Proposition 39.7.11. Let $G$ be a group scheme over a field $k$. There exists a canonical closed subgroup scheme $G^0 \subset G$ with the following properties

1. $G^0 \to G$ is a flat closed immersion,

2. $G^0 \subset G$ is the connected component of the identity,

3. $G^0$ is geometrically irreducible, and

4. $G^0$ is quasi-compact.

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. $\square$

Comment #5819 by Zhenhua Wu on

To verifiy a closed subgroup scheme, the condition mentioned above is not enough. We still need to verify that $i|_{G^0}:G^0\to G^0$. The verification is similar, using 04PW and the fact that $i(G^0)$ is a connected subset containing the identity section so contained in $G^0$. Moreover, I suggest that we add a criterion of open/closed subgroup scheme test.

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