topos theory

# Contents

## Definition

Fix some scheme $S$.

###### Definition

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

• whose underlying category is the category $\mathrm{Aff}/S$ of affine schemes over $S$;

• whose coverage has as covering families $\left\{f:{U}_{i}\to X\right\}$ those families of morphisms that are such that

• each ${f}_{i}$ is a flat morphism;

• for every affine open $W↪X$ there exists $n\ge 0$, a function $a:\left\{1,\cdots ,n\right\}\to I$ and affine opens ${V}_{j}↪{T}_{a\left(j\right)}$ with

${\cup }_{j=1}^{n}{f}_{a\left(j\right)}\left({V}_{j}\right)=W\phantom{\rule{thinmathspace}{0ex}}.$\cup_{j = 1}^{n} f_{a(j)}(V_j) = W \,.

This appears as (de Jong, def. 27.8.1).

###### Remark

The last condition does imply that ${\cup }_{i}{f}_{i}\left({U}_{i}\right)=X$.

###### 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, 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

Revised on September 5, 2011 09:41:46 by Urs Schreiber (89.204.153.80)