The Stacks project

Lemma 39.7.3. Let $G$ be a group scheme over a field. Then $G$ is a separated scheme.

Proof. Say $S = \mathop{\mathrm{Spec}}(k)$ with $k$ a field, and let $G$ be a group scheme over $S$. By Lemma 39.6.1 we have to show that $e : S \to G$ is a closed immersion. By Morphisms, Lemma 29.20.2 the image of $e : S \to G$ is a closed point of $G$. It is clear that $\mathcal{O}_ G \to e_*\mathcal{O}_ S$ is surjective, since $e_*\mathcal{O}_ S$ is a skyscraper sheaf supported at the neutral element of $G$ with value $k$. We conclude that $e$ is a closed immersion by Schemes, Lemma 26.24.2. $\square$

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