Lemma 39.7.2. If (G, m) is a group scheme over a field k. Let U \subset G open and T \to G a morphism of schemes. Then the image of the composition T \times _ k U \to G \times _ k G \to G is open.
Proof. For any field extension K/k the morphism G_ K \to G is open (Morphisms, Lemma 29.23.4). Every point \xi of T \times _ k U is the image of a morphism (t, u) : \mathop{\mathrm{Spec}}(K) \to T \times _ k U for some K. Then the image of T_ K \times _ K U_ K = (T \times _ k U)_ K \to G_ K contains the translate t \cdot U_ K which is open. Combining these facts we see that the image of T \times _ k U \to G contains an open neighbourhood of the image of \xi . Since \xi was arbitrary we win. \square
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