Lemma 83.5.11 (048Y)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 83: Quotients of Groupoids
Section 83.5: Quotients as orbit spaces
Lemma 83.5.11 (cite)

Lemma 83.5.11. In the situation of Definition 83.5.8. Let $\phi : U \to X$ be a morphism of algebraic spaces over $B$ . Assume

$\phi$ is set-theoretically $R$ -invariant,
$R$ is reduced, and
$X$ is locally separated over $B$ .

Then $\phi$ is $R$ -invariant.

Proof. Consider the equalizer

$Z = R \times _{(\phi , \phi ) \circ j, X \times _ B X, \Delta _{X/B}} X$

algebraic space. Then $Z \to R$ is an immersion by assumption (3). By assumption (1) $|Z| \to |R|$ is surjective. This implies that $Z \to R$ is a bijective closed immersion (use Schemes, Lemma 26.10.4) and by assumption (2) we conclude that $Z = R$ . $\square$

The Stacks project

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