The Stacks project

Remark 29.12.3. In Properties, Lemma 28.26.7 we see that a scheme which has an ample invertible module is separated. This is wrong for schemes having an ample family of invertible modules. Namely, let $X$ be as in Schemes, Example 26.14.3 with $n = 1$, i.e., the affine line with zero doubled. We use the notation of that example except that we write $x$ for $x_1$ and $y$ for $y_1$. There is, for every integer $n$, an invertible sheaf $\mathcal{L}_ n$ on $X$ which is trivial on $X_1$ and $X_2$ and whose transition function $U_{12} \to U_{21}$ is $f(x) \mapsto y^ n f(y)$. The global sections of $\mathcal{L}_ n$ are pairs $(f(x), g(y)) \in k[x] \oplus k[y]$ such that $y^ n f(y) = g(y)$. The sections $s = (1, y)$ of $\mathcal{L}_1$ and $t = (x, 1)$ of $\mathcal{L}_{-1}$ determine an open affine cover because $X_ s = X_1$ and $X_ t = X_2$. Therefore $X$ has an ample family of invertible modules but it is not separated.

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Your email address will not be published.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