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Suppose $F:Sch^\text{op}\to Set$ is a sheaf in the fpqc topology, has quasi-compact representable diagonal, and has an fpqc cover by a scheme. Must $F$ be an algebraic space? That is, must $F$ have an étale cover by a scheme?

Note that it is enough to find an fppf cover of $F$ by a scheme, at which point you know that $F$ is an algebraic stack ("Artin's slice theorem", 10.1 of Laumon-Moret-Bailly) fibered in sets, so it's an algebraic space.

The analogous question, "is every fpqc stack an algebraic stack?" has a negative answer, but the counterexample is "purely stacky".

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A. J. de Jong has a post about this: – Akhil Mathew Oct 11 '11 at 23:01
To clarify, by an "fpqc cover", I meant a morphism which is representable by fpqc morphisms of schemes, which is the case de Jong's example focuses on. – Anton Geraschenko Oct 12 '11 at 3:12

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