topos theory

# Contents

## Definition

Fix some scheme $S$.

###### Definition

the fppf-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;

• each $f_i$ is locally of finite presentation;

• we have that $X = \cup_i f_i(U_i)$.

This appears as (de Jong, def. 27.7.1, def 27.7.6, lemma 27.7.11).

###### Remark

The abbreviation “fppf” is for fidèlement plat de présentation finie : faithfully flat and of finite presentation.

But notice that it is common practice, as in the above definition, to require only local finite presentability.

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

## References

Chaper 27.7 in

Revised on September 5, 2011 09:40:52 by Urs Schreiber (89.204.153.80)