In
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 ...
, Tate duality or Poitou–Tate duality is a duality theorem for
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 ...
groups of modules over the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
or
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
, introduced by and .
Local Tate duality
For a ''p''-adic local field
, local Tate duality says there is a
perfect pairing
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
of the
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s arising from Galois cohomology:
:
where
is a finite group scheme,
its dual
, and
is the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to ...
.
For a local field of characteristic
, the statement is similar, except that the pairing takes values in
. The statement also holds when
is an
Archimedean field
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
The property, typicall ...
, though the definition of the cohomology groups looks somewhat different in this case.
Global Tate duality
Given a finite group scheme
over a global field
, global Tate duality relates the cohomology of
with that of
using the local pairings constructed above. This is done via the localization maps
:
where
varies over all places of
, and where
denotes a restricted product with respect to the unramified cohomology groups. Summing the local pairings gives a canonical perfect pairing
:
One part of Poitou-Tate duality states that, under this pairing, the image of
has annihilator equal to the image of
for
.
The map
has a finite kernel for all
, and Tate also constructs a canonical perfect pairing
:
These dualities are often presented in the form of a nine-term exact sequence
:
:
:
Here, the asterisk denotes the Pontryagin dual of a given locally compact abelian group.
All of these statements were presented by Tate in a more general form depending on a set of places
of
, with the above statements being the form of his theorems for the case where
contains all places of
. For the more general result, see e.g.
.
Poitou–Tate duality
Among other statements, Poitou–Tate duality establishes a perfect pairing between certain
Shafarevich group
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. ...
s. Given a global field
, a set ''S'' of primes, and the maximal extension
which is unramified outside ''S'', the Shafarevich groups capture, broadly speaking, those elements in the cohomology of
which vanish in the Galois cohomology of the local fields pertaining to the primes in ''S''.
[See for a precise statement.]
An extension to the case where the ring of ''S''-integers
is replaced by a regular scheme of finite type over
was shown by .
See also
*
Artin–Verdier duality
In mathematics, Artin–Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by , that generalizes Tate duality.
It shows that, as far as etale (or flat) cohomolo ...
*
Tate pairing
In mathematics, Tate pairing is any of several closely related bilinear pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings introduced by and extended by . applied t ...
References
*
*
*
*
*{{Citation , last1=Tate , first1=John , author1-link=John Tate (mathematician) , title=Proceedings of the International Congress of Mathematicians (Stockholm, 1962) , chapter-url=http://mathunion.org/ICM/ICM1962.1/ , publisher=Inst. Mittag-Leffler , location=Djursholm , mr=0175892 , year=1963 , chapter=Duality theorems in Galois cohomology over number fields , pages=288–295 , url-status=dead , archiveurl=https://web.archive.org/web/20110717144510/http://mathunion.org/ICM/ICM1962.1/ , archivedate=2011-07-17
Algebraic number theory
Galois theory
Duality theories