Lemma 31.16.8. Let $X$ be a quasi-compact, regular scheme with affine diagonal. Then $X$ has an ample family of invertible modules (Morphisms, Definition 29.12.1.

**Proof.**
Observe that $X$ is a finite disjoint union of integral schemes (Properties, Lemmas 28.9.4 and 28.7.6). Thus we may assume that $X$ is integral as well as Noetherian, regular, and having affine diagonal. Let $x \in X$. Choose an affine open neighbourhood $U \subset X$ of $x$. Since $X$ is integral, $U$ is dense in $X$. By More on Algebra, Lemma 15.118.2 the local rings of $X$ are UFDs. Hence by Lemma 31.16.7 we can find an effective Cartier divisor $D \subset X$ whose complement is $U$. Then the canonical section $s = 1_ D$ of $\mathcal{L} = \mathcal{O}_ X(D)$, see Definition 31.14.1, vanishes exactly along $D$ hence $U = X_ s$. Thus both conditions in Morphisms, Definition 29.12.1 hold and we are done.
$\square$

## 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Your email address will not be published.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)