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 Template:Harvs and Template:Harvs.

Local Tate duality

Template:Main For a p-adic local field $k$, local Tate duality says there is a perfect pairing of the finite groups arising from Galois cohomology:

$H^r(k,M)\times H^{2-r}(k,M^{\prime })\to H^2(k,{𝔾}_m)=\mathbb{Q}/\mathbb{Z}$

where $M$ is a finite group scheme, $M^{\prime }$ its dual $\mathrm{Hom}(M,{𝔾}_m)$, and ${𝔾}_m$ is the multiplicative group. For a local field of characteristic $p>0$, the statement is similar, except that the pairing takes values in $H^2(k,\mu )=\bigcup _{p\nmid n}\frac{1}{n}\mathbb{Z}/\mathbb{Z}$. 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 $M^{\prime }=\mathrm{Hom}(M,{𝔾}_m)$ using the local pairings constructed above. This is done via the localization maps

${\alpha }_{r,M}:H^r(k,M)\to {\prod _v}^{\prime }{H}^r(k_v,M),$

where $v$ varies over all places of $k$, and where ${\prod }^{\prime }$ denotes a restricted product with respect to the unramified cohomology groups. Summing the local pairings gives a canonical perfect pairing

$\prod _v\text{'}{H}^r(k_v,M)\times {\prod _v}^{\prime }{H}^{2-r}(k_v,M^{\prime })\to \mathbb{Q}/\mathbb{Z}.$

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 $H^{2-r}(k,M^{\prime })$ 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

$\text{ker}({\alpha }_{1,M})\times \ker ({\alpha }_{2,M^{\prime }})\to \mathbb{Q}/\mathbb{Z}.$

These dualities are often presented in the form of a nine-term exact sequence

$0\to H^0(k,M)\to {\prod _v}^{\prime }{H}^0(k_v,M)\to H^2(k,M^{\prime }{)}^{\ast }$

$\to H^1(k,M)\to {\prod _v}^{\prime }{H}^1(k_v,M)\to H^1(k,M^{\prime }{)}^{\ast }$

$\to H^2(k,M)\to {\prod _v}^{\prime }{H}^2(k_v,M)\to H^0(k,M^{\prime }{)}^{\ast }\to 0.$

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. Template:Harvtxt.

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 $\mathrm{Gal}(k_S/k)$ 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 ${𝒪}_S$ is replaced by a regular scheme of finite type over $\mathrm{Spec}{𝒪}_S$ was shown by Template:Harvtxt. 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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