# Tate duality

(Redirected from Poitou–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 Tate (1962) and Poitou (1967).

## Local Tate duality

Main article: local Tate duality

Local Tate duality says there is a perfect pairing of finite groups

$\displaystyle H^r(k,M)\times H^{2-r}(k,M')\rightarrow H^2(k,G_m)=Q/Z$

where M is a finite group scheme and M′ its dual Hom(M,Gm).