Flat topology
Encyclopedia
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 has played a fundamental role in the theory of descent
(faithfully flat descent). (The term flat here comes from flat module
s.)
Strictly, there is no single definition of the flat topology, because, technically speaking, different finiteness conditions may be applied.
with each Xα affine and each uα flat
, finitely presented, and quasi-finite
. This generates a pretopology: for X arbitrary, we define an fppf cover of X to be a family
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 Xα and took covering families to be jointly surjective families of flat, finitely presented, and quasi-finite 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, and quasi-finite.) 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.
. 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.
s 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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the flat topology is a Grothendieck topology
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
used in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. It is used to define the theory of flat cohomology; it also has played a fundamental role in the theory of descent
Descent (category theory)
In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated...
(faithfully flat descent). (The term flat here comes from flat module
Flat module
In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...
s.)
Strictly, there is no single definition of the flat topology, because, technically speaking, different finiteness conditions may be applied.
The big and small fppf sites
Let X be an affine scheme. We define an fppf cover of X to be a finite and jointly surjective family of morphisms- {uα : Xα → X}
with each Xα affine and each uα flat
Flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,is a flat map for all P in X...
, finitely presented, and quasi-finite
Quasi-finite morphism
In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is finite type and satisfies any of the following equivalent conditions:...
. This generates a pretopology: for X arbitrary, we define an fppf cover of X to be a family
- {uα : Xα → 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 Xα and took covering families to be jointly surjective families of flat, finitely presented, and quasi-finite 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, and quasi-finite.) 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 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 uα flatFlat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,is a flat map for all P in X...
. 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 functorDerived functor
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...
s 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
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
.
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 cohomologyGalois cohomologyIn mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...