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Let $X = \mathbb{A}^1_{\mathbb{C}}/\mathbb{Z}$, where $\mathbb{Z}$ acts on $\mathbb{A}^1$ via translation. [To clarify, $X$ is an \'etale sheaf with a smooth presentation $\mathbb{A}^1_{\mathbb{C}}\to X$ by a scheme.]

Note that $X$ is a finite type algebraic space over $\mathbb{C}$ (in the sense of the stacks project). This algebraic space has appeared several times on MO (see for instance Why is this not an algebraic space?).

Also, as Jason Starr explains in the comments, the diagonal $X\to X\times X$ is a morphism of finite type algebraic spaces, but it is not of finite type (because it is not quasi-compact). Thus, $X$ is not quasi-separated over $\mathbb{C}$, and therefore $X$ is not of finite presentation over $\mathbb{C}$.

My previous question about the stack $\mathbb{A}^1_{\mathbb{Z}}/\mathbb{Z}$ was Question 2 before, and was answered by Laurent Moret-Bailly in the comments. This leaves one last question:

Question. The torsor $\mathbb{A}^1\to X$ is an example of a $\mathbb{Z}$-torsor whose total space is an algebraic curve. Are there other similar examples of $G$-torsors $C\to Y$ over a finite type algebraic space $Y$, where $G$ is an infinite discrete group and $C$ is an algebraic curve? Preferably with $C$ not $\mathbb{A}^1$ or $\mathbb{G}_m$.

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    $\begingroup$ The general rule here is: one question per post. I'm not flagging this for closure, but I'm writing this in case your question gets closed, and you might not know why. $\endgroup$
    – Alex M.
    Commented Nov 3, 2017 at 19:51
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    $\begingroup$ @Bernd Let $E$ be an elliptic curve, and let $p$ be a point of infinite order. Let $G$ be the subgroup generated by $p$. This acts freely on $E$. Thus, the sheaf $E/G$ is an algebraic space, and the morphism $E\to E/G$ is a $\mathbb{Z}$-torsor. This is another example. [Also, I don't think there are any "other" real examples, because the automorphism group of a curve is finite, in general.] $\endgroup$
    – Ariyan Javanpeykar
    Commented Nov 5, 2017 at 13:11
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    $\begingroup$ @JasonStarr: the notions of "finite type" and "locally finite type" shouldn't require quasi-separatedness, but rather one should instead just be more attentive than is necessary for schemes to the distinction between "finite type" and "finite presentation" (and even the meaning of "noetherian") when considering general algebraic spaces (which need not be Zariski-locally quasi-separated). See Tag 03E9 in the Stacks Project for a nice discussion of this. Since even with schemes we don't "spread out" in the loc. finite type context, focusing on "finite type" vs. "finite presentation" seems best. $\endgroup$
    – nfdc23
    Commented Nov 5, 2017 at 13:25
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    $\begingroup$ @nfdc23. I do not hold any dogmatic views about what one "should" do. I try to help confused students navigate contradictory hypotheses in different parts of the literature. $\endgroup$
    – Jason Starr
    Commented Nov 5, 2017 at 13:27
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    $\begingroup$ @JasonStarr: I agree about clarifying seemingly contradictory hypotheses in the literature. One reason it seems better not to impose quasi-separatedness in the definition of "locally finite type" (admittedly a non-issue when one only works with quasi-separated algebraic spaces) is that the notion is then etale-local on the source as it is for schemes, whereas otherwise it isn't since quasi-separatedness isn't etale-local on the source (which seems to lie at the core of what makes the quotients in the question posed so disorienting at first sight). $\endgroup$
    – nfdc23
    Commented Nov 5, 2017 at 13:51

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