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 $f:X\to Y$ 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 $V_{i}$ 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 $x\in X$ has a neighborhood $U$ such that $f(U)$ is open and $f:U\to f(U)$ is quasi-compact.
4. Each point $x\in X$ 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:

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

See also

• flat topology
• fppf morphism

References

• Angelo Vistoli, "Notes on Grothendieck topologies, fibered categories and descent theory." arXiv:math.AG/0412512v4