On the noncommutative geometry of semi-graded rings

En esta tesis, establecemos diversas caracterizaciones topológicas del espectro no conmutativo de anillos semi-graduados al considerar la noción de topología débil de Zariski. Con este propósito, formulamos condiciones necesarias o suficientes para garantizar que familias de estos anillos definidos...

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Autores:: Chacón Capera, Andrés

Tipo de recurso:: Doctoral thesis

Fecha de publicación:: 2022

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: eng

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dc.title.eng.fl_str_mv	On the noncommutative geometry of semi-graded rings
dc.title.translated.spa.fl_str_mv	Sobre la geometría no conmutativa de anillos semi graduados
title	On the noncommutative geometry of semi-graded rings
spellingShingle	On the noncommutative geometry of semi-graded rings 510 - Matemáticas::512 - Álgebra Geometría algebraíca Topología Polinomios Geometry, algebraic Topology Polynomials Anillo semi-graduado Anillo de polinomios torcidos Extensión PBW torcida Álgebra esquemática Geometría algebraica no conmutativa Esquema no conmutativo Semi-graded ring Skew polynomial ring Skew PBW extension Schematic algebra Noncommutative algebraic geometry Noncommutative scheme
title_short	On the noncommutative geometry of semi-graded rings
title_full	On the noncommutative geometry of semi-graded rings
title_fullStr	On the noncommutative geometry of semi-graded rings
title_full_unstemmed	On the noncommutative geometry of semi-graded rings
title_sort	On the noncommutative geometry of semi-graded rings
dc.creator.fl_str_mv	Chacón Capera, Andrés
dc.contributor.advisor.none.fl_str_mv	Reyes Villamil, Milton Armando
dc.contributor.author.none.fl_str_mv	Chacón Capera, Andrés
dc.contributor.researchgroup.spa.fl_str_mv	Sac2
dc.subject.ddc.spa.fl_str_mv	510 - Matemáticas::512 - Álgebra
topic	510 - Matemáticas::512 - Álgebra Geometría algebraíca Topología Polinomios Geometry, algebraic Topology Polynomials Anillo semi-graduado Anillo de polinomios torcidos Extensión PBW torcida Álgebra esquemática Geometría algebraica no conmutativa Esquema no conmutativo Semi-graded ring Skew polynomial ring Skew PBW extension Schematic algebra Noncommutative algebraic geometry Noncommutative scheme
dc.subject.lemb.spa.fl_str_mv	Geometría algebraíca Topología Polinomios
dc.subject.lemb.eng.fl_str_mv	Geometry, algebraic Topology Polynomials
dc.subject.proposal.spa.fl_str_mv	Anillo semi-graduado Anillo de polinomios torcidos Extensión PBW torcida Álgebra esquemática Geometría algebraica no conmutativa Esquema no conmutativo
dc.subject.proposal.eng.fl_str_mv	Semi-graded ring Skew polynomial ring Skew PBW extension Schematic algebra Noncommutative algebraic geometry Noncommutative scheme
description	En esta tesis, establecemos diversas caracterizaciones topológicas del espectro no conmutativo de anillos semi-graduados al considerar la noción de topología débil de Zariski. Con este propósito, formulamos condiciones necesarias o suficientes para garantizar que familias de estos anillos definidos por endomorfismos y derivaciones sean anillos NI o anillos NJ. Presentamos resultados sobre la caracterización de diferentes tipos de elementos de anillos no conmmutativos tales como idempotentes, unidades, von Neumann regulares, π-regulares, y elementos limpios. También investigamos las nociones de anillo fuertemente armónico y de Gelfand sobre dichas familias de anillos semi-graduados. Nuestros resultados generalizan tratamientos desarrollados para anillos conmutativos, anillos de polinomios torcidos, y variadas familias de anillos N-graduados, y contribuyen a la investigación sobre estos temas que ha sido llevada a cabo parcialmente en la literatura. Por otra parte, investigamos la esquematicidad y el teorema de Serre-Artin-Zhang-Verevkin para anillos semi-graduados. Más exactamente, para los polinomios de Ore de orden superior generados por relaciones homogéneas y las extensiones torcidas de Poincaré-Birkhoff-Witt, formulamos condiciones necesarias o suficientes para garantizar la esquematicidad de estas familias de anillos. Desarrollamos una teoría de esquemas no conmutativa para anillos semi-graduados que no son necesariamente conexos y N-graduados. Con esta teoría, demostramos el teorema de Serre-Artin-Zhang-Verevkin para diversas familias de álgebras no N-graduadas que incluyen diferentes clases de anillos no conmutativos que surgen en la teoría de anillos y la geometría algebraica no conmutativa. Nuestro tratamiento contribuye a la investigación sobre este teorema desarrollada en la literatura. (Texto tomado de la fuente)
publishDate	2022
dc.date.issued.none.fl_str_mv	2022
dc.date.accessioned.none.fl_str_mv	2023-11-02T19:38:40Z
dc.date.available.none.fl_str_mv	2023-11-02T19:38:40Z
dc.type.spa.fl_str_mv	Trabajo de grado - Doctorado
dc.type.driver.spa.fl_str_mv	info:eu-repo/semantics/doctoralThesis
dc.type.version.spa.fl_str_mv	info:eu-repo/semantics/acceptedVersion
dc.type.coar.spa.fl_str_mv	http://purl.org/coar/resource_type/c_db06
dc.type.content.spa.fl_str_mv	Text
dc.type.redcol.spa.fl_str_mv	http://purl.org/redcol/resource_type/TD
format	http://purl.org/coar/resource_type/c_db06
status_str	acceptedVersion
dc.identifier.uri.none.fl_str_mv	https://repositorio.unal.edu.co/handle/unal/84866
dc.identifier.instname.spa.fl_str_mv	Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv	Repositorio Institucional Universidad Nacional de Colombia
dc.identifier.repourl.spa.fl_str_mv	https://repositorio.unal.edu.co/
url	https://repositorio.unal.edu.co/handle/unal/84866 https://repositorio.unal.edu.co/
identifier_str_mv	Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv	eng
language	eng
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dc.format.extent.spa.fl_str_mv	vii, 113 páginas
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dc.publisher.spa.fl_str_mv	Universidad Nacional de Colombia
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spelling	Reconocimiento 4.0 Internacionalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Reyes Villamil, Milton Armando767f31307c790697ee2bf5d2c4f57583Chacón Capera, Andrés50222393337edcac1a8141d1517f5264Sac22023-11-02T19:38:40Z2023-11-02T19:38:40Z2022https://repositorio.unal.edu.co/handle/unal/84866Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/En esta tesis, establecemos diversas caracterizaciones topológicas del espectro no conmutativo de anillos semi-graduados al considerar la noción de topología débil de Zariski. Con este propósito, formulamos condiciones necesarias o suficientes para garantizar que familias de estos anillos definidos por endomorfismos y derivaciones sean anillos NI o anillos NJ. Presentamos resultados sobre la caracterización de diferentes tipos de elementos de anillos no conmmutativos tales como idempotentes, unidades, von Neumann regulares, π-regulares, y elementos limpios. También investigamos las nociones de anillo fuertemente armónico y de Gelfand sobre dichas familias de anillos semi-graduados. Nuestros resultados generalizan tratamientos desarrollados para anillos conmutativos, anillos de polinomios torcidos, y variadas familias de anillos N-graduados, y contribuyen a la investigación sobre estos temas que ha sido llevada a cabo parcialmente en la literatura. Por otra parte, investigamos la esquematicidad y el teorema de Serre-Artin-Zhang-Verevkin para anillos semi-graduados. Más exactamente, para los polinomios de Ore de orden superior generados por relaciones homogéneas y las extensiones torcidas de Poincaré-Birkhoff-Witt, formulamos condiciones necesarias o suficientes para garantizar la esquematicidad de estas familias de anillos. Desarrollamos una teoría de esquemas no conmutativa para anillos semi-graduados que no son necesariamente conexos y N-graduados. Con esta teoría, demostramos el teorema de Serre-Artin-Zhang-Verevkin para diversas familias de álgebras no N-graduadas que incluyen diferentes clases de anillos no conmutativos que surgen en la teoría de anillos y la geometría algebraica no conmutativa. Nuestro tratamiento contribuye a la investigación sobre este teorema desarrollada en la literatura. (Texto tomado de la fuente)ilustraciones, diagramasIn this thesis, we establish several topological characterizations of the noncommutative spectrum of semi-graded rings by considering the notion of weak Zariski topology. With this aim, necessary or sufficient conditions to guarantee that families of these rings defined by endomorphisms and derivations are NI or NJ rings are formulated. We present results about the characterization of different types of elements of noncommutative rings such as idempotents, units, von Neumann regular, π-regular, and clean elements. We also investigate the notions of strongly harmonic and Gelfand rings over such families of semi-graded rings. Our results generalize treatments developed for commutative rings, skew polynomial rings, and several families of N-graded rings, and contribute to the research on these topics that has been partially carried out in the literature. On the other hand, we investigate the schematicness and the Serre-Artin-Zhang-Verevkin theorem for semi-graded rings. More exactly, for the Ore polynomials of higher order generated by homogeneous relations and skew Poincaré-Birkhoff-Witt extensions, we formulate necessary or sufficient conditions to guarantee the schematicness of these families of rings. We develop a noncommutative scheme theory for semi-graded rings that are not necessarily connected and N-graded. With this theory, we prove the Serre-Artin-ZhangVerevkin theorem for several families of non-N-graded algebras that include different kinds of noncommutative rings appearing in ring theory and noncommutative algebraic geometry. Our treatment contributes to the research on this theorem developed in the literature.DoctoradoDoctor en Ciencias - Matemáticasvii, 113 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Doctorado en Ciencias - MatemáticasFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::512 - ÁlgebraGeometría algebraícaTopologíaPolinomiosGeometry, algebraicTopologyPolynomialsAnillo semi-graduadoAnillo de polinomios torcidosExtensión PBW torcidaÁlgebra esquemáticaGeometría algebraica no conmutativaEsquema no conmutativoSemi-graded ringSkew polynomial ringSkew PBW extensionSchematic algebraNoncommutative algebraic geometryNoncommutative schemeOn the noncommutative geometry of semi-graded ringsSobre la geometría no conmutativa de anillos semi graduadosTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TDM. 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Phys., 89(2):1146–1157, 1991LICENSElicense.txtlicense.txttext/plain; charset=utf-85879https://repositorio.unal.edu.co/bitstream/unal/84866/1/license.txteb34b1cf90b7e1103fc9dfd26be24b4aMD51ORIGINAL1032455420.2022.pdf1032455420.2022.pdfTesis deDoctorado en Ciencias - Matemáticasapplication/pdf1374511https://repositorio.unal.edu.co/bitstream/unal/84866/2/1032455420.2022.pdf1b36c8bcd299265f879557c4fcd4ec0aMD52THUMBNAIL1032455420.2022.pdf.jpg1032455420.2022.pdf.jpgGenerated Thumbnailimage/jpeg4071https://repositorio.unal.edu.co/bitstream/unal/84866/3/1032455420.2022.pdf.jpg62f21032f0ce3b6d1dfc7492fb8d1740MD53unal/84866oai:repositorio.unal.edu.co:unal/848662024-08-19 23:10:45.679Repositorio Institucional Universidad Nacional de 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On the noncommutative geometry of semi-graded rings

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