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The Stacks project

Lemma 83.5.16. Let S be a scheme, and let B be an algebraic space over S. Let j : R \to U \times _ B U be a pre-relation over B.

  1. If j is a pre-equivalence relation, then j is a set-theoretic pre-equivalence relation. This holds in particular when j comes from a groupoid in algebraic spaces, or from an action of a group algebraic space on U.

  2. If j is an equivalence relation, then j is a set-theoretic equivalence relation.

Proof. Omitted. \square


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