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.


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