The Stacks project

Definition 83.7.1. 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 83.3.4. In this case we say condition (1), (2), (3), or (4) holds uniformly or universally.

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View Definition 83.7.1 as pdf