The Stacks project

Lemma 101.4.9. Let $f : \mathcal{X} \to \mathcal{T}$ be a morphism of algebraic stacks. Let $s : \mathcal{T} \to \mathcal{X}$ be a morphism such that $f \circ s$ is $2$-isomorphic to $\text{id}_\mathcal {T}$. Then

  1. $s$ is representable by algebraic spaces and locally of finite type,

  2. if $f$ is DM, then $s$ is unramified,

  3. if $f$ is quasi-DM, then $s$ is locally quasi-finite,

  4. if $f$ is separated, then $s$ is proper, and

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

Proof. This is a special case of Lemma 101.4.8 applied to $g = s$ and $\mathcal{Y} = \mathcal{T}$ in which case $i : \mathcal{T} \to \mathcal{T} \times _\mathcal {T} \mathcal{X}$ is $2$-isomorphic to $s$. $\square$

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