fpqc morphism

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, 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 <a href="/facts/French_language/qe1DnaQh">French</a> 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 <a href="/facts/Faithfully_flat_morphism/CLM7e8d6">faithfully flat morphism</a> that satisfies the following equivalent conditions:

<ol><li>Every <a href="/facts/Quasi-compact/d0cgXJH7">quasi-compact</a> open subset of Y is the image of a quasi-compact open subset of X.</li>
<li>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.</li>
<li>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 <a href="/facts/Quasi-compact_morphism/sUCP0wez">quasi-compact</a>.</li>
<li>Each point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 has a quasi-compact neighborhood such that 
 
 
 
 f
 (
 U
 )
 
 
 {\displaystyle f(U)}
 
 is open affine.</li></ol>
Examples: An open faithfully flat morphism is fpqc.
An fpqc morphism satisfies the following properties:

<ul><li>The composite of fpqc morphisms is fpqc.</li>
<li>A base change of an fpqc morphism is fpqc.</li>
<li>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.</li>
<li>A faithfully flat morphism that is locally of finite presentation (i.e., fppf) is fpqc.</li>
<li>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.</li></ul>

fpqc morphism open-in-new

fpqc morphism