Tate 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 field , local Tate duality says there is a perfect pairing of the finite groups arising from Galois cohomology:

where  is a finite group scheme,  its dual , and  is the multiplicative group. 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, 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 groups. 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.

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
Tate pairing

References

 

Algebraic number theory
Galois theory
Duality theories