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 the finite groups arising from Galois cohomology: where formula_3 is a finite group scheme, formula_4 its dual formula_5, and formula_6 is the multiplicative group. For a local field of characteristic formula_7, the statement is similar, except that the pairing takes values in formula_8. The statement also  holds when formula_1 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 formula_3 over a global field formula_1, global Tate duality relates the cohomology of formula_3 with that of formula_13 using the local pairings constructed above. This is done via the localization maps where formula_15 varies over all places of formula_1, and where formula_17 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_19 has annihilator equal to the image of formula_20 for formula_21. The map formula_22 has a finite kernel for all formula_23, 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_28 of formula_1, with the above statements being the form of his theorems for the case where formula_30 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_33 which is unramified outside "S", the Shafarevich groups capture, broadly speaking, those elements in the cohomology of formula_34 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_35 is replaced by a regular scheme of finite type over formula_36 was shown  by .