Poitou-Tate without restrictions on the order

TL;DR

This paper generalizes the Poitou-Tate sequence to include arbitrary ramification sets, removing previous restrictions on the order of the Galois module, and connects local cohomologies with derived functor cohomology of adeles.

Contribution

It extends the Poitou-Tate sequence to cases with arbitrary ramification sets and introduces a derived functor perspective using adelic cohomology.

Findings

Generalization of Poitou-Tate sequence to arbitrary ramification sets

Identification of local cohomologies with derived functor cohomology of adeles

Proof leveraging properties of a natural topology on adelic cohomology

Abstract

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.