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Background/motivation

One of the main reason to introduce (algebraic) stacks is build "fine moduli spaces" for functors which, strictly speaking, are not representable. The yoga is more or less as follows.

One notices that a representable functor on the category of schemes is a sheaf in the fpqc topology. In particular it is a sheaf in coarser topologies, like the fppf or étale topologies. Now some naturally defined functors (for instance the functor $\mathcal{M}_{1,1}$ of elliptic curves) are not sheaves in the fpqc topology

(actually $\mathcal{M}_{1,1}$ is not even an étale sheaf) so there is no hope to represent them.

Enters the $2$-categorical world and we introduce fibered categories and stacks. Many functors which are not sheaves arise by collapsing fibered categories which ARE stacks, so not all hope is lost. But, as not every fpqc sheaf is representable, we should not expect that every fpqc stack is in some sense "represented by a generalized space", so we make a definition of what we mean by an algebraic stack.

Let me stick with the Deligne-Mumford case. Then a DM stack is a fibered category (in groupoids) over the category of schemes, which

1) is a stack in the étale topology

2) has a "nice" diagonal

3) is in some sense étale locally similar to a scheme.

I don't need to make precise what 2) and 3) mean.

By the preceding philosophy we should expect that DM stacks generalize schemes in the same way that stacks generalize sheaves. In particular I would expect that DM stacks turn out to be stacks in finer topologies, just as schemes are sheaves not only in the Zariski topology (which is trivial) but also in the fpqc topology (which is a theorem of Grothendieck).

Question

Is it true that DM stacks are actually stacks in the fpqc topology? And if not, did someone propose a notion of "generalized space" in the context of stacks, so that this result holds?

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1 Answer 1

up vote 12 down vote accepted

The rule of thumb is this: Your DM (or Artin) stack will be a sheaf in the fppf/fpqc topology if the condition imposed on its diagonal is fppf/fpqc local on the target ("satisfies descent").

In other words, in condition 2 you asked that the diagonal be a relative scheme/relative algebraic space perhaps with some extra properties. If there if fppf descent for morphisms of this type (e.g., "relative algebraic space", "relative monomorphism of schemes"), you'll have something satisfying fppf descent. If there is fpqc descent for morphisms of this type (e.g., "relative quasi-affine scheme"), then you'll have something satisfying fpqc descent.

See for instance LMB (=Laumon, Moret-Bailly. Champs algebriques), Corollary 10.7. Alternatively: earlier this year I wrote up some notes (PDF link) that included an Appendix collecting in one place the equivalences of some standard definitions of stacks, including statements of the type above.

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The suggested references appeal to fpqc-sheafifiction, a somewhat far-out operation. (I generally view appeal to universes as a bit of laziness when it can be avoided with a bit more effort to unravel what is actually going on.) I think it is a very instructive exercise to unravel the arguments at this step in the suggested references to see that fpqc-sheafification is not needed at all. It gives one a more "hands-on" understanding of what makes the argument work to make it more explicit in this way. Give it a try! –  BCnrd Feb 20 '10 at 18:11
    
Thank you very much. Sorry that it took me so much time to accept the answer; I didn't have time to read your notes before. –  Andrea Ferretti Mar 2 '10 at 21:33
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