Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.05838 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916362161487872 |
|---|---|
| author | Moses, Milo |
| author_facet | Moses, Milo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_05838 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pontryagin Duality for Modules over Compact Discrete Valuation Rings Moses, Milo Commutative Algebra 13H05 (Primary), 22B99 (Secondary) 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. |
| title | Pontryagin Duality for Modules over Compact Discrete Valuation Rings |
| topic | Commutative Algebra 13H05 (Primary), 22B99 (Secondary) |
| url | https://arxiv.org/abs/2210.05838 |