nLab fppf site

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.

Related concepts

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

References

Chaper 27.7 in

• Aise Johan de Jong, The Stacks Project