal{I}_1 \mathcal{L}_1^{\otimes m})$ with $x \in (Z_1)_ t$ by construction and $(Z_1)_ t$ is affine by Properties, Lemma 28.26.4. $\square$


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