Mathematical Research Letters
Volume 22 (2015)
Poitou–Tate without restrictions on the order
Pages: 1621 – 1666
The Poitou–Tate sequence relates Galois cohomology with restricted ramification of a finite Galois module $M$ over a global field to that of the dual module under the assumption that $\# M$ is a unit away from the allowed ramification set. We remove the assumption on $\# M$ by proving a generalization that allows arbitrary “ramification sets” that contain the archimedean places. We also prove that restricted products of local cohomologies that appear in the Poitou–Tate sequence may be identified with derived functor cohomology of an adele ring. In our proof of the generalized sequence we adopt this derived functor point of view and exploit properties of a natural topology carried by cohomology of the adeles.
arithmetic duality, global field, Poitou–Tate, Cassels–Poitou–Tate, adeles
2010 Mathematics Subject Classification
Primary 11R99. Secondary 11R34, 11R56.