Definition 64.13.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism.

1. We say $F$ is separated over $S$ if $\Delta$ is a closed immersion.

2. We say $F$ is locally separated over $S$1 if $\Delta$ is an immersion.

3. We say $F$ is quasi-separated over $S$ if $\Delta$ is quasi-compact.

4. We say $F$ is Zariski locally quasi-separated over $S$2 if there exists a Zariski covering $F = \bigcup _{i \in I} F_ i$ such that each $F_ i$ is quasi-separated.

[1] In the literature this often refers to quasi-separated and locally separated algebraic spaces.
[2] This definition was suggested by B. Conrad.

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Your email address will not be published.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 02X5. Beware of the difference between the letter 'O' and the digit '0'.