Lemma 70.10.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying (a), (b), and (c) of Lemma 70.10.8. Then every invertible $\mathcal{O}_ X$-module $\mathcal{L}$ has a regular meromorphic section.

**Proof.**
With notation as in Lemma 70.10.8 the stalk $\mathcal{L}_\eta $ of $\mathcal{L}$ at is defined for all $\eta \in X^0$ and it is a rank $1$ free $\mathcal{O}_{X, \eta }$-module. Pick a generator $s_\eta \in \mathcal{L}_\eta $ for all $\eta \in X^0$. It follows immediately from the description of $\mathcal{K}_ X$ and $\mathcal{K}_ X(\mathcal{L})$ in Lemma 70.10.8 that $s = \prod s_\eta $ is a regular meromorphic section of $\mathcal{L}$.
$\square$

## 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Your email address will not be published.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.

## Comments (0)