The Stacks project

Lemma 67.4.7. Let $S$ be a scheme. Let $f : X \to T$ be a morphism of algebraic spaces over $S$. Let $s : T \to X$ be a section of $f$ (in a formula $f \circ s = \text{id}_ T$). Then

1. $s$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism,

2. if $f$ is locally separated, then $s$ is an immersion,

3. if $f$ is separated, then $s$ is a closed immersion, and

4. if $f$ is quasi-separated, then $s$ is quasi-compact.