The Stacks project

Lemma 33.38.14. Let $f : X \to Y$ be a morphism of schemes. Assume $Y$ is Noetherian of dimension $\leq 1$, $f$ is finite, and there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is a closed immersion. Then every invertible $\mathcal{O}_ X$-module is the pullback of an invertible $\mathcal{O}_ Y$-module.

Proof. We factor $f$ as $X \to Z \to Y$ where $Z$ is the scheme theoretic image of $f$. Then $X \to Z$ is an isomorphism over $V \cap Z$ and Lemma 33.38.7 applies. On the other hand, Lemma 33.38.11 applies to $Z \to Y$. Some details omitted. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 33.38: One dimensional Noetherian schemes

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Your email address will not be published.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 0C0U. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 33.38.14 as pdf