A reference for "an algebraic space is a scheme iff its reduction is"?

Question

It seems to be a known fact that an algebraic space is a scheme if and only if its associated reduced closed subspace is a scheme. For instance, this is used in Chai-Faltings in proving that the dual abelian scheme always exists. Is there a reference containing a proof of this fact?

Answer 1

For arbitrary algebraic spaces (no quasi-separatedness hypothesis) this is Corollary 3.1.12 of "Nagata compactification for algebraic spaces" in Volume 11 (no. 4) of Journal of the Institute of Mathematics of Jussieu (2012), pp. 747--814, deduced from the noetherian case by approximation methods. But for the needs of the Faltings-Chai book this is very much overkill, as in their setup one can easily reduce to the noetherian case by more elementary limit considerations (if one begins with an abelian scheme over a scheme, for example).

Note that to prove this in the noetherian case one can reduce to when $S_{\rm{red}}$ is even affine, so then the diagonal of $S$ is a proper monomorphism, hence a closed immersion, so $S$ is even separated (let alone "locally separated"; i.e., with diagonal a quasi-compact immersion), so D. Knutson's result applies.

Answer 2

If $X$ is noetherian and locally separated, this is Corollary III.3.6 of D. Knutson's book Algebraic Spaces.