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In [[algebraic geometry]], an '''fpqc morphism''' <math>f: X \to Y</math> 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 <math>V_i</math> of Y by open affine subschemes such that each <math>V_i</math> is the image of a quasi-compact open subset of X.
# Each point <math>x \in X</math> has a neighborhood <math>U</math> such that <math>f(U)</math> is open and <math>f: U \to f(U)</math> is [[quasi-compact morphism|quasi-compact]].
# Each point <math>x \in X</math> has a quasi-compact neighborhood such that <math>f(U)</math> 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 <math>f: X \to Y</math> is a morphism of schemes and if there is an open covering <math>V_i</math> of ''Y'' such that the <math>f: f^{-1}(V_i) \to V_i</math> is fpqc, then ''f'' is fpqc.
* A faithfully flat morphism that is locally of finite presentation (i.e., fppf) is fpqc.
* If <math>f:X \to Y</math> 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]]
* [[quasi-compact morphism]]
 
== References ==
*Angelo Vistoli, "Notes on Grothendieck topologies, fibered categories and descent theory." {{arxiv|id=0412512v4|archive=math.AG}}
 
{{geometry-stub}}
[[Category:Morphisms of schemes]]

Latest revision as of 04:10, 16 November 2014

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