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 and.

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:
 * $$\displaystyle H^r(k,M)\times H^{2-r}(k,M')\rightarrow H^2(k,\mathbb{G}_m)=\Q/ \Z$$

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 $$H^2(k, \mu) = \bigcup_{p \nmid n} \tfrac{1}{n} \Z/\Z$$. 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
 * $${\prod_v}' H^r(k_v, M) \times {\prod_v}' H^{2- r}(k_v, M') \rightarrow \Q/\Z .$$

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
 * $$ \text{ker}(\alpha_{1, M}) \times \ker(\alpha_{2, M'}) \rightarrow \Q/\Z .$$

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..

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.