Flat topology

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). (The term flat here comes from flat modules.)

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

(φ_a : X_a → X)

with each X_a affine and each φ_a 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 : 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, 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 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 {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.

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.

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.

See also

• fpqc morphism

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

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