In this post I mentioned a theorem usually attributed to Michael Artin which basically says that an fppf sheaf which has a flat, finitely presented cover by a scheme is an algebraic space. I am still working on adding this to the stacks project, and more or less all the preliminary work is done.
But what I wanted to say here is that to prove this one does not have to use “Artin’s trick”. What I mean is the argument in Artin’s versal deformations paper that rests on the following fact: Given a morphism f : X –> Y which is flat and of finite presentation then the space H_n(X/Y) of length n complete intersections in fibers of f is smooth over Y, and moreover \coprod_n H_n(X/Y) –> Y is surjective.
Instead one can use a slicing argument to go down to relative dimension zero (see Lemma Tag 0461) and etale localization of groupoids (see Lemma Tag 03FM) to get an etale covering by a scheme (by dividing out by the P-part of the groupoid scheme). Note that the last lemma is a version of Keel-Mori, Proposition 4.2 and that they in their proof use some form of Hilbert schemes also… but they needn’t have and standard etale locailzation techniques would have sufficed.
Morally speaking it is clear that Hilbert schemes needn’t be considered when proving this result since the original flat finitely presented covering X –> Y might have had relative dimension zero with connected fibres, and then only one H_n(X/Y) is nonempty (locally on Y), namely that one where n is the relative degree and H_n(X/Y) = Y. In other words you are just directly proving that Y is an algebraic space!
On the other hand, as Jarod Alper pointed out, when we try to prove the analogous result for algebraic stacks, then we have to construct a smooth cover which will have in general a positive relative dimension over the stack and the remark in the preceding paragraph doesn’t apply. Of course this was the point of Artin’s trick and this is how he used it in his paper.
I haven’t yet read the proof of the slicing Lemma carefully but it seems to suggest that any algebraic stack with locally quasi-finite diagonal has a locally quasi-finite flat presentation. Is this correct? I have only been able to prove this for quasi-separated stacks (i.e., stacks such that the diagonal is quasi-finite and quasi-separated) or more generally for stacks such that every point is algebraic.
As this class of stacks (with (locally) quasi-finite diagonal) is so important, it certainly merits a dedicated name and a general equivalence:
qff-presentation (locally qf flat presentation) locally quasi-finite diagonal
étale presentation unramified diagonal
is welcome indeed!
Yes, I think you are right. But a strange consequence of building the material on algebraic stacks from the ground up is that I only feel sure about results I was able to document in the stacks project, and this one isn’t yet.
Do you have a suggestion as to what we should name algebraic stacks with locally quasi-finite diagonal if anything?
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