Lemma 79.9.3 (06E0)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 79: More on Groupoids in Spaces
Section 79.9: Properties of groups over fields and groupoids on fields
Lemma 79.9.3 (cite)

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Lemma 79.9.3. In Situation 79.9.2 the composition morphism $c : R \times _{s, U, t} R \to R$ is flat and universally open. In Situation 79.9.1 the group law $m : G \times _ k G \to G$ is flat and universally open.

Proof. The composition is isomorphic to the projection map $\text{pr}_1 : R \times _{t, U, t} R \to R$ by Diagram (79.3.0.2). The projection is flat as a base change of the flat morphism $t$ and open by Morphisms of Spaces, Lemma 67.6.6. The second assertion follows immediately from the first because $m$ matches $c$ in (79.9.2.1). $\square$

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