On the noncommutative geometry of semi-graded rings

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dc.contributor.advisorReyes Villamil, Milton Armando
dc.contributor.authorChacón Capera, Andrés
dc.contributor.researchgroupSac2spa
dc.date.accessioned2023-11-02T19:38:40Z
dc.date.available2023-11-02T19:38:40Z
dc.date.issued2022
dc.description.abstractEn 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)spa
dc.description.abstractIn 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.eng
dc.description.abstractilustraciones, diagramasspa
dc.description.degreelevelDoctoradospa
dc.description.degreenameDoctor en Ciencias - Matemáticasspa
dc.format.extentvii, 113 páginasspa
dc.format.mimetypeapplication/pdfspa
dc.identifier.instnameUniversidad Nacional de Colombiaspa
dc.identifier.reponameRepositorio Institucional Universidad Nacional de Colombiaspa
dc.identifier.repourlhttps://repositorio.unal.edu.co/spa
dc.identifier.urihttps://repositorio.unal.edu.co/handle/unal/84866
dc.language.isoengspa
dc.publisherUniversidad Nacional de Colombiaspa
dc.publisher.branchUniversidad Nacional de Colombia - Sede Bogotáspa
dc.publisher.facultyFacultad de Cienciasspa
dc.publisher.placeBogotá, Colombiaspa
dc.publisher.programBogotá - Ciencias - Doctorado en Ciencias - Matemáticasspa
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dc.rights.accessrightsinfo:eu-repo/semantics/openAccessspa
dc.rights.licenseReconocimiento 4.0 Internacionalspa
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/spa
dc.subject.ddc510 - Matemáticas::512 - Álgebraspa
dc.subject.lembGeometría algebraícaspa
dc.subject.lembGeometry, algebraiceng
dc.subject.lembTopologíaspa
dc.subject.lembTopologyeng
dc.subject.lembPolinomiosspa
dc.subject.lembPolynomialseng
dc.subject.proposalAnillo semi-graduadospa
dc.subject.proposalAnillo de polinomios torcidosspa
dc.subject.proposalExtensión PBW torcidaspa
dc.subject.proposalÁlgebra esquemáticaspa
dc.subject.proposalGeometría algebraica no conmutativaspa
dc.subject.proposalEsquema no conmutativospa
dc.subject.proposalSemi-graded ringeng
dc.subject.proposalSkew polynomial ringeng
dc.subject.proposalSkew PBW extensioneng
dc.subject.proposalSchematic algebraeng
dc.subject.proposalNoncommutative algebraic geometryeng
dc.subject.proposalNoncommutative schemeeng
dc.titleOn the noncommutative geometry of semi-graded ringseng
dc.title.translatedSobre la geometría no conmutativa de anillos semi graduadosspa
dc.typeTrabajo de grado - Doctoradospa
dc.type.coarhttp://purl.org/coar/resource_type/c_db06spa
dc.type.coarversionhttp://purl.org/coar/version/c_ab4af688f83e57aaspa
dc.type.contentTextspa
dc.type.driverinfo:eu-repo/semantics/doctoralThesisspa
dc.type.redcolhttp://purl.org/redcol/resource_type/TDspa
dc.type.versioninfo:eu-repo/semantics/acceptedVersionspa
oaire.accessrightshttp://purl.org/coar/access_right/c_abf2spa

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