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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 $k$?

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    $\begingroup$ 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). $\endgroup$ – nfdc23 Feb 13 '18 at 13:54
  • $\begingroup$ How neat! (In the last line you meant to write $\mathrm{dim}(G) >0$ right?) $\endgroup$ – Gerard Feb 13 '18 at 14:08
  • $\begingroup$ "locally finite type" means "locally of finite type"? $\endgroup$ – YCor Feb 13 '18 at 14:18
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    $\begingroup$ 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. $\endgroup$ – nfdc23 Feb 13 '18 at 15:12
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    $\begingroup$ 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)! $\endgroup$ – YCor Feb 13 '18 at 17:25
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The comment of nfdc23 answers the question:

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).

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