Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme?


My feeling is that the answer is "yes" because an algebraic space group which is not a scheme would be too awesome. Any group homomorphism from such a G to an algebraic group (a scheme group) would have to have infinite kernel since [an algebraic space which is quasi-finite over a scheme is itself a scheme](https://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme/4574#4574). In particular, G would have no faithful representations or [faithful actions on projective varieties (probably)](https://mathoverflow.net/questions/8812/why-do-automorphism-groups-of-algebraic-varieties-have-natural-algebraic-group-st/8820#8820). There can't be a surjective group homomorphism from an algebraic group H to G, since that would identify G with H/K, [which is a scheme](https://mathoverflow.net/questions/8809/is-every-subgroup-of-an-algebraic-group-a-stabilizer-for-some-action). In particular, you cannot put a group structure on any étale cover of G.



Is an algebraic space group always a scheme?
I think the answer is yes. If S is a noetherian scheme and G is a relative algebraic group space over S, then there is a stratification of S such that over each stratum, G is a group scheme (see K. Behrend, Derived \ell-adic categories for Algebraic Stacks, 5.1.1). When S is Spec k there is no non-trivial stratification.


