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 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:
:$\displaystyle H^r(k,M)\times H^(k,M')\rightarrow H^2(k,\mathbb_m)=\Q/ \Z$
where $M$ is a finite group scheme, $M'$ its dual $\operatorname(M,G_m)$, and $\mathbb_m$ is the multiplicative group.
For a local field of characteristic $p>0$, the statement is similar, except that the pairing takes values in $H^2(k, \mu) = \bigcup_ \tfrac \Z/\Z$. 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 $M' = \operatorname(M,G_m)$ using the local pairings constructed above. This is done via the localization maps
:$\alpha_: H^r(k, M) \rightarrow ' H^r(k_v, M),$
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
:$' H^r(k_v, M) \times ' H^(k_v, M') \rightarrow \Q/\Z .$
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 $H^(k, M')$ for $r = 0, 1, 2$.

The map $\alpha_$ has a finite kernel for all $r$, and Tate also constructs a canonical perfect pairing
:$\text(\alpha_) \times \ker(\alpha_) \rightarrow \Q/\Z .$

These dualities are often presented in the form of a nine-term exact sequence
:$0 \rightarrow H^0(k, M) \rightarrow ' H^0(k_v, M) \rightarrow H^2(k, M')^*$
:$\rightarrow H^1(k, M) \rightarrow ' H^1(k_v, M) \rightarrow H^1(k, M')^*$
:$\rightarrow H^2(k, M) \rightarrow ' H^2(k_v, M) \rightarrow H^0(k, M')^* \rightarrow 0.$
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 $\operatorname(k_S/k)$ 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 $\mathcal_S$ is replaced by a regular scheme of finite type over $\operatorname \mathcal_S$ was shown by . Another generalisation is due to Česnavičius, who relaxed the condition on the localising set ''S'' by using flat cohomology on smooth proper curves.

See also

* Artin–Verdier duality
* Tate pairing

References

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*{{Citation , last1=Tate , first1=John , author1-link=John Tate (mathematician) , title=Proceedings of the International Congress of Mathematicians (Stockholm, 1962) , chapter-url=http://mathunion.org/ICM/ICM1962.1/ , publisher=Inst. Mittag-Leffler , location=Djursholm , mr=0175892 , year=1962 , chapter=Duality theorems in Galois cohomology over number fields , pages=288–295 , url-status=dead , archiveurl=https://web.archive.org/web/20110717144510/http://mathunion.org/ICM/ICM1962.1/ , archivedate=2011-07-17

Algebraic number theory
Galois theory
Duality theories