Contents

topos theory

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Definition

Fix some scheme $S$.

###### Definition

The fpqc-site (over $S$) is the site

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

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

• each $f_i$ is a flat morphism;

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

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

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

###### Remark

The last condition does imply that $\cup_i f_i(U_i) = X$.

###### 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

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