Pontryagin duality and sheaves of profinite modules

TL;DR

This paper extends Pontryagin duality to include sheaves of profinite modules, providing a dual construction for cohomology with discrete modules, thus enriching the homological framework for profinite groups.

Contribution

It introduces a dual construction for cohomology with discrete modules, complementing existing profinite direct sum constructions in the theory of profinite groups.

Findings

Develops a duality framework for sheaves of profinite modules.

Establishes a new dual construction for cohomology with discrete coefficients.

Enhances the understanding of homology and cohomology interplay in profinite groups.

Abstract

The well-known theory of Pontryagin duality provides a strong connection between the homology and cohomology theories of a profinite group in appropriate categories. A construction for taking the `profinite direct sum' of an infinite family of profinite modules indexed over a profinite space has been found to be useful in the study of homology of profinite groups, but hitherto the appropriate dual construction for studying cohomology with coefficients in discrete modules has not been studied. This paper remedies this gap in the theory.