Nekovar duality over p-adic Lie extensions of global fields

TL;DR

This paper extends Nekovar's duality results to noncommutative Iwasawa theory, establishing a duality between Iwasawa cohomology complexes and their compactly supported variants over p-adic Lie extensions.

Contribution

It generalizes Nekovar's derived duality to noncommutative Iwasawa algebras, linking Iwasawa cohomology and its dual in a new setting.

Findings

Established duality in the derived category for noncommutative Iwasawa algebras.

Extended Nekovar's duality to p-adic Lie extensions.

Provided a framework for duality in noncommutative Iwasawa theory.

Abstract

Poitou-Tate duality for the Galois group of an extension of a global field with appropriately restricted ramification can be seen as taking place between the cohomology of a compact or discrete module and the compactly-supported cohomology of its Pontryagin dual. Nekovar proved a derived variant of this in which the module is replaced by a bounded complex of modules over the group ring of R that are finitely generated over R, where R is a complete commutative local Noetherian ring with finite residue field. Here, the Pontryagin dual is replaced by the Grothendieck dual, which is itself a bounded complex of modules with finitely generated cohomology. Nekovar's duality then takes place in the derived category of R-modules. We extend Nekovar's result to the setting of noncommutative Iwasawa theory. That is, we exhibit a duality between complexes computing Iwasawa cohomology and its compactly supported variant in the derived category of finitely generated modules over the Iwasawa algebra over R of a p-adic Lie extension.