Lemma 40.10.1 (04LM)—The Stacks project

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Lemma 40.10.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$ . If $U$ is the spectrum of a field, then the composition morphism $c : R \times _{s, U, t} R \to R$ is open.

Proof. The composition is isomorphic to the projection map $\text{pr}_1 : R \times _{t, U, t} R \to R$ by Diagram (40.3.0.2). The projection is open by Morphisms, Lemma 29.23.4. $\square$

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