The Stacks project

Lemma 78.8.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \to U$ be a morphism of algebraic spaces over $B$. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$.

  1. If $s, t$ are locally of finite type and $g$ is locally of finite type, then $s', t'$ are locally of finite type.

  2. If $s, t$ are locally of finite presentation and $g$ is locally of finite presentation, then $s', t'$ are locally of finite presentation.

  3. If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat.

  4. Add more here.

Proof. The property of being locally of finite type is stable under composition and arbitrary base change, see Morphisms of Spaces, Lemmas 66.23.2 and 66.23.3. Hence (1) is clear from Diagram (78.8.0.1). For the other cases, see Morphisms of Spaces, Lemmas 66.28.2, 66.28.3, 66.30.3, and 66.30.4. $\square$


Comments (0)


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Your email address will not be published.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 04RP. Beware of the difference between the letter 'O' and the digit '0'.