The Stacks project

Proof. Let $G^0$ be the connected component of the identity with its canonical scheme structure (Morphisms, Definition 29.26.3). To show that $G^0$ is a closed subsgroup scheme we will use the criterion of Lemma 39.4.4. The morphism $e : \mathop{\mathrm{Spec}}(k) \to G$ factors through $G^0$ as we chose $G^0$ to be the connected component of $G$ containing $e$. Since $i : G \to G$ is an automorphism fixing $e$, we see that $i$ sends $G^0$ into itself. By Varieties, Lemma 33.7.13 the scheme $G^0$ is geometrically connected over $k$. Thus $G^0 \times _ k G^0$ is connected (Varieties, Lemma 33.7.4). Thus $m(G^0 \times _ k G^0) \subset G^0$ set theoretically. Thus $m|_{G^0 \times _ k G^0} : G^0 \times _ k G^0 \to G$ factors through $G^0$ by Morphisms, Lemma 29.26.1. Hence $G^0$ is a closed subgroup scheme of $G$. By Lemma 39.7.10 we see that $G^0$ is irreducible. By Lemma 39.7.4 we see that $G^0$ is geometrically irreducible. By Lemma 39.7.9 we see that $G^0$ is quasi-compact. $\square$