In the discussion of groupoid stacks [U/R] it turns out that given objects x, y of [U/R] over some scheme T, then Isom(x, y) is fppf locally on T an algebraic space. Thus it makes sense to go back to algebraic spaces and prove a result characterizing algebraic spaces. Namely, an fppf sheaf of sets F for which there exists an algebraic space X and a map f : X –> F which is

  • representable by algebraic spaces, and
  • surjective, flat and locally of finite presentation

is an algebraic space. The only ingredient missing for the proof is an analogue of Keel-Mori, Lemma 3.3. Hopefully we will have some time to write this in the near future.

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