Flat is not enough

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

  1. F is a sheaf for the etale topology,
  2. the diagonal of F is representable by schemes, and
  3. 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.)