Definition 26.4.1. Let $i : Z \to X$ be a morphism of locally ringed spaces. We say that $i$ is a closed immersion if:

1. The map $i$ is a homeomorphism of $Z$ onto a closed subset of $X$.

2. The map $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective; let $\mathcal{I}$ denote the kernel.

3. The $\mathcal{O}_ X$-module $\mathcal{I}$ is locally generated by sections.

There are also:

• 14 comment(s) on Section 26.4: Closed immersions of locally ringed spaces

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.

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 01HK. Beware of the difference between the letter 'O' and the digit '0'.