mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the flat topology is a

Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...

used in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of

descent Descent may refer to: As a noun Genealogy and inheritance * Common descent, concept in evolutionary biology * Kinship, one of the major concepts of cultural anthropology **Pedigree chart or family tree ** Ancestry ** Lineal descendant **Heritag ...

(faithfully flat descent). The term ''flat'' here comes from

flat module In 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. A module is faithfully flat if taking the tensor product with a sequence produces an exact seq ...

s. There are several slightly different flat topologies, the most common of which are the fppf topology and the fpqc topology. ''fppf'' stands for ', 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 ', 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. In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover. These topologies are closely related to

. 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.*

The big and small fppf sites

Let ''X'' be an

affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...

. We define an fppf cover of ''X'' to be a finite and jointly surjective family of morphisms :(''φ''_a : ''X''_a → ''X'') with each ''X''_a affine and each ''φ''_a

flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...

, finitely presented. This generates a

pretopology In general topology, a pretopological space is a generalization of the concept of topological space. A pretopological space can be defined in terms of either filters or a preclosure operator. The similar, but more abstract, notion of a Grothendie ...

: for ''X'' arbitrary, we define an fppf cover of ''X'' to be a family :(''φ'_a : ''X''_a → ''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''_a 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''(''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.) 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 with each ''X''_α affine and each ''u''_α

. This generates a pretopology: For ''X'' arbitrary, we define an fpqc cover of ''X'' to be a family 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. 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 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 It was noted in vari ...

s of the functor taking the

sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...

of a

sheaf of abelian groups In mathematics, a sheaf is a tool for systematically tracking data (such as Set (mathematics) , sets, abelian groups, Ring (mathematics) , rings) attached to the open sets of a topological space and defined locally with regard to them. For exam ...

. 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 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 conjecture ...

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 ''R''_''x'' 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(''R''_''x'') 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.

Notes

References

*''

Éléments de géométrie algébrique The ''Éléments de géométrie algébrique'' ("Elements of algebraic geometry, Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or ''EGA'' for short, is a rigorous treatise, in French language, French, on algebraic ge ...

'', Vol. IV. 2 * Milne, James S. (1980), ''Étale Cohomology'', Princeton University Press, {{ISBN, 978-0-691-08238-7 *

Michael Artin Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.

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 In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by and . Local Tate duality For a ''p''-adic local f ...

Galois cohomology In 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. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...