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 abbreviation 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 ${\displaystyle f:X\to Y}$ 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 ${\displaystyle V_{i}}$ of Y by open affine subschemes such that each ${\displaystyle V_{i}}$ is the image of a quasi-compact open subset of X.
3. Each point ${\displaystyle x\in X}$ has a neighborhood ${\displaystyle U}$ such that ${\displaystyle f(U)}$ is open and ${\displaystyle f:U\to f(U)}$ is quasi-compact.
4. Each point ${\displaystyle x\in X}$ has a quasi-compact neighborhood such that ${\displaystyle f(U)}$ 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 ${\displaystyle f:X\to Y}$ is a morphism of schemes and if there is an open covering ${\displaystyle V_{i}}$ of Y such that the ${\displaystyle f:f^{-1}(V_{i})\to V_{i}}$ is fpqc, then f is fpqc.
• A faithfully flat morphism that is locally of finite presentation (i.e., fppf) is fpqc.
• If ${\displaystyle f:X\to Y}$ 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.