Lemma 39.7.1. If $(G, m)$ is a group scheme over a field $k$, then the multiplication map $m : G \times _ k G \to G$ is open.

Proof. The multiplication map is isomorphic to the projection map $\text{pr}_0 : G \times _ k G \to G$ because the diagram

$\xymatrix{ G \times _ k G \ar[d]^ m \ar[rrr]_{(g, g') \mapsto (m(g, g'), g')} & & & G \times _ k G \ar[d]^{(g, g') \mapsto g} \\ G \ar[rrr]^{\text{id}} & & & G }$

is commutative with isomorphisms as horizontal arrows. The projection is open by Morphisms, Lemma 29.23.4. $\square$

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