nLab
fpqc site

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 $\{f:U_i\to X\}$ those families of morphisms that are such that

  • each $f_i$ is a flat morphism;

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

    $${\cup }_{j=1}^n{f}_{a(j)}(V_j)=W.$$\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 ${\cup }_i{f}_i(U_i)=X$.

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, 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).

Related concepts

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

References

Chaper 27.8 in

• Aise Johan de Jong, The Stacks Project

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