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 finite groups

\displaystyle H^{r}(k,M)\times H^{2-r}(k,M')\rightarrow H^{2}(k,G_{m})=\mathbb {Q} /\mathbb {Z}

where M is a finite group scheme and M′ its dual Hom(M,G_m). For a local field of characteristic p>0, the statement is similar, except that the pairing takes values in $H^{2}(k,\mu )=\bigcup _{p\nmid n}{\frac {1}{n}}\mathbb {Z} /\mathbb {Z} .$ ^[1]

Global Tate duality

For a global field k, a similar statement holds true if μ is replaced by the S-idele class group, where S is a set of primes of k. See 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 and 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 $Gal(k_{S}/k)$ 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 ${\mathcal {O}}_{S}$ is replaced by a regular scheme of finite type over $Spec{\mathcal {O}}_{S}$ was shown by Geisser & Schmidt (2017).

References

↑ Neukirch, Schmidt & Wingberg (2000, Theorem 7.2.6)
↑ See Neukirch, Schmidt & Wingberg (2000, Theorem 8.6.8) for a precise statement.

Geisser, Thomas H.; Schmidt, Alexander (2017), Poitou-Tate duality for arithmetic schemes, arXiv:1709.06913, Bibcode:2017arXiv170906913G
Haberland, Klaus (1978), Galois cohomology of algebraic number fields, VEB Deutscher Verlag der Wissenschaften, 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, 13, Paris: Dunod, pp. 255–277, MR 0219591
Tate, John (1963), "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

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[1] Neukirch, Schmidt & Wingberg (2000, Theorem 7.2.6)

[2] See Neukirch, Schmidt & Wingberg (2000, Theorem 8.6.8) for a precise statement.

Tate duality

Local Tate duality

Global Tate duality

Poitou–Tate duality

See also

References