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 formula_1, local Tate duality says there is a perfect pairing of finite groups
where formula_3 is a finite group scheme and formula_4 its dual formula_5.
For a local field of characteristic formula_6, the statement is similar, except that the pairing takes values in formula_7. The statement also holds for Archimedean fields, though the definition of the cohomology groups looks somewhat different in this case.
Global Tate duality.
Given a finite group scheme formula_3 over a global field formula_1, global Tate duality relates the cohomology of formula_3 with that of formula_11 using the local pairings constructed above. This is done via the localization maps
where formula_13 varies over all places of formula_1, and where formula_15 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 formula_17 has annihilator equal to the image of formula_18 for formula_19.
The map formula_20 has a finite kernel for all formula_21, 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 formula_26 of formula_1, with the above statements being the form of his theorems for the case where formula_28 contains all places of formula_1. 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 formula_1, a set "S" of primes, and the maximal extension formula_31 which is unramified outside "S", the Shafarevich groups capture, broadly speaking, those elements in the cohomology of formula_32 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 formula_33 is replaced by a regular scheme of finite type over formula_34 was shown by .