There are several slightly different flat topologies, the most common of which are the fppf topology and the fpqc topology. fppf stands for fidèlement plate de présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation. fpqc stands for fidèlement plate et quasi-compacte, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat. In both categories, a covering family is defined be a family which is a cover on Zariski open subsets.[2] In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover.[3] These topologies are closely related to descent. The "pure" faithfully flat topology without any further finiteness conditions such as quasi compactness or finite presentation is not used much as is not subcanonical; in other words, representable functors need not be sheaves.
Unfortunately the terminology for flat topologies is not standardized. Some authors use the term "topology" for a pretopology, and there are several slightly different pretopologies sometimes called the fppf or fpqc (pre)topology, which sometimes give the same topology.
Flat cohomology was introduced by Grothendieck in about 1960.[4]
Let X be an affine scheme. We define an fppf cover of X to be a finite and jointly surjective family of morphisms
(φa : Xa → X)
with each Xa affine and each φaflat, finitely presented. This generates a pretopology: for X arbitrary, we define an fppf cover of X to be a family
(φ'a : Xa → X)
which is an fppf cover after base changing to an open affine subscheme of X. This pretopology generates a topology called the fppf topology. (This is not the same as the topology we would get if we started with arbitrary X and Xa and took covering families to be jointly surjective families of flat, finitely presented morphisms.) We write Fppf for the category of schemes with the fppf topology.
The small fppf site of X is the category O(Xfppf) whose objects are schemes U with a fixed morphism U → X which is part of some covering family. (This does not imply that the morphism is flat, finitely presented.) The morphisms are morphisms of schemes compatible with the fixed maps to X. The large fppf site of X is the category Fppf/X, that is, the category of schemes with a fixed map to X, considered with the fppf topology.
"Fppf" is an abbreviation for "fidèlement plate de présentation finie", that is, "faithfully flat and of finite presentation". Every surjective family of flat and finitely presented morphisms is a covering family for this topology, hence the name. The definition of the fppf pretopology can also be given with an extra quasi-finiteness condition; it follows from Corollary 17.16.2 in EGA IV4 that this gives the same topology.
The big and small fpqc sites
Let X be an affine scheme. We define an fpqc cover of X to be a finite and jointly surjective family of morphisms {uα : Xα → X} with each Xα affine and each uαflat. This generates a pretopology: For X arbitrary, we define an fpqc cover of X to be a family {uα : Xα → X} which is an fpqc cover after base changing to an open affine subscheme of X. This pretopology generates a topology called the fpqc topology. (This is not the same as the topology we would get if we started with arbitrary X and Xα and took covering families to be jointly surjective families of flat morphisms.) We write Fpqc for the category of schemes with the fpqc topology.
The small fpqc site of X is the category O(Xfpqc) whose objects are schemes U with a fixed morphism U → X which is part of some covering family. The morphisms are morphisms of schemes compatible with the fixed maps to X. The large fpqc site of X is the category Fpqc/X, that is, the category of schemes with a fixed map to X, considered with the fpqc topology.
"Fpqc" is an abbreviation for "fidèlement plate quasi-compacte", that is, "faithfully flat and quasi-compact". Every surjective family of flat and quasi-compact morphisms is a covering family for this topology, hence the name.
Flat cohomology
The procedure for defining the cohomology groups is the standard one: cohomology is defined as the sequence of derived functors of the functor taking the sections of a sheaf of abelian groups.
While such groups have a number of applications, they are not in general easy to compute, except in cases where they reduce to other theories, such as the étale cohomology.
Example
The following example shows why the "faithfully flat topology" without any finiteness conditions does not behave well. Suppose X is the affine line over an algebraically closed field k. For each closed point x of X we can consider the local ring Rx at this point, which is a discrete valuation ring whose spectrum has one closed point and one open (generic) point. We glue these spectra together by identifying their open points to get a scheme Y. There is a natural map from Y to X. The affine line X is covered by the sets Spec(Rx) which are open in the faithfully flat topology, and each of these sets has a natural map to Y, and these maps are the same on intersections. However they cannot be combined to give a map from X to Y, because the underlying spaces of X and Y have different topologies.
Michael Artin and J. S. Milne, "Duality in the flat cohomology of curves", Inventiones Mathematicae, Volume 35, Number 1, December, 1976
External links
Arithmetic Duality Theorems (PDF), online book by James Milne, explains at the level of flat cohomology duality theorems originating in the Tate–Poitou duality of Galois cohomology
January 07, 2023
flat, topology, mathematics, flat, topology, grothendieck, topology, used, algebraic, geometry, used, define, theory, flat, cohomology, also, plays, fundamental, role, theory, descent, faithfully, flat, descent, term, flat, here, comes, from, flat, modules, th. In mathematics the flat topology is a Grothendieck topology used in algebraic geometry It is used to define the theory of flat cohomology it also plays a fundamental role in the theory of descent faithfully flat descent 1 The term flat here comes from flat modules There are several slightly different flat topologies the most common of which are the fppf topology and the fpqc topology fppf stands for fidelement plate de presentation finie and in this topology a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation fpqc stands for fidelement plate et quasi compacte and in this topology a morphism of affine schemes is a covering morphism if it is faithfully flat In both categories a covering family is defined be a family which is a cover on Zariski open subsets 2 In the fpqc topology any faithfully flat and quasi compact morphism is a cover 3 These topologies are closely related to descent The pure faithfully flat topology without any further finiteness conditions such as quasi compactness or finite presentation is not used much as is not subcanonical in other words representable functors need not be sheaves Unfortunately the terminology for flat topologies is not standardized Some authors use the term topology for a pretopology and there are several slightly different pretopologies sometimes called the fppf or fpqc pre topology which sometimes give the same topology Flat cohomology was introduced by Grothendieck in about 1960 4 Contents 1 The big and small fppf sites 2 The big and small fpqc sites 3 Flat cohomology 4 Example 5 See also 6 Notes 7 References 8 External linksThe big and small fppf sites EditLet X be an affine scheme We define an fppf cover of X to be a finite and jointly surjective family of morphisms fa Xa X with each Xa affine and each fa flat finitely presented This generates a pretopology for X arbitrary we define an fppf cover of X to be a family f a Xa X which is an fppf cover after base changing to an open affine subscheme of X This pretopology generates a topology called the fppf topology This is not the same as the topology we would get if we started with arbitrary X and Xa and took covering families to be jointly surjective families of flat finitely presented morphisms We write Fppf for the category of schemes with the fppf topology The small fppf site ofX is the category O Xfppf whose objects are schemes U with a fixed morphism U X which is part of some covering family This does not imply that the morphism is flat finitely presented The morphisms are morphisms of schemes compatible with the fixed maps to X The large fppf site ofX is the category Fppf X that is the category of schemes with a fixed map to X considered with the fppf topology Fppf is an abbreviation for fidelement plate de presentation finie that is faithfully flat and of finite presentation Every surjective family of flat and finitely presented morphisms is a covering family for this topology hence the name The definition of the fppf pretopology can also be given with an extra quasi finiteness condition it follows from Corollary 17 16 2 in EGA IV4 that this gives the same topology The big and small fpqc sites EditLet X be an affine scheme We define an fpqc cover of X to be a finite and jointly surjective family of morphisms ua Xa X with each Xa affine and each ua flat This generates a pretopology For X arbitrary we define an fpqc cover of X to be a family ua Xa X which is an fpqc cover after base changing to an open affine subscheme of X This pretopology generates a topology called the fpqc topology This is not the same as the topology we would get if we started with arbitrary X and Xa and took covering families to be jointly surjective families of flat morphisms We write Fpqc for the category of schemes with the fpqc topology The small fpqc site ofX is the category O Xfpqc whose objects are schemes U with a fixed morphism U X which is part of some covering family The morphisms are morphisms of schemes compatible with the fixed maps to X The large fpqc site ofX is the category Fpqc X that is the category of schemes with a fixed map to X considered with the fpqc topology Fpqc is an abbreviation for fidelement plate quasi compacte that is faithfully flat and quasi compact Every surjective family of flat and quasi compact morphisms is a covering family for this topology hence the name Flat cohomology EditThe procedure for defining the cohomology groups is the standard one cohomology is defined as the sequence of derived functors of the functor taking the sections of a sheaf of abelian groups While such groups have a number of applications they are not in general easy to compute except in cases where they reduce to other theories such as the etale cohomology Example EditThe following example shows why the faithfully flat topology without any finiteness conditions does not behave well Suppose X is the affine line over an algebraically closed field k For each closed point x of X we can consider the local ring Rx at this point which is a discrete valuation ring whose spectrum has one closed point and one open generic point We glue these spectra together by identifying their open points to get a scheme Y There is a natural map from Y to X The affine line X is covered by the sets Spec Rx which are open in the faithfully flat topology and each of these sets has a natural map to Y and these maps are the same on intersections However they cannot be combined to give a map from X to Y because the underlying spaces of X and Y have different topologies See also Editfpqc morphismNotes Edit Form of an algebraic structure Encyclopedia of Mathematics EMS Press 2001 1994 SGA III1 IV 6 3 SGA III1 IV 6 3 Proposition 6 3 1 v Grothendieck Alexander Raynaud Michele 2003 1971 Revetements etales et groupe fondamental SGA 1 Documents Mathematiques Paris Mathematical Documents Paris vol 3 Paris Societe Mathematique de France p XI 4 8 arXiv math 0206203 Bibcode 2002math 6203G ISBN 978 2 85629 141 2 MR 2017446References EditElements de geometrie algebrique Vol IV 2 Milne James S 1980 Etale Cohomology Princeton University Press ISBN 978 0 691 08238 7 Michael Artin and J S Milne Duality in the flat cohomology of curves Inventiones Mathematicae Volume 35 Number 1 December 1976External links EditArithmetic Duality Theorems PDF online book by James Milne explains at the level of flat cohomology duality theorems originating in the Tate Poitou duality of Galois cohomology Retrieved from https en wikipedia org w index php title Flat topology amp oldid 1095993725, wikipedia, wiki, book, books, library,
