Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms
Autor(a) principal: | |
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Data de Publicação: | 2022 |
Tipo de documento: | Tese |
Idioma: | por |
Título da fonte: | Biblioteca Digital de Teses e Dissertações da UFPB |
Texto Completo: | https://repositorio.ufpb.br/jspui/handle/123456789/24034 |
Resumo: | This thesis deals with equigenerated Gorenstein ideals of nite colength in a standard graded ring R = k[x1; : : : ; xn] over an in nite eld k. We focus especially on such ideals of codimension 3, by looking at properties involving the Macaulay inverse system, the degree of socle, the reduction number, and the Cohen-Macaulayness of the associated Rees algebra. A special attention is devoted to the classical problem of general forms, as in the well-known conjecture of Fröberg. Our interest is to understand the sparsity of Gorenstein ideals generated by general forms. We conjecture that if I R is an ideal generated by a general set of r n+2 forms of degree d 2, then I is Gorenstein if and only if d = 2 and r = n+1 2 1. We prove this conjecture for n = 3 and one of its implications for arbitrary n. Another theme considered in this thesis is what we called the colon problem, a subject related to the presentation of a Gorenstein ideal as a link I = (xm1 ; : : : ; xmn ) : f, for a form f 2 R. If I has nite colength and linear resolution, we establish under what conditions the form f is uniquely determined, in addition to determining its degree. As we show, this problem is related to the so-called Newton dual introduced by Costa Simis and further studied by various recent authors. |
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Equigenerated Gorenstein ideals of codimension 3: with a chapter on general formsIdeais de Gorenstein equigeradosSistema inverso de MacaulayProblema do quocienteFormas geraisEquigenerated Gorenstein idealsMacaulay inverse systemColon problemGeneral fomsCNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICAThis thesis deals with equigenerated Gorenstein ideals of nite colength in a standard graded ring R = k[x1; : : : ; xn] over an in nite eld k. We focus especially on such ideals of codimension 3, by looking at properties involving the Macaulay inverse system, the degree of socle, the reduction number, and the Cohen-Macaulayness of the associated Rees algebra. A special attention is devoted to the classical problem of general forms, as in the well-known conjecture of Fröberg. Our interest is to understand the sparsity of Gorenstein ideals generated by general forms. We conjecture that if I R is an ideal generated by a general set of r n+2 forms of degree d 2, then I is Gorenstein if and only if d = 2 and r = n+1 2 1. We prove this conjecture for n = 3 and one of its implications for arbitrary n. Another theme considered in this thesis is what we called the colon problem, a subject related to the presentation of a Gorenstein ideal as a link I = (xm1 ; : : : ; xmn ) : f, for a form f 2 R. If I has nite colength and linear resolution, we establish under what conditions the form f is uniquely determined, in addition to determining its degree. As we show, this problem is related to the so-called Newton dual introduced by Costa Simis and further studied by various recent authors.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPESEsta tese versa sobre ideais de Gorenstein equigerados de co-comprimento nito em um anel de polinômios graduado standard R = k[x1; : : : ; xn] sobre um corpo in nito k. Focalizamos especialmente o caso de codimensão 3, estudando propriedades envolvendo o sistema inverso de Macaulay, o grau do socle, o número de redução, e a Cohen-Macaulicidade da álgebra de Rees associada. Uma atenção especial é dedicada ao problema clássico de formas gerais, no espírito da conjectura de Fröberg. Nosso interesse é entender a rarefação de ideais de Gorenstein gerados por formas gerais. Nessa direção conjecturamos que se I R é um ideal gerado por r n + 2 formas de grau d 2, então I é Gorenstein se, e somente se, d = 2 e r = n+1 2 1. Provamos esta conjectura para n = 3 e uma das implicações para n arbitrário. Outro tema abordado é o aqui denominado problema do quociente, relacionado à apresentação de um ideal Gorenstein na forma I = (xm1 ; : : : ; xmn ) : f, para certa forma f 2 R. Se I tem co-comprimento nito e resolução linear, estabelecemos sob quais condições a forma f é unicamente determinada e qual é seu grau. Mostramos também que esse problema está relacionado com a noção de dual de Newton, introduzido por Costa Simis e posteriormente estudado por vários autores recentes.Universidade Federal da ParaíbaBrasilMatemáticaPrograma Associado de Pós-Graduação em MatemáticaUFPBSimis, Aronhttp://lattes.cnpq.br/8415377033264469Ramos, Zaqueu Alveshttp://lattes.cnpq.br/9937925412759644Lira, Dayane Santos de2022-07-27T20:57:22Z2022-06-082022-07-27T20:57:22Z2022-05-27info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesishttps://repositorio.ufpb.br/jspui/handle/123456789/24034porAttribution-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nd/3.0/br/info:eu-repo/semantics/openAccessreponame:Biblioteca Digital de Teses e Dissertações da UFPBinstname:Universidade Federal da Paraíba (UFPB)instacron:UFPB2022-08-09T12:08:35Zoai:repositorio.ufpb.br:123456789/24034Biblioteca Digital de Teses e Dissertaçõeshttps://repositorio.ufpb.br/PUBhttp://tede.biblioteca.ufpb.br:8080/oai/requestdiretoria@ufpb.br|| diretoria@ufpb.bropendoar:2022-08-09T12:08:35Biblioteca Digital de Teses e Dissertações da UFPB - Universidade Federal da Paraíba (UFPB)false |
dc.title.none.fl_str_mv |
Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms |
title |
Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms |
spellingShingle |
Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms Lira, Dayane Santos de Ideais de Gorenstein equigerados Sistema inverso de Macaulay Problema do quociente Formas gerais Equigenerated Gorenstein ideals Macaulay inverse system Colon problem General foms CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
title_short |
Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms |
title_full |
Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms |
title_fullStr |
Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms |
title_full_unstemmed |
Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms |
title_sort |
Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms |
author |
Lira, Dayane Santos de |
author_facet |
Lira, Dayane Santos de |
author_role |
author |
dc.contributor.none.fl_str_mv |
Simis, Aron http://lattes.cnpq.br/8415377033264469 Ramos, Zaqueu Alves http://lattes.cnpq.br/9937925412759644 |
dc.contributor.author.fl_str_mv |
Lira, Dayane Santos de |
dc.subject.por.fl_str_mv |
Ideais de Gorenstein equigerados Sistema inverso de Macaulay Problema do quociente Formas gerais Equigenerated Gorenstein ideals Macaulay inverse system Colon problem General foms CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
topic |
Ideais de Gorenstein equigerados Sistema inverso de Macaulay Problema do quociente Formas gerais Equigenerated Gorenstein ideals Macaulay inverse system Colon problem General foms CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
description |
This thesis deals with equigenerated Gorenstein ideals of nite colength in a standard graded ring R = k[x1; : : : ; xn] over an in nite eld k. We focus especially on such ideals of codimension 3, by looking at properties involving the Macaulay inverse system, the degree of socle, the reduction number, and the Cohen-Macaulayness of the associated Rees algebra. A special attention is devoted to the classical problem of general forms, as in the well-known conjecture of Fröberg. Our interest is to understand the sparsity of Gorenstein ideals generated by general forms. We conjecture that if I R is an ideal generated by a general set of r n+2 forms of degree d 2, then I is Gorenstein if and only if d = 2 and r = n+1 2 1. We prove this conjecture for n = 3 and one of its implications for arbitrary n. Another theme considered in this thesis is what we called the colon problem, a subject related to the presentation of a Gorenstein ideal as a link I = (xm1 ; : : : ; xmn ) : f, for a form f 2 R. If I has nite colength and linear resolution, we establish under what conditions the form f is uniquely determined, in addition to determining its degree. As we show, this problem is related to the so-called Newton dual introduced by Costa Simis and further studied by various recent authors. |
publishDate |
2022 |
dc.date.none.fl_str_mv |
2022-07-27T20:57:22Z 2022-06-08 2022-07-27T20:57:22Z 2022-05-27 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufpb.br/jspui/handle/123456789/24034 |
url |
https://repositorio.ufpb.br/jspui/handle/123456789/24034 |
dc.language.iso.fl_str_mv |
por |
language |
por |
dc.rights.driver.fl_str_mv |
Attribution-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nd/3.0/br/ info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Attribution-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nd/3.0/br/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal da Paraíba Brasil Matemática Programa Associado de Pós-Graduação em Matemática UFPB |
publisher.none.fl_str_mv |
Universidade Federal da Paraíba Brasil Matemática Programa Associado de Pós-Graduação em Matemática UFPB |
dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da UFPB instname:Universidade Federal da Paraíba (UFPB) instacron:UFPB |
instname_str |
Universidade Federal da Paraíba (UFPB) |
instacron_str |
UFPB |
institution |
UFPB |
reponame_str |
Biblioteca Digital de Teses e Dissertações da UFPB |
collection |
Biblioteca Digital de Teses e Dissertações da UFPB |
repository.name.fl_str_mv |
Biblioteca Digital de Teses e Dissertações da UFPB - Universidade Federal da Paraíba (UFPB) |
repository.mail.fl_str_mv |
diretoria@ufpb.br|| diretoria@ufpb.br |
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1801842997384445952 |