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).^[1] (The term flat here comes from flat modules.) Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... 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. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in topology. ... In abstract algebra, 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. ...

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 In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...

{u_α : X_α → X}

with each X_α affine and each u_α flat, finitely presented, and quasi-finite. This generates a pretopology: for X arbitrary, we define an fppf cover of X to be a family 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. ... A morphism of of schemes) is called quasi-finite if for every point the fibre (where is the residue field of and is the canonical morphism) has only a finite number of points. ... A pretopological space (X, cl ) is a set X with a function cl : P(X) → P(X) , where P(X) denotes the power set of X. This function has to be extensive and finitely additive; that is, it must satisfy the following conditions for all subsets A and B of...

{u_α : X_α → 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 X_α 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(X_fppf) 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 {u_α : X_α → X} with each X_α affine and each u_α flat. This generates a pretopology: For X arbitrary, we define an fpqc cover of X to be a family {u_α : X_α → 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 X_α and took covering families to be jointly surjective families of flat morphisms.) We write Fpqc for the category of schemes with the fpqc topology. 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. ...

The small fpqc site of X is the category O(X_fpqc) 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. In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...

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. In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...

References

• Éléments de géométrie algébrique, Vol. IV.2
• Milne, James S. (1980), Étale 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, 1976

The Éléments de géométrie algébrique (Elements of Algebraic Geometry) by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, are an unfinished 1500-page treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut... Michael Artin Michael Artin (born 1934) is an American mathematician and a professor at MIT, known for his contributions to algebraic geometry. ...

Notes

1. ^ Springer EoM article

External link

• Arithmetic Duality Theorems (PDF), online book by Jim Milne, explains at the level of flat cohomology duality theorems originating in the Tate-Poitou duality of Galois cohomology