78.10 Group algebraic spaces over fields

There exists a nonseparated group algebraic space over a field, namely $\mathbf{G}_ a/\mathbf{Z}$ over a field of characteristic zero, see Examples, Section 109.48. In fact any group scheme over a field is separated (Lemma 78.9.4) hence every nonseparated group algebraic space over a field is nonrepresentable. On the other hand, a group algebraic space over a field is separated as soon as it is decent, see Lemma 78.9.4. In this section we will show that a separated group algebraic space over a field is representable, i.e., a scheme.

Lemma 78.10.1. Let $k$ be a field with algebraic closure $\overline{k}$. Let $G$ be a group algebraic space over $k$ which is separated1. Then $G_{\overline{k}}$ is a scheme.

Proof. By Spaces over Fields, Lemma 71.10.2 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 65.13.1. By Properties of Spaces, Proposition 65.13.3 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

$G_ K \longrightarrow G_ K,\quad h \longmapsto g(g')^{-1}h$

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

Lemma 78.10.2. Let $k$ be a field. Let $G$ be a group algebraic space over $k$. If $G$ is separated and locally of finite type over $k$, then $G$ is a scheme.

Proof. This follows from Lemma 78.10.1, Groupoids, Lemma 39.8.6, and Spaces over Fields, Lemma 71.10.7. $\square$

Proposition 78.10.3. Let $k$ be a field. Let $G$ be a group algebraic space over $k$. If $G$ is separated, then $G$ is a scheme.

Proof. This lemma generalizes Lemma 78.10.2 (which covers all cases one cares about in practice). The proof is very similar to the proof of Spaces over Fields, Lemma 71.10.7 used in the proof of Lemma 78.10.2 and we encourage the reader to read that proof first.

By Lemma 78.10.1 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 71.10.5 it suffices to show that $G_ K$ is a scheme. Let $K^{perf}$ be the perfect closure of $K$. By Spaces over Fields, Lemma 71.10.1 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

\begin{align*} T & = \mathop{\mathrm{Spec}}(L) \times _{G_{K^{perf}}} G_{\overline{K}} \\ & = \mathop{\mathrm{Spec}}(L) \times _{\mathop{\mathrm{Spec}}(K^{perf})} \mathop{\mathrm{Spec}}(\overline{K}) \\ & = \mathop{\mathrm{Spec}}(L \otimes _{K^{perf}} \overline{K}) \end{align*}

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 71.10.2 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 71.10.3 and 71.10.4 we conclude. $\square$

 It is enough to assume $G$ is decent, e.g., locally separated or quasi-separated by Lemma 78.9.4.

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