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

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

However it is more common to define an fpqc morphism f : X → Y {\displaystyle f:X\to 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 {\displaystyle V_{i}} of Y by open affine subschemes such that each V i {\displaystyle V_{i}} is the image of a quasi-compact open subset of X.
3. Each point x ∈ X {\displaystyle x\in X} has a neighborhood U {\displaystyle U} such that f ( U ) {\displaystyle f(U)} is open and f : U → f ( U ) {\displaystyle f:U\to f(U)} is quasi-compact.
4. Each point x ∈ X {\displaystyle x\in X} has a quasi-compact neighborhood such that f ( U ) {\displaystyle f(U)} is open affine.

Examples: An open faithfully flat morphism is fpqc.

An fpqc morphism satisfies the following properties:

• The composite of fpqc morphisms is fpqc.
• A base change of an fpqc morphism is fpqc.
• If f : X → Y {\displaystyle f:X\to Y} is a morphism of schemes and if there is an open covering V i {\displaystyle V_{i}} of Y such that the f : f − 1 ( V i ) → V i {\displaystyle f:f^{-1}(V_{i})\to V_{i}} is fpqc, then f is fpqc.
• A faithfully flat morphism that is locally of finite presentation (i.e., fppf) is fpqc.
• If f : X → Y {\displaystyle f:X\to Y} is an fpqc morphism, a subset of Y is open in Y if and only if its inverse image under f is open in X.