Tate duality

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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[edit]

For a p-adic local field k, local Tate duality says there is a perfect pairing of finite groups

where M is a finite group scheme and M′ its dual Hom(M,Gm). For a local field of characteristic p>0, the statement is similar, except that the pairing takes values in [1]

Global Tate duality[edit]

For a global field k, a similar statement holds true if μ is replaced by the S-idele class group, where S is a set of primes of k. See Neukirch, Schmidt & Wingberg (2000, Theorem 8.4.4).

Poitou–Tate duality[edit]

Among other statements, Poitou–Tate duality establishes a perfect pairing between certain Shafarevich groups. Given a global field k and 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 (2017).

See also[edit]


  1. ^ Neukirch, Schmidt & Wingberg (2000, Theorem 7.2.6)
  2. ^ See Neukirch, Schmidt & Wingberg (2000, Theorem 8.6.8) for a precise statement.