Pontryagin Duality for Modules over Compact Discrete Valuation Rings

TL;DR

This paper develops a Pontryagin duality theory for modules over compact discrete valuation rings, establishing duality, double dual isomorphisms, and a canonical topology, bridging topological and algebraic module theory.

Contribution

It introduces a duality framework for modules over compact discrete valuation rings, including a canonical topology and a fully faithful functor, extending Pontryagin duality.

Findings

Double dual map is an isomorphism for locally compact modules.

Canonical topology makes the double dual map injective and continuous.

The functor is fully faithful, linking topological and algebraic modules.

Abstract

We establish an analogue of Pontryagin duality for modules over compact discrete valuation rings $R$. Namely, we define the dual of a topological $R$ module to be its continuous $R$-module homomorphisms into $K/R$, the quotient module of the fraction field by its ring of integers. It is established that for locally compact $R$-modules the double dual map is an isomorphism and homeomorphism. Additionally, given a non-topological $R$-module a canonical topology is constructed, uniquely defined so that the double dual map will be injective and continuous. Finally, the functor assigning a module to itself equipped with canonical topology is shown to be fully faithful, allowing one to recontextualize the topological statements in purely algebraic forms.