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On a scheme, the coherent sheaves that are invertible objects for the tensor product (monoid) operation are precisely the coherent sheaves that are (Zariski) locally free of rank one. Is the same true for algebraic spaces? (I believe that this follows from a theorem of Nisnevich since there is simultaneously an etale local section and a Nisnevich local section, but hopefully somebody has a nice answer.)

This is motivated by my (incomplete!) answer to the following MO post: Quotients of schemes by connected groups

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  • $\begingroup$ Sorry for my confusion. If you have something like $\mathcal M\otimes_{\mathcal O_X}\mathcal N\cong\mathcal O_X$ for quasicoherent sheaves $\mathcal M,\mathcal N$, you can always pullback along a map $\operatorname{Spec}(R)\to X$, and invoking the affine result? $\endgroup$
    – Z. M
    Commented Sep 1, 2022 at 20:12
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    $\begingroup$ @Z.M An algebraic space has a Zariski cover by algebraic spaces of the form $\text{Spec}(R)$ if and only if the algebraic space is a scheme. $\endgroup$
    – Jason Starr
    Commented Sep 1, 2022 at 20:43
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    $\begingroup$ Presumably this is motivated by your answer to Quotients of schemes by connected groups? $\endgroup$
    – LSpice
    Commented Sep 1, 2022 at 21:06
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    $\begingroup$ @LSpice Yes, indeed. I will add a remark. $\endgroup$
    – Jason Starr
    Commented Sep 1, 2022 at 23:52

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There are counterexamples by Stefan Schröer. One of them is not locally separated (a bug-eyed cover, as Kollár calls it), another is a (non-normal) proper surface. See the paper here.

About the link: some characters do not display properly, but one can click "View PDF" on top of the page, or use the arXiv version. Thanks to Jason Starr and Sasha for the links.

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