# Stacks in the fpqc topology

This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme.

Suppose one developed the theory of stacks in the fpqc topology and by fpqc-algebraic meant only (a condition on the diagonal plus) "admitting an fpqc cover by a scheme" (rather than the usual definition of algebraic, which I'd like to call fppf-algebraic). One seems to give up the ability to make sense of notions that can be checked locally on smooth covers but not on fpqc covers. But some notions still seem to survive for fpqc-algebraic stacks (e.g. "locally noetherian," cf. EGA IV 2.2.14).

Question: do people principally work with fppf-algebraic stacks rather than fpqc-algebraic stacks simply because one gets a nicer package that still applies to most examples one comes across in practice?

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It is important that properties move both up and down through the covering maps. Though noetherianness descends from an fpqc cover, it is not inherited upstairs from the property on the base. Also, smooth and even fppf maps are open for the Zariski topology, whereas general fpqc maps are not better than topologically quotient maps. It seems doubtful that one could have a useful geometric theory of fpqc-stacks (as distinct from the fpqc topology, which is convenient in various situations as you know). Maybe someone could grind out categorical formalism for it, but likely it has no beef. –  Boyarsky Jun 23 '10 at 19:13
Thanks, point well taken. Still, there are interesting properties that move both up and down under fpqc base change in the target: see EGA IV, 2.7.1. So at least these would make sense for representable morphisms. I suppose that's not saying very much, however... –  Thomas Nevins Jun 30 '10 at 21:44
@Thomas: Sure, and also more properties in EGA IV, sections 2.6 and 17.7. But since we have to allow to work Zariski-locally, and it is rather awkward to describe the topology generated by fpqc and Zariski (since fpqc maps are generally not open), it is hard to do much of anything that really uses existence of an fpqc scheme cover in an interesting way. –  Boyarsky Jul 2 '10 at 7:02
I also wonder if someone has developed a theory of fpqc-stacks. @Boyarski: In his notes on descent theory, Angelo Vistoli defines a notion of fqpc coverings which includes Zariski coverings. He uses a certain weaking of quasi-compactnes. –  Martin Brandenburg Sep 15 '13 at 8:23