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Specialization and Complexity of Integral Closure of IdealsRachel Von Arb Lynn (10725408) 05 May 2021 (has links)
<div> This dissertation is based on joint work with Lindsey Hill. There are two main parts, which are linked by the common theme of the integral closure of the Rees algebra.</div><div> </div><div> In the first part of this dissertation, comprised of Chapter 3 and Chapter 4, we study the integral closure of the Rees algebra directly. In Chapter 3 we identify a bound for the multiplicity of the Rees algebra R[It] of a homogeneous ideal I generated in the same degree, and combine this result with theorems of Ulrich and Vasconcelos to obtain upper bounds on the number of generators of the integral closure of the Rees algebra as a module over R[It]. We also find various other upper bounds for this number, and compare them in the case of a monomial ideal generated in the same degree. In Chapter 4, inspired by the large depth assumption on the integral closure of R[It] in the results of Chapter 3, we obtain a lower bound for the depth of the associated graded ring and the Rees algebra of the integral closure filtration in terms of the dimension of the Cohen-Macaulay local ring R and the equimultiple ideal I. We finish the first part of this dissertation with a characterization of when the integral closure of R[It] is Cohen-Macaulay for height 2 ideals. </div><div> </div><div> In the second part of this dissertation, Chapter 5, we use the integral closure of the Rees algebra as a tool to discuss specialization of the integral closure of an ideal I. We prove that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo general elements of I. This result is analogous to a result of Itoh and an extension by Hong and Ulrich which show that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo generic elements of I. We then discuss specialization modulo a general element of the maximal ideal, rather than modulo a general element of the ideal I itself. In general it is not the case that the operations of integral closure and specialization modulo a general element of the maximal ideal are compatible, even under the assumptions of our main theorem. We prove that the two operations are compatible for local excellent algebras over fields of characteristic zero whenever R/I is reduced with depth at least 2, and conclude with a class of ideals for which the two operations appear to be compatible based on computations in Macaulay2.</div>
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Algebras de ReesMacedo, Ricardo Burity Croccia 15 March 2013 (has links)
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Previous issue date: 2013-03-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we present the notion of Rees algebra of an ideal and some of its basic properties.
Such concept is related to the normality of rings and ideals, and to reductions of ideals as well.
Finally, we shall exhibit the Rees algebra of a module, proving some generalizations of results in
the case of ideals. / Neste trabalho, apresentaremos a noção de álgebra de Rees de um ideal e propriedades básicas.
Tal conceito será relacionado com normalidade de anéis e ideais, e redução de ideais. Por fim,
exibiremos a álgebra de Rees de um módulo, mostrando algumas generalizações de resultados do
caso de ideais.
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Álgebras simétrica e de Rees do módulo de diferenciais de KählerSousa, Fraciélia Limeira de 16 July 2015 (has links)
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Previous issue date: 2015-07-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation, we initially present an overview about the symmetric and the
Rees algebras in the wide context of modules, and we consider particularly the special
situation in which the given module possesses a linear presentation. In the sequel, the
main goal is the study of such blowup algebras in the case where the module is the
celebrated module of K ahler di erentials, the focus being given on the investigation
of an interesting version of the long-standing Berger's Conjecture for the symmetric
algebra, as well as on the study of fundamental properties such as: integrality, Cohen-
Macaulayness and normality; these properties are also investigated in a special way in
the case of the Rees algebra (of the di erential module), highlighting the connection
to the so-called Fitting conditions. / Nesta disserta c~ao, inicialmente apresentamos no c~oes gerais sobre a algebra sim etrica e
a algebra de Rees no contexto amplo de m odulos, e consideramos particularmente a
situa c~ao especial na qual o dado m odulo possui apresenta c~ao linear. Na sequ^encia, o
principal objetivo e o estudo de tais algebras de blowup no caso em que o m odulo e
o celebrado m odulo de diferenciais de K ahler, tendo como foco a investiga c~ao de uma
interessante vers~ao da persistente Conjectura de Berger para a algebra sim etrica, bem
como o estudo de propriedades fundamentais como: integridade, Cohen-Macaulicidade
e normalidade; tais propriedades s~ao tamb em investigadas de forma especial no caso
da algebra de Rees (do m odulo de diferenciais), evidenciando inclusive a conex~ao com
as chamadas condi c~oes de Fitting.
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Parametrizações de JonquièresAndrade, Pêdra Daricléa Santos 10 August 2015 (has links)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation, we will present a projective parameterization class that resemble
the classic maps of Jonquières. We will have the aim to show how the main properties
of your ideal base, such as the structure of sizígias and the free presentation of this
ideal. In this sense, we obtain the implicit equation that defines the your image and we
explicit the formula the degree of the equation. Finally, we will determine Rees algebra
equations associated with the base ideal of the parameterization, using computational
methods. This parameterization, called parameterization of Jonquières, is constructed
from a Cremona and defines a birational map P^n in a hypersurface W of P^n+1. / Nesta dissertação apresentaremos uma classe de parametrizações projetivas que se assemelham aos mapas clássicos de Jonquières e teremos como objetivos mostrar as principais propriedades do seu ideal base, tais como a estrutura das sizígias e a apresentação livre desse ideal, consequentemente obteremos a equação implícita que define a sua imagem, bem como explicitar a fórmula do grau dessa equação. Por fim, determinaremos as equações da álgebra de Rees associada ao ideal base da parametrização, com o uso de métodos computacionais. Tal parametrização, denominada parametrização de Jonquières, é construída a partir de uma Cremona e define um mapa birracional de P^n em uma hipersuperfície W de P^{n+1}.
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The Equations Defining Rees Algebras of Ideals and Modules over Hypersurface RingsMatthew J Weaver (11108382) 26 July 2022 (has links)
<p>The defining equations of Rees algebras provide a natural pathway to study these rings. However, information regarding these equations is often elusive and enigmatic. In this dissertation we study Rees algebras of particular classes of ideals and modules over hypersurface rings. We extend known results regarding Rees algebras of ideals and modules to this setting and explore the properties of these rings.</p>
<p><br></p>
<p>The majority of this thesis is spent studying Rees algebras of ideals in hypersurface rings, beginning with perfect ideals of grade two. After introducing certain constructions, we arrive in a setting similar to the one encountered by Boswell and Mukundan in [3]. We establish a similarity between Rees algebras of ideals with linear presentation in hypersurface rings and Rees algebras of ideals with <em>almost</em> linear presentation in polynomial rings. Hence we adapt the methods developed by Boswell and Mukundan in [3] to our setting and follow a path parallel to theirs. We introduce a recursive algorithm of <em>modified Jacobian dual iterations</em> which produces a minimal generating set for the defining ideal of the Rees algebra.</p>
<p><br></p>
<p>Once success has been achieved for perfect ideals of grade two, we consider perfect Gorenstein ideals of grade three in hypersurface rings and their Rees algebras. We follow a path similar to the one taken for the previous class of ideals. A recursive algorithm of <em>gcd-iterations</em> is introduced and it is shown that this method produces a minimal generating set of the defining ideal of the Rees algebra. </p>
<p><br></p>
<p>Lastly, we extend our techniques regarding Rees algebras of ideals to Rees algebras of modules. Using <em>generic Bourbaki ideals</em> we study Rees algebras of modules with projective dimension one over hypersurface rings. For such a module $E$, we show that there exists a generic Bourbaki ideal $I$, with respect to $E$, which is perfect of grade two in a hypersurface ring. We then adapt the techniques used by Costantini in [9] to our setting in order to relate the defining ideal of $\mathcal{R}(E)$ to the defining ideal of $\mathcal{R}(I)$, which is known from the earlier work mentioned above.</p>
<p><br></p>
<p>In all three situations above, once the defining equations have been determined, we investigate certain properties of the Rees algebra. The depth, Cohen-Macaulayness, relation type, and Castelnuovo-Mumford regularity of these rings are explored.</p>
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Sylvester forms and Rees algebrasMacêdo, Ricado Burity croccia 24 July 2015 (has links)
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Previous issue date: 2015-07-24 / This work is about the Rees algebra of a nite colength almost complete intersection ideal
generated by forms of the same degree in a polynomial ring over a eld. We deal with two
situations which are quite apart from each other: in the rst the forms are monomials in an
unrestricted number of variables, while the second is for general binary forms. The essential
goal in both cases is to obtain the depth of the Rees algebra. It is known that for such ideals the
latter is rarely Cohen{Macaulay (i.e., of maximal depth). Thus, the question remains as to how
far one is from the Cohen{Macaulay case. In the case of monomials one proves under certain
restriction a conjecture of Vasconcelos to the e ect that the Rees algebra is almost Cohen{
Macaulay. At the other end of the spectrum, one proposes a proof of a conjecture of Simis
on general binary forms, based on work of Huckaba{Marley and on a theorem concerning the
Ratli {Rush ltration. Still within this frame, one states a couple of stronger conjectures that
imply Simis conjecture, along with some solid evidence. / Este trabalho versa sobre a algebra de Rees de um ideal quase intersec cão completa, de cocomprimento
nito, gerado por formas de mesmo grau em um anel de polinômios sobre um
corpo. Considera-se duas situa c~oes inteiramente diversas: na primeira, as formas s~ao mon^omios
em um n umero qualquer de vari aveis, enquanto na segunda, s~ao formas bin arias gerais. O
objetivo essencial em ambos os casos e obter a profundidade da algebra de Rees. E conhecido
que tal algebra e raramente Cohen{Macaulay (isto e, de profundidade m axima). Assim, a quest~ao
que permanece e qua o distante são do caso Cohen{Macaulay. No caso de monômios prova-se,
mediante certa restri cão, uma conjectura de Vasconcelos no sentido de que a algébra de Rees e
quase Cohen {Macaulay. No outro caso extremo, estabelece-se uma prova de uma conjectura de
Simis sobre formas bin arias gerais, baseada no trabalho de Huckaba{Marley e em um teorema
sobre a ltera cão de Ratli {Rush. Al em disso, apresenta-se um par de conjecturas mais fortes
que implicam a conjectura de Simis, juntamente com uma evidência s olida.
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Potências simbólicas e suas interaçõesSantos, Diego Cardoso dos 29 February 2016 (has links)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The notion of symbolic power dates back to W. Krull, who used it in the proof of
the famous theorem of principal ideal, this a crucial milestone in the short history of
commutative algebra. Later, O. Zariski, M. Nagata, D. Rees and others have shown
how this purely algebraic notion has important signi cance in algebraic geometry.
In this paper we study the symbolic powers showing some of its most fundamental
properties and their connections with various aspects of algebraic geometry and
commutative algebra. / A no ção de potência simb ólica remonta a W. Krull, que a usou na prova do
c élebre teorema do ideal principal, este um marco crucial na curta hist ória da álgebra
comutativa. Mais adiante, O. Zariski, M. Nagata, D. Rees e outros mostraram como
esta no ção puramente alg ébrica tem importante signi ficado em geometria alg ébrica.
Neste trabalho estudaremos as potências simb ólicas evidenciando algumas de suas
propriedades mais fundamentais e suas conexões com aspectos variados da geometria
alg ébrica e álgebra comutativa.
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On the Defining Ideals of Rees Rings for Determinantal and Pfaffian Ideals of Generic HeightEdward F Price (9188318) 04 August 2020 (has links)
<div>This dissertation is based on joint work with Monte Cooper and is broken into two main parts, both of which study the defining ideals of the Rees rings of determinantal and Pfaffian ideals of generic height. In both parts, we attempt to place degree bounds on the defining equations.</div><div> </div><div> The first part of the dissertation consists of Chapters 3 to 5. Let $R = K[x_{1},\ldots,x_{d}]$ be a standard graded polynomial ring over a field $K$, and let $I$ be a homogeneous $R$-ideal generated by $s$ elements. Then there exists a polynomial ring $\mathcal{S} = R[T_{1},\ldots,T_{s}]$, which is also equal to $K[x_{1},\ldots,x_{d},T_{1},\ldots,T_{s}]$, of which the defining ideal of $\mathcal{R}(I)$ is an ideal. The polynomial ring $\mathcal{S}$ comes equipped with a natural bigrading given by $\deg x_{i} = (1,0)$ and $\deg T_{j} = (0,1)$. Here, we attempt to use specialization techniques to place bounds on the $x$-degrees (first component of the bidegrees) of the defining equations, i.e., the minimal generators of the defining ideal of $\mathcal{R}(I)$. We obtain degree bounds by using known results in the generic case and specializing. The key tool are the methods developed by Kustin, Polini, and Ulrich to obtain degree bounds from approximate resolutions. We recover known degree bounds for ideals of maximal minors and submaximal Pfaffians of an alternating matrix. Additionally, we obtain $x$-degree bounds for sufficiently large $T$-degrees in other cases of determinantal ideals of a matrix and Pfaffian ideals of an alternating matrix. We are unable to obtain degree bounds for determinantal ideals of symmetric matrices due to a lack of results in the generic case; however, we develop the tools necessary to obtain degree bounds once similar results are proven for generic symmetric matrices.</div><div> </div><div> The second part of this dissertation is Chapter 6, where we attempt to find a bound on the $T$-degrees of the defining equations of $\mathcal{R}(I)$ when $I$ is a nonlinearly presented homogeneous perfect Gorenstein ideal of grade three having second analytic deviation one that is of linear type on the punctured spectrum. We restrict to the case where $\mathcal{R}(I)$ is not Cohen-Macaulay. This is a natural next step following the work of Morey, Johnson, and Kustin-Polini-Ulrich. Based on extensive computation in Macaulay2, we give a conjecture for the relation type of $I$ and provide some evidence for the conjecture. In an attempt to prove the conjecture, we obtain results about the defining ideals of general fibers of rational maps, which may be of independent interest. We end with some examples where the bidegrees of the defining equations exhibit unusual behavior.</div>
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Redução de um idealSantos, Maxwell da Paixão de Jesus 22 February 2018 (has links)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, under the view of commutative algebra, we will study reductions of an ideal,
the concept was introduced by Northcott and Rees. First of all, we will give preliminary no-
tions about dimension theory, Hilbert’s polynomial, Hilbert-Samuel’s polynomial, regularity
of modules and superficial elements. Next we will discuss the main theme of this dissertation,
where we will talk about integral closure of ideal, reduction and the Rees algebra, moreover,
we will establish connections between these concepts. Finally, we will discuss some applica-
tions in Hilbert-Samuel's polynomial and multiplicity theory, in which some recent results
will be presented. / Neste trabalho, sob a luz da álgebra comutativa, estudaremos reduções de um ideal, tal
conceito foi introduzido por Northcott e Rees. Em um primeiro momento, daremos noções
preliminares sobre teoria de dimensão, polinômio de Hilbert, polinômio de Hilbert-Samuel,
regularidade de módulos e elementos superficiais. Na sequência discutiremos o tema principal
da dissertação, no qual falaremos de fecho integral de um ideal, redução e a álgebra de
Rees, além disso, estabeleceremos conexões entre esses conceitos. Por fim, discutiremos
algumas aplicações na teoria de multiplicidade e polinômio de Hilbert-Samuel, no qual será apresentado alguns resultados recentes. / São Cristóvão, SE
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