The Stacks project

A (local) trivialisation of a linebundle is the same as a (local) nonvanishing section.

Lemma 17.24.10. Let $X$ be a ringed space. Assume that each stalk $\mathcal{O}_{X, x}$ is a local ring with maximal ideal $\mathfrak m_ x$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. For any section $s \in \Gamma (X, \mathcal{L})$ the set

\[ X_ s = \{ x \in X \mid \text{image }s \not\in \mathfrak m_ x\mathcal{L}_ x\} \]

is open in $X$. The map $s : \mathcal{O}_{X_ s} \to \mathcal{L}|_{X_ s}$ is an isomorphism, and there exists a section $s'$ of $\mathcal{L}^{\otimes -1}$ over $X_ s$ such that $s' (s|_{X_ s}) = 1$.

Proof. Suppose $x \in X_ s$. We have an isomorphism

\[ \mathcal{L}_ x \otimes _{\mathcal{O}_{X, x}} (\mathcal{L}^{\otimes -1})_ x \longrightarrow \mathcal{O}_{X, x} \]

by Lemma 17.24.5. Both $\mathcal{L}_ x$ and $(\mathcal{L}^{\otimes -1})_ x$ are free $\mathcal{O}_{X, x}$-modules of rank $1$. We conclude from Algebra, Nakayama's Lemma 10.20.1 that $s_ x$ is a basis for $\mathcal{L}_ x$. Hence there exists a basis element $t_ x \in (\mathcal{L}^{\otimes -1})_ x$ such that $s_ x \otimes t_ x$ maps to $1$. Choose an open neighbourhood $U$ of $x$ such that $t_ x$ comes from a section $t$ of $\mathcal{L}^{\otimes -1}$ over $U$ and such that $s \otimes t$ maps to $1 \in \mathcal{O}_ X(U)$. Clearly, for every $x' \in U$ we see that $s$ generates the module $\mathcal{L}_{x'}$. Hence $U \subset X_ s$. This proves that $X_ s$ is open. Moreover, the section $t$ constructed over $U$ above is unique, and hence these glue to give the section $s'$ of the lemma. $\square$

Comments (5)

Comment #2443 by Junho Won on

(Typo) I think there should be no in in the fourth sentence of the proof.

Comment #2594 by Rogier Brussee on

Suggested slogan: A (local) trivialisation of a linebundle is the same as a (local) non vanishing section.

Comment #4898 by Florian Stümpfl on

There is a typo in the last sentence: "... To give the section...".

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Your email address will not be published.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 01CY. Beware of the difference between the letter 'O' and the digit '0'.