The Stacks project

Proof. By Spaces over Fields, Lemma 72.10.2 it suffices to show that $G_ K$ is a scheme for some field extension $K/k$. Denote $G_ K' \subset G_ K$ the schematic locus of $G_ K$ as in Properties of Spaces, Lemma 66.13.1. By Properties of Spaces, Proposition 66.13.3 we see that $G_ K' \subset G_ K$ is dense open, in particular not empty. Choose a scheme $U$ and a surjective étale morphism $U \to G$. By Varieties, Lemma 33.14.2 if $K$ is an algebraically closed field of large enough transcendence degree, then $U_ K$ is a Jacobson scheme and every closed point of $U_ K$ is $K$-rational. Hence $G_ K'$ has a $K$-rational point and it suffices to show that every $K$-rational point of $G_ K$ is in $G_ K'$. If $g \in G_ K(K)$ is a $K$-rational point and $g' \in G_ K'(K)$ a $K$-rational point in the schematic locus, then we see that $g$ is in the image of $G_ K'$ under the automorphism

of $G_ K$. Since automorphisms of $G_ K$ as an algebraic space preserve $G_ K'$, we conclude that $g \in G_ K'$ as desired. $\square$

Proof. This follows from Lemma 79.10.1, Groupoids, Lemma 39.8.6, and Spaces over Fields, Lemma 72.10.7. $\square$

Proof. This lemma generalizes Lemma 79.10.2 (which covers all cases one cares about in practice). The proof is very similar to the proof of Spaces over Fields, Lemma 72.10.7 used in the proof of Lemma 79.10.2 and we encourage the reader to read that proof first.

By Lemma 79.10.1 the base change $G_{\overline{k}}$ is a scheme. Let $K/k$ be a purely transcendental extension of very large transcendence degree. By Spaces over Fields, Lemma 72.10.5 it suffices to show that $G_ K$ is a scheme. Let $K^{perf}$ be the perfect closure of $K$. By Spaces over Fields, Lemma 72.10.1 it suffices to show that $G_{K^{perf}}$ is a scheme. Let $K \subset K^{perf} \subset \overline{K}$ be the algebraic closure of $K$. We may choose an embedding $\overline{k} \to \overline{K}$ over $k$, so that $G_{\overline{K}}$ is the base change of the scheme $G_{\overline{k}}$ by $\overline{k} \to \overline{K}$. By Varieties, Lemma 33.14.2 we see that $G_{\overline{K}}$ is a Jacobson scheme all of whose closed points have residue field $\overline{K}$.

Since $G_{\overline{K}} \to G_{K^{perf}}$ is surjective, it suffices to show that the image $g \in |G_{K^{perf}}|$ of an arbitrary closed point of $G_{\overline{K}}$ is in the schematic locus of $G_ K$. In particular, we may represent $g$ by a morphism $g : \mathop{\mathrm{Spec}}(L) \to G_{K^{perf}}$ where $L/K^{perf}$ is separable algebraic (for example we can take $L = \overline{K}$). Thus the scheme

is the spectrum of a $\overline{K}$-algebra which is a filtered colimit of algebras which are finite products of copies of $\overline{K}$. Thus by Groupoids, Lemma 39.7.13 we can find an affine open $W \subset G_{\overline{K}}$ containing the image of $g_{\overline{K}} : T \to G_{\overline{K}}$.

Choose a quasi-compact open $V \subset G_{K^{perf}}$ containing the image of $W$. By Spaces over Fields, Lemma 72.10.2 we see that $V_{K'}$ is a scheme for some finite extension $K'/K^{perf}$. After enlarging $K'$ we may assume that there exists an affine open $U' \subset V_{K'} \subset G_{K'}$ whose base change to $\overline{K}$ recovers $W$ (use that $V_{\overline{K}}$ is the limit of the schemes $V_{K''}$ for $K' \subset K'' \subset \overline{K}$ finite and use Limits, Lemmas 32.4.11 and 32.4.13). We may assume that $K'/K^{perf}$ is a Galois extension (take the normal closure Fields, Lemma 9.16.3 and use that $K^{perf}$ is perfect). Set $H = \text{Gal}(K'/K^{perf})$. By construction the $H$-invariant closed subscheme $\mathop{\mathrm{Spec}}(L) \times _{G_{K^{perf}}} G_{K'}$ is contained in $U'$. By Spaces over Fields, Lemmas 72.10.3 and 72.10.4 we conclude. $\square$