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