This thesis consists of two parts. Part one revolves around a construction for homogeneous Gorenstein ideals and properties of these ideals. Part two focuses on the behavior of the Boij-Söderberg decomposition of lex ideals. Gorenstein ideals are known for their nice duality properties. For codimension two and three, the structures of Gorenstein ideals have been established by Hilbert-Burch and Buchsbaum-Eisenbud, respectively. However, although some important results have been found about Gorenstein ideals of higher codimension, there is no structure theorem proven for higher codimension cases. Kustin and Miller showed how to construct a Gorenstein ideals in local Gorenstein rings starting from smaller such ideals. A modification of their construction in the case of graded rings is discussed. In a Noetherian ring, for a given two homogeneous Gorenstein ideals, we construct another homogeneous Gorenstein ideal and so we describe the resulting ideal in terms of the initial homogeneous Gorenstein ideals. Gorenstein liaison theory plays a central role in this construction. Using liaison properties, we examine structural relations between the constructed homogeneous ideal and the starting ideals. Boij-Söderberg theory is a very recent theory. It arose from two conjectures given by Boij and Söderberg and their proof by Eisenbud and Schreyer. It establishes a unique decomposition for Betti diagram of graded modules over polynomial rings. In the second part of this thesis, we focus on Betti diagrams of lex ideals which are the ideals having the largest Betti numbers among the ideals with the same Hilbert function. We describe Boij-Söderberg decomposition of a lex ideal in terms of Boij-Söderberg decompositions of some related lex ideals.

Orientador: Marcos Benevenuto Jardim / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T05:02:36Z (GMT). No. of bitstreams: 1 Prata_DanielaMoura_D.pdf: 839425 bytes, checksum: 24b3dac76766d8c843d040b951a4376a (MD5) Previous issue date: 2012 / Resumo: Neste trabalho relacionamos algumas classes de fibrados vetoriais...Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: In this work we relate some classes of vector bundles on...Note: The complete abstract is available with the full electronic document / Doutorado / Matematica / Doutor em Matemática

Tropical geometry is an area of mathematics that interfaces algebraic geometry and combinatorics. The main object of study in tropical geometry is the tropical variety, which is the combinatorial counterpart of a classical variety. A classical variety is converted into a tropical variety by a process called tropicalization, thus reducing the problems of algebraic geometry to problems of combinatorics. This new tropical variety encodes several useful information about the original variety, for example an algebraic variety and its tropical counterpart have the same dimension. In this thesis, we look at the some of the computational aspects of tropical algebraic geometry. We study a generalization of Grobner basis theory of modules which unlike the standard Grobner basis also takes the valuation of coefficients into account. This was rst introduced in (Maclagan & Sturmfels, 2009) in the settings of polynomial rings and its computational aspects were first studied in (Chan & Maclagan, 2013) for the polynomial ring case. The motivation for this comes from tropical geometry as it can be used to compute tropicalization of varieties. We further generalize this to the case of modules. But apart from that it has many other computational advantages. For example, in the standard case the size of the initial submodule generally grows with the increase in degree of the generators. But in this case, we give an example of a family of submodules where the size of the initial submodule remains constant. We also develop an algorithm for computation of Grobner basis of submodules of modules over Z=p`Z[x1; : : : ; xn] that works for any weight vector. We also look at some of the important applications of this new theory. We show how this can be useful in efficiently solving the submodule membership problem. We also study the computation of Hilbert polynomials, syzygies and free resolutions.

Dado um grupo G, a definição dos grupos de cohomologia com coeficientes em um ZG-módulo M podem ser dadas usando as técnicas usuais da Álgebra Homológica, que garantem a existência de resoluções projetivas P de Z como um ZG-módulo trivial, a equivalência entre resoluções distintas etc. Podemos também construir o produto cup em cohomologia, cuja definição depende de uma aproximação da diagonal para a resolução projetiva P. Entretanto, o cálculo explicito de tais resoluções e dos grupos de cohomologia pode ser bastante difícil na prática, e ainda mais difícil a obtenção de uma aproximação da diagonal. Nesta tese, obteremos resoluções livres e aproximações da diagonal para os grupos fundamentais das superfícies que são espaços K(G,1) e também para o grupo fundamental de fibrados do toro com base S^1, bem como a estrutura de anel de cohomologia de tais grupos. Ainda, para certos grupos virtualmente cíclicos G, obteremos o anel de cohomologia calculando diretamente uma resolução livre e uma aproximação da diagonal, ou então usando a sequência espectral de Lyndon-Hochschild-Serre. A motivação para o estudo da primeira família de grupos vem do fato de representarem variedades de dimensão 2 e 3, e da segunda família por ser constituída de grupos que atuam em esferas de homotopia. / Given a group G, a definition for its cohomology groups with coefficients in a given ZG-module M can be given using the standard techniques of Homological Algebra, that ensure the existence of projective resolutions P of Z as a trivial ZG-module, the equivalence between two such resolutions etc . We can also construct the cup product, whose definition depends on a diagonal approximation for a given projective resolution P. However, the explicit computation of such resolutions and of the cohomology groups may be very hard in practice, and even worse may be the task of constructing a diagonal approximation. In this thesis, we obtain free resolutions and diagonal approximations for the fundamental groups of surfaces that are K(G,1) spaces and for the fundamental group of the torus bundle with the circle as the base space, as well as the structure of the cohomology ring of these groups. Also, for some virtually cyclic groups, we obtain the cohomology ring by an explicit computation of a free resolution and a diagonal approximation, or by the Lyndon-Hochschild-Serre spectral sequence. The motivation for the study of the first family of groups comes from the fact that such groups represent manifolds of dimension 2 and 3, and the groups of the second family act on homotopy spheres.

Dado um grupo G, a definição dos grupos de cohomologia com coeficientes em um ZG-módulo M podem ser dadas usando as técnicas usuais da Álgebra Homológica, que garantem a existência de resoluções projetivas P de Z como um ZG-módulo trivial, a equivalência entre resoluções distintas etc. Podemos também construir o produto cup em cohomologia, cuja definição depende de uma aproximação da diagonal para a resolução projetiva P. Entretanto, o cálculo explicito de tais resoluções e dos grupos de cohomologia pode ser bastante difícil na prática, e ainda mais difícil a obtenção de uma aproximação da diagonal. Nesta tese, obteremos resoluções livres e aproximações da diagonal para os grupos fundamentais das superfícies que são espaços K(G,1) e também para o grupo fundamental de fibrados do toro com base S^1, bem como a estrutura de anel de cohomologia de tais grupos. Ainda, para certos grupos virtualmente cíclicos G, obteremos o anel de cohomologia calculando diretamente uma resolução livre e uma aproximação da diagonal, ou então usando a sequência espectral de Lyndon-Hochschild-Serre. A motivação para o estudo da primeira família de grupos vem do fato de representarem variedades de dimensão 2 e 3, e da segunda família por ser constituída de grupos que atuam em esferas de homotopia. / Given a group G, a definition for its cohomology groups with coefficients in a given ZG-module M can be given using the standard techniques of Homological Algebra, that ensure the existence of projective resolutions P of Z as a trivial ZG-module, the equivalence between two such resolutions etc . We can also construct the cup product, whose definition depends on a diagonal approximation for a given projective resolution P. However, the explicit computation of such resolutions and of the cohomology groups may be very hard in practice, and even worse may be the task of constructing a diagonal approximation. In this thesis, we obtain free resolutions and diagonal approximations for the fundamental groups of surfaces that are K(G,1) spaces and for the fundamental group of the torus bundle with the circle as the base space, as well as the structure of the cohomology ring of these groups. Also, for some virtually cyclic groups, we obtain the cohomology ring by an explicit computation of a free resolution and a diagonal approximation, or by the Lyndon-Hochschild-Serre spectral sequence. The motivation for the study of the first family of groups comes from the fact that such groups represent manifolds of dimension 2 and 3, and the groups of the second family act on homotopy spheres.