fpqc morphism
In algebraic geometry, an fpqc morphism of schemes is a faithfully flat morphism that satisfies the following equivalent conditions:
- Every quasi-compact open subset of Y is the image of a quasi-compact open subset of X.
- There exists a covering
of Y by open affine subschemes such that each
is the image of a quasi-compact open subset of X.
- Each point
has a neighborhood
such that
is open and
is quasi-compact.
- Each point
has a quasi-compact neighborhood such that
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
is a morphism of schemes and if there is an open covering
of Y such that the
is fpqc, then f is fpqc.
- A faithfully flat morphism that is locally of finite presentation (i.e., fppf) is fpqc.
- If
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.
Fpqc stands for the French phrase "fidèlement plat et quasi-compact", meaning "faithfully flat and quasi-compact".
See also
References
- Angelo Vistoli, "Notes on Grothendieck topologies, fibered categories and descent theory." arXiv:math.AG/0412512v4
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