Take the tour ×  Let X be an algebraic space such that there
exists an etale surjective map f: X -> Y, where Y is a scheme.
Is it true that this implies X is also a scheme?  share|improve
this question add comment  1 Answer  The point must be to avoid
separatedness hypotheses on $f$. (D. Knutson proved algebraic
spaces locally quasi-finite and separated over schemes are
schemes; he may have had noetherian hypotheses, in which case
those hypotheses are removed in the appendix to Champs
Algebriques.)  Anyway, in the absence of separatedness the
answer is negative. In the Introduction to Knutson's book
there's a procedure beginning with smooth algebraic variety $Y$
of dimension $> 1$ over a field and "replacing" a smooth
hypersurface in $Y$ with an irreducible etale double cover while
remaining smooth and irreducible. This resulting $X$ is an
algebraic space etale over $Y$, and the rank-jumping of $f:X
\rightarrow Y$ is really weird: it jumps up along a closed set
rather than down, contrary to Zariski's Main The