Poitou–Tate without restrictions on the order

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.

Keywords

arithmetic duality, global field, Poitou–Tate, Cassels–Poitou–Tate, adeles

2010 Mathematics Subject Classification

Primary 11R99. Secondary 11R34, 11R56.

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Accepted 7 September 2015

Published 23 May 2016