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](https://mathoverflow.net/q/429489)



Are the tensor-invertible coherent sheaves on an algebraic space (Zariski) locally free of rank one?
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](https://doi.org/10.1016/S0022-4049(02)00012-9).


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](https://arxiv.org/abs/math/0110243). Thanks to Jason Starr and Sasha for the links.


