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 Tate (1962) and Poitou (1967).

Local Tate duality

Local Tate duality says there is a perfect pairing of finite groups

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

where M is a finite group scheme and M′ its dual Hom(M,G_m).

See also

• Artin–Verdier duality
• Tate pairing

References

• Haberland, Klaus (1978), Galois cohomology of algebraic number fields, VEB Deutscher Verlag der Wissenschaften, MR 519872
• 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