# Flat topology

(Redirected from Flat cohomology)

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 two flat topologies, 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, of finite presentation, and is quasi-finite. 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 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

(φa : XaX)

with each Xa affine and each φa flat, finitely presented, and quasi-finite. 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, 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 UX 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 definition of the fppf topology is generally given without the quasi-finiteness condition; its equivalence with the above definition follows from Corollary 17.16.2 in EGA IV4.

## 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 : XαX} with each Xα affine and each uα flat. This generates a pretopology: For X arbitrary, we define an fpqc cover of X to be a family {uα : XαX} 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 UX 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.

## Notes

1. ^ Springer EoM article
2. ^ SGA III1, IV 6.3.
3. ^ SGA III1, IV 6.3, Proposition 6.3.1(v).