The Stacks project

81.7 Topological properties

Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation. We say a subset $T \subset |U|$ is $R$-invariant if $s^{-1}(T) = t^{-1}(T)$ as subsets of $|R|$. Note that if $T$ is closed, then it may not be the case that the corresponding reduced closed subspace of $U$ is $R$-invariant (as in Groupoids in Spaces, Definition 76.17.1) because the pullbacks $s^{-1}(T)$, $t^{-1}(T)$ may not be reduced. Here are some conditions that we can consider for an invariant morphism $\phi : U \to X$.

Definition 81.7.1. Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation. Let $\phi : U \to X$ be an $R$-invariant morphism of algebraic spaces over $B$.

  1. The morphism $\phi $ is submersive.

  2. For any $R$-invariant closed subset $Z \subset |U|$ the image $\phi (Z)$ is closed in $|X|$.

  3. Condition (2) holds and for any pair of $R$-invariant closed subsets $Z_1, Z_2 \subset |U|$ we have

    \[ \phi (Z_1 \cap Z_2) = \phi (Z_1) \cap \phi (Z_2) \]
  4. The morphism $(t, s) : R \to U \times _ X U$ is universally submersive.

For each of these properties we can also require them to hold after any flat base change, or after any base change, see Definition 81.3.4. In this case we say condition (1), (2), (3), or (4) holds uniformly or universally.

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Your email address will not be published.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 04A3. Beware of the difference between the letter 'O' and the digit '0'.