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arXiv · 2210.05838

Pontryagin Duality for Modules over Compact Discrete Valuation Rings

Milo Moses
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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.

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Milo Moses. 2022-10-11. Pontryagin Duality for Modules over Compact Discrete Valuation Rings. https://arxiv.org/abs/2210.05838

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