Flat topology


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 has played a fundamental role in the theory of descent (faithfully flat descent). [ [http://eom.springer.de/f/f040800.htm Springer EoM article] ] (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

:{"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

:{"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.

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.

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

Notes

External links

* [http://www.jmilne.org/math/Books/adt.html "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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Topology — (Greek topos , place, and logos , study ) is the branch of mathematics that studies the properties of a space that are preserved under continuous deformations. Topology grew out of geometry, but unlike geometry, topology is not concerned with… …   Wikipedia

  • Flat Neighborhood Network — (FNN) is a topology for distributed computing and other computer networks. Each node connects to two or more switches which, ideally, entirely cover the node collection, so that each node can connect to any other node in two hops (jump up to one… …   Wikipedia

  • 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 …   Wikipedia

  • Nisnevich topology — In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K theory, A¹ homotopy theory, and the theory of motives. It …   Wikipedia

  • List of geometric topology topics — This is a list of geometric topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of algebraic topology topics Publications in topology Contents 1 Low dimensional topology 1.1 …   Wikipedia

  • Asymptotically flat spacetime — An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski… …   Wikipedia

  • List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

  • Glossary of arithmetic and Diophantine geometry — This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of… …   Wikipedia

  • Cotangent complex — In mathematics the cotangent complex is a roughly a universal linearization of a morphism of geometric or algebraic objects. Cotangent complexes were originally defined in special cases by a number of authors. Luc Illusie, Daniel Quillen, and M.… …   Wikipedia

  • Shape of the Universe — Edge of the Universe redirects here. For the Bee Gees song, see Edge of the Universe (song). The local geometry of the universe is determined by whether Omega is less than, equal to or greater than 1. From top to bottom: a spherical universe, a… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.