nLab fpqc topology

Idea

The fpqc topology is a Grothendieck topology on the category of (commutative) affine schemes. It is one of the main Grothendieck topologies used in algebraic geometry.

Definition

Let $Aff$ be the category opposite to the category of commutative algebras. A family of maps $\{f_i : U_i\to X\}_{i\in I}$ in that category is a cover in the fpqc topology if the induced morphism $\coprod U_i \to X$ is faithfully flat and quasicompact. In particular, the union of images then cover $X$ as a topological space. The French for this is fidèlement plat et quasicompact (fpqc).

Related concepts

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

category: algebraic geometry