# What is your picture of the flat topology?

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?

• 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). Apr 22, 2012 at 1:01
• 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. Apr 24, 2012 at 10:04