The Stacks project

Properties of the graph of a morphism of algebraic spaces as a consequence of separation properties of the target.

Lemma 66.4.6. Let $S$ be a scheme. Let $T$ be an algebraic space over $S$. Let $g : X \to Y$ be a morphism of algebraic spaces over $T$. Consider the graph $i : X \to X \times _ T Y$ of $g$. Then

  1. $i$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism,

  2. if $Y \to T$ is locally separated, then $i$ is an immersion,

  3. if $Y \to T$ is separated, then $i$ is a closed immersion, and

  4. if $Y \to T$ is quasi-separated, then $i$ is quasi-compact.

Proof. This is a special case of Lemma 66.4.5 applied to the morphism $X = X \times _ Y Y \to X \times _ T Y$. $\square$

Comments (1)

Comment #912 by Matthieu Romagny on

Suggested slogan: Graph of a morphism of algebraic spaces vs separation properties of the target.

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Your email address will not be published.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 03KO. Beware of the difference between the letter 'O' and the digit '0'.