# 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 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 the finite groups arising from Galois cohomology:
    $\displaystyle H^{r}(k,M)\times H^{2-r}(k,M')\rightarrow H^{2}(k,\mathbb {G} _{m})=\mathbb {Q} /\mathbb {Z}$
where $M$ is a finite group scheme, $M'$ its dual $\operatorname {Hom} (M,G_{m})$, and $\mathbb {G} _{m}$ is the multiplicative group. For a local field of characteristic $p>0$, the statement is similar, except that the pairing takes values in $H^{2}(k,\mu )=\bigcup _{p\nmid n}{\tfrac {1}{n}}\mathbb {Z} /\mathbb {Z}$. The statement also holds when $k$ 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 $M$ over a global field $k$, global Tate duality relates the cohomology of $M$ with that of $M'=\operatorname {Hom} (M,G_{m})$ using the local pairings constructed above. This is done via the localization maps
    $\alpha _{r,M}:H^{r}(k,M)\rightarrow {\prod _{v}}'H^{r}(k_{v},M),$
where $v$ varies over all places of $k$, and where $\prod '$ denotes a restricted product with respect to the unramified cohomology groups. Summing the local pairings gives a canonical perfect pairing
    ${\prod _{v}}'H^{r}(k_{v},M)\times {\prod _{v}}'H^{2-r}(k_{v},M')\rightarrow \mathbb {Q} /\mathbb {Z} .$
One part of Poitou-Tate duality states that, under this pairing, the image of $H^{r}(k,M)$ has annihilator equal to the image of $H^{2-r}(k,M')$ for $r=0,1,2$.
The map $\alpha _{r,M}$ has a finite kernel for all $r$, and Tate also constructs a canonical perfect pairing
    ${\text{ker}}(\alpha _{1,M})\times \ker(\alpha _{2,M'})\rightarrow \mathbb {Q} /\mathbb {Z} .$
These dualities are often presented in the form of a nine-term exact sequence
    $0\rightarrow H^{0}(k,M)\rightarrow {\prod _{v}}'H^{0}(k_{v},M)\rightarrow H^{2}(k,M')^{*}$
    $\rightarrow H^{1}(k,M)\rightarrow {\prod _{v}}'H^{1}(k_{v},M)\rightarrow H^{1}(k,M')^{*}$
    $\rightarrow H^{2}(k,M)\rightarrow {\prod _{v}}'H^{2}(k_{v},M)\rightarrow H^{0}(k,M')^{*}\rightarrow 0.$
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 $S$ of $k$, with the above statements being the form of his theorems for the case where $S$ contains all places of $k$. For the more general result, see e.g. 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 $k$, a set S of primes, and the maximal extension $k_{S}$ which is unramified outside S, the Shafarevich groups capture, broadly speaking, those elements in the cohomology of $\operatorname {Gal} (k_{S}/k)$ 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 ${\mathcal {O}}_{S}$ is replaced by a regular scheme of finite type over $\operatorname {Spec} {\mathcal {O}}_{S}$ 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.