The Stacks project

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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