Fpqc morphism

In algebraic geometry, there are two slightly different definitions of an fpqc morphism, both variations of faithfully flat morphisms.

Sometimes a fpqc morphism means one that is faithfully flat and quasicompact. This is where the abbrviation fpqc comes from: fpqc stands for the French phrase "fidèlement plat et quasi-compact", meaning "faithfully flat and quasi-compact".

However it is more common to define an fpqc morphism f : X → Y of schemes to be a faithfully flat morphism that satisfies the following equivalent conditions:

1. Every quasi-compact open subset of Y is the image of a quasi-compact open subset of X.
2. There exists a covering V i of Y by open affine subschemes such that each V i is the image of a quasi-compact open subset of X.
3. Each point x ∈ X has a neighborhood U such that f ( U ) is open and f : U → f ( U ) is quasi-compact.
4. Each point x ∈ X has a quasi-compact neighborhood such that f ( U ) is open affine.

Examples: An open faithfully flat morphism is fpqc.

An fpqc morphism satisfies the following properties: