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fpqc morphism

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In algebraic geometry, 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 French phrase "fidèlement plat et quasi-compact", meaning "faithfully flat and quasi-compact".

However it is more common to define an fpqc morphism 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 of Y by open affine subschemes such that each is the image of a quasi-compact open subset of X.
  3. Each point has a neighborhood such that is open and is quasi-compact.
  4. 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.

See also

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References

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  • Vistoli, Angelo (2004). "Notes on Grothendieck topologies, fibered categories and descent theory" (PDF). arXiv:math/0412512. Bibcode:2004math.....12512V.
  • Stacks Project, "The fpqc Topology." http://stacks.math.columbia.edu/tag/03NV