higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
Fix some scheme .
The fpqc-site (over ) is the site
whose underlying category is the category of affine schemes over ;
whose coverage has as covering families those families of morphisms that are such that
each is a flat morphism;
for every affine open there exists , a function and affine opens with
This appears as (Stacks Project, Tag 022B).
The last condition does imply that .
The abbreviation “fpqc” is for fidèlement plat quasi-compacte : faithfully flat and quasi-compact.
Because the collection of fpqc covers of a scheme does not have a small collection of refinements (Stacks project, Tag 0BBK), working with the fpqc topology can be set-theoretically tricky. Indeed, in 1975, Waterhouse gave an example of a functor on schemes that admits no fpqc sheafification. This contradicts many claims in the literature that fpqc sheafification and stackification is functorial (and such claims continue to be made).
fpqc-site fppf-site syntomic site étale site Nisnevich site Zariski site
Chaper 27.8 in
Last revised on January 8, 2019 at 21:44:07. See the history of this page for a list of all contributions to it.