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I recently tried to explain the fppf site to a differential geometer. I started with the etale site, where I had two motivating claims:

  1. If X is a smooth projective variety over the complexes, the etale topology "recovers" the traditional topology. More precisely, the category of sheaves on the etale site and the category of sheaves on the complex-analytic site are equivalent. (EDIT: I have led my friend astray! See below.)

  2. With just three words scare-quoted: The etale topology is (by "definition") the coarsest "topology" that makes the inverse function theorem "true."

I don't know how to make a statement analogous to (1) or (2) for the fpqc (or fppf, or ...) topology. I don't even really have a feeling for the difference between these topologies. Is there an analogue of (1) or (2) or anything else "purely geometric" to hang my hat on?

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The first claim is false: for instance, the subobjects of the terminal object in the etale topos are the Zariski opens, whereas in the analytic topos they're the analytic opens... or again the etale topos is coherent and the analytic topos is not... or again there can be no action of Aut(C) on H^2(P^1;Z)=Z inducing the usual (cyclotomic) action on H^2(P^1;Z/n)=Z/n(1). –  Dustin Clausen Apr 22 '12 at 1:01
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I think the main observation is, that even for this very fine and "un-intutive" topologies one can proof descend theorems e.g. for quasi-coherent sheaves and morphisms of schemes: $Hom(?,X)$ is a sheaf. If you look in the proof of these theorems you see precisely where you need "fp" and "qc" or "pf" properties. –  Heinrich Hartmann Apr 24 '12 at 10:04

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So, I actually kind of agree with you on the \'etale topology. So let's say your friend has a rough idea of that one. Or below you can even replace \'etale with Zariski and the thing works too.

Then, next you can try to give them an idea of what a surjective, finite locally free morphism of schemes is. Surjective finite locally free morphisms have many good topological properties: they are universally closed, have finite fibres, are quasi-compact, etc, etc. I guess topologically this is like having a surjective proper map of locally compact spaces with finite fibres. You can explain to your friend how these (in topology) can be used (via hypercoverings, woohoo!) to compute cohomology, etc.

Then the fppf topology just combines the \'etale topology (or the Zariski one) with the coverings coming from surjective finite locally free morphisms. See Section Tag 05WM.

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"don't forget to buy my products" means "vote for my answer" ? –  Niels Dec 21 '13 at 20:53

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