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

\[ t : (X/Y, \sigma )_{fin} \longrightarrow (X/Y)_{fin}. \]

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.

Proof. Omitted. Hint: Given a pair $(a, Z)$ over $T$ as in (79.12.0.1) the inverse image of $Z$ by $(1_ T, \sigma \circ a) : T \to T \times _ Y X$ is the open subscheme of $T$ we are looking for. $\square$

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