# 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 $\mathrm{Aff}$ be the category opposite to the category of commutative algebras. A family of maps $\left\{{f}_{i}:{U}_{i}\to X{\right\}}_{i\in I}$ in that category is a cover in the fpqc topology if the union of images cover $X$ as a topological space and all morphisms ${f}_{i}$ are faithfully? flat and quasicompact. The French for this is fidèlement plat et quasicompact (fpqc).