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
, local Tate duality says there is a perfect pairing of the finite groups arising from Galois cohomology:
![{\displaystyle \displaystyle H^{r}(k,M)\times H^{2-r}(k,M')\rightarrow H^{2}(k,\mathbb {G} _{m})=\mathbb {Q} /\mathbb {Z} }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is a finite group scheme,
its dual
, and
is the multiplicative group.
For a local field of characteristic
, the statement is similar, except that the pairing takes values in
.[1] The statement also holds when
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
over a global field
, global Tate duality relates the cohomology of
with that of
using the local pairings constructed above. This is done via the localization maps
![{\displaystyle \alpha _{r,M}:H^{r}(k,M)\rightarrow {\prod _{v}}'H^{r}(k_{v},M),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
varies over all places of
, and where
denotes a restricted product with respect to the unramified cohomology groups. Summing the local pairings gives a canonical perfect pairing
![{\displaystyle {\prod _{v}}'H^{r}(k_{v},M)\times {\prod _{v}}'H^{2-r}(k_{v},M')\rightarrow \mathbb {Q} /\mathbb {Z} .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
One part of Poitou-Tate duality states that, under this pairing, the image of
has annihilator equal to the image of
for
.
The map
has a finite kernel for all
, and Tate also constructs a canonical perfect pairing
![{\displaystyle {\text{ker}}(\alpha _{1,M})\times \ker(\alpha _{2,M'})\rightarrow \mathbb {Q} /\mathbb {Z} .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
These dualities are often presented in the form of a nine-term exact sequence
![{\displaystyle 0\rightarrow H^{0}(k,M)\rightarrow {\prod _{v}}'H^{0}(k_{v},M)\rightarrow H^{2}(k,M')^{*}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \rightarrow H^{1}(k,M)\rightarrow {\prod _{v}}'H^{1}(k_{v},M)\rightarrow H^{1}(k,M')^{*}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \rightarrow H^{2}(k,M)\rightarrow {\prod _{v}}'H^{2}(k_{v},M)\rightarrow H^{0}(k,M')^{*}\rightarrow 0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
of
, with the above statements being the form of his theorems for the case where
contains all places of
. 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
, a set S of primes, and the maximal extension
which is unramified outside S, the Shafarevich groups capture, broadly speaking, those elements in the cohomology of
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
is replaced by a regular scheme of finite type over
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.[3]
See also
References
- ^ Neukirch, Schmidt & Wingberg (2000, Theorem 7.2.6)
- ^ See Neukirch, Schmidt & Wingberg (2000, Theorem 8.6.8) for a precise statement.
- ^ Česnavičius, Kęstutis (2015). "Poitou–Tate without restrictions on the order" (PDF). Mathematical Research Letters. 22 (6): 1621–1666.
- Geisser, Thomas H.; Schmidt, Alexander (2018), "Poitou-Tate duality for arithmetic schemes", Compositio Mathematica, 154 (9): 2020–2044, arXiv:1709.06913, Bibcode:2017arXiv170906913G, doi:10.1112/S0010437X18007340, S2CID 119735104
- Haberland, Klaus (1978), Galois cohomology of algebraic number fields, VEB Deutscher Verlag der Wissenschaften, ISBN 9780685872048, MR 0519872
- Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of number fields, Springer, ISBN 3-540-66671-0, MR 1737196
- Poitou, Georges (1967), "Propriétés globales des modules finis", Cohomologie galoisienne des modules finis, Séminaire de l'Institut de Mathématiques de Lille, sous la direction de G. Poitou. Travaux et Recherches Mathématiques, vol. 13, Paris: Dunod, pp. 255–277, MR 0219591
- Tate, John (1962), "Duality theorems in Galois cohomology over number fields", Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 288–295, MR 0175892, archived from the original on 2011-07-17