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 John Tate (1962) and Georges Poitou (1967).
, local Tate duality says there is a perfect pairing of the finite groups arising from Galois cohomology: where
is a finite group scheme,
For a local field of characteristic
, the statement is similar, except that the pairing takes values in
is an Archimedean field, though the definition of the cohomology groups looks somewhat different in this case.
Given a finite group scheme
, global Tate duality relates the cohomology of
using the local pairings constructed above.
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
, 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
, with the above statements being the form of his theorems for the case where
For the more general result, see e.g. Neukirch, Schmidt & Wingberg (2000, Theorem 8.4.4).
Among other statements, Poitou–Tate duality establishes a perfect pairing between certain Shafarevich groups.
, 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.[2] An extension to the case where the ring of S-integers
is replaced by a regular scheme of finite type over
was shown by Geisser & Schmidt (2018).
Another generalisation is due to Česnavičius, who relaxed the condition on the localising set S by using flat cohomology on smooth proper curves.