Structure of Finite-Dimensional Protori
A <i>Structure Theorem for Protori</i> is derived for the category of finite-dimensional <i>protori</i> (compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional proto...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-08-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/8/3/93 |
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| author | Wayne Lewis |
| author_facet | Wayne Lewis |
| author_sort | Wayne Lewis |
| collection | DOAJ |
| description | A <i>Structure Theorem for Protori</i> is derived for the category of finite-dimensional <i>protori</i> (compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are parameterized in the structure theorem by the dual category of finite rank torsion-free abelian groups. A consequence is a <i>universal resolution</i> for a finite-dimensional protorus, independent of a choice of a particular subgroup. A resolution is also given strictly in terms of the path component of the identity and the union of all zero-dimensional subgroups. The structure theorem is applied to show that a morphism of finite-dimensional protori lifts to a product morphism between products of periodic locally compact groups and real vector spaces. |
| first_indexed | 2024-04-12T08:18:30Z |
| format | Article |
| id | doaj.art-1f296c7a431647e5b28330eb1c1f060b |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-04-12T08:18:30Z |
| publishDate | 2019-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-1f296c7a431647e5b28330eb1c1f060b2022-12-22T03:40:41ZengMDPI AGAxioms2075-16802019-08-01839310.3390/axioms8030093axioms8030093Structure of Finite-Dimensional ProtoriWayne Lewis0University of Hawaii, 874 Dillingham Blvd., Honolulu, HI 96817, USAA <i>Structure Theorem for Protori</i> is derived for the category of finite-dimensional <i>protori</i> (compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are parameterized in the structure theorem by the dual category of finite rank torsion-free abelian groups. A consequence is a <i>universal resolution</i> for a finite-dimensional protorus, independent of a choice of a particular subgroup. A resolution is also given strictly in terms of the path component of the identity and the union of all zero-dimensional subgroups. The structure theorem is applied to show that a morphism of finite-dimensional protori lifts to a product morphism between products of periodic locally compact groups and real vector spaces.https://www.mdpi.com/2075-1680/8/3/93compact abelian grouptorustorus-freeperiodicprotorusprofinite subgrouptorus quotienttorsion-free abelian groupfinite rank |
| spellingShingle | Wayne Lewis Structure of Finite-Dimensional Protori Axioms compact abelian group torus torus-free periodic protorus profinite subgroup torus quotient torsion-free abelian group finite rank |
| title | Structure of Finite-Dimensional Protori |
| title_full | Structure of Finite-Dimensional Protori |
| title_fullStr | Structure of Finite-Dimensional Protori |
| title_full_unstemmed | Structure of Finite-Dimensional Protori |
| title_short | Structure of Finite-Dimensional Protori |
| title_sort | structure of finite dimensional protori |
| topic | compact abelian group torus torus-free periodic protorus profinite subgroup torus quotient torsion-free abelian group finite rank |
| url | https://www.mdpi.com/2075-1680/8/3/93 |
| work_keys_str_mv | AT waynelewis structureoffinitedimensionalprotori |