The Stacks project

Lemma 79.8.1. 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.