# Group algebraic spaces that are locally of finite type and have only finitely many points

Let $k$ be an algebraically closed field of characteristic zero. Let $G$ be a group algebraic space over $k$ such that $G\to$ Spec $k$ is locally of finite type.

Suppose that $G(k)$ is finite.

Does it follow that $G\to$ Spec $k$ is finite (and thus $G$ is a scheme)?

If we assume $G$ is quasi-separated over $k$, then $G$ is in fact a scheme (see Commutative group algebraic spaces). It follows then that $G$ is finite.

But what if $G$ is not quasi-separated over $k$? Is $G$ still finite over $k$?

• No: for any algebraically closed field $k$, let $C$ be any commutative $k$-group scheme of finite type with positive dimension and $G = C/H$ where $H \to C$ is the $k$-subgroup functor given by the (etale) constant group on $C(k)$. In other words, $G$ is the quotient of $C$ modulo the etale equivalence relation $\delta:H\times C \rightarrow C \times C$ defined by $(h,c)\mapsto (c,hc)$. This is a locally finite type algebraic space group with $G(k)=1$ and dimension $\dim(C) > 0$ and it is not quasi-separated (since $\delta$ is not quasi-compact). – nfdc23 Feb 13 '18 at 13:54
• How neat! (In the last line you meant to write $\mathrm{dim}(G) >0$ right?) – Gerard Feb 13 '18 at 14:08
• "locally finite type" means "locally of finite type"? – YCor Feb 13 '18 at 14:18
• I wrote what I meant for the dimension. In more detail (as you recognize): the dimension is equal to $\dim(C)$ since $\dim(H)=0$, and $\dim(C)$ is positive by hypothesis. – nfdc23 Feb 13 '18 at 15:12
• OK. "locally finite" is widely used in group theory and "locally finite type group" makes it hard to decipher (esp inside a huge expression adverb+adjective+noun+noun+adjective+noun, without any dash nor preposition)! – YCor Feb 13 '18 at 17:25

No: for any algebraically closed field $k$, let $C$ be any commutative $k$-group scheme of finite type with positive dimension and $G=C/H$ where $H\to C$ is the $k$-subgroup functor given by the (etale) constant group on $C(k)$. In other words, $G$ is the quotient of $C$ modulo the etale equivalence relation $\delta:H\times C \to C\times C$ defined by $(h,c)\mapsto (c,hc)$ This is a locally finite type algebraic space group with $G(k)=1$ and dimension $\dim(G)=dim(C)>0$. Note that $G$ is not quasi-separated (since $\delta$ is not quasi-compact).