The title of this blog post is the opposite of this post. But don’t click through yet, because it may be more fun to read this one first.

I claim there exists a functor F on the category of schemes such that

- F is a sheaf for the etale topology,
- the diagonal of F is representable by schemes, and
- there exists a scheme U and a surjective, finitely presented, flat morphism U —> F

but F is not an algebraic space. Namely, let k be a field of characteristic p > 0 and let k ⊂ k’ be a nontrivial finite purely inseparable extension. Define

F(S) = {f : S —> Spec(k), f factors through Spec(k’) etale locally on S}

It is easy to see that F satisfies (1). It satisfies (2) as F —> Spec(k) is a monomorphism. It satisfies (3) because U = Spec(k’) —> F works. But F is not an algebraic space, because if it were, then F would be isomorphic to Spec(k) by Lemma Tag 06MG.

Ok, now go back and read the other blog post I linked to above. Conclusion: to get Artin’s result as stated in that blog post you definitively need to work with the fppf topology.

(Thanks to Bhargav for a discussion.)