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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09NX. Beware of the difference between the letter 'O' and the digit '0'.