The Stacks project

Lemma 26.24.2. Let $f : X \to Y$ be a morphism of schemes. Suppose that

  1. $f$ induces a homeomorphism of $X$ with a closed subset of $Y$, and

  2. $f^\sharp : \mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is surjective.

Then $f$ is a closed immersion of schemes.

Proof. Assume (1) and (2). By (1) the morphism $f$ is quasi-compact (see Topology, Lemma 5.12.3). Conditions (1) and (2) imply conditions (1) and (2) of Lemma 26.23.7. Hence $f : X \to Y$ is a monomorphism. In particular, $f$ is separated, see Lemma 26.23.3. Hence Lemma 26.24.1 above applies and we conclude that $f_*\mathcal{O}_ X$ is a quasi-coherent $\mathcal{O}_ Y$-module. Therefore the kernel of $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is quasi-coherent by Lemma 26.7.8. Since a quasi-coherent sheaf is locally generated by sections (see Modules, Definition 17.10.1) this implies that $f$ is a closed immersion, see Definition 26.4.1. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 26.24: Functoriality for quasi-coherent modules

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Your email address will not be published.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



View Lemma 26.24.2 as pdf