The Stacks project

Lemma 64.4.7. 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.

Proof. This is a special case of Lemma 64.4.6 applied to $g = s$ so the morphism $i = s : T \to T \times _ T X$. $\square$

Comments (2)

Comment #451 by Kestutis Cesnavicius on

---> 'Let S be a scheme.'

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