nLab fpqc site

Redirected from "fpqc-site".
Note: fpqc site and fpqc site both redirect for "fpqc-site".
Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Geometry

Contents

Definition

Fix some scheme SS.

Definition

The fpqc-site (over SS) is the site

  • whose underlying category is the category Aff/SAff/S of affine schemes over SS;

  • whose coverage has as covering families {f:U iX}\{f : U_i \to X\} those families of morphisms that are such that

    • each f if_i is a flat morphism;

    • for every affine open WXW \hookrightarrow X there exists n0n \geq 0, a function a:{1,,n}Ia : \{1, \cdots, n\} \to I and affine opens V jU a(j)V_j \hookrightarrow U_{a(j)} with

      j=1 nf a(j)(V j)=W. \cup_{j = 1}^{n} f_{a(j)}(V_j) = W \,.

This appears as (Stacks Project, Tag 022B).

Remark

The last condition does imply that if i(U i)=X\cup_i f_i(U_i) = X.

Remark

The abbreviation “fpqc” is for fidèlement plat quasi-compacte : faithfully flat and quasi-compact.

Remark

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 \to fppf-site \to syntomic site \to étale site \to Nisnevich site \to Zariski site

References

Chaper 27.8 in

  • W. C. Waterhouse, Basically bounded functors and flat sheaves, Pacific Journal of Mathematics 57 (1975), no. 2, 597–610 MR396578, euclid

Last revised on January 9, 2019 at 02:44:07. See the history of this page for a list of all contributions to it.