The Stacks project

Lemma 79.12.15. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\sigma : Y \to X$ be a section of $f$. Consider the transformation of functors

defined above. Then

1. $t$ is representable by open immersions,

2. if $f$ is separated, then $t$ is representable by open and closed immersions,

3. if $(X/Y)_{fin}$ is an algebraic space, then $(X/Y, \sigma )_{fin}$ is an algebraic space and an open subspace of $(X/Y)_{fin}$, and

4. if $(X/Y)_{fin}$ is a scheme, then $(X/Y, \sigma )_{fin}$ is an open subscheme of it.