# Tate duality

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

## Local Tate duality

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

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

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 ${\displaystyle H^{2}(k,\mu )=\bigcup _{p\nmid n}{\frac {1}{n}}\mathbb {Z} /\mathbb {Z} .}$[1]

## Global Tate duality

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

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 ${\displaystyle k_{S}}$ which is unramified outside S, the Shafarevich groups capture, broadly speaking, those elements in the cohomology of ${\displaystyle Gal(k_{S}/k)}$ 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 ${\displaystyle {\mathcal {O}}_{S}}$ is replaced by a regular scheme of finite type over ${\displaystyle Spec{\mathcal {O}}_{S}}$ was shown by Geisser & Schmidt (2017).